NDARC - NASA Design and Analysis of Rotorcraft

Chapter 1 Introduction The NASA Design and Analysis of Rotorcraft (NDARC) software is an aircraft system analysis tool that supports both conceptual design efforts and technology impact assessments. The principal tasks are to design (or size) a rotorcraft to meet specified requirements, including vertical takeoff and landing (VTOL) operation, and then analyze the performance of the aircraft for a set of conditions. For broad and lasting utility, it is important that the code have the capability to model general rotorcraft configurations, and estimate the performance and weights of advanced rotor concepts. The architecture of the NDARC code accommodates configuration flexibility, a hierarchy of models, and ultimately multidisciplinary design, analysis, and optimization. Initially the software is implemented with low-fidelity models, typically appropriate for the conceptual design environment.

An NDARC job consists of one or more cases, each case optionally performing design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions.

The analysis tasks can include off-design mission performance calculation, flight performance calcula- tion for point operating conditions, and generation of subsystem or component performance maps. For analysis tasks, the aircraft description can come from the sizing task, from a previous case or a previous NDARC job, or be independently generated (typically the description of an existing aircraft).

The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion.

For each component, attributes such as performance, drag, and weight can be calculated; and the aircraft attributes are obtained from the sum of the component attributes. Description and analysis of conven- tional rotorcraft configurations is facilitated, while retaining the capability to model novel and advanced concepts. Specific rotorcraft configurations considered are single-main-rotor and tail-rotor helicopter, tandem helicopter, coaxial helicopter, and tiltrotor. The architecture of the code accommodates addition of new or higher-fidelity attribute models for a component, as well as addition of new components.

1–1 Background The definition and development of NDARC requirements benefited substantially from the ex- periences and computer codes of the preliminary design team of the U.S. Army Aeroflightdynamics Directorate (AFDD) at Ames Research Center.

In the early 1970s, the codes SSP-1 and SSP-2 were developed by the Systems Research Integration Office (SRIO, in St. Louis) of the U.S. Army Air Mobility Research and Development Laboratory.

SSP-1 performed preliminary design to meet specified mission requirements, and SSP-2 estimated the performance for known geometry and engine characteristics, both for single-main-rotor helicopters (ref. 1). Although similar tools were in use in the rotorcraft community, these computer programs were independently developed, to meet the requirements of government analysis. The Advanced Systems Research Office (ASRO, at Ames Research Center) of USAAMRDL produced in 1974 two Preliminary 2 Introduction Systems Design Engineering (PSDE) studies (refs. 2 and 3) using SSP-1 and SSP-2. These two codes were combined into one code called PSDE by Ronald Shinn.

The MIT Flight Transportation Laboratory created design programs for helicopters (ref. 4) and tiltrotors (ref. 5). Michael Scully, who wrote the helicopter design program and was significantly involved in the development of the tiltrotor design program, joined ASRO in 1975, and ideas from the MIT programs began to be reflected in the continuing development of PSDE. An assessment of design trade-offs for the Advanced Scout Helicopter (ASH) used a highly modified version of PSDE (ref. 6).

A DoD Joint Study Group was formed in April 1975 to perform an Interservice Helicopter Com- monality Study (HELCOM) for the Director of Defense Research and Engineering. The final HELCOM study report was published in March 1976 (ref. 7). A result of this study was an assessment by ASRO that PSDE needed substantial development, including better mathematical models and better technical substantiation, more flexible mission analysis, and improved productivity for both design and analysis tasks. Thus began an evolutionary improvement of the code, eventually named RASH (after the devel- oper Ronald A. Shinn, as a consequence of the computer system identification of output by the first four characters of the user name). RASH included improvements in flight performance modeling, output depth, mission analysis, parametric weight estimation, design sensitivity studies, off-design cases, and coding style. The code was still only for single-main-rotor helicopters.

In the early 1980s, tool development evolved in two separate directions with the Preliminary Design Team at ASRO. RASH was developed into the HELO (or PDPAC) code, for conventional and compound single-main-rotor helicopters. With the addition of conversion models and wing weight estimation methods (refs. 8 and 9), RASH became the TR code, for tiltrotor aircraft. The JVX Joint Technology Assessment of 1982 utilized the HELO and TR codes. A special version called PDABC, including a weight estimation model for lift-offset rotors (ref. 10), was used to analyze the Advancing Blade Concept. The JVX JTA report (ref. 11) documented the methodology implemented in these codes.

Work in support of the LHX program from 1983 on led to a requirement for maneuver analysis of helicopters and tiltrotors, implemented in the MPP (Maneuver Performance Program) code by John Davis. The core aircraft model in MPP was similar to that in TR and HELO, but the trim strategy in particular was new. A design code does not require extensive maneuver analysis capability, but MPP had an impact on the design code development, with the MPP performance and trim methods incorporated into TR87. The sizing analysis of TR88 and the aircraft flight model from MPP were combined into the VAMP (VSTOL Design and Maneuver Program) code. VAMP combined the capability to analyze helicopters and tiltrotors in a single tool, although the capability of HELO to analyze compound helicopters was not replicated.

In the early 1990s, the RC (RotorCraft) code emerged from the evolution of VAMP, with John Preston as the lead developer (refs. 12 and 13). Some maneuver analysis capabilities from MPP were added, and the analysis capability extended to helicopters. The models were confirmed by comparison with results from TR and HELO. RC was operational by 1994, although HELO and TR continued to be used into the mid-1990s. RC97 was a major version, unifying the tiltrotor and helicopter analyses. The RC code introduced new features and capabilities, and productivity enhancements, as well as coding standards and software configuration control. Special versions of RC were routinely produced to meet the unique requirements of individual projects (such as ref. 14).

NASA, with support from the U.S. Army, in 2005 conducted the design and in-depth analysis of rotorcraft configurations that could satisfy the Vehicle Systems Program technology goals (ref. 15). These Introduction 3 technology goals and accompanying mission were intended to identify enabling technology for civil application of heavy-lift rotorcraft. The emphasis was on efficient cruise and hover, efficient structures, and low noise. The mission specified was to carry 120 passengers for 1200 nm, at a speed of 350 knots and 30,000 ft altitude. The configurations investigated were a Large Civil Tiltrotor (LCTR), a Large Civil Tandem Compound (LCTC), and a Large Advancing Blade Concept (LABC). The results of the NASA Heavy Lift Rotorcraft Systems Investigation subsequently helped define the content and direction of the Subsonic Rotary Wing project in the NASA Fundamental Aeronautics program. The design tool used was the AFDD RC code. This investigation was an example of the role of a rotorcraft sizing code within NASA. The investigation also illustrated the difficulties involved in adapting or modifying RC for configurations other than conventional helicopters and tiltrotors, supporting the requirement for a new tool.

1–2 Requirements Out of this history, the development of NDARC began in early 2007. NDARC is entirely new software, built on a new architecture for the design and analysis of rotorcraft. From the RC theoretical basis, the parametric weight equations and the Referred Parameter Turboshaft Engine Model were used with only minor changes. Use was also made of the RC component aerodynamic models and rotor performance model. The current users of RC, informed by past and recent applications, contributed significantly to the requirements definition.

The principal tasks are to design (size) rotorcraft to meet specified requirements, and then analyze the performance of the aircraft for a set of flight conditions and missions. Multiple design requirements, from specific flight conditions and various missions, must be used in the sizing task. The aircraft performance analysis must cover the entire spectrum of aircraft capabilities, and allow general and flexible definition of conditions and missions.

For government applications and to support research, it is important to have the capability to model general rotorcraft configurations, including estimates of the performance and weights of advanced rotor concepts. In such an environment, software extensions and modifications are routinely required to meet the unique requirements of individual projects, including introduction of special weight and performance models for particular concepts.

Thus the code architecture must accommodate configuration flexibility and alternate models, in- cluding a hierarchy of model fidelity. Although initially implemented with low-fidelity models, typical of the conceptual design environment, ultimately the architecture must allow multidisciplinary design, analysis, and optimization. The component performance and engine models must cover all operat- ing conditions. The software design and architecture must facilitate extension and modification of the software.

Complete and thorough documentation of the theory and its software implementation is essential, to support development and maintenance, and to enable effective use and modification. Most of the history described above supports this requirement by the difficulties encountered in the absence of good documentation. Documentation of the methodology was often prompted only by the need to substantiate conclusions of major technology assessments, and occasionally by the introduction of new users and developers. For a new software implementation of a new architectures, documentation is required from the beginning of the development.

4 Introduction fixed model or previous job or previous case

DESIGN ANALYZE

Airframe Aerodynamics Map Sizing Task Engine Aircraft size iteration Performance Map Description Mission Analysis design design Flight conditions missions Performance Analysis Mission Flight Condition adjust & fuel wt iteration max GW max takeoff GW each segment Flight State max effort / trim aircraft / flap equations Figure 1-1. Outline of NDARC tasks.

1–3 Overview The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include off-design mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. Figure 1-1 illustrates the tasks. The principal tasks (sizing, mission analysis, and flight performance analysis) are shown in the figure as boxes with heavy borders.

Heavy black arrows show control of subordinate tasks.

The aircraft description (fig. 1-1) consists of all the information, input and derived, that defines the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. This information can be the result of the sizing task; can come entirely from input, for a fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description information is available to all tasks and all solutions (indicated by light green arrows).

Introduction 5 The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight, weight empty, rotor radius, and engine power available. The relationships between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized.

Missions are defined for the sizing task and for the mission performance analysis. A mission consists of a number of mission segments, for which time, distance, and fuel burn are evaluated. For the sizing task, certain missions are designated to be used for engine sizing, for design gross weight calculations, for transmission sizing, and for fuel tank sizing. The mission parameters include mission takeoff gross weight and useful load. For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is on fuel weight or energy.

Flight conditions are specified for the sizing task and for the flight performance analysis. For the sizing task, certain flight conditions are designated to be used for engine sizing, for design gross weight calculations, for transmission sizing, for maximum takeoff weight calculations, and for antitorque or auxiliary-thrust rotor sizing. The flight condition parameters include gross weight and useful load.

For flight conditions and mission takeoff, the gross weight can be maximized, such that the power required equals the power available.

A flight state is defined for each mission segment and each flight condition. The aircraft performance can be analyzed for the specified state, or a maximum effort performance can be identified. The maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion that produce equilibrium in the specified flight state. Different trim solution definitions are required for various flight states. Evaluating the rotor hub forces may require solution of the blade flap equations of motion.

1–4 Terminology The following terminology is introduced as part of the development of the NDARC theory and software. Relationships among these terms are reflected in figure 1-1.

a) Job: An NDARC job consists of one or more cases.

b) Case: Each case performs design and/or analysis tasks. The analysis tasks can include off-design mission performance calculation, flight performance calculation for point operating conditions, and generation of airframe aerodynamics or engine performance maps.

c) Design Task: Size rotorcraft to satisfy specified set of design flight conditions and/or design missions.

Key aircraft design variables are adjusted until all criteria are met. The resulting aircraft description can be basis for the mission analysis and flight performance analysis tasks.

d) Mission Analysis Task: Calculate aircraft performance for one off-design mission.

e) Flight Performance Analysis Task: Calculate aircraft performance for point operating condition.

6 Introduction f) Mission: Ordered set of mission segments, for which time, distance, and fuel burn are evaluated.

Gross weight and useful load are specified for the beginning of the mission, and adjusted for fuel burn and useful load changes at each segment. Missions are defined for the sizing task and for the mission performance analysis.

g) Flight Condition: Point operating condition, with specified gross weight and useful load. Flight conditions are specified for the sizing task and for the flight performance analysis.

h) Flight State: Aircraft flight condition, part of definition of each flight condition and each mission segment. Flight state solution involves rotor blade motion, aircraft trim, and perhaps a maximum-effort calculation.

i) Component: The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. For each component, attributes such as performance, drag, and weight are calculated.

j) Propulsion: A propulsion group is a set of components and engine groups, connected by a drive system. An engine group consists of one or more engines of a specific type. The components define the power required. The engine groups define the power available. A jet group consists of one or more systems that produce a force on the aircraft. A charge group consists of systems that generate energy for the aircraft. Fuel tank systems are associated with the engine groups, jet groups, and charge groups.

Fuel quantity is measured as either weight or energy.

1–5 Analysis Units The code can use either English or SI units for input, output, and internal calculations. A consistent mass-length-time-temperature system is used, except for weight and power: length mass time temperature weight power ◦ English: foot slug second F pound horsepower ◦ SI: meter kilogram second C kilogram kiloWatt Weight in the design description is actually mass, with pounds converted to slugs using the reference gravitational acceleration. Gravitational force is the product of the mass and the actual acceleration due to gravity. In addition, the default units for flight conditions and missions are: speed in knots, time in minutes, distance in nautical miles, and rate of climb in feet-per-minute. The user can specify alternate units for these and other quantities.

1–6 Outline of Report This document provides a complete description of the NDARC theoretical basis and architecture.

Chapters 3–5 describe the tasks and solution procedures, the cost model is described in chapter 6, and chapters 7–19 present the models for the aircraft and its components. The propulsion system models are described in chapters 14–19. The engine, jet, and charger models are described in chapters 20–26; and the weight model in chapter 27. The accompanying NDARC Input Manual describes the use of the code.

Introduction 7 1–7 References 1) Schwartzberg, M.A.; Smith, R.L.; Means, J.L.; Law, H.Y.H.; and Chappell, D.P. “Single-Rotor Helicopter Design and Performance Estimation Programs.” USAAMRDL Report SRIO 77-1, June 1977.

2) Wheatley, J.B., and Shinn, R.A. “Preliminary Systems Design Engineering for a Small Tactical Aerial Reconnaissance System-Visual.” USAAMRDL, June 1974.

3) Shinn, R.A. “Preliminary Systems Design Engineering for an Advanced Scout Helicopter.” US- AAMRDL, August 1974.

4) Scully, M., and Faulkner, H.B. “Helicopter Design Program Description.” MIT FTL Technical Memo 71-3, March 1972.

5) Faulkner, H.B. “A Computer Program for the Design and Evaluation of Tilt Rotor Aircraft.” MIT FTL Technical Memo 74-3, September 1974.

6) Scully, M.P., and Shinn, R.A. “Rotor Preliminary Design Trade-Offs for the Advanced Scout Heli- copter.” American Helicopter Society National Specialists’ Meeting on Rotor System Design, Philadel- phia, Pennsylvania, October 1980.

7) “Interservice Helicopter Commonality Study, Final Study Report.” Director of Defense Research and Engineering, Office of the Secretary of Defense, March 1976.

8) Chappell, D.P. “Tilt-rotor Aircraft Wing Design.” ASRO-PDT-83-1, 1983.

9) Chappell, D., and Peyran, R. “Methodology for Estimating Wing Weights for Conceptual Tilt-Rotor and Tilt-Wing Aircraft.” SAWE Paper No. 2107, Category No. 23, May 1992.

10) “Weight Trend Estimation for the Rotor Blade Group, Rotor Hub Group, and Upper Rotor Shaft of the ABC Aircraft.” ASRO-PDT-83-2, 1983.

11) “Technology Assessment of Capability for Advanced Joint Vertical Lift Aircraft (JVX), Summary Report.” U.S. Army Aviation Research and Development Command, AVRADCOM Report, May 1983.

12) Preston, J., and Peyran, R. “Linking a Solid-Modeling Capability with a Conceptual Rotorcraft Sizing Code.” American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco, California, January 2000.

13) Preston, J. “Aircraft Conceptual Design Trim Matrix Selection.” American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco, California, January 2006.

14) Sinsay, J.D. “The Path to Turboprop Competitive Rotorcraft: Aerodynamic Challenges.” American Helicopter Society Specialists’ Conference on Aeromechanics, San Francisco, California, January 2008.

15) Johnson, W.; Yamauchi, G.K.; and Watts, M.E. “NASA Heavy Lift Rotorcraft Systems Investiga- tion.” NASA TP 2005-213467, December 2005.

8 Introduction

Chapter 2 Nomenclature The nomenclature for geometry and rotations employs the following conventions. A vector x is a column matrix of three elements, measuring the vector relative to a particular basis (or axes, or frame).

The basis is indicated as follows: A a) x is a vector measured in axes A; EF/A b) x is a vector from point F to point E, measured in axes A.

A rotation matrix C is a three-by-three matrix that transforms vectors from one basis to another: BA B BA A c) C transforms vectors from basis A to basis B, so x = C x .

BA The matrix C defines the orientation of basis B relative to basis A, so it also may be viewed as rotating the axes from A to B. For a vector u , a cross-product matrix ˜ u is defined as follows: ⎡ ⎤ 0 − u u 3 2 ⎣ ⎦ ˜ u = u 0 − u 3 1 − u u 0 2 1 such that ˜ uv is equivalent to the vector cross-product u × v . The cross-product matrix enters the rela- tionship between angular velocity and the time derivative of a rotation matrix: AB AB/A AB AB BA/B ˙ C = − ˜ ω C = C ˜ ω (the Poisson equations). For rotation by an angle α about the x , y , or z axis (1, 2, or 3 axis), the following notation is used: ⎡ ⎤ 1 0 0 ⎣ ⎦ X = 0 cos α sin α α 0 − sin α cos α ⎡ ⎤ cos α 0 − sin α ⎣ ⎦ Y = 0 1 0 α sin α 0 cos α ⎡ ⎤ cos α sin α 0 ⎣ ⎦ Z = − sin α cos α 0 α 0 0 1 BA Thus for example, C = X Y Z means that the axes B are located relative to the axes A by first φ θ ψ rotating by angle ψ about the z -axis, then by angle θ about the y -axis, and finally by angle φ about the x -axis.

10 Nomenclature Acronyms AFDD U.S. Army Aeroflightdynamics Directorate ASM available seat mile CAS calibrated airspeed CG charge group CPI consumer price index CTM Cost Too Much (cost model) EG engine group GW gross weight IGE in ground effect IRP intermediate rated power IRS infrared suppressor ISA International Standard Atmosphere ISO International Organization for Standardization JG jet group MCP maximum continuous power MCT maximum continuous thrust MJ Mega-Joule MRP maximum rated power NDARC NASA Design and Analysis of Rotorcraft OEI one engine inoperative OGE out of ground effect PG propulsion group RPJEM referred parameter jet engine model RPTEM referred parameter turboshaft engine model SDGW structural design gross weight SI Syst` eme International d’Unit´ es (International System of Units) SLS sea level standard TAS true airspeed WMTO maximum takeoff weight Weights W design gross weight D W empty weight E W maximum takeoff weight M T O W structural design gross weight SD W gross weight, W = W + W = W + W + W G G E U L O pay fuel W operating weight, W = W + W O O E F U L W useful load, W = W + W + W U L U L F U L pay fuel W payload pay W fuel weight fuel W fixed useful load F U L W mission fuel burn burn W vibration control weight vib W contingency weight cont χ technology factor Nomenclature 11 Fuel Tanks W fuel capacity, maximum usable fuel weight fuel − cap E fuel capacity, maximum usable fuel energy fuel − cap V fuel capacity, volume fuel − cap N number of auxiliary fuel tanks auxtank W auxiliary fuel tank capacity (weight) aux − cap E auxiliary fuel tank capacity (energy) aux − cap Power P power required, propulsion group; P + P + P reqP G comp xmsn acc P power required, engine group reqEG P power required, charge group reqCG ∑ P power available, propulsion group; min( f P , (Ω / Ω ) P ) avP G P avEG prim ref DS limit P power available, engine group; ( N − N ) P avEG eng inop av P power available, charge group; ( N − N ) P avCG chrg inop av P component power required comp P transmission losses xmsn P accessory power acc N number of inoperative systems, engine group or jet group or charge group inop P drive system torque limit (specified as power limit at reference rotor speed) DS limit P engine shaft limit ES limit P rotor shaft limit RS limit Engine P sea level static power available per engine at specified takeoff rating eng N number of engines in engine group eng P power available, installed; min( P − P , P ) av a loss mech P power available, uninstalled a P power required, installed; P − P req q loss P power required, uninstalled q P installation losses loss P mechanical power limit mech SP specific power, P/ ˙ m (conventional units) sfc specific fuel consumption, ˙ w/P (conventional units) ˙ m mass flow (conventional units) ˙ w fuel flow (conventional units) ˙ E energy flow F net jet thrust N D momentum drag aux N specification turbine speed SW specific weight, P/W 12 Nomenclature Jet T sea level static thrust available per jet at specified takeoff rating eng N number of jets in jet group jet T thrust available, jet group; ( N − N ) T avCG jet inop av T thrust available, installed; min( T η, T ) av a mech T thrust available, uninstalled a T thrust required, jet group reqJG T thrust required, installed; T η req q T thrust required, uninstalled q η installation losses (efficiency) T mechanical thrust limit mech ST specific thrust, T / ˙ m (conventional units) sfc specific fuel consumption, ˙ w/T (conventional units) ˙ m mass flow (conventional units) ˙ w fuel flow (conventional units) D momentum drag aux SW specific weight, T /W Charger P sea level static power available per charger at specified takeoff rating chrg N number of chargers in charge group chrg η installation losses (efficiency) Tip Speed and Rotation V reference tip speed, propulsion group primary rotor; each drive state tip − ref r gear ratio; Ω / Ω for rotor, Ω / Ω for engine dep prim spec prim Ω primary rotor rotational speed, Ω = V /R prim tip − ref Ω dependent rotor rotational speed, Ω = V /R dep tip − ref Ω specification engine turbine speed spec N specification engine turbine speed (rpm) spec Mission T mission segment time D mission segment distance dR mission segment range contribution E endurance R range ˙ w fuel flow Nomenclature 13 Environment g gravitational acceleration h altitude c speed of sound s ρ density ν kinematic viscosity μ viscosity ◦ ◦ T temperature, R or K ◦ ◦ τ temperature, F or C V wind speed w Axis Systems I inertial F aircraft A component aerodynamic B component V velocity Geometry SL, BL, WL fixed input position (station line, buttline, waterline) positive aft, right, up; arbitrary origin x/L , y/L , z/L scaled input position; positive aft, right, up; origin at reference point L reference length (fuselage length, rotor radius, or wing span) x , y , z calculated position, aircraft axes; positive forward, right, down; origin at reference point for geometry, origin at center of gravity for motion and loads F z component position vector, in aircraft axes, relative reference point length S wetted area wet 14 Nomenclature Motion φ , θ , ψ roll, pitch, yaw angles; orientation airframe axes F relative inertial axes F F F ˙ ψ turn rate F θ , ψ climb, sideslip angles; orientation velocity axes V relative inertial axes V V F v aircraft velocity AC F ω aircraft angular velocity AC F a aircraft linear acceleration AC n load factor V aircraft velocity magnitude V horizontal velocity h V forward velocity f V sideward velocity s V climb velocity c V calibrated airspeed cal Aerodynamics and Loads v component velocity relative air (including interference) q dynamic pressure, / ρ | v | α angle of attack, component axes B relative aerodynamic axes A β sideslip angle, component axes B relative aerodynamic axes A ratio flap chord to airfoil chord, c /c f f δ flap deflection f F force M moment D , Y , L aerodynamic drag, side, lift forces (component aerodynamic axes A) M , M , M aerodynamic roll, pitch, yaw moments (component aerodynamic axes A) x y z c , c section drag, lift coefficients d  C , C , C component drag, side, lift force coefficients D Y L C , C , C component roll, pitch, yaw moment coefficients  M N D/q drag area, SC ( S = reference area of component) D Aircraft DL disk loading, W /A D ref ∑ A reference rotor area, f A ; typically projected area of lifting rotors ref A WL wing loading, W /S D ref ∑ S reference wing area, S ; sum area all wings ref c aircraft control AC T control matrix c component control, c = ST c + c AC 0 α tilt control variable tilt √ M aircraft hover figure of merit, W W/ 2 ρA /P ref D aircraft effective drag, P/V e L/D aircraft effective lift-to-drag ratio, W V /P e Nomenclature 15 Rotor W/A disk loading, W = f W W D C /σ design blade loading, W/ρAV σ ( V = hover tip speed) W tip tip R blade radius A disk area σ solidity (ratio blade area to disk area) T design thrust of antitorque or auxiliary-thrust rotor design r direction of rotation ( 1 for counter-clockwise, − 1 for clockwise) r blade span coordinate ψ blade azimuth coordinate μ advance ratio λ inflow ratio M advancing tip Mach number at ν blade flap frequency (per-rev) γ blade Lock number C /σ thrust coefficient divided by solidity, T /ρA (Ω R ) σ T β , β longitudinal, lateral flapping (tip-path plane tilt relative shaft) c s θ blade collective pitch angle (at 75% radius) 0 . 75 θ , θ lateral, longitudinal blade pitch angle) c s H , Y , T drag, side, thrust force on hub (shaft axes) M , M roll, pitch moment on hub x y Q shaft torque P , P , P , P induced, interference, profile, parasite power i t o p κ induced power factor, P = κP i ideal c profile power mean drag coefficient, C = ( σ/ 8) c F d mean P o d mean P M rotor hover figure of merit, T f v/P D L/D rotor effective lift-to-drag ratio, V L/ ( P + P ) e i o η propulsive efficiency, T V /P Wing W/S wing loading, W = f W W D S area b span c chord, S/b A R aspect ratio, b /S 16 Nomenclature

Chapter 3 Tasks The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps.

3–1 Size Aircraft for Design Conditions and Missions 3-1.1 Sizing Method The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight ( W ) or weight empty ( W ), rotor radius ( R ), and engine power available ( P ).

D E eng The relationships between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized: a) Engine power: Determine P , for fixed R . The engine power is the maxi- eng mum of the power required for all designated sizing flight conditions and sizing missions (typically including vertical flight, forward flight, and one-engine inopera- tive). Hence the engine power is changed by the ratio max( P /P ) (exclud- reqP G avP G ing flight states for which zero power margin is calculated, such as maximum gross weight or maximum effort). This approach is the one most commonly used for the sizing task.

b) Rotor radius: Determine R for input P . The maximum power required for all eng designated sizing flight conditions and sizing missions is calculated, and then the rotor radius determined such that the power required equals the input power available.

√ The change in radius is estimated as R = R P /P (excluding flight old reqP G avP G states for which zero power margin is calculated, such as maximum gross weight or maximum effort). For multi-rotor aircraft, the radius can be fixed rather than sized for some rotors.

Alternatively, P and R can be input rather than sized. For each jet group, the design thrust can be eng sized: Determine T . The design thrust is the maximum of the thrust required for all jet designated sizing flight conditions and sizing missions. Hence the design thrust 18 Tasks is changed by the ratio max( T /T ) (excluding flight states for which zero reqJG avJG thrust margin is calculated).

For each charge group, the design power can be sized: Determine P . The design power is the maximum of the power required for all chrg designated sizing flight conditions and sizing missions. Hence the design power is changed by the ratio max( P /P ) (excluding flight states for which zero reqCG avCG power margin is calculated).

Aircraft parameters can be determined by a subset of the design conditions and missions: a) Design gross weight W : maximum gross weight from designated conditions D and missions (for which gross weight is not fixed).

b) Maximum takeoff gross weight W : maximum gross weight from designated M T O conditions (for which gross weight is not fixed).

c) Drive system torque limit P : maximum torque from designated conditions DS limit and missions (for each propulsion group; specified as power limit at reference rotor speed).

d) Fuel tank capacity: maximum fuel weight W or energy E from fuel − cap fuel − cap designated missions (without auxiliary tanks).

e) Antitorque or auxiliary thrust rotor design thrust T : maximum rotor thrust design from designated conditions and missions.

Alternatively, these parameters can be fixed at input values. The design gross weight ( W ) can be fixed.

D The weight empty can be fixed, which is implemented by changing the contingency weight.

A successive substitution method is used for the sizing iteration, with an input tolerance  . Re- laxation is applied to P or R , T , P , W , W , P , W or E , and T .

eng jet chrg D M T O DS limit fuel − cap fuel − cap design Convergence is tested in terms of these parameters, and the aircraft weight empty W . Two successive E substitution loops are used. The outer loop is an iteration on performance: engine power or rotor radius, jet thrust, charger power. The inner loop is an iteration on parameters: W , W , P , W D M T O DS limit fuel − cap or E , and T . Either loop can be absent, depending on the definition of the size task.

fuel − cap design For each flight condition and each mission, the gross weight and useful load are specified. The gross weight can be input, maximized, or fallout. For flight conditions, the payload or fuel weight can be specified, and the other calculated; or both payload and fuel weight specified, with gross weight fallout.

For missions, the payload or fuel weight can be specified, the other fallout, and then time or distance of mission segments adjusted; or fuel weight calculated from mission, and payload fallout; or both payload and fuel weight specified (or payload specified and fuel weight calculated from mission), with gross weight fallout. For each flight condition and mission segment, the following checks are performed: a) The power required does not exceed the power available: P ≤ (1 +  ) P reqP G avP G (for each propulsion group).

b) The torque required does not exceed the drive system limit: P / Ω ≤ reqP G (1 +  ) P / Ω (for each propulsion group). Rotor shaft torque and engine DS limit prim shaft torque are also checked.

c) The jet thrust required does not exceed the thrust available: T ≤ (1+  ) T reqJG avJG (for each jet group).

Tasks 19 d) The charger power required does not exceed the power available: P ≤ reqCG (1 +  ) P (for each charge group).

avCG e) The fuel weight does not exceed the fuel capacity: W ≤ (1 +  )( W + fuel fuel − cap ∑ N W ) (including auxiliary tanks).

auxtank aux − cap These checks are performed using an input tolerance  .

Sizing flight conditions typically include takeoff (hover or specified vertical rate of climb), one- engine inoperative, cruise or dash, perhaps transmission, and perhaps mission midpoint hover. Sizing missions typically include a design mission and a mission to determine fuel tank capacity.

3-1.2 Component Sizing 3-1.2.1 Propulsion System The engine size is described by the power P , which is the sea-level static power available per eng engine at a specified takeoff rating. The number of engines N is specified for each engine group.

eng If the sizing task determines the engine power for a propulsion group, the power P of at least one eng ∑ engine group is found (including the first engine group). The total power required is P = r N P , P G eng eng ∑ where r = max( P /P ) . The sized power is P = P − N P , where the sum is reqP G avP G sized P G eng eng fixed over the engine groups for which the power is fixed. Then the sized engine power is P = f P /N eng n sized eng ∑ for the n -th engine group (with f an input ratio and f = f for the first group).

n 1 n n  =1 , sized The jet size is described by the thrust T , which is the sea-level static thrust available per jet at jet a specified takeoff rating. The number of jets N is specified for each jet group. If the sizing task jet determines the jet thrust for a jet group, the thrust T is scaled by the factor r = max( T /T ) .

jet reqJG avJG The charger size is described by the power P , which is the sea-level static power available. The chrg number of systems N is specified for each charge group. If the sizing task determines the charger chrg power for a charge group, the power P is scaled by the factor r = max( P /P ) .

chrg reqCG avCG 3-1.2.2 Main Rotor The main-rotor size is defined by the radius R or disk loading W/A , thrust-weighted solidity σ , hover tip speed V , and blade loading C /σ = W/ρAV σ . With more than one main-rotor, the disk tip W tip loading and blade loading are obtained from an input fraction of design gross weight, W = f W . The W D air density ρ for C /σ is obtained from a specified takeoff condition.

W If the rotor radius is fixed for the sizing task, three of ( R or W/A ), C /σ , V , and σ are input, and W tip the other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius of another rotor.

If the sizing task determines the rotor radius ( R and W/A ), then two of C /σ , V , and σ are input, W tip and the other parameter is derived. The radius can be sized for just a subset of the rotors, with fixed radius for the others. The radii of all sized rotors are changed by the same factor.

3-1.2.3 Antitorque or Auxiliary Thrust Rotor For antitorque and auxiliary thrust rotors, three of ( R or W/A ), C /σ , V , and σ are input, and the W tip other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius of another rotor. Optionally the radius can be scaled with the main-rotor radius. The disk loading and 20 Tasks blade loading are based on f T , where f is an input factor and T is the maximum thrust from T design T design designated design conditions and missions.

3-1.2.4 Wing The wing size is defined by the wing area S or wing loading W/S , span (perhaps calculated from other geometry), chord, and aspect ratio. With more than one wing, the wing loading is obtained from an input fraction of design gross weight, W = f W .

W D Two of the following parameters are input: area (or wing loading), span, chord, and aspect ratio; the other parameters are derived. Optionally the span can be calculated from the rotor radius, fuselage width, and clearance (typically used for tiltrotors). Optionally the span can be calculated from a specified ratio to the span of another wing.

3-1.2.5 Fuel Tank The fuel tank capacity W (maximum usable fuel weight) or E (maximum usable fuel fuel − cap fuel − cap energy) is determined from designated sizing missions. The maximum mission fuel required, W fuel − miss or E (excluding reserves and any fuel in auxiliary tanks), gives fuel − miss W = max( f W , W + W ) fuel − cap fuel − cap fuel − miss fuel − miss reserve E = max( f E , E + E ) fuel − cap fuel − cap fuel − miss fuel − miss reserve where f ≥ 1 is an input factor. Alternatively, the fuel tank capacity W or E can fuel − cap fuel − cap fuel − cap be input. Optionally the maximum mission battery discharge power gives P , from which E = cap fuel − cap max( E , ( e /π ) P ) (MJ from kW).

fuel − cap tank tank cap 3-1.2.6 Weights The structural design gross weight W and maximum takeoff weight W can be input, or SD M T O specified as an increment d plus a fraction f of a weight W : { d + f W SDGW SDGW D W = d + f W = SD SDGW SDGW d + f ( W − W + f W ) SDGW SDGW D fuel fuel fuel − cap { d + f W W M T O W M T O D W = d + f W = M T O W M T O W M T O d + f ( W − W + W ) W M T O W M T O D fuel fuel − cap This convention allows the weights to be input directly ( f = 0 ), or scaled with W . For W , W is D SD the design gross weight W , or W adjusted for a specified fuel state (input fraction of fuel capacity).

D D Alternatively, W can be calculated as the gross weight at a designated sizing flight condition. For SD W , W is the design gross weight W , or W adjusted for maximum fuel capacity. Alternatively, M T O D D W can be calculated as the maximum gross weight possible at a designated sizing flight condition.

M T O 3-1.2.7 Drive System Limit The drive system limit is defined as a power limit, P . The limit is properly a torque limit, DS limit Q = P / Ω , but is expressed as a power limit for clarity. The drive system limit can be DS limit DS limit ref specified as follows (with f an input factor): limit a) Input P .

DS limit ∑ b) From the engine takeoff power limit, P = f N P (summed over DS limit limit eng eng Tasks 21 all engine groups).

c) From the power available at the transmission sizing conditions and missions, ∑ P = f (Ω / Ω ) N P (largest of all conditions and segments).

DS limit limit ref prim eng av d) From the power required at the transmission sizing conditions and missions, ∑ P = f (Ω / Ω ) N P (largest of all conditions and segments).

DS limit limit ref prim eng req The drive system limit is a limit on the entire propulsion system. To account for differences in the distribution of power through the drive system, limits are also used for the torque of each rotor shaft ( P ) and of each engine group ( P ). The engine shaft limit is calculated as for the drive system RS limit ES limit limit, without the sum over engine groups. The rotor shaft limit is either input or calculated from the rotor power required at the transmission sizing flight conditions. The power limit is associated with a reference rotational speed, and when applied the limit is scaled with the rotational speed of the flight state. The rotation speed for the drive system limit P is the hover speed of the primary rotor of DS limit the propulsion group (for the first drive state). The rotation speed for the engine shaft limit P ES limit is the corresponding engine turbine speed. The rotation speed for the rotor shaft limit P is the RS limit corresponding speed of that rotor.

The drive system limits can be specified for several levels, analogous to engine ratings. The limit P is associated with the maximum continuous rating (MCQ or MCP). An alternate rating changes DS limit the torque limit by the factor x . Typically x > 1 for ratings associated with short duration operation.

The torque limit is calculated from Q = Q/x for the flight condition or mission segment. The torque limit limit is applied as Q = xQ .

limit 3–2 Mission Analysis For the mission analysis, the fuel weight or payload weight is calculated. Power required, torque (drive system, engine shaft, and rotor shaft), and fuel weight are then verified to be within limits.

Missions can be fixed or adjustable.

3–3 Flight Performance Analysis For each performance flight condition, the power required is calculated or maximum gross weight is calculated. Power required, torque (drive system, engine shaft, and rotor shaft), and fuel weight are then verified to be within limits.

3–4 Maps 3-4.1 Engine Performance The engine performance can be calculated for a specified range of power, altitude, and speed.

3-4.2 Airframe Aerodynamics The airframe aerodynamic loads can be calculated for a specified range of angle of attack, sideslip angle, and control angles. The aerodynamic analysis evaluates the component lift, drag, and moments F F A T given the velocity. The aircraft velocity is v = C ( v 0 0) ; interference velocity from the rotors is AC not considered. From the angle of attack α and sideslip angle β , the transformation from wind axes to F A F A airframe axes is C = Y Z (optionally C = Z Y can be used, for better behavior in sideward α − β − β α 22 Tasks flight). The loads are summed in the airframe axes (with and without tail loads), and then the wind axis loads are: ⎛ ⎞ ⎛ ⎞ − D M x A AF F A AF F ⎝ ⎠ ⎝ ⎠ F = Y = C F M = M = C M y − L M z The center of action for the total loads is the fuselage location z . The ratio of the loads to dynamic fuse pressure is required, so a nominal velocity v = 100 (ft/sec or m/sec) and sea level standard density are used.

Chapter 4 Operation 4–1 Flight Condition Flight conditions are specified for the sizing task and for the flight performance analysis. For each condition, a flight state is also defined. For the sizing task, certain flight conditions are designated for engine sizing, design gross weight calculations, transmission sizing, maximum takeoff weight calcula- tions, or rotor thrust sizing. The flight condition parameters include gross weight and useful load. The gross weight can be specified as follows, consistent with the sizing method: a) Design gross weight, W (calculated or input).

D b) Structural design gross weight, W , or maximum takeoff weight, W (which SD M T O may depend on W ).

D c) Function of W : W = d + f W (with d an input weight and f an input factor).

D D d) Function of W ( W = d + f W ); or function of W ( W = d + f W ).

SD SD M T O M T O e) Input W .

f) Gross weight from specified mission segment or flight condition.

g) Gross weight maximized, such that power required equals specified power: P = f P + d , with d an input power and f an input factor; in general, reqP G avP G min(( f P + d ) − P ) = 0 , minimum over all propulsion groups; default avP G reqP G d = 0 and f = 1 gives zero power margin, min( P − P ) = 0 .

avP G reqP G h) Gross weight maximized, such that thrust required equals specified thrust: T = f T + d , with d an input thrust and f an input factor; in general, reqJG avJG min(( f T + d ) − T ) = 0 , minimum over all jet groups; default d = 0 and avJG reqJG f = 1 gives zero thrust margin, min( T − T ) = 0 .

avJG reqJG i) Gross weight maximized, such that transmission torque equals limit: zero torque margin, min( P − P ) = 0 (mininum over all propulsion groups, engine groups, limit req and rotors).

j) Gross weight maximized, such that power required equals specified power, or thrust required equals specified thrust, or transmission torque equals limit (most restrictive).

k) Gross weight fallout from input payload and fuel weights: W = W + W + G O pay W .

fuel Only the last five options are available for W design conditions in the sizing task. The gross weight can D be obtained from a mission segment only for the sizing task. Optionally the altitude can be obtained from the specified mission segment or flight condition. The secant method or the method of false position is used to solve for the maximum gross weight. A tolerance  and a perturbation Δ are specified.

24 Operation The useful load can be specified as follows, consistent with the sizing method and the gross weight specification.

a) Input payload weight W , fuel weight fallout: W = W − W − W .

pay fuel G O pay b) Input fuel weight W , payload weight fallout: W = W − W − W .

fuel pay G O fuel c) Input payload and fuel weights, gross weight fallout (must match gross weight option): W = W + W + W .

G O pay fuel ∑ The input fuel weight is W = min( d + f W , W ) + N W . For fallout fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap fuel weight, N is changed (optionally only increased). If the auxiliary tank weight is greater than auxtank the increment in fuel weight needed, then the fallout fuel weight W = W − W − W can not be fuel G O pay achieved; in such a case, the fuel weight is capped at the maximum fuel capacity and the payload weight changed instead. The fixed useful load can have increments, including crew weight increment; equipment weight increment; and installed folding, wing, wing extension, and other kits. These increments are reflected in the fallout weight. If the motive device burns energy not weight, then the fuel weight is zero ∑ and the input fuel energy is E = min( d + f E , E ) + N E .

fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap 4–2 Mission Missions are defined for the sizing task and for the mission performance analysis. A mission consists of a specified number of mission segments. A flight state is defined for each mission segment.

For the sizing task, certain missions are designated for engine sizing, design gross weight calculations, transmission sizing, or fuel tank sizing. The mission parameters include mission takeoff gross weight and useful load. The gross weight can be specified as follows, consistent with the sizing method: a) Design gross weight, W (calculated or input).

D b) Structural design gross weight, W , or maximum takeoff weight, W (which SD M T O may depend on W ).

D c) Function of W : W = d + f W (with d an input weight and f an input factor).

D D d) Function of W , W = d + f W ; or function of W , W = d + f W .

SD SD M T O M T O e) Input W .

f) Gross weight maximized at specified mission segments, such that power required equals specified power: P = f P + d , with d an input power and f an input reqP G avP G factor; in general, min(( f P + d ) − P ) = 0 , minimum over all propulsion avP G reqP G groups; default d = 0 and f = 1 gives zero power margin, min( P − P ) = 0 .

avP G reqP G g) Gross weight maximized at specified mission segments, such that thrust required equals specified thrust: T = f T + d , with d an input power and f an input reqJG avJG factor; in general, min(( f T + d ) − T ) = 0 , minimum over all jet groups; avJG reqJG default d = 0 and f = 1 gives zero thrust margin, min( T − T ) = 0 .

avJG reqJG h) Gross weight maximized at specified mission segments, such that transmission torque equals limit: zero torque margin, min( P − P ) = 0 (mininum over all limit req propulsion groups, engine groups, and rotors).

i) Gross weight maximized at specified mission segments, such that power required equals specified power, or thrust required equals specified thrust, or transmission torque equals limit (most restrictive).

j) Gross weight fallout from input initial payload and fuel weights: W = W + G O Operation 25 W + W .

pay fuel k) Gross weight fallout from input initial payload weight and calculated mission fuel weight: W = W + W + W .

G O pay fuel If maximum gross weight is specified for more than one mission segment, then the minimum takeoff gross weight increment is used; so the power or torque margin is zero for the critical segment and positive for other designated segments. Only the last six options are available for W design conditions in the D sizing task. The secant method or the method of false position is used to solve for the maximum gross weight. A tolerance  and a perturbation Δ are specified.

The useful load can be specified as follows, consistent with the sizing method and the gross weight specification: a) Input initial payload weight W , fuel weight fallout: W = W − W − W .

pay fuel G O pay b) Input fuel weight W , initial payload weight fallout: W = W − W − W .

fuel pay G O fuel c) Calculated mission fuel weight, initial payload weight fallout: W = W − pay G W − W .

O fuel d) Input payload and fuel weights, takeoff gross weight fallout (must match gross weight option): W = W + W + W .

G O pay fuel e) Input payload weight and calculated mission fuel weight, takeoff gross weight fallout (must match gross weight option): W = W + W + W .

G O pay fuel ∑ The input fuel weight is W = min( d + f W , W ) + N W ; if the fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap fuel weight is fallout, then this is the initial value for the mission iteration. If the fuel weight is not calculated from the mission, then the mission is changed. The fixed useful load can have increments, including installed folding kits; other increments are specified for individual mission segments. If the motive device burns energy not weight, then the fuel weight is zero and the input fuel energy is ∑ E = min( d + f E , E ) + N E .

fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap The takeoff gross weight is evaluated at the start of the mission, perhaps maximized for zero power margin at a specified mission segment (either takeoff conditions or midpoint). Then the aircraft is flown for all segments. For calculated mission fuel weight, the fuel weight at takeoff is set equal to the fuel required for the mission (burned plus reserve). For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is thus on mission fuel weight or energy.

Range credit segments (defined below) can also require an iteration. A successive substitution method is used if an iteration is required, with a tolerance  specified. The iteration to maximize takeoff gross weight could be an outer loop around the mission iteration, but instead it is executed as part of the mission iteration. At the specified mission segment, the gross weight is maximized for zero power margin, and the resulting gross weight increment added to the takeoff gross weight for the next mission iteration.

Thus takeoff gross weight is also a variable of the mission iteration.

Each mission consists of a specified number of mission segments. The following segment types can be specified: a) Taxi or warm-up (fuel burned but no distance added to range).

b) Distance: fly segment for specified distance (calculate time).

c) Time: fly segment for specified time (calculate distance).

d) Hold: fly segment for specified time (loiter, so fuel burned but no distance added 26 Operation to range).

e) Climb: climb or descend from present altitude to next segment altitude (calculate time and distance).

f) Spiral: climb or descend from present altitude to next segment altitude (fuel burned but no distance added to range).

g) Fuel: use or replace specified fuel amount (calculate time and distance).

h) Burn: use or replace specified fuel amount (calculate time but no distance added to range).

For each mission segment a payload weight can be specified; or a payload weight change can be specified, as an increment from the initial payload or as a fraction of the initial payload. If the payload is calculated from the number of passengers, then for each mission segment a change in the number of passengers can be specified.

The number of auxiliary fuel tanks can change with each mission segment: N is changed auxtank based on the fuel weight (optionally only increased relative to the input number at takeoff, optionally fixed during mission). For input fuel weight, N is specified at takeoff. For fallout fuel weight, the takeoff auxtank fuel weight is changed for the auxiliary fuel tank weight given N (fixed W − W = W + W ).

auxtank G pay O fuel If the auxiliary tank weight is greater than the increment in fuel weight needed, then the fallout fuel weight W = W − W − W can not be achieved; in such a case, the fuel weight is capped at the fuel G O pay maximum fuel capacity and the takeoff payload weight changed instead. For fuel tank design missions, N and fuel tank capacity is determined from W . Optionally the aircraft can refuel (either on the auxtank fuel ground or in the air) at the start of a mission segment, by either filling all tanks to capacity or adding a specified fuel weight. Optionally fuel can be dropped at the start of a mission segment. The fixed useful load can have changes, including crew weight increment, equipment weight increment, and installed wing extension and other kits.

For calculation of the time or distance in a mission segment, a headwind or tailwind can be specified.

The wind velocity is a linear function of altitude h : V = ± (max(0 , d + f h )) , with the plus sign w wind wind th for a headwind and the minus sign for a tailwind. For example, California-to-Hawaii 85 percentile winter quartile headwind profile is V = 9 . 59 + 0 . 00149 h (with altitude h in ft).

w Mission fuel reserves can be specified in several ways for each mission. Fuel reserves can be defined in terms of specific mission segments, for example 200 miles plus 20 minutes at speed for best endurance. Fuel reserves can be an input fraction of the fuel burned by all (except reserve) mission segments, so W = (1 + f ) W or E = (1 + f ) E . Fuel reserves can be an input fraction fuel res burn fuel res burn of the fuel capacity, so W = W + f W or E = E + f E . If more than fuel burn res fuel − cap fuel burn res fuel − cap one criterion for reserve fuel is specified, the maximum reserve is used. Time and distance in reserve segments are not included in endurance and range.

To facilitate specification of range, range calculated for a group of segments (typically climb and descent segments) can be credited to a designated distance segment. For mission analysis, missions can be fixed or adjustable. In an adjustable mission, the fuel is input, so the time or distance in specified segments is adjusted based on the calculated fuel burned. If more than one segment is adjusted, all must be distance or all must be time or hold. Each segment can have only one special designation: reserve, adjustable, or range credit.

A segment with a large distance, time, or altitude change can be split into several segments, for more accurate calculation of the performance and fuel burned. The number of segments n can be input, Operation 27 or calculated from an input increment Δ : n = [ x/ Δ] + 1 , where the brackets indicate integer truncation, and x is the total distance, time, or altitude change. Then the change for each split segment is Δ = x/n .

Table 4-1 summarizes the time T , distance D , and range dR calculations for each segment. The ˙ ˙ segment fuel burned is dW = T ˙ w , where ˙ w is the fuel flow; and dE = T E , where E is the energy burn burn flow. The horizontal velocity is V , and the vertical velocity (climb or descent) is V . The altitude at the h c start of the segment is h , and at the end of the segment (start of next segment) h . The wind speed is end V , and the ground speed is V − V . The air distance is calculated from the time and speed ( D/V ), w h w h without the wind speed.

To use or replace fuel (weight or energy), the increment is specified in terms of the capacity ( dW = d + f W ) or the current fuel ( dW = d + f W ). Alternatively, a burn tank tank fuel − cap burn tank tank fuel target fuel is specified and the increment calculated from the current fuel. The tank is charged if the ˙ energy rate is negative. The segment time T is the minimum of dW / ˙ w or dE / E for all fuel tank burn burn systems.

In an adjusted mission, the distances or times are changed at the end of the mission such that the sum of the fuel burned increments will equal the difference between takeoff fuel weight (plus any ∑ ∑ ∑ added fuel) and the calculated mission fuel: dW = ˙ w dT = ˙ w dD/ ( V − V ) = Δ W . The burn h w fuel increments are apportioned among the adjusted segments by the factor f , determined from the ratio of ( ∑ ) the input distances or times: dD = f Δ D or dT = f Δ T . Hence Δ D = Δ W / f ˙ w/ ( V − V ) or fuel h w ( ∑ ) Δ T = Δ W / f ˙ w . The approach is similar if fuel energy is used, not weight. For a segment that fuel is a source of range credit, the range increment is set to zero and the distance D is added to D of other the destination segment. For the destination segment, the range contribution remains fixed at the input value, but the time and hence fuel burned are calculated from (dist − D ) . It is necessary to separately other accumulate D from earlier segments and D from later segments; D from later segments other other other are estimated initially from the last iteration. At the end of the mission, the times and fuel burned are recalculated for all range credit destination segments.

Table 4-1. Mission segment calculations.

segment kind time T distance D range dR taxi time 0 D distance D/ ( V − V ) dist D h w time time T ( V − V ) D h w hold time 0 D climb ( h − h ) /V T ( V − V ) D end c h w spiral ( h − h ) /V 0 D end c ˙ fuel dW / ˙ w or dE / E T ( V − V ) D burn burn h w ˙ burn dW / ˙ w or dE / E 0 D burn burn range credit source T D 0 destination D/ ( V − V ) dist − D dist h w other adjusted distance T + dD/ ( V − V ) D + dD = D + f Δ D D h w new time T + dT = T + f Δ T D + dT ( V − V ) D h w new hold T + dT = T + f Δ T 0 D new 28 Operation The segment time, distance, and fuel burned are evaluated by integrating over the segment duration.

This integration can be performed by using the horizontal velocity, climb velocity, and fuel flow obtained for the flight state with the gross weight and altitude at the start of the segment; or at the middle of the segment; or the average of the segment start and segment end values (trapezoidal integration). The gross weight at the segment middle equals the gross weight at the segment start, less half the segment fuel burned (obtained from the previous mission iteration). The gross weight at the segment end equals the gross weight at the segment start, less the segment fuel burned. With trapezoidal integration, for the output the flight state is finally evaluated at the segment middle.

∑ ∑ The mission endurance (block time), range, and fuel burned are E = T , R = dR , W = burn ∑ dW (sum over all non-reserve segments). The reserve fuel from mission segments is W = burn res ∑ dW (sum over all reserve segments). Optionally the reserve fuel is the maximum of that from burn mission segments and the fraction f W , or the fraction f W . The calculated mission fuel res burn res fuel − cap is then W = W + W .

fuel burn res A fuel efficiency measure for the mission is the product of the payload and range, divided by the fuel weight: e = W R/W (ton-nm/lb or ton-nm/kg). A productivity measure for the mission is pay burn p = W V /W (ton-kt/lb or ton-kt/kg), where W is the operating weight and V the block speed; or pay O O p = W V /W (ton-kt/lb or ton-kt/kg). The Br´ eguet range equation R = RF ln( W /W ) is obtained pay burn 0 1 by integrating dR = − RF ( dW/W ) for constant range factor L/D W V /P e RF = = sfc sfc (√ ) √ 3 / 2 The endurance E = EF 2 W /W − 1 is obtained by integrating dE = − EF W ( dW/W ) for 0 1 0 constant endurance factor √ √ L/D W/W W/P W/W e 0 0 EF = = sfc V sfc Constant RF implies operation at constant L/D = W V /P . Constant EF implies operation at constant e √ 3 / 2 3 / 2 ( L/D ) W /V = W /P (or constant C /C for an airplane). It follows that overall range and e D L endurance factors can be calculated from the mission performance: R RF = ln W /W 0 1 E ( ) EF = √ 2 W /W − 1 0 1 where W = W is the takeoff weight, and W = W − W . If energy is burned, not weight, the 0 to 1 to burn efficiency metrics can be based on the equivalent fuel burned E /e and the range factor can be burn ref based on the equivalent specific fuel consumption. But since the weight does not change as energy is used, the equation dR = − RF ( dW/W ) integrates to R = RF ( W /W ) .

burn 4–3 Takeoff Distance The takeoff distance can be calculated, either as ground run plus climb to clear an obstacle or accelerate-stop distance in case of engine failure. The obstacle height h is typically 35 ft for commercial o transport aircraft, or 50 ft for military aircraft and general aviation. This calculation allows determination of the balanced field length: engine failure at critical speed, such that the distance to clear the obstacle Operation 29 horizontal obstacle h γ O ground slope G ground x inertial z inertial climb R TR γ γ relative ground ground run rotation h transition TR s s s s distance G R TR CL V V V V V V V=0 ground EF 1 R LO TR CL or climb start obstacle climb transition liftoff rotation decision engine failure speed accelerate stop s s distance A S V=0 ground V V V=0 EF 1 speed Figure 4-1. Takeoff distance and accelerate-stop distance elements.

equals the distance to stop. Landing and VTOL takeoff calculations are not implemented, as these are best solved as an optimal control problem.

The takeoff distance consists of a ground run, from zero ground speed to liftoff speed V , perhaps LO including engine failure at speed V ; then rotation, transition, and climb; or decelerate to stop. Figure EF 4-1 describes the elements of the takeoff distance and the accelerate-stop distance, with the associated speeds. The ground is at angle γ relative to the horizontal (inertial axes), with γ positive for takeoff up G G hill. The takeoff profile is defined in terms of ground speed or climb speed, input as calibrated airspeed (CAS). The aircraft speed relative to the air is obtained from the ground speed, wind, and ground slope.

The aircraft acceleration as a function of ground speed is integrated to obtain the ground distance, as well as the time, height, and fuel burned. Usually the speed increases from the start to liftoff (or engine failure), but the calculated acceleration depends on the flight state specification. The analysis checks for consistency of the input velocity and the calculated acceleration (on the ground), and for consistency of the input height and input or calculated climb angle (during climb).

The takeoff profile consists of a set of mission segments. The first segment is at the start of the takeoff, V = 0 . Subsequent segments correspond to the ends of the integration intervals. The last segment has the aircraft at the required obstacle height, or stopped on the ground. The mission can consist of just one takeoff, more than one takeoff, or both takeoff and non-takeoff segments. Takeoff segments contribute to the mission fuel burned, but do not contribute to the mission time, distance, or range. The takeoff distance calculation is performed for a set of adjacent segments, the first segment specified as the takeoff start, and the last segment identified as before a non-takeoff segment or before 30 Operation another takeoff start. The takeoff distance is calculated if a liftoff segment (with V ) is specified; LO otherwise the accelerate-stop distance is calculated. Table 4-2 summarizes the mission segments for takeoff calculations. There can be only one liftoff, engine failure, rotation, and transition segment (or none). The engine failure segment must occur before the liftoff segment. Rotation and transition segments must occur after liftoff. All ground run segments must be before liftoff, and all climb segments must be after liftoff. Takeoff segments (except start, rotation, and transition) can be split, in terms of height for climb and in terms of velocity for other segments. Splitting the takeoff or engine failure segment produces additional ground run segments. Separately defining multiple ground run, climb, or brake segments allows configuration variation during the takeoff.

Each takeoff segment requires that the flight state specify the appropriate configuration, trim option, and maximum effort. In particular, the number of inoperative engines for a segment is part of the flight state specification, regardless of whether or not an engine failure segment is defined. The engine failure segment (if present) serves to implement a delay in decision after failure: for a time t after engine failure, the engine rating, power fraction, and friction of the engine failure segment are used (so the engine failure segment corresponds to conditions before failure). The number of inoperative engines specified must be consistent with the presence of the engine failure segment. The takeoff is assumed to occur at fixed altitude (so the maximum-effort variable can not be altitude). The flight state velocity specification is superseded by the ground or climb speed input for the takeoff segment. The flight state specification of height above ground level is superseded by the height input for the takeoff segment.

The ground distance, time, height, and fuel burned are calculated for each takeoff segment. The takeoff distance or accelerate-stop distance is the sum of the ground distance of all segments. Takeoff segments do not contribute to mission time, distance, or range.

Table 4-2. Mission segments for takeoff calculation.

takeoff distance accelerate-stop distance start V = 0 start V = 0 ground run V ground run V engine failure V engine failure V EF EF ground run V brake V liftoff V brake V = 0 LO rotation V R transition V T R climb, to h V CL climb, to h V o CL 4-3.1 Ground Run The takeoff starts at zero ground speed and accelerates to liftoff ground speed V (input as CAS).

LO Possibly an engine failure speed V < V is specified. Start, liftoff, and engine failure segments EF LO designate events, but otherwise are analyzed as ground run segments. The decision speed V is t 1 1 seconds after engine failure (typically t = 1 to 2 sec). Up to t after engine failure, conditions of the 1 1 engine failure segment are used (so the engine failure segment corresponds to conditions before failure).

The aircraft acceleration is obtained from the thrust minus drag ( T − D in airplane notation), plus a Operation 31 friction force proportional to the weight on wheels ( W − L in airplane notation): ∑ ∑ M a = T − D − μ ( W − L ) = F − μ F x z from the force components in ground axes (rotated by the ground slope angle γ from inertial axes).

G Table 4-3 gives typical values of the friction coefficient μ . The velocity of the aircraft relative to the air is obtained from the ground velocity V , wind velocity V (assumed parallel to the ground here), w and the ground slope: V = ( V + V ) cos γ and V = ( V + V ) sin γ . The takeoff configuration is h w G c w G specified, including atmosphere, in ground effect, gear down, power rating, nacelle tilt, flap setting, and number of inoperative engines. An appropriate trim option is specified, typically fixed attitude with longitudinal force trimmed using collective, for a given longitudinal acceleration. Perhaps the net aircraft yaw moment is trimmed with pedal. The maximum-effort condition is specified: maximum longitudinal acceleration (ground axes) for zero power margin. The aircraft acceleration as a function of ground speed is integrated to obtain the segment time, ground distance, height, and fuel burned: ∫ ∫ ∫ ( ) ∑ ∑ dt dv 1 1 1 t = dt = dv = = + ( v − v ) = Δ t G 2 1 dv a 2 a a 2 1 seg seg ∫ ∫ ∫ ∫ ( ) ( ) ∑ ∑ dt v dv d ( v ) 1 1 1 v + v 2 1 2 2 s = v dt = v dv = = = + ( v − v ) = Δ t G 2 1 dv a 2 a 2 2 a 2 a 2 2 1 seg seg h = 0 G ∫ ( ) ∑ ˙ w + ˙ w f 2 f 1 w = ˙ w dt = Δ t G f seg Trapezoidal integration is used; each segment corresponds to the end of an integration integral.

Table 4-3. Typical friction coefficient μ .

surface rolling braking dry and hard 0.03–0.05 0.30–0.50 grass 0.08 0.20 ice 0.02 0.06–0.10 4-3.2 Brake After engine failure, the aircraft can decelerate to a stop. The operating engines are at idle. Reverse thrust is not permitted for the accelerate-stop distance calculation. The braking configuration is specified.

Typically no trim option is executed; rather the aircraft has fixed attitude with controls for zero rotor thrust (such as zero collective and pedal). The aircraft acceleration as a function of ground speed is integrated, as for ground run.

4-3.3 Rotation Rotation occurs at speed V ; usually V = V is used. The duration t is specified, then s = V t , R R LO R R R R h = 0 , and w = ˙ w t are the ground distance, height, and fuel burned. Typically t = 1 to 3 sec.

R R f R R 32 Operation 4-3.4 Transition Transition from liftoff to climb is modeled as a constant load factor pull-up to the specified climb ∼ angle γ , at speed V . Usually V = V is used, and typically n . 2 . From the load factor = 1 T R T R LO T R 2 2 ˙ n = 1+ V /gR , the flight path radius is R = V / ( g ( n − 1)) and the pitch rate is θ = V /R .

T R T R T R T R T R T R T R T R Then R V T R T R ˙ t = γ/ θ = γ = γ T R V g ( n − 1) T R T R s = R sin γ T R T R h = R (1 − cos γ ) T R T R w = ˙ w t T R f T R are the time, ground distance, height, and fuel burned.

4-3.5 Climb Climb occurs at an angle γ relative to the ground and air speed V , from the transition height CL h to the obstacle height h (perhaps in several climb segments). The climb configuration is specified, T R o including atmosphere, in ground effect, gear down or retracted, power rating, nacelle tilt, flap setting, and number of inoperative engines. An appropriate trim option is specified, typically aircraft force and moment trimmed with attitude and controls. The climb angle and air speed can be fixed or a maximum- effort condition can be specified. The maximum-effort options are fixed air speed and maximum rate of climb for zero power margin; or airspeed for best climb rate or best climb angle with maximum rate of climb for zero power margin. Not implemented is a maximum-effort calculation of maximum flight path acceleration for zero power margin, for specified climb angle; this calculation would require integration of the acceleration as a function of flight speed. For the climb segment, the input V is the magnitude CL of the aircraft velocity relative to the air, and the climb angle relative to the horizon is θ = γ + γ .

V G Hence from the maximum-effort calculation, the climb angle relative to the ground is γ = θ − γ and V G the ground speed is V = V cos γ − V (from the wind speed V ). Then ground CL w w t = s /V CL CL ground s = ( h − h ) / tan γ CL last h = h CL w = ˙ w t CL f CL are the time, ground distance, height, and fuel burned.

4–4 Flight State A flight state is defined for each flight condition (sizing task design conditions and flight performance analysis), and for each mission segment. The following parameters are required: a) Speed: flight speed and vertical rate of climb, with the following options: 1) Specify horizontal speed (or forward speed or velocity magnitude), rate of climb (or climb angle), and sideslip angle.

2) Hover or vertical flight (input vertical rate of climb; climb angle 0 or ± 90 deg).

3) Left or right sideward flight (input velocity and rate of climb; sideslip Operation 33 angle ± 90 deg).

4) Rearward flight (input velocity and rate of climb; sideslip angle 180 deg).

b) Aircraft motion: 1) Pitch and roll angles (Aircraft values or flight state input; initial values for trim variables, fixed otherwise).

2) Turn, pull-up, or linear acceleration.

c) Altitude: For mission segment, optionally input, or from last mission segment; climb segment end altitude from next segment.

d) Atmosphere: 1) Standard day, polar day, tropical day, or hot day at specified altitude.

2) Standard day, polar day, or tropical day plus temperature increment.

3) Standard day, polar day, or tropical day and specified temperature.

4) Input density and temperature.

5) Input density, speed of sound, and viscosity.

e) Height of landing gear above ground level. Landing gear state (extended or retracted).

f) Aircraft control state: input, or conversion schedule.

g) Aircraft control values (Aircraft values or flight state input; initial values for trim variables, fixed otherwise).

h) Aircraft center-of-gravity position (increment or input value).

For each propulsion group, the following parameters are required: i) Drive system state.

j) Rotor tip speed for primary rotor: 1) Input.

2) Reference.

3) Conversion schedule or function speed.

4) Default for hover, cruise, maneuver, one engine inoperative (OEI), or transmission sizing condition.

5) From input C /σ = t − μt , or μ , or M (where μ is the rotor advance T 0 1 at ratio, and M is the rotor advancing tip Mach number).

at For each engine group (which is associated with a propulsion group): k) Number of inoperative engines.

l) Infrared suppressor state: off (hot exhaust) or on (suppressed exhaust).

m) Engine rating, fraction of rated engine power available, and drive system rating.

For each jet group: n) Number of inoperative jets.

o) Jet rating, fraction of rated thrust available.

34 Operation For each charge group: p) Number of inoperative chargers.

q) Charger rating, fraction of rated power available.

Aircraft and rotor performance parameters for each flight state: r) Aircraft drag: forward flight drag increment, accounting for payload aerodynam- ics.

s) Rotor performance: induced power factor κ and profile power mean c .

d The aircraft trim state and trim targets are also specified.

The aircraft performance can be analyzed for the specified state, or a maximum-effort performance can be identified. For the maximum effort, a quantity and variable are specified. The available maximum- effort quantities include: a) Best endurance: maximum 1 / ˙ w .

b) Best range: 99% maximum V / ˙ w (high side), or low side, or 100%.

c) Best climb or descent rate: maximum V or 1 /P .

z d) Best climb or descent angle: maximum V /V or V /P .

z h e) Ceiling: maximum altitude.

f) Power limit: zero power margin, min( P − P ) = 0 (minimum over all avP G reqP G propulsion groups).

g) Torque limit: zero torque margin, min( Q − Q ) = 0 (minimum over all limit req propulsion groups, engine groups, and rotors; Q expressed as power at reference limit rotation speed).

j) Jet thrust limit: zero thrust margin, min( T − T ) = 0 (minimum over all avJG reqJG jet groups).

i) Power limit or torque limit or thrust limit: most restrictive.

˙ j) Battery limit: zero power margin, min( P − | E | ) = 0 (minimum over all fuel max batt tanks).

k) Rotor stall: zero rotor thrust margin, ( C /σ ) − C /σ = 0 (for designated rotor, T max T steady or transient limit).

l) Wing stall: zero wing lift margin, C − C = 0 (for designated wing).

L max L ˙ Here ˙ w is the aircraft fuel flow (or equivalent fuel flow E/e ), and P is the aircraft power. The available ref maximum-effort variables include: a) Horizontal velocity V or vertical rate of climb V (times an input factor).

h z b) Aircraft sideslip angle.

c) Altitude.

˙ ˙ d) Aircraft angular rate, θ (pull-up) or ψ (turn).

e) Aircraft linear acceleration (airframe, inertial, or ground axes).

f) Aircraft controls.

g) Aircraft orientation, θ (pitch) or φ (roll).

h) Propulsion group tip speed or engine speed.

If the variable is velocity, first the velocity is found for the specified maximum effort; then the performance is evaluated at that velocity times an input factor. For endurance, range, or climb, the slope of the quantity Operation 35 to be maximized must be zero; hence in all cases the target is zero. The slope of the quantity is evaluated by first-order backward difference. For the range, first the variable is found such that V / ˙ w is maximized (slope zero), then the variable is found such that V / ˙ w equals 99% of that maximum. Two maximum- effort quantity/variable pairs can be specified, and solved in nested iterations. The secant method or the method of false position is used to solve for the maximum effort. The task of finding maximum endurance, range, or climb is usually solved using the golden-section or curve-fit method. A tolerance  and a perturbation Δ are specified.

Given the gross weight and useful load (from the flight condition or mission specification), the performance is calculated for this flight state. The calculated state information includes weight, speed and velocity orientation (climb and sideslip), aircraft Euler angles, rotor tip speeds, and aircraft controls.

A number of performance metrics are calculated for each flight state. The aircraft effective drag is D = P/V , hence the effective lift-to-drag ratio is L/D = W V /P . For these metrics, the aircraft power e e is the sum of the engine group power and jet group propulsive power: P = P + V T . The specific req jet range is the ratio of the speed to the fuel flow: V / ˙ w (nm/lb or nm/kg). From the Br´ eguet range equation, it follows that the range for which the fuel equals 1% of the gross weight is ( ) L/D 1 e R = ln 1% GW sfc . 99 A fuel efficiency measure is the product of the payload and specific range: e = W ( V / ˙ w ) (ton- pay nm/lb or ton-nm/kg). A productivity measure is p = W V /W (ton-kt/lb or ton-kt/kg), where W is pay O O the operating weight. If fuel energy is used, not weight, these performance metrics are based on the equivalent ˙ w and equivalent sfc .

The aircraft weight statement defines the fixed useful load and operating weight for the design configuration. For each flight state, the fixed useful load may be different from the design configuration, because of changes in auxiliary fuel tank weight or kit weights or increments in crew or equipment weights. Thus the fixed useful load weight is calculated for the flight state; and from it the useful load weight and operating weight. The gross weight, payload weight, and usable fuel weight (in standard and auxiliary tanks) completes the weight information for the flight state.

4–5 Environment and Atmosphere The aerodynamic environment is defined by the speed of sound c , density ρ , and kinematic viscosity s ν = μ/ρ of the air (or other fluid). These quantities can be obtained from the standard day (International Standard Atmosphere), or input directly. Polar day, tropical day, and hot day atmospheres can also be used. The following options are implemented: a) Input the altitude h and a temperature increment Δ T . Calculate the temper- geom ature from altitude and pressure from temperature for the standard day (or polar day, tropical day, hot day), add Δ T , and then calculate the density from the equa- tion of state for a perfect gas. Calculate the speed of sound and viscosity from the temperature.

◦ ◦ b) Input the pressure altitude h and the temperature τ ( F or C). Calculate the geom pressure (from temperature vs. altitude) for the standard day (or polar day, tropical day, hot day), and then calculate the density from the equation of state for a perfect gas. Calculate the speed of sound and viscosity from the temperature.

36 Operation ◦ ◦ c) Input the density ρ and the temperature τ ( F or C). Calculate the speed of sound and viscosity from the temperature.

d) Input the density ρ , sound speed c , and viscosity μ . Calculate the temperature s from the sound speed.

Here h is the geometric altitude above mean sea level. The sources of the atmosphere descriptions geom are references 1–6.

The gravitational acceleration g can have the standard value or an input value.

The International Standard Atmosphere (ISA) is a model for the variation with altitude of pressure, temperature, density, and viscosity, published as International Standard ISO 2533 by the International Organization for Standardization (ISO) (ref. 1). The ISA is intended for use in calculations and design of flying vehicles, to present the test results of flying vehicles and their components under identical conditions, and to allow unification in the field of development and calibration of instruments. The ISA is defined up to 80 km geopotential altitude and is identical to the ICAO Standard Atmosphere up to 32 km.

Dry air is modeled in the ISA as a perfect gas with a mean molecular weight, and hence a gas constant R , defined by adopted values for sea level pressure, temperature, and density ( p , T , ρ ). The 0 0 0 speed of sound at sea level c is defined by an adopted value for the ratio of specific heats γ . The variation s 0 of temperature with geopotential altitude is defined by adopted values for vertical temperature gradients (lapse rates, L ) and altitudes ( h ). The variation of pressure with geopotential altitude is further defined b b by an adopted value for the standard acceleration of free fall ( g ). The variation of dynamic viscosity μ with temperature is defined by adopted values for Sutherland’s empirical coefficients β and S .

The required parameters are given in table 4-4, including the acceleration produced by gravity, g .

◦ ◦ ◦ The temperature T is in K, while τ is C (perhaps input as F); T = T + τ . The gas constant is zero R = p /ρ T , and μ is actually obtained from S and β . The standard atmosphere is defined in SI units.

0 0 0 0 Although table 4-4 gives values in both SI and English units, all the calculations for the aerodynamic environment are performed in SI units. As required, the results are converted to English units using the exact conversion factors for length and force.

The ISA consists of a series of altitude ranges with constant lapse rate L (linear temperature change b with altitude). Thus at altitude h , the standard day temperature is g T = T + L ( h − h ) std b b g b for h > h . The altitude ranges and lapse rates are given in table 4-5. Note that h is sea level, and h is g b 0 1 the boundary between the troposphere and the stratosphere. This altitude h is the geopotential height, g calculated assuming constant acceleration due to gravity. The geometric height h is calculated using an inverse square law for gravity. Hence h = rh/ ( r + h ) , where r is the nominal radius of the Earth. The g standard day pressure is obtained from hydrostatic equilibrium ( dp = − ρg dh ) and the equation of state g for a perfect gas ( p = ρRT , so dp/p = − ( g/RT ) dh ). In isothermal regions ( L = 0 ) the standard day g b pressure is p std − ( g/RT )( h − h ) g b = e p b and in gradient regions ( L  = 0 ) b ( ) − g/RL b p T std = p T b b Operation 37 where p is the pressure at h , obtained from these equations by working up from sea level. Let T , p , ρ , b b 0 0 0 c , μ be the temperature, pressure, density, sound speed, and viscosity at sea level standard conditions.

s 0 0 Then the density, sound speed, and viscosity are obtained from ( ) ( ) − 1 p T ρ = ρ p T 0 0 ( ) 1 / 2 T c = c s s 0 T ( ) 3 / 2 ( T /T ) μ = μ α ( T /T ) + 1 − α 3 / 2 where μ = βT / ( T + S ) and α = T / ( T + S ) . For the cases using input temperature, T = T + τ .

0 0 0 0 zero The density altitude and pressure altitude are calculated for reference. From the density and the standard day (troposphere only), the density altitude is: ( ) ( ) 1 / ( g/R | L |− 1) T ρ h = 1 − d | L | ρ 0 0 From the pressure p = ρRT and the standard day, ( ) ( ) ( ) ( ) 1 / ( g/R | L | ) 1 / ( g/R | L | ) 0 0 T p T ρ T 0 0 h = 1 − = 1 − p | L | p | L | ρ T 0 0 0 0 0 is the pressure altitude.

The polar and tropical days are atmospheric models that describe realistic profiles of extremes of temperature and density, needed to calculate performance in near-worst-case conditions (ref. 6).

These atmospheres are hydrodynamically balanced and can be used in calculations involving engine performance and aerodynamic characteristics, including calculations of true vertical velocity (ref. 4).

Tables of air properties for the polar and tropical days are given in MIL-STD-3013A, based on MIL- STD-210A. The break points in the lapse rates are evident in MIL-STD-210A, in terms of degrees C as a function of altitude in ft (geopotential altitude in MIL-C-5011B). The pressure ratio at sea level is δ = 30 . 268 / 29 . 92 = 1 . 0116 for the polar day, and δ = 1 for the tropical day. The altitude ranges and lapse rates are given in table 4-6 for the polar day and in table 4-7 for the tropical day. Figure 4-2 compares the temperature profiles. The air properties for the polar and tropical days are calculated from h , L , b b and T using the equations of hydrostatic equilibrium, as for the standard day. The primary data for the b ◦ polar and tropical days are the altitude in ft and temperature in C; h in km and L are derived using b b the exact conversion factor for length.

The conditions of the standard day, polar day, and tropical day are applicable to free air conditions.

Temperatures close to the surface of the earth, even at high elevations, can be considerably higher than those for free air. The hot day is a model of ground-level atmospheric conditions, to be used for takeoff and other ground operations at elevations up to 15,000 ft (ref. 6). The hot day properties are statistically sampled and are not hydrodynamically balanced, hence should be used for approximately constant altitude conditions (ref. 4).

◦ ◦ The origins of the hot day are in ref. 2, which has 103 F for sea level, and a lapse rate − 3 . 7 F per 1000 ft (geometric) to 40,000 ft. The pressure in the table is from the equations for equilibrium, which gives the pressure altitude; these data are not used further. According to MIL-STD-210A (paragraph 38 Operation 3.1.1), the pressure (hence pressure altitude) as a function of altitude was obtained from statistics. Up to 15,000 ft, the ratio of geometric altitude to pressure altitude is 1.050. The − 3 . 7 geometric lapse rate ◦ plus the mapping of geometric to pressure altitude gave F vs. pressure altitude, rounded to 1 decimal ◦ ◦ place. Then C was calculated from F, rounded to 1 decimal place. The pressure ratio δ followed from the pressure altitude. The geometric altitude (to 35,000 ft) in MIL-STD-210A was calculated from the temperature and the − 3 . 7 lapse rate, rounded to 100s. MIL-C-5011B has the same data as MIL- STD-210A, but does not give geometric altitude, and truncates the table at 15,000 ft. MIL-STD-3013A ◦ took C (already rounded to 1 decimal place) vs. pressure altitude (ft) from MIL-STD-210A as the ◦ temperature profile up to 15,000 ft, and calculated F (2 decimal places, so no more loss of information) ◦ ◦ ◦ from C. The geopotential altitude was calculated from C and − 3 . 7 F lapse rate, rounded to 100s; but this altitude information is not meaningful, since the atmosphere is not in equilibrium; only the pressure ◦ ◦ altitude is used. Thus the hot day model has a sea level temperature of 39 . 4 C ( 102 . 92 F). Curve fitting ◦ ◦ the data to 15,000 ft gives a lapse rate of − 7 . 065 C per 1000 m = − 2 . 1534 C per 1000 ft (table 4-8).

Given the pressure altitude h , the hot day temperature and pressure ratio are T = T + L ( h − h ) hot b b b ( ) − g/RL b p L hot 0 = 1 + h p T 0 0 using the standard day L = − 6 . 5 . A constant lapse rate fits the tabular data for the hot day atmosphere ◦ to only about 0 . 1 C. Thus the tabular data can be used directly instead (table 4-9, from MIL-C-5011B), with linear interpolation to the specified pressure altitude. MIL-STD-3013A has the same data, but only for altitudes a multiple of 1000 ft.

4–6 References 1) International Organization for Standardization. “Standard Atmosphere.” ISO 2533-1975(E), May 1975.

2) Theiss, E.C. “Proposed Standard Cold and Hot Atmospheres for Aeronautical Design.” Wright Air Development Center, United States Air Force, Technical Memorandum Report WCSE 141, June 1952.

3) Department of Defense Military Specification. “Climatic Extremes for Military Equipment.” MIL- STD-210A, August 1957.

4) Department of Defense Military Specification. “Charts: Standard Aircraft Characteristics and Perfor- mance, Piloted Aircraft (Fixed Wing). Appendix IC, Atmospheric Tables.” MIL-C-005011B(USAF), June 1977.

5) Department of Defense Military Specification. “Climatic Information to Determine Design and Test Requirements for Military Systems and Equipment.” MIL-STD-210C, January 1987.

6) Department of Defense Military Specification. “Glossary of Definitions, Ground Rules, and Mission Profiles to Define Air Vehicle Performance Capability.” MIL-STD-3013A, September 2008.

Operation 39 standard day 40.

polar day tropical day 20.

hot day 0.

-20.

-40.

temperature (deg C) -60.

-80.

0. 5. 10. 15. 20. 25. 30.

altitude (km) Figure 4-2. Temperature as a function of altitude.

Table 4-4. Constants adopted for calculation of the ISA.

parameter SI units English units units h m ft ◦ ◦ units τ C F m per ft 0.3048 kg per lbm 0.45359237 ◦ ◦ T 288.15 K 518.67 R ◦ ◦ T 273.15 K 459.67 R zero 2 2 p 101325.0 N/m 2116.22 lb/ft 3 3 ρ 1.225 kg/m 0.002377 slug/ft c 340.294 m/sec 1116.45 ft/sec s 0 μ 1.7894E-5 kg/m-sec 3.7372E-7 slug/ft-sec ◦ S 110.4 K β 1.458E-6 γ 1.4 2 2 g 9.80665 m/sec 32.17405 ft/sec r 6356766 m 20855531 ft 40 Operation Table 4-5. Temperatures and vertical temperature gradients: standard day.

level base altitude h lapse rate L temperature T b b b ◦ ◦ ◦ ◦ km K/km K C F troposphere –2 –6.5 301.15 28 82.4 0 troposphere 0 –6.5 288.15 15 59 1 tropopause 11 0 216.65 –56.5 –69.7 2 stratosphere 20 +1.0 216.65 –56.5 –69.7 3 stratosphere 32 +2.8 228.65 –44.5 –48.1 4 stratopause 47 0 270.65 –2.5 27.5 5 mesosphere 51 –2.8 270.65 –2.5 27.5 6 mesosphere 71 –2.0 214.65 –58.5 –73.3 7 mesopause 80 0 196.65 –76.5 –105.7 Table 4-6. Temperatures and vertical temperature gradients: polar day.

base altitude h lapse rate L temperature T pressure ratio b b b ◦ ◦ ◦ ◦ ft km * K/km * K C F * δ = p/p 0 0 6.326 246.15 –27 –16.6 30.268/29.92 3111.871 0.948 –1.083 252.15 –21 –5.8 9172.604 2.796 –5.511 250.15 –23 –9.4 28224.543 8.603 –0.476 218.15 –55 –67.0 83363.393 25.409 0 210.15 –63 –81.4 * derived Table 4-7. Temperatures and vertical temperature gradients: tropical day.

base altitude h lapse rate L temperature T pressure ratio b b b ◦ ◦ ◦ ◦ ft km * K/km * K C F * δ = p/p 0 0 –6.687 305.25 32.1 89.78 1 55000 16.764 4.374 193.15 –80 –112.00 70000 21.336 2.401 213.15 –60 –76.00 100745 30.707 235.65 –37.5 –35.50 * derived Table 4-8. Temperatures and vertical temperature gradients: hot day.

base altitude h lapse rate L temperature T pressure ratio b b b ◦ ◦ ◦ ◦ ft km * K/km K C F * δ = p/p 0 0 –7.065 312.55 39.4 102.92 1 15000 4.572 280.25 * 7.10 * 44.78 * derived Operation 41 Table 4-9. Temperature table for hot day (from MIL-C-5011B).

pressure altitude temperature ◦ ◦ ◦ ft km * K C F * 0 0 312.55 39.4 102.92 500 0.152 311.55 38.4 101.12 1000 0.305 310.45 37.3 99.14 1500 0.457 309.45 36.3 97.34 2000 0.610 308.35 35.2 95.36 2500 0.762 307.25 34.1 93.38 3000 0.914 306.25 33.1 91.58 3500 1.067 305.15 32.0 89.60 4000 1.219 304.05 30.9 87.62 4500 1.372 302.95 29.8 85.64 5000 1.524 301.85 28.7 83.66 5500 1.676 300.75 27.6 81.68 6000 1.829 299.65 26.5 79.70 6500 1.981 298.55 25.4 77.72 7000 2.134 297.45 24.3 75.74 7500 2.286 296.35 23.2 73.76 8000 2.438 295.25 22.1 71.78 8500 2.591 294.15 21.0 69.80 9000 2.743 293.05 19.9 67.82 9500 2.896 291.95 18.8 65.84 10000 3.048 290.85 17.7 63.86 10500 3.200 289.85 16.7 62.06 11000 3.353 288.85 15.7 60.26 11500 3.505 287.75 14.6 58.28 12000 3.658 286.75 13.6 56.48 12500 3.810 285.65 12.5 54.50 13000 3.962 284.55 11.4 52.52 13500 4.115 283.55 10.4 50.72 14000 4.267 282.45 9.3 48.74 14500 4.420 281.35 8.2 46.76 15000 4.572 280.35 7.2 44.96 * derived 42 Operation

Chapter 5 Solution Procedures The NDARC code performs design and analysis tasks. The design task involves sizing the rotorcraft to satisfy specified design conditions and missions. The analysis tasks can include off-design mission performance analysis, flight performance calculation for point operating conditions, and generation of subsystem or component performance maps. Figure 5-1 illustrates the tasks. The principal tasks (sizing, mission analysis, and flight performance analysis) are shown in the figure as boxes with dark borders.

Dark black arrows show control of subordinate tasks.

The aircraft description (fig. 5-1) consists of all the information, input and derived, that defines the aircraft. The aircraft consists of a set of components, including fuselage, rotors, wings, tails, and propulsion. This information can be the result of the sizing task; can come entirely from input, for a fixed model; or can come from the sizing task in a previous case or previous job. The aircraft description information is available to all tasks and all solutions (indicated by light green arrows).

Missions are defined for the sizing task and for the mission performance analysis. A mission consists of a specified number of mission segments, for which time, distance, and fuel burn are evaluated. For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is on fuel weight or energy.

Flight conditions are specified for the sizing task and for the flight performance analysis.

For flight conditions and mission takeoff, the gross weight can be maximized such that the power required equals the power available.

A flight state is defined for each mission segment and each flight condition. The aircraft performance can be analyzed for the specified state, or a maximum-effort performance can be identified. The maximum effort is specified in terms of a quantity such as best endurance or best range, and a variable such as speed, rate of climb, or altitude. The aircraft must be trimmed, by solving for the controls and motion that produce equilibrium in the specified flight state. Different trim solution definitions are required for various flight states. Evaluating the rotor hub forces may require solution of the blade flap equations of motion.

The sizing task is described in more detail in chapter 3. The flight condition, mission, and flight state calculations are described in chapter 4. The solution of the blade flap equations of motion is described in chapter 11. The present chapter provides details of the solution procedures implemented for each iteration of the analysis.

The nested iteration loops involved in the solution process are indicated by the subtitles in the boxes of figure 5-1, and illustrated in more detail in figure 5-2. The flight state solution involves up 44 Solution Procedures fixed model or previous job or previous case

DESIGN ANALYZE

Airframe Aerodynamics Map Sizing Task Engine Aircraft size iteration Performance Map Description Mission Analysis design design Flight conditions missions Performance Analysis Mission Flight Condition adjust & fuel wt iteration max GW max takeoff GW each segment Flight State max effort / trim aircraft / flap equations Figure 5-1. Outline of NDARC tasks.

to three loops. The innermost loop is the solution of the blade flap equations of motion, needed for an accurate evaluation of the rotor hub forces. The next loop is the trim solution, which is required for most flight states. The flight state optionally has one or two maximum-effort iterations. The flight state solution is executed for each flight condition and for each mission segment. A flight condition solution or any mission segment solution can optionally maximize the aircraft gross weight. The mission usually requires an iterative solution, for fuel weight or for adjustable segment time or distance. Thus each flight condition solution involves up to four nested iterations: maximum gross weight (outer), maximum effort, trim, and blade motion (inner). Each mission solution involves up to five nested iterations: mission (outer), and then for each segment maximum gross weight, maximum effort, trim, and blade motion (inner). Finally, the design task introduces a sizing iteration, which is the outermost loop of the process.

Solution Procedures 45 Sizing Task Flight Condition Size Iteration Maximum GW method: successive method: secant or false position substitution Flight State Missions Mission Flight Conditions Mission Iteration fuel weight, adjust time/distrance Mission Analysis method: successive substitution Missions Maximum GW method: secant or false Flight Performance Analysis position Flight State Flight Conditions Flight State Maximum Effort method: golden section search for maximum endurance, range, or climb; otherwise secant or false position Trim method: Newton-Raphson Component Performance Blade Flapping method: Newton-Raphson Figure 5-2. Design and analysis tasks, with nested loops and solution methods.

46 Solution Procedures 5–1 Iterative Solution Tasks 5-1.1 Tolerance and Perturbation For each solution procedure, a tolerance  and a perturbation Δ may be required. Single values are specified for the task and then scaled for each element tested or perturbed.

The scaling is based on a reference weight W (design gross weight, or derived from aircraft C /σ = 0 . 07 ), a reference length L (fuselage length, rotor radius, or wing span), and a reference power T √ P (aircraft installed power, or derived from P = W W/ 2 ρA ). Then the force reference is F = W , the moment reference is M = W L/ 10 , and the angle reference is A = 1 deg. The velocity reference is V = 400 knots. The angular velocity reference is Ω = V /L (in deg/sec). The coefficient reference is C = 0 . 6 for wings and C = 0 . 1 for rotors. Altitude scale is H = 10000 ft. Acceleration scale is G = g (acceleration due to gravity). The range scale is X = 100 nm. These scaling variables are referred to in the subsections that follow, and in tables 5-1 to 5-4.

5-1.2 Size Aircraft The sizing task determines the dimensions, power, and weight of a rotorcraft that can perform a specified set of design conditions and missions. The aircraft size is characterized by parameters such as design gross weight, weight empty, rotor radius, and engine power available. The relationships between dimensions, power, and weight generally require an iterative solution. From the design flight conditions and missions, the task can determine the total engine power or the rotor radius (or both power and radius can be fixed), as well as the design gross weight, maximum takeoff weight, drive system torque limit, and fuel tank capacity. For each propulsion group, the engine power or the rotor radius can be sized.

A successive substitution method is used for the sizing iteration, with an input tolerance  . Relax- ation is applied to P or R , T , P , W , W , P , W or E , and T . Two eng jet chrg D M T O DS limit fuel − cap fuel − cap design successive substitution loops are used. The outer loop is an iteration on performance: engine power or rotor radius, jet thrust, charger power. The inner loop is an iteration on parameters: W , W , D M T O P , W or E , and T . Either loop can be absent, depending on the definition of the DS limit fuel − cap fuel − cap design size task. Convergence is tested in terms of these parameters, and the aircraft weight empty W . The E tolerance is 0 . 1 P  for engine power and drive system limit; 0 . 01 W  for gross weight, maximum takeoff weight, fuel weight, jet thrust, and design rotor thrust; and 0 . 1 L for rotor radius.

5-1.3 Mission Missions consist of a specified number of segments, for which time, distance, and fuel burn are evaluated. For calculated mission fuel weight, the fuel weight at takeoff is adjusted to equal the fuel required for the mission (burned plus reserve). For specified takeoff fuel weight with adjustable segments, the mission time or distance is adjusted so the fuel required for the mission (burned plus reserve) equals the takeoff fuel weight. The mission iteration is thus on fuel weight or energy. Range credit segments can also require an iteration.

A successive substitution method is used if an iteration is required, with a tolerance  specified.

The principal iteration variable is takeoff fuel weight, for which the tolerance is 0 . 01 W  . For calculated mission fuel weight, the relaxation is applied to the mission fuel value used to update the takeoff fuel weight. For specified takeoff fuel weight, the relationship is applied to the fuel weight increment used to adjust the mission segments. The tolerance for the distance flown in range credit segments is X . The relaxation is applied to the distance flown in the destination segments for range credit.

Solution Procedures 47 5-1.4 Maximum Gross Weight Flight conditions are specified for the sizing task and for the flight performance analysis. Mission takeoff conditions are specified for the sizing task and for the mission analysis. Optionally for flight conditions and mission takeoff, the gross weight can be maximized, such that the power required equals the power available, min( P − P ) = 0 (zero power margin, minimum over all propulsion groups); avP G reqP G or such that the power required equals an input power, min(( d + f P ) − P ) = 0 (minimum over avP G reqP G all propulsion groups, with d an input power and f an input factor; this convention allows the power to be input directly, f = 0 , or scaled with power available). Similarly, the gross weight can be maximized for zero jet thrust margin, or zero torque margin.

The secant method or the method of false position is used to solve for the maximum gross weight.

A tolerance  and a perturbation Δ are specified. The variable is gross weight, with initial increment of W Δ , and tolerance of 0 . 01 W  . Note that the convergence test is applied to the magnitude of the gross weight increment.

5-1.5 Maximum Effort The aircraft performance can be analyzed for the specified state or a maximum-effort performance can be identified. The secant method or the method of false position is used to solve for the maximum effort. The task of finding maximum endurance, range, or climb is usually solved using the golden- section or curve-fit method. A tolerance  and a perturbation Δ are specified.

A quantity and variable are specified for the maximum-effort calculation. Tables 5-1 and 5-2 summarize the available choices, with the tolerance and initial increment used for the variables. Note that the convergence test is applied to the magnitude of the variable increment. Optionally two quantity/ variable pairs can be specified, solved in nested iterations. The two variables must be unique. The two variables can maximize the same quantity (endurance, range, or climb). If the variable is velocity, first the velocity is found for the specified maximum effort; the performance is then evaluated at that velocity times an input factor. For endurance, range, or climb, the slope of the quantity to be maximized must be zero; hence in all cases the target is zero. The slope of the quantity is evaluated by first-order backward difference. For the range, first the variable is found such that V / ˙ w is maximized (slope zero), and then the variable is found such that V / ˙ w equals 99% of that maximum; for the latter the variable perturbation is increased by a factor of 4 to ensure that the solution is found on the correct side of the maximum.

Table 5-1. Maximum-effort solution.

maximum-effort variable initial increment tolerance horizontal velocity V 0 . 1 V Δ 0 . 1 V  h vertical rate of climb V 0 . 1 V Δ 0 . 1 V  z aircraft velocity β (sideslip) 100 A Δ 100 A altitude H Δ H ˙ ˙ aircraft angular rate θ (pullup), ψ (turn) ΩΔ Ω  aircraft linear acceleration a , a , a G Δ G x y z aircraft control angle 100 A Δ 100 A aircraft orientation θ (pitch), φ (roll) 100 A Δ 100 A propulsion group tip speed V V Δ V  tip propulsion group engine speed N 10ΩΔ 10Ω  spec 48 Solution Procedures Table 5-2. Maximum-effort solution.

maximum-effort quantity best endurance maximum 1 / ˙ w best range 99% maximum V / ˙ w high or low side, or 100% best climb or descent rate maximum V or 1 /P z best climb or descent angle maximum V /V or V /P z ceiling maximum altitude power limit power margin, min( P − P ) = 0 over all propulsion groups avP G reqP G torque limit torque margin, min( Q − Q ) = 0 over all limits limit req thrust limit thrust margin, min( T − T ) = 0 over all jet groups avJG reqJG power, torque, thrust limit power, torque, thrust margin most restrictive ˙ battery limit power margin, min( P − | E | ) = 0 over all fuel tanks max batt rotor stall thrust margin, ( C /σ ) − C /σ = 0 for designated rotor T max T wing stall lift margin, C − C = 0 for designated wing L max L Table 5-3. Trim solution.

trim quantity target tolerance aircraft total force x , y , z components 0 F  aircraft total moment x , y , z components 0 M  aircraft load factor x , y , z components Flight State  propulsion group power Flight State P  power margin P − P Flight State P  avP G reqP G torque margin P − P Flight State P  DS limit reqP G engine group power Flight State P  power margin P − P Flight State P  avEG reqEG jet group thrust Flight State F  thrust margin T − T Flight State F  avJG reqJG charge group power Flight State P  charge power margin P − P Flight State P  avCG reqCG ˙ fuel tank energy flow E Flight State P  batt ˙ battery power margin P − | E | Flight State P  max batt rotor force lift, vertical, propulsive Flight State, component schedule F  rotor thrust C /σ Flight State, component schedule C T rotor thrust margin ( C /σ ) − C /σ Flight State C T max T rotor flapping β , β Flight State A c s rotor hub moment x (roll), y (pitch) Flight State M  rotor torque Flight State M  wing force lift Flight State, component schedule F  wing lift coefficient C Flight State, component schedule C L wing lift margin C − C Flight State C L max L tail force lift Flight State F  Solution Procedures 49 Table 5-4. Trim solution.

trim variable perturbation aircraft control angle 100 A Δ aircraft orientation θ (pitch), φ (roll) 100 A Δ aircraft velocity V (horizontal velocity) V Δ h aircraft velocity V (vertical velocity) V Δ z aircraft velocity β (sideslip) 100 A Δ ˙ ˙ aircraft angular rate θ (pullup), ψ (turn) ΩΔ propulsion group tip speed V 20 V Δ tip propulsion group engine speed N 10ΩΔ spec 5-1.6 Trim The aircraft trim operation solves for the controls and motion that produce equilibrium in the specified flight state. A Newton–Raphson method is used for trim. The derivative matrix is obtained by numerical perturbation. A tolerance  and a perturbation Δ are specified.

Different trim solution definitions are required for various flight states. Therefore one or more trim states are defined for the analysis, and the appropriate trim state selected for each flight state of a performance condition or mission segment. For each trim state, the trim quantities, trim variables, and targets are specified. Tables 5-3 and 5-4 summarize the available choices, with the tolerances and perturbations used.

5-1.7 Rotor Flap Equations Evaluating the rotor hub forces may require solution of the flap equations E ( v ) = 0 . For tip- T path plane command, the thrust and flapping are known, so v = ( θ θ θ ) . For no-feathering plane 0 . 75 c s T command, the thrust and cyclic pitch are known, so v = ( θ β β ) . A Newton–Raphson solution 0 . 75 c s ∼ method is used: from E ( v ) = E ( v ) + ( dE/dv )( v − v ) = 0 , the iterative solution is n +1 n n +1 n v = v − C E ( v ) n +1 n n − 1 where C = f ( dE/dv ) , including the relaxation factor f . The derivative matrix for axial flow can be used. Alternatively, the derivative matrix dE/dv can be obtained by numerical perturbation. Convergence of the Newton–Raphson iteration is tested in terms of | E | <  for each equation, where  is an input tolerance.

5–2 Theory The analysis uses several methods to solve nonlinear algebraic equations. Such equations may be written in two forms: (a) fixed point x = G ( x ) , and (b) zero point f ( x ) = 0 ; where x , G , and f are vectors.

The analysis provides operations that implement the function G or f . Solution procedures appropriate for the zero point form can be applied to equations in fixed point form, by defining f ( x ) = x − G ( x ) . In this context, f can be considered the iteration error.

Efficient and convergent methods are required to find the solution x = α of these equations. Note ′ ′ that f ( α ) = 0 or G ( α ) = 1 means that α is a higher-order root. For nonlinear problems, the method will 50 Solution Procedures successive substitution iteration save: x = x old evaluate x relax: x = λx + (1 − λ ) x old test convergence: error = ‖ x − x ‖ ≤ λ tolerance × weight old Figure 5-3. Outline of successive substitution method.

be iterative: x = F ( x ) . The operation F depends on the solution method. The solution error is: n +1 n ′ ′ ∼  = α − x = F ( α ) − F ( x ) = ( α − x ) F ( ξ ) =  F ( α ) n +1 n +1 n n n n ′ Thus the iteration will converge if F is not too sensitive to errors in x : | F ( α ) | < 1 for scalar x . For x a vector, the criterion is that all the eigenvalues of the derivative matrix ∂F/∂x have magnitude less than one. The equations in this section are generally written for scalar x ; the extension to vector x ′ ′ is straightforward. Convergence is linear for F nonzero, quadratic for F = 0 . Iterative methods have a relaxation factor (and other parameters) to improve convergence, and a tolerance to measure convergence.

The following subsections describe the solution methods used for the various iterations, as shown in figure 5-2.

5-2.1 Successive Substitution Method The successive substitution method (with relaxation) is an example of a fixed point solution. A ′ direct iteration is simply x = G ( x ) , but | G | > 1 for many practical problems. A relaxed iteration n +1 n uses F = (1 − λ ) x + λG : x = (1 − λ ) x + λG ( x ) = x − λf ( x ) n +1 n n n n with relaxation factor λ . The convergence criterion is then ′ ′ | F ( α ) | = | 1 − λ + λG | < 1 ′ so a value of λ can be found to ensure convergence for any finite G . Specifically, the iteration converges ′ ′ if the magnitude of λ is less than the magnitude of 2 / (1 − G ) = 2 /f (and λ has the same sign as ′ ′ ′ ′ ′ 1 − G = f ). Quadratic convergence ( F = 0 ) is obtained with λ = 1 / (1 − G ) = 1 /f . Over-relaxation ′ ( λ > 1 ) can be used if | G | < 1 . Since the correct solution x = α is not known, convergence must be tested by comparing the values of two successive iterations: error = ‖ x − x ‖ ≤ tolerance n +1 n where the error is some norm of the difference between iterations (typically absolute value for scalar x ).

Note that the effect of the relaxation factor is to reduce the difference between iterations: ( ) x − x = λ G ( x ) − x n +1 n n n Hence the convergence test is applied to ( x − x ) /λ , in order to maintain the definition of tolerance n +1 n independent of relaxation. The process for the successive substitution method is shown in figure 5-3.

Solution Procedures 51 initialize evaluate h test convergence: error = | h − h | ≤ tolerance × weight j target j j initialize derivative matrix D to input matrix − 1 calculate gain matrix: C = λD iteration identify derivative matrix optional perturbation identification perturb each element of x : δx = Δ × weight i i evaluate h calculate D − 1 calculate gain matrix: C = λD increment solution: δx = − C ( h − h ) target evaluate h test convergence: error = | h − h | ≤ tolerance × weight j target j j Figure 5-4. Outline of Newton–Raphson method.

5-2.2 Newton–Raphson Method The Newton–Raphson method (with relaxation and identification) is an example of a zero point ′ solution. The Taylor series expansion of f ( x ) = 0 leads to the iteration operator F = x − f /f : − 1 ′ x = x − [ f ( x )] f ( x ) n +1 n n n which gives quadratic convergence. The behavior of this iteration depends on the accuracy of the ′ ′ derivative f . Here it is assumed that the analysis can evaluate directly f , but not f . It is necessary to ′ evaluate f by numerical perturbation of f , and for efficiency the derivatives may not be evaluated for each x . These approximations compromise the convergence of the method, so a relaxation factor λ is n introduced to compensate. Hence a modified Newton–Raphson iteration is used, F = x − Cf : − 1 x = x − Cf ( x ) = x − λD f ( x ) n +1 n n n n ′ where the derivative matrix D is an estimate of f . The convergence criterion is then ′ ′ − 1 ′ | F ( α ) | = | 1 − Cf | = | 1 − λD f | < 1 ′ since f ( α ) = 0 . The iteration converges if the magnitude of λ is less than the magnitude of 2 D/f (and λ ′ ′ has the same sign as D/f ). Quadratic convergence is obtained with λ = D/f (which would require λ to change during the iteration however). The Newton–Raphson method ideally uses the local derivative ′ in the gain factor, C = 1 /f , so has quadratic convergence: ′′ f f ′ F ( α ) = = 0 ′ 2 f ′ ′′ ′ ′ since f ( α ) = 0 (if f  = 0 and f is finite; if f = 0 , then there is a multiple root, F = / , and the convergence is only linear). A relaxation factor is still useful, since the convergence is only quadratic sufficiently close to the solution. A Newton–Raphson method has good convergence when x is suf- ficiently close to the solution, but frequently has difficulty converging elsewhere. Hence the initial 52 Solution Procedures initialize evaluate f at x , f at x = x + Δ x , f at x = x + Δ x 0 0 1 1 0 2 2 1 iteration ′ calculate derivative f secant: from f and f 0 1 false position: from f , and f or f (opposite sign from f ) 0 1 2 0 ′ calculate gain: C = λ/f increment solution: δx = − Cf shift: f = f , f = f 2 1 1 0 evaluate f test convergence Figure 5-5. Outline of secant method or method of false position.

estimate x that starts the iteration is an important parameter affecting convergence. Convergence of the solution for x may be tested in terms of the required value (zero) for f : error = ‖ f ‖ ≤ tolerance where the error is some norm of f (typically absolute value for scalar f ).

The derivative matrix D is obtained by an identification process. The perturbation identification can be performed at the beginning of the iteration, and optionally every M iterations thereafter. The PID derivative matrix is calculated from a one-step finite-difference expression (first order). Each element x of the vector x is perturbed, one at a time, giving the i -th column of D : i [ ] [ ] ∂f f ( x + δx ) − f ( x ) i i i D = · · · · · · = · · · · · · ∂x δx i i Alternatively, a two-step finite-difference expression (second order) can be used: [ ] [ ] ∂f f ( x + δx ) − f ( x − δx ) i i i i D = · · · · · · = · · · · · · ∂x 2 δx i i With this procedure, the accuracy of D (hence convergence) can be affected by both the magnitude and sign of the perturbation (only the magnitude for a two-step difference).

The process for the Newton–Raphson method is shown in figure 5-4. A problem specified as h ( x ) = h becomes a zero point problem with f = h − h . A successive substitution problem, target target x = G ( x ) , becomes a zero point problem with f = x − G . At the beginning of the solution, x has an initial value. The perturbation identification can optionally never be performed (so an input matrix is required), be performed at the beginning of the iteration, or be performed at the beginning and every M iterations thereafter.

PID 5-2.3 Secant Method The secant method (with relaxation) is developed from the Newton–Raphson method. The modified Newton–Raphson iteration is: − 1 x = x − Cf ( x ) = x − λD f ( x ) n +1 n n n n Solution Procedures 53 initialize evaluate f at x , f at x = x + Δ x , f at x = x + Δ x 0 0 1 1 0 2 2 1 bracket maximum: while not f ≥ f , f 1 0 2 if f > f , then x = x + ( x − x ) ; 1,2,3 → 0,1,2 2 0 3 2 2 1 if f > f , then x = x − ( x − x ) ; 3,0,1 → 0,1,2 0 2 3 0 1 0 iteration (search) if x − x > x − x , then x = x + W ( x − x ) 2 1 1 0 3 1 2 1 if f < f , then 0,1,3 → 0,1,2 3 1 if f > f , then 1,3,2 → 0,1,2 3 1 if x − x > x − x , then x = x − W ( x − x ) 1 0 2 1 3 1 1 0 if f < f , then 3,1,2 → 0,1,2 3 1 if f > f , then 0,3,1 → 0,1,2 3 1 test convergence Figure 5-6. Outline of golden-section search.

′ where the derivative matrix D is an estimate of f . In the secant method, the derivative of f is evaluated numerically at each step: f ( x ) − f ( x ) n n − 1 ′ ∼ f ( x ) = n x − x n n − 1 It can be shown that then the error reduces during the iteration according to: ′′ ′ ′′ ′ . 62 1 . 62 ∼ ∼ |  | = | f / 2 f | |  | |  | = | f / 2 f | |  | n +1 n n − 1 n which is slower than the quadratic convergence of the Newton–Raphson method (  ), but still better than n linear convergence. In practical problems, whether the iteration converges at all is often more important than the rate of convergence. Limiting the maximum amplitude of the derivative estimate may also be ′ appropriate. Note that with f = x − G ( x ) , the derivative f is dimensionless, so a universal limit (say ′ maximum | f | = 0 . 3 ) can be specified. A limit on the maximum increment of x (as a fraction of the x value) can also be imposed. The process for the secant method is shown in figure 5-5.

5-2.4 Method of False Position The method of false position is a derivative of the secant method, based on calculating the derivative with values that bracket the solution. The iteration starts with values of x and x such that f ( x ) and 0 1 0 ′ f ( x ) have opposite signs. Then the derivative f and new estimate x are 1 n +1 f ( x ) − f ( x ) n k ′ ∼ f ( x ) = n x − x n k − 1 x = x − λD f ( x ) n +1 n n using k = n − 1 or k = n − 2 such that f ( x ) and f ( x ) have opposite signs. The convergence is n k slower (roughly linear) than for the secant method, but by keeping the solution bracketed convergence is guaranteed. The process for the method of false position is shown in figure 5-5.

5-2.5 Golden-Section Search The golden-section search method can be used to find the solution x that maximizes f ( x ) . The problem of maximizing f ( x ) can be attacked by applying the secant method or method of false position to 54 Solution Procedures initialize evaluate f at x , f at x = x + Δ x , f at x = x + Δ x 0 0 1 1 0 2 2 1 bracket maximum: while not f ≥ f , f 1 0 2 if f > f , then x = x + ( x − x ) ; 1,2,3 → 0,1,2 2 0 3 2 2 1 if f > f , then x = x − ( x − x ) ; 3,0,1 → 0,1,2 0 2 3 0 1 0 f = f max 1 curve fit f = f , x = x max 1 max 1 evaluate f for x = x + n Δ x and x = x − n Δ x max max least-squared error solution for polynomial coefficients solve polynomial for x at peak f Figure 5-7. Outline of curve-fit method.

′ the derivative f ( x ) = 0 , but that approach is often not satisfactory as it depends on numerical evaluation of the second derivative. The golden-section search method begins with a set of three values x < x < x 0 1 2 and the corresponding functions f , f , f . The x value is incremented until the maximum is bracketed, 0 1 2 f ≥ f , f . Then a new value x is selected in the interval x to x ; f evaluated; and the new set of 1 0 2 3 0 2 3 x < x < x determined such that the maximum is still bracketed. The new value x is a fraction 0 1 2 3 √ ∼ W = (3 − 5) / 2 . 38197 from x into the largest segment. The process for the golden-section search = 0 is shown in figure 5-6.

5-2.6 Curve-Fit Method The curve-fit method can be used to find the solution x that maximizes f ( x ) , by fitting the solution to a polynomial. If the function f is flat around the maximum and the inner loop tolerances are not tight enough, the golden-section search can become erratic, particularly for best range and best endurance calculations. Curve fitting the evaluated points and then solving the curve for the maximum has the potential to improve the behavior. The curve-fit method begins with a set of three values x < x < x 0 1 2 and the corresponding functions f , f , f . The x value is incremented until the maximum is bracketed, 0 1 2 f ≥ f , f , giving a course maximum f at x . Next a set of x and f values are generated by 1 0 2 max max incrementing x above and below x , until f < r f is found (typically r = 0 . 98 for best range).

max fit max fit 3 2 This set of points is fit to the cubic polynomial f = c z + c z + c z + c , z = x/x − 1 (or to a 3 2 1 0 max ( ) ( ) T T 2 3 quadradic polynomial). Let c = c c c c and ξ = 1 z z z . Then the least-squared-error solution 0 1 2 3 for the coefficients is (∑ ) (∑ ) − 1 T c = ξ ξ f ξ i i i i i i where the sums are over the set of points to be fit. For a quadratic polynomial fit, the solution is then ( ) √ √ x = x 1 − c / 2 c ± 1 − r ( c / 2 c ) − c /c max 1 2 1 2 0 2 where r = 1 for the maximum, or r = 0 . 99 for 99% best range. For a cubic polynomial fit, the maximum is at ( √ ) ( ) c 3 c c c 1 3 c c 2 3 1 1 3 1 ∼ z = − 1 − 1 − − 1 + = 2 2 3 c c 2 c 4 c 3 2 2 2 It is simplest to search the cubic for the peak ( z where df /dz = 0 ), and then if necessary search for the 99% range point ( f = 0 . 99 f ) The process for the golden-section search is shown in figure 5-7.

peak

Chapter 6 Cost Costs are estimated using statistical models based on historical aircraft price and maintenance cost data, with appropriate factors to account for technology impact and inflation. The aircraft purchase price ( C , in dollars) covers airframe, mission equipment package (MEP), and flight control electronics AC (FCE) costs. The direct operating cost (DOC, in cents per available seat mile (ASM)) is the sum of maintenance cost ( C , in dollars per flight hour), flight crew salary and expenses, fuel and oil cost, maint depreciation, insurance cost, and finance cost.

Inflation factors can be input, or internal factors used. Table 6-1 gives the internal inflation factors for DoD (ref. 1) and CPI (ref. 2). For years beyond the data in the table, optionally the inflation factor is extrapolated based on the last yearly increase.

6–1 CTM Rotorcraft Cost Model The CTM rotorcraft cost model (refs. 3–6) gives an estimate of aircraft purchase price, maintenance cost, and direct operating cost. The model was developed for shaft-driven helicopters and turboprop aircraft. If the aircraft shaft power is zero, P/W = 0 . 25 is used in the equation for purchase price.

AF 6-1.1 Aircraft Purchase Price Aircraft purchase price is estimated from the statistical relationship of Harris and Scully (ref. 3, updated in 2001), based on a 1994 database of mostly civil aircraft (plus EH-101, UH-60L, CH-47D, CH- 53E, and MV-22). The model starts with a function of aircraft weight and power; has several complexity factors; a factor for rotorcraft or turboprop aircraft; and a country or industry factor (specifically U.S.

military). The model includes (as $/lb) separate calculations of a composite construction increment (increase or decrease), mission equipment package cost, and flight control electronics cost. The model accounts for inflation and includes an overall technology factor. With these equations, the purchase price is predicted within 20% for 96% of 128 rotorcraft (figs. 6-1 and 6-2), implying a standard deviation of 10%. For the five military aircraft, price is predicted within ± 10% or less.

The basic statistical relationship for airframe purchase price is: 1 . 0619 0 . 5887 0 . 1465 c = 739 . 91 K K K K W ( P/W ) N AF ET EN LG R AF AF blade with W = W + Δ W − W − W , including airframe kits Δ W (the wing and wing extension AF E kit MEP FCE kit kits, and optionally the folding kit). The configuration factor K = K K K K has the config ET EN LG R 56 Cost factors: K = 1.0 for turbine aircraft 0.557 for piston aircraft ET K = 1.0 for multi-engine aircraft 0.736 for single-engine aircraft EN K = 1.0 for retractable landing gear 0.884 for fixed landing gear LG K = 1.0 for single main-rotor 1.057 for twin main-rotors, 1.117 for four main-rotors R The number of blades and the configuration factor are essentially measures of complexity. In particular, retractable/fixed landing gear is a surrogate for general complexity. The term C = r W comp comp comp accounts for additional costs for composite construction (negative for cost savings); W is the com- comp posite structure weight, obtained as an input fraction of the component weight, with separate fractions for body, tail, pylon, and wing weight. The MEP and FCE costs are obtained from input cost-per-weight factors: C = r W and C = r W .

MEP MEP MEP FCE FCE FCE The statistical cost equation for c is based on 1994 dollars and current technology levels. In- AF cluding an inflation factor F and technology factor χ gives the purchase price C : i AF AC C = χ ( F c ) + C + C + C AC AF i AF comp MEP FCE In addition to technology, χ accounts for calibration and industry factors; for example, χ = 0 . 87 AF for U.S. Military (ref. 3). This equation also estimates turboprop airliner purchase price by setting N = N = 1 and using the additional factor 0 . 8754 (pressurized) or 0 . 7646 (unpressurized). The rotor blade purchase price in $/lb or $/kg is r = ( χ ( F c )) /W AF AF i AF AF r = C / ( W + Δ W ) AC AC E kit for the airframe and the aircraft. Parameters are defined in table 6-2, including units as used in these equations.

6-1.2 Maintenance Cost Total maintenance cost per hour (dollars per flight hour) is estimated from the statistical relationship of Harris and Scully (ref. 3, updated in 2001). The maintenance cost per hour is: 0 . 3746 0 . 4635 c = 0 . 49885 W P maint E This equation is based on 1994 dollars and current technology levels. Including an inflation factor F i and technology factor χ gives the maintenance cost per flight hour C : maint maint C = χ ( F c ) maint maint i maint Parameters are defined in table 6-2, including units as used in these equations.

Alternatively, the maintenance cost c can be calculated from separate estimates of labor, parts maint (airframe, engine, and avionics), engine overhaul, and major periodic maintenance costs. The equations for these maintenance cost components are from Harris (ref. 6), based on a 2011 civil database. The contributions to the dollars per flight hour are: C = r (MMH / FH) labor labor 6 0 . 68 c = M ( C / 10 ) parts parts AC 0 . 67 c = M P engine engine c = M ( C / 10 ) major major AC Cost 57 where r is the maintenance labor rate (dollars per hour). The maintenance-man-hours per flight-hour labor is estimated from 0 . 78 MMH / FH = M W labor E or specified directly. These equations are based on 2011 dollars and current technology levels. For current best practice (bottom of data), the constants are M = 0 . 0017 , M = 34 , M = 1 . 45 , labor parts engine M = 18 ; while for current average practice M = 0 . 0027 , M = 56 , M = 1 . 74 , M = major labor parts engine major 28 . Finally, the maintenance cost per flight hour is C = χ F ( c + c + c ) + C maint maint i parts engine major labor including an inflation factor F and technology factor χ .

i maint 6-1.3 Direct Operating Cost Given specifics of the operation (including available block hours, non-flight time, spares fraction, and financial numbers), the direct operating cost is calculated for each mission. The direct operating cost includes maintenance, fuel, crew, depreciation, insurance, and finance costs. The contributions to the yearly operating cost are: C = G ( W /ρ ) N or GE N fuel fuel fuel dep fuel dep 0 . 4 C = 2 . 84 F K W B crew i crew M T O 1 + S C = C (1 − V ) dep AC D C = 0 . 0056 C ins AC 1 + S 2 L + 1 i C = C fin AC D 4 100 There is no credit for mission energy generation ( E < 0 ). The crew factor K = 1 corresponds to fuel crew low-cost, domestic airlines (1994 dollars). The fuel burn W , block time T , and block range R fuel miss miss are obtained for a designated mission. The number of departures per year is N = B/T . The flight dep miss time per trip is T = T − T . The flight hours per year are T = T N . Alternatively, the sum trip miss N F F trip dep of the crew, insurance, and depreciation costs can be estimated from the purchase price: ( ) C + C + C = K 175000 + 29 . 8( C / 1000) crew dep ins cdi AC where K is a calibration factor (ref. 6). The yearly operating cost C and DOC (cents per available cdi OP seat mile) are then: C = T C + C + C + C + C + C OP F maint fuel crew dep ins fin DOC = 100 C / ASM OP where the available seat miles per year are ASM = 1 . 1508 N R N (range in nm).

pass miss dep 6–2 References 1) “National Defense Budget Estimates for FY 1998/2015.” Office of the Under Secretary of Defense (Comptroller), March 1997/2014. Department of Defense Deflators, for Total Obligational Authority (TOA), Procurement.

2) “Consumer Price Index.” U.S. Department of Labor, Bureau of Labor Statistics, 2014. All Urban Consumers (CPI-U), U.S. city average, All items.

58 Cost 3) Harris, F.D., and Scully, M.P. “Rotorcraft Cost Too Much.” Journal of the American Helicopter Society, Vol. 43, No. 1, January 1998.

4) Harris, F.D. “An Economic Model of U.S. Airline Operating Expenses.” NASA CR 2005-213476, December 2005.

5) Coy, J.J. “Cost Analysis for Large Civil Transport Rotorcraft.” American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco, California, January 2006.

6) Harris, F.D. “Introduction to Autogyros, Helicopters, and Other V/STOL Aircraft.” NASA SP 2011-215959, volume 2, 2012.

Cost 59 Table 6-1. DoD and CPI inflation factors.

inflation factors inflation factors inflation factors year DoD CPI year DoD CPI year DoD CPI 1911 1951 17.02 17.54 1991 93.99 91.90 1912 1952 16.26 17.88 1992 96.21 94.67 1913 6.68 1953 16.31 18.02 1993 98.16 97.50 1914 6.75 1954 16.02 18.15 1994 100.00 100.00 1915 6.82 1955 17.29 18.08 1995 101.70 102.83 1916 7.35 1956 17.58 18.35 1996 103.18 105.87 1917 8.64 1957 18.17 18.96 1997 104.32 108.30 1918 10.19 1958 18.02 19.50 1998 105.40 109.99 1919 11.67 1959 18.11 19.64 1999 106.81 112.42 1920 13.50 1960 18.34 19.97 2000 108.39 116.19 1921 12.08 1961 18.31 20.18 2001 109.92 119.50 1922 11.34 1962 18.84 20.38 2002 111.62 121.39 1923 11.54 1963 19.09 20.65 2003 113.95 124.16 1924 11.54 1964 19.76 20.92 2004 116.92 127.46 1925 11.81 1965 20.36 21.26 2005 120.10 131.78 1926 11.94 1966 21.97 21.86 2006 123.06 136.03 1927 11.74 1967 22.77 22.54 2007 125.56 139.91 1928 11.54 1968 24.03 23.48 2008 127.43 145.28 1929 11.54 1969 25.10 24.76 2009 128.95 144.76 1930 11.27 1970 25.76 26.18 2010 131.66 147.14 1931 10.26 1971 27.24 27.33 2011 133.84 151.78 1932 9.24 1972 28.98 28.21 2012 136.07 154.92 1933 8.77 1973 31.34 29.96 2013 138.34 157.19 1934 9.04 1974 34.13 33.27 2014 140.81 159.74 1935 9.24 1975 38.10 36.30 2015* 143.49 1936 9.38 1976 42.14 38.39 2016* 146.33 1937 9.72 1977 43.75 40.89 2017* 149.26 1938 9.51 1978 47.82 43.99 2018* 152.25 1939 9.38 1979 53.02 48.99 2019* 155.29 1940 9.45 1980 58.52 55.60 1941 9.92 1981 63.80 61.34 1942 11.00 1982 68.35 65.11 1943 11.67 1983 71.92 67.21 1944 11.88 1984 74.57 70.11 1945 12.15 1985 76.86 72.60 1946 13.16 1986 79.19 73.95 1947 15.05 1987 81.87 76.65 1948 16.26 1988 85.01 79.82 1949 16.06 1989 88.19 83.67 1950 15.37 16.26 1990 91.30 88.19 * projected 60 Cost Table 6-2. Cost model parameters.

parameter definition units W weight empty lb E W maximum takeoff weight lb M T O N number of blades per rotor blade P rated takeoff power (all engines) hp W fixed useful load weight, mission equipment package lb or kg MEP W fixed useful load weight, flight control electronics lb or kg FCE r cost factor, mission equipment package $/lb or $/kg MEP r cost factor, flight control electronics $/lb or $/kg FCE r additional cost for composite construction $/lb or $/kg comp F inflation factor, relative 1994 i W mission fuel burned lb or kg fuel T mission time hr miss R mission range nm miss G fuel cost $/gallon or $/liter; or $/MJ B available block hours hr S spares per aircraft (fraction purchase price) D depreciation period yr V residual value (fraction) L loan period yr i interest rate % T non-flight time per trip hr N F N number of passengers pass ρ fuel density (weight per volume) lb/gal or kg/liter fuel N number of departures per year dep K crew factor crew T flight hours per year F Cost 61 1000.

no error ± 10% 900.

± 20% aircraft 800.

700.

600.

500.

400.

predicted base price (1994$/lb) 300.

200.

100.

0.

0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000.

actual base price (1994$/lb) Figure 6-1. Statistical estimation of rotorcraft purchase price ($/lb).

62 Cost 100.0 10.0 1.0 predicted base price (1994 $M) 0.1 0.1 1.0 10.0 100.0 actual base price (1994 $M) Figure 6-2. Statistical estimation of rotorcraft purchase price ($M).

Chapter 7 Aircraft The aircraft consists of a set of components, including rotors, wings, tails, fuselage, and propul- sion. For each component, attributes such as performance, drag, and weight can be calculated. The aircraft attributes are obtained from the sum of the component attributes. Description and analysis of conventional rotorcraft configurations is facilitated, while retaining the capability to model novel and ad- vanced concepts. Specific rotorcraft configurations considered include: single-main-rotor and tail-rotor helicopter, tandem helicopter, coaxial helicopter, and tiltrotor.

The following components form the aircraft: a) Systems: The systems component contains weight information (fixed useful load, vibration, contin- gency, and systems and equipment) for the aircraft.

b) Fuselage: There is one fuselage for the aircraft.

c) Landing Gear: There is one landing gear for the aircraft.

d) Rotors: The aircraft can have one or more rotors, or no rotors. In addition to main-rotors, the component can model tail-rotors, propellers, proprotors, and ducted fans.

e) Wings: The aircraft can have one or more wings, or no wings.

f) Tails: The aircraft can have one or more horizontal or vertical tail surfaces, or no tails.

g) Fuel Tanks: There are one or more fuel tank systems for the aircraft. Fuel tank systems are associated with the engine groups, jet groups, and charge groups. Fuel quantity is measured as either weight or energy. There can be one or more sizes of auxiliary fuel tanks.

h) Propulsion Groups: The aircraft can have one or more propulsion groups, or none. Each propulsion group is a set of components (rotors) and engine groups, connected by a drive system. The components define the power required. The engine groups define the power available.

i) Engine Groups: An engine group consists of one or more engines of a specific type. An engine group transfers power by shaft torque, so it is associated with a propulsion group. For each engine type an engine model is defined. The engine model describes a particular engine, used in one or more engine groups.

j) Jet Groups: The aircraft can have one or more jet groups, or none. A jet group produces a force on the aircraft. A jet model describes a particular jet, used in one or more jet groups.

k) Charge Groups: The aircraft can have one or more charge groups, or none. A charge group generates energy for the aircraft. A charge model describes a particular charger, used in one or more charge groups.

64 Aircraft 7–1 Loading The aircraft disk loading is the ratio of the design gross weight and a reference rotor area: DL = ∑ W /A . The reference area is a sum of specified fractions of the rotor areas, A = f A (typically D ref ref A the projected area of the lifting rotors). The disk loading of a rotor is the ratio of a specified fraction of the design gross weight and the rotor area: T f W f W W D W D ( DL ) = = = rotor A A A/A A ref ref ∑ where probably f = 1 , and the lifting rotors are all rotors not designated antitorque or auxiliary- W rotor thrust. If there are N lifting rotors of the same area, with no overlap, then f = 1 , A = N A , A ref f = A/A = 1 /N , and ( DL ) = DL . For rotors designated antitorque or auxiliary-thrust, the disk W ref rotor loading is calculated from the design rotor thrust: ( DL ) = T /A .

rotor design 1 1 For coaxial rotors, the default reference area is the area of one rotor: f = / , A = A , f = / , A 2 ref W 2 and ( DL ) = / DL . For tandem rotors, the default reference area is the projected area: A = rotor 2 ref 2 − m (2 − m ) A , where mA is the overlap area ( m = 0 for no overlap, m = 1 for coaxial). Then f = , A 2 − m f = / , and ( DL ) = DL . Optionally, the reference area for tandem rotors can be total rotor W 2 rotor area instead: A = 2 A .

ref The aircraft wing loading is the ratio of the design gross weight and a reference wing area: WL = ∑ W /S . The reference area is a sum of the wing areas, S = S . The wing loading of an individual D ref ref wing is the ratio of a specified fraction of the design gross weight and the wing area: W f W f W W D W D ( WL ) = = = wing S S S/S S ref ref ∑ where probably f = 1 . If there are N wings of the same area, then f = S/S = 1 /N , and W W ref wing ( WL ) = WL .

wing The aircraft power loading is the ratio of the design gross weight and the total installed takeoff ∑ power: W/P = W / N P eng , where the sum is over all engine groups.

D eng 7–2 Controls A set of aircraft controls c are defined, and these aircraft controls are connected to the component AC controls. The connection to the component control c is typically of the form c = ST c + c , where T AC 0 is an input matrix and c the component control for zero aircraft control. The connection (matrix T ) is defined for a specified number of control system states (allowing change of control configuration with flight state). The factor S is available for internal scaling of the matrix. The control state and initial control values are specified for each flight state. Figure 7-1 illustrates the control relationships.

Typical (default) aircraft controls are the pilot’s controls: collective stick, lateral and longitudinal cyclic sticks, pedal, and tilt. Units and sign convention of the pilot’s controls are contained in the matrix T . For the single-main-rotor and tail-rotor configuration, it is often convenient for the collective and cyclic stick motion to equal the collective and cyclic pitch input of the main-rotor, and the pedal motion to equal the collective pitch input of the tail-rotor. The aircraft controls should be scaled to approximately the same amplitude, by appropriate definition of the matrix T and scale factor S .

These aircraft controls are available for trim of the aircraft. Any aircraft controls not selected for trim will remain fixed at the values specified for the flight state. Thus by defining additional aircraft controls, component controls can be specified as required for a flight state.

Aircraft 65 component controls aircraft controls c c = T c + c AC AC 0 c = zero, constant, f( V ) collective lateral cyclic ROTOR long cyclic collective pedal lateral cyclic tilt long cyclic

T

other incidence (e.g. flap, cant elevator, diameter rudder, gear) gear TAIL control state control (or conversion (elevator WING schedule) or rudder) flap incidence flight state value flaperon or zero, constant, f( V ) aileron or conversion schedule incidence JET or trim GROUP amplitude mode ENGINE incidence GROUP trim option yaw amplitude mode incidence CHARGE yaw GROUP amplitude mode incidence yaw PROPULSION GROUP rotational speed Figure 7-1. Aircraft and component controls.

Each aircraft control variable c can be zero, constant, or a function of flight speed (piecewise AC linear input). The flight state input can override this value of the aircraft control. The input value is an initial value if the control is a trim variable.

Each component control variable c (value for zero aircraft control) can be zero, constant, or a function of flight speed (piecewise linear input). Optionally the use of c can be suppressed for a flight state. The component control from aircraft control ( T c ) is a fixed value, or a function of speed, or a AC linear function of another control (perhaps a trim variable).

66 Aircraft The tilt control variable α is intended for nacelle tilt angle or conversion control, particularly for tilt tiltrotors. The convention is α = 0 for cruise, and α = 90 deg for helicopter mode. If α exists as tilt tilt tilt a control, it can be zero, constant, or a function of flight speed (piecewise linear input).

An optional control conversion schedule is defined in terms of conversion speeds: hover and helicopter mode for speeds below V , cruise mode for speeds above V , and conversion mode C hover C cruise between. The nacelle angle is α = 90 in helicopter mode, α = 0 in airplane mode, and it varies tilt tilt linearly with speed in conversion mode. The tip speed is V in helicopter and conversion mode, tip − hover and V in airplane mode. Control states and drive system states are defined for helicopter, cruise, tip − cruise and conversion mode flight. The flight state specifies the nacelle tilt angle, tip speeds, control state, and drive system state, including the option to obtain any or all of these quantities from the conversion schedule.

The flight speed used for control scheduling is usually the calibrated airspeed (CAS), hence variation with dynamic pressure. Velocity schedules are used for conversion, controls and motion, rotor tip speed, landing gear retraction, and trim targets. Optionally these velocity schedules use either calibrated airspeed V or the true airspeed V .

cal The control matrices T can be defined based on the configuration. Let c , c , c , c be AC 0 ACc ACs ACp the pilot’s controls (collective, lateral cyclic, longitudinal cyclic, and pedal). For the helicopter, the first rotor is the main-rotor and the second rotor is the tail-rotor; then ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ T 1 0 0 0 c Mcoll AC 0 ⎜ T ⎟ ⎢ 0 − r 0 0 ⎥ ⎜ c ⎟ Mlat ACc = ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ T 0 0 − 1 0 c Mlng ACs T 0 0 0 − r c Tcoll ACp where r is the main-rotor direction of rotation ( r = 1 for counter-clockwise rotation, r = − 1 for clockwise rotation). For the tandem configuration, the first rotor is the front rotor and the second rotor is the rear rotor; then ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ T 1 0 − 1 0 c Fcoll AC 0 ⎜ T ⎟ ⎢ 0 − r 0 − r ⎥ ⎜ c ⎟ Flat F F ACc = ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ T 1 0 1 0 c Rcoll ACs T 0 − r 0 r c Rlat R R ACp For the coaxial configuration: ⎛ ⎞ ⎡ ⎤ T 1 0 0 r 1coll 1 ⎛ ⎞ ⎜ T ⎟ ⎢ 0 − r 0 0 ⎥ c 1lat 1 AC 0 ⎜ ⎟ ⎢ ⎥ ⎜ T ⎟ ⎢ 0 0 − 1 0 ⎥ ⎜ c ⎟ 1lng ACc ⎜ ⎟ = ⎢ ⎥ ⎝ ⎠ ⎜ T ⎟ ⎢ 1 0 0 r ⎥ c 2coll 2 ACs ⎝ ⎠ ⎣ ⎦ T 0 − r 0 0 c 2lat 2 ACp T 0 0 − 1 0 2lng For the tiltrotor, the first rotor is the right rotor and the second rotor is the left rotor; then ⎛ ⎞ ⎡ ⎤ T 1 − 1 0 0 Rcoll ⎜ T ⎟ ⎢ 0 0 − 1 1 ⎥ ⎛ ⎞ Rlng c ⎜ ⎟ ⎢ ⎥ AC 0 T 1 1 0 0 ⎜ ⎟ ⎢ ⎥ Lcoll c ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ACc T = 0 0 − 1 − 1 ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ Llng c ⎜ ⎟ ⎢ ⎥ ACs T 0 − 1 0 0 ⎜ ⎟ ⎢ ⎥ ail c ⎝ ⎠ ⎣ ⎦ ACp T 0 0 1 0 elev T 0 0 0 1 rud Aircraft 67 with cyclic stick and pedal connected to rotor controls only for helicopter mode. The sign conventions for the pilot’s controls are collective stick positive up, lateral cyclic stick positive right, longitudinal cyclic stick positive forward, and pedal positive nose right. The rotor controls are a positive Fourier series, with azimuth measured in the direction of rotation.

7–3 Trim The aircraft trim operation solves for the controls and motion that produce equilibrium in the specified flight state. In steady flight (including hover, level flight, climb and descent, and turns), equilibrium implies zero net force and moment on the aircraft. In general, there can be additional quantities that at equilibrium must equal target values. In practice, the trim solution can deal with a subset of these quantities. Usually it is at least necessary to achieve equilibrium in the aircraft lift and drag forces, as well as in yaw moment for torque balance. The basic purpose of the trim solution is to determine the component states, including aircraft drag and rotor thrust, sufficient to evaluate the aircraft performance.

Different trim solution definitions are required for various flight states. Therefore one or more trim states are defined for the analysis, and the appropriate trim state selected for each flight state of a performance condition or mission segment. For each trim state, the trim quantities, trim variables, and targets are specified. The available trim quantities include: aircraft total force and moment; aircraft load factor; propulsion group power; power margin P − P ; torque margin P − P ; avP G reqP G DS limit reqP G engine group power; power margin P − P ; avEG reqEG jet group thrust; thrust margin T − T ; avJG reqJG charge group power; charge power margin P − P ; avCG reqCG ˙ ˙ fuel tank energy flow E ; battery power margin, P − | E | ; batt max batt rotor force (lift, vertical, or propulsive); rotor thrust C /σ ; rotor thrust margin ( C /σ ) − C /σ ; T T max T rotor flapping β , β ; rotor hub moment, roll and pitch; rotor torque; c s wing force; wing lift coefficient C ; wing lift margin C − C ; L L max L tail force.

Targets for aircraft total force and total moment (including inertial loads in turns) are always zero. The available trim variables include: aircraft controls; aircraft orientation, θ (pitch), φ (roll); aircraft horizontal velocity V ; h aircraft vertical rate of climb V ; aircraft sideslip angle; c ˙ ˙ aircraft angular rate, θ (pullup), ψ (turn); propulsion group tip speed or engine speed.

The aircraft orientation variables are the Euler angles of the body axes relative to inertial axes. The aircraft controls (appropriately scaled) are connected to the component controls.

A Newton–Raphson method is used for trim. The derivative matrix is obtained by numerical perturbation. A tolerance  and a perturbation Δ are specified.

68 Aircraft axes for description of aircraft coordinate aircraft geometry system (arbitrary reference) (origin at CG) starboard WL BL y forward SL x aft port z Figure 7-2. Aircraft geometry.

7–4 Geometry The aircraft coordinate system has the x -axis forward, y -axis to the right, and z -axis down, measured from the center of gravity (fig. 7-2). These aircraft axes are body axes ( x is not aligned with the wind), the orientation determined by the convention used for the input geometry. The center of gravity is the appropriate origin for describing the motion of the aircraft, and summing the forces and moments acting on the aircraft.

Layout of the geometry is typically in terms of station line (SL, positive aft), buttline (BL, positive right), and waterline (WL, positive up), measured relative to some arbitrary origin (fig. 7-2). The x - y - z axes are parallel to the SL-BL-WL directions. One or more locations are defined for each component of the aircraft. Each component will at least have a location that is the point where component forces and moments act on the aircraft. Each location is input in fixed or scaled form. The fixed form input is SL/BL/WL (dimensional). The scaled form input is x/L (positive aft), y/L (positive right), and z/L (positive up), based on a reference length L , from a reference point. The reference length is the rotor radius or wing span of a designated component, or the fuselage length. The reference point can optionally be input, or the location (hub) of a designated rotor, or the location (center of action) of a designated wing component, or the location (center of action) of the fuselage, or the location of the center of gravity. Fixed input can be used for the entire aircraft, or just for certain components.

From this fixed or scaled input and the current aircraft size, the actual geometry ( x , y , z ) can be calculated for each location. There are also options to calculate geometry from other parameters (such as tiltrotor span from rotor radius and clearance). This calculated geometry has the sign convention of the aircraft axes ( x positive forward, y positive right, z positive down), but has the origin at the reference point (which may or may not be the center of gravity). All input uses the same sign convention; all internal calculations use the same sign conventions. Table 7-1 summarizes the conventions.

Aircraft 69 The location of the aircraft center of gravity is specified for a baseline configuration. With tilting rotors, this location is in helicopter mode. For each flight state the aircraft center of gravity is calculated, from the baseline location plus any shift due to nacelle tilt, plus an input center-of-gravity increment.

Alternatively, the aircraft center-of-gravity location for the flight state can be input. Any change of the center-of-gravity position with fuel burn during a mission is not automatically calculated, but could be accounted for using the flight state input.

The aircraft operating length and width are calculated from the component positions and dimensions: = x − x and w = y − y , where the maximum and minimum dimensions are for the total max min total max min fuselage and all rotors, wings, and tails. The corresponding footprint area is then S = w .

total total total Table 7-1. Geometry conventions.

layout scaled input calculated motion and loads origin arbitrary reference point reference point center of gravity x SL (+ aft) x/L (+ aft) x (+ forward) x (+ forward) y BL (+ right) y/L (+ right) y (+ right) y (+ right) z WL (+ up) z/L (+ up) z (+ down) z (+ down) 7–5 Aircraft Motion The aircraft velocity and orientation are defined by the following parameters: flight speed V ; turn rate; orientation of the body frame relative to inertial axes (Euler angles); and orientation of the velocity frame relative to inertial axes (flight path angles). Aircraft conventions are followed for the direction and orientation of axes: the z -axis is down, the x -axis forward, and the y -axis to the right; and a yaw- pitch-roll sequence is used for the Euler angles. However, the airframe axes are body axes (fixed to the airframe, regardless of the flight direction) rather than wind axes (which have the x -axis in the direction of the flight speed). The orientation of the body frame F relative to inertial axes I is defined by yaw, pitch, and roll Euler angles, which are rotations about the z , y , and x axes, respectively: F I C = X Y Z φ θ ψ F F F So yaw is positive to the right, pitch is positive nose up, and roll is positive to the right. The flight path is specified by the velocity V , in the positive x -axis direction of the velocity axes. The orientation of the velocity axes V relative to inertial axes I is defined by yaw (sideslip) and pitch (climb) angles: V I C = Y Z Z θ ψ ψ V V F Sideslip is positive for the aircraft moving to the right, and climb is positive for the aircraft moving up.

Then F V F I IV C = C C = X Y Z Y φ θ − ψ − θ F F V V In straight flight, all these angles and matrices are constant. In turning flight at a constant yaw rate, the ˙ ˙ yaw angle is ψ = ψ t ; the turn radius is R = V / ψ ; and the nominal bank angle and load factor are F F T h F √ 2 ˙ tan φ = n − 1 = ψ V /g . Then the forward, sideward, and climb velocities are: F F h V = V cos θ cos ψ = V cos ψ f V V h V V = V cos θ sin ψ = V sin ψ s V V h V V = V sin θ = V tan θ c V h V 70 Aircraft where V = V cos θ is the horizontal velocity component. The velocity components in airframe axes are h V F F I/F F V T v = v = C ( V 0 0) (aircraft velocity relative to the air). The calibrated airspeed is calculated AC from the true airspeed V : √ [ ] 2 / 7 [ ] 2 7 / 2 √ δ ((1 + 0 . 2 M ) − 1) + 1 − 1 √ 1 3 2 2 4 ∼ V = V σ = V σ 1 + (1 − δ ) M + (1 − 10 δ + 9 δ ) M cal 0 . 2 M δ 8 640 where σ = ρ/ρ is the density ratio, δ = p/p is the pressure ratio, and M is the Mach number. The 0 0 aircraft angular velocity is ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ˙ ˙ φ 1 0 − sin θ φ F F F F F I/F ˙ ˙ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ω = ω = R θ = 0 cos φ sin φ cos θ θ F F F F F AC ˙ ˙ ψ 0 − sin φ cos φ cos θ ψ F F F F F ˙ ˙ ˙ For steady state flight, θ = φ = 0 ; ψ is nonzero in a turn.

F F F F F I/F Accelerated flight is also considered, in terms of linear acceleration a = ˙ v = gn and pitch L AC ˙ ˙ rate θ . The nominal pullup load factor is n = 1 + θ V /g . For accelerated flight, the instantaneous F F h equilibrium of the forces and moments on the aircraft is evaluated, for specified acceleration and angular velocity; the equations of motion are not integrated to define a maneuver. Note that the fuselage and wing aerodynamic models do not include all roll and yaw moment terms needed for general unsteady flight (notably derivatives L , L , L , N , N , N ).

v p r v p r The aircraft pitch and roll angles are available for trim of the aircraft. Any motion not selected for trim will remain fixed at the values specified for the flight state. The pitch and roll angles each can be zero, constant, or a function of flight speed (piecewise linear input). The flight state input can override this value of the aircraft motion. The input value is an initial value if the motion is a trim variable.

7–6 Loads and Performance For each component, the power required and the net forces and moments acting on the aircraft can be calculated. The aerodynamic forces F and moments M are typically calculated in wind axes and then resolved into body axes ( x , y , z ), relative to the origin of the body axes (the aircraft center of gravity).

The power and loads of all components are summed to obtain the aircraft power and loads. Typically the trim solution drives the net forces and moments on the aircraft to zero.

The aircraft equations of motion, in body axes F with origin at the aircraft center of gravity, are the equations of force and moment equilibrium: F I/F F I/F F I/F F F m ( ˙ v + ˜ ω v ) = F + F grav F F I/F F I/F F F I/F F I ˙ ω + ˜ ω I ω = M F F I I F I T where m = W/g is the aircraft mass; the gravitational force is F = mC g = mC (0 0 g ) ; and grav the moment of inertia matrix is ⎡ ⎤ I − I − I xx xy xz F ⎣ ⎦ I = − I I − I yx yy yz − I − I I zx zy zz F I/F F I/F F I/F T ˙ For steady flight, ˙ ω = ˙ v = 0 , and ω = R (0 0 ψ ) is nonzero only in turns. For accelerated F F I/F F I/F T ˙ ˙ flight, ˙ v can be nonzero, and ω = R (0 θ ψ ) . The equations of motion are thus F F F F F F F m ( a + ˜ ω v ) = F + F AC AC AC grav F F F F ˜ ω I ω = M AC AC Aircraft 71 F I I F F F F The body axis load factor is n = ( C g − ( a + ˜ ω v )) /g . The a term is absent for steady flight.

AC AC AC AC The forces and moments are the sum of loads from all components of the aircraft: ∑ ∑ ∑ ∑ ∑ ∑ F F F F F F F F F F = F + F + F + F + F + F + F + F fus rotor wing tail tank engine jet charge ∑ ∑ ∑ ∑ ∑ ∑ F F F F F F F F F M = M + M + M + M + M + M + M + M fus rotor wing tail tank engine jet charge I IF F I IF F Forces and moments in inertial axes are also of interest ( F = C F and M = C M ). A particular component can have more than one source of loads; for example, the rotor component produces hub F forces and moments, but also includes hub and pylon drag. The equations of motion are E = F + f F F F F F − F = 0 and E = M − M = 0 .

m grav inertial inertial The component power required P is evaluated for all components (rotors, motors, and com- comp pressors) of the propulsion group. The total power required for the propulsion group P is ob- reqP G tained by adding the transmission losses and accessory power. The power required for the propulsion group must be distributed to the engine groups. The fuel flow is calculated from the engine power, jet thrust, and charger power required. The total fuel flow is the sum from all components of the aircraft: ∑ ∑ ∑ ˙ w = ˙ w + ˙ w + ˙ w .

reqEG reqJG reqCG 7–7 Aerodynamics F Each component has a position z in aircraft axes F, relative to the reference point; and orientation BF of component axes B relative to aircraft axes given by the rotation matrix C . It is expected that the component axes are (roughly) x forward and z down (or in negative lift direction). The aerodynamic model must be consistent with the convention for component orientation. Acting at the component F are interference velocities v (velocity of air, in F axes), from all other components. Then the total int component velocity relative to the air is ∑ F F F F F v = v + ˜ ω Δ z − v AC AC int F F F B BF F where Δ z = z − z . Then v = C v is the velocity in component axes. The aerodynamic cg B 2 environment is defined in the component axes: velocity magnitude v = | v | , dynamic pressure q = / ρv , angle of attack α , and sideslip angle β . The angle of attack and sideslip angle provide the transformation between airframe axes and velocity axes: BA C = Y Z α − β This is the conventional aircraft definition, corresponding to yaw-then-pitch of the airframe axes relative B BA T to the velocity vector. By definition, the velocity is along the x -axis in the A axes, v = C ( v 0 0) ; B from which the angle of attack and sideslip in terms of the components of v are obtained: − 1 B B α = tan v /v 3 1 − 1 B B β = sin v / | v | B B This definition is not well behaved for v = 0 (it gives α = 90 sign( v ) ), so for sideward flight a 1 3 BA pitch-then-yaw definition can be useful: C = Z Y . Then − β α − 1 B B α = sin v / | v | − 1 B B β = tan v /v 2 1 B B which gives β = 90 sign( v ) for v = 0 .

2 1 72 Aircraft The component aerodynamic model may include coefficient values for sideward flight, but not have equations for a continuous variation of the coefficients with sideslip angle. For such cases, sideward flight is defined as | β | = 80 to 100 degrees.

From v , q , α , and β , the aerodynamic model calculates the component force and moment, in wind F axes acting at z : ⎛ ⎞ ⎛ ⎞ − D M x A A ⎝ ⎠ ⎝ ⎠ F = Y M = M y − L M z where D , Y , and L are the drag, side force, and lift; M , M , and M are the roll, pitch, and yaw moments.

x y z The aerodynamic loads in aircraft axes acting at the center of gravity are then: F F B BA A F = C C F F F B BA A F ˜ F M = C C M + Δ z F F F F I T IF F where Δ z = z − z . In hover and low speed, the download is calculated: F = k ( C F ) , the cg z downward component of the aerodynamic force in inertial axes. Download can be expressed as a fraction of the total rotor vertical force or as a fraction of gross weight. The aerodynamic model also calculates F F B B the interference velocities caused by this component at all other components: v = C v .

int int Equations for the aerodynamics models are defined for all angles in radians. Input values of angles are, however, in degrees.

The aircraft neutral point is calculated from the airframe aerodynamics with all controls set to zero.

The neutral point is here defined as the longitudinal position about which the derivative of the pitch moment with lift is zero. Hence SL = SL − Δ M/ Δ L , with the change in lift and moment calculated na cg from the loads at angles of attack of 0 and 5 deg.

7–8 Trailing-Edge Flaps The lifting surfaces have controls in the form of trailing-edge flaps: flap, flaperon, and aileron for wings; elevator or rudder for tails. The aerodynamic loads generated by flap deflection δ (radians) are f estimated based on two-dimensional aerodynamic data (as summarized in refs. 1 and 2). Let = c /c f f be the ratio of the flap chord to the wing chord. The lift coefficient is c = c ( α + τ ηδ ) , where  α f ∼ η = 0 . 85 − 0 . 43 δ is an empirical correction for viscous effects (ref. 1, eq. 3.54 and fig. 3.36). Thin f airfoil theory gives ( ) n θ − sin θ π f f ∼ τ = 1 − = sin( ) f π 2 − 1 with θ = cos (2 − 1) (ref. 1, eq. 3.56 and fig. 3.35; ref. 2, eq. 5.40). The last expression is an f f approximation that is a good fit to the thin airfoil theory result for n = / , and a good approximation 2 2 including the effects of real flow for n = / (ref. 2, fig. 5.18); the last expression with n = / is used 3 3 here. The increase of maximum lift coefficient caused by flap deflection is less than the increase in lift coefficient, so the stall angle of attack is decreased. Approximately Δ c  max 2 3 ∼ − ) (1 + − 5 + 3 ) = (1 f f f f Δ c  (ref. 1, fig. 3.37). Thin airfoil theory gives the moment coefficient increment about the quarter chord: √ ( ) ( ) Δ c = − 0 . 85 (1 − ) sin θ δ = − 0 . 85 (1 − )2 (1 − ) δ m f f f f f f f Aircraft 73 (ref. 1, eq. 3.57; ref. 2, eq. 5.41); with the factor of 0 . 85 accounting for real flow effects (ref. 2, fig. 5.19).

The drag increment is estimated using S f 1 . 38 2 Δ C = 0 . 9 sin δ D f f S for slotted flaps (ref. 1, eq. 3.51). In summary, the section load increments are: c f Δ c = c L η δ  α f f f c Δ c = X Δ c  max f  c f Δ c = M δ m f f c The decrease in angle of attack for maximum lift is Δ c − Δ c Δ c   max  Δ α = − = − (1 − X ) max f c c α α The coefficients η = 0 . 85 − 0 . 43 | δ | = η − η | δ | f f 0 1 f ( ) 2 / 3 1 π L = sin( ) f f f 2 3 X = (1 − ) (1 + − 5 + 3 ) f f f f f √ ( ) M = − 0 . 85 (1 − )2 (1 − ) f f f f f 1 . 38 D = 0 . 9 f f follow from these equations.

For three-dimensional aerodynamic loads, these two-dimensional coefficients are corrected by using the three-dimensional lift-curve slope, and multiplying by the ratio of flap span to wing span b /b .

f Then the wing load increments caused by flap deflection, in terms of coefficients based on the wing area, are: S f Δ C = C L η δ L Lα f f f S Δ C = X Δ C L max f L S f Δ C = M δ Δ C M f f L S Δ α = − (1 − X ) max f C Lα S f Δ C = D sin δ D f f S where S /S is the ratio of flap area to wing area.

f 7–9 Drag Each component can contribute drag to the aircraft. A fixed drag can be specified as a drag area D/q ; or the drag can be scaled, specified as a drag coefficient C based on an appropriate area S . There D may also be other ways to define a scaled drag value. For fixed drag, the coefficient is C = ( D/q ) /S D (the aerodynamic model is formulated in terms of drag coefficient). For scaled drag, the drag area is D/q = SC . For all components, the drag ( D/q ) or C is defined for forward flight or D comp D comp cruise; typically this is the minimum drag value. For some components, the vertical drag ( ( D/q ) or V comp C ) or sideward drag ( ( D/q ) or C ) is defined. For some components, the aerodynamic DV comp S comp DS comp model includes drag due to lift, angle of attack, or stall.

74 Aircraft Table 7-2 summarizes the component contributions to drag, and the corresponding reference areas.

If no reference area is indicated, then the input is only drag area D/q . An appropriate drag reference area is defined for each component, and either input or calculated. Wetted area is calculated for each component, even if it is not the reference area. The component wetted areas are summed to obtain the aircraft wetted area. Some of the weight models also require the wetted area. The component drag contributions must be consistent. In particular, a rotor with a spinner (such as on a tiltrotor aircraft) would likely not have hub drag. The pylon is the rotor support and the nacelle is the engine support. The drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag), since the pylon is connected to the rotor shaft axes; with non-tilting engines it would use the nacelle drag as well.

Table 7-2. Component contributions to drag.

component drag contribution reference area fuselage fuselage fuselage wetted area fuselage vertical fuselage projected area fittings and fixtures fuselage wetted area rotor-body interference fuselage wetted area contingency (aircraft) — payload increment (flight state) — landing gear landing gear — rotor hub, hub vertical rotor disk area duct, duct vertical duct wetted area pylon, pylon vertical pylon wetted area spinner spinner wetted area wing wing, wing vertical wing planform area wing-body interference wing planform area tail tail, tail vertical tail planform area fuel tank auxiliary tank (flight state) — engine nacelle, nacelle vertical nacelle wetted area momentum drag — jet nacelle, nacelle vertical nacelle wetted area momentum drag — charger nacelle, nacelle vertical nacelle wetted area momentum drag — Optionally the aircraft drag can be fixed. The quantity specified is the sum (over all components) of the drag area D/q (minimum drag, excluding drag due to lift and angle of attack), without accounting for interference effects on dynamic pressure. The input parameter can be D/q ; or the drag can be scaled, specified as a drag coefficient based on the rotor disk area, so D/q = A C ( A is the reference rotor ref D ref 2 / 3 disk area); or the drag can be estimated based on the gross weight, D/q = k ( W / 1000) ( W M T O M T O 2 2 / 3 2 2 / 3 is the maximum takeoff gross weight; units of k are ft /k-lb or m /Mg ). Based on historical data, the drag coefficient C = 0 . 02 for old helicopters, C = 0 . 008 for current low-drag helicopters. Based D D on historical data, k = 9 for old helicopters, k = 2 . 5 for current low-drag helicopters, k = 1 . 6 for current tiltrotors, and k = 1 . 4 for turboprop aircraft (English units). If the aircraft drag is input, then the fuselage contingency drag is adjusted so the total aircraft D/q equals the input value.

Aircraft 75 Table 7-3. Component contributions to nominal drag area.

component drag contribution cruise helicopter vertical fuselage fuselage D D D V fittings and fixtures D D D rotor-body int D D D landing gear landing gear D D 0 retractable 0 D 0 2 2 rotor hub D D cos i + D sin i 0 V 2 2 duct D D cos i + D sin i 0 V 2 2 pylon D D cos i + D sin i 0 V spinner D D 0 2 2 2 2 wing wing D D cos i + D sin i D sin i + D cos i V V wing-body int D D D tail horizontal tail D D D cos φ V tail vertical tail D D D sin φ V 2 2 engine nacelle D D cos i + D sin i 0 V 2 2 jet nacelle D D cos i + D sin i 0 V 2 2 charger nacelle D D cos i + D sin i 0 V contingency D D D V Optionally the aircraft vertical drag (download fraction) can be fixed. The quantity specified is the sum over all components of the vertical drag area ( D/q ) . The input parameter can be ( D/q ) , or V V k = ( D/q ) /A ( A is reference rotor disk area). Approximating the dynamic pressure in the wake V ref ref as q = / ρ (2 v ) = T /A , the download fraction is DL/T = q ( D/q ) /T = k . If the aircraft vertical 2 h ref V drag is input, then the fuselage contingency vertical drag is adjusted so the total aircraft ( D/q ) equals V the input value.

The nominal drag areas of the components and the aircraft are part of the aircraft description and are used when the aircraft drag is fixed. The nominal drag area is calculated for low-speed helicopter flight, for high-speed cruise flight, and for vertical flight. An incidence angle i is specified for the rotors, wings, and nacelles, to be used solely to calculate the nominal helicopter and vertical drag areas.

The convention is that i = 0 if the component does not tilt. Table 7-3 summarizes the contributions to the nominal drag areas, with D for the drag in normal flow and D for the drag in vertical flow.

V While vertical drag parameters are part of the aerodynamic model for the hub, duct, pylon, and nacelle, aerodynamic interference at the rotor and at the propulsion components is not considered, so these terms do not contribute to download. In the context of download, only the fuselage, wing, tail, and contingency contribute to the nominal vertical drag.

From the input and the current aircraft size, the drag areas D/q and coefficients C are calculated.

D The aerodynamic analysis is usually in terms of coefficients. If the aircraft drag is fixed for the aircraft model, then the fuselage contingency drag is set: ∑ ( D/q ) = ( D/q ) − ( D/q ) cont fixed comp 76 Aircraft and similarly for fixed vertical drag. Note that this adjustment ignores changes caused by interference in the dynamic pressure and the velocity direction, which will affect the actual component drag.

The component aerodynamic model calculates the drag, typically from a drag coefficient C , a D ∑ reference area, and the air velocity of the component. The drag force is then D = q S C , where comp ref D the dynamic pressure q includes interference. From the aerodynamic forces and moments in wind comp F F axes, the total force and moment in body axes ( F and M ) are calculated. For reference, the aircraft total drag and total drag area are ∑ T F D = e F AC d aero ( D/q ) = D /q AC AC F F where the aircraft velocity (without interference) gives the direction e = − v / | v | and dynamic d AC AC F 2 F pressure q = / ρ | v | ; and F is the component aerodynamic force. An overall skin friction AC aero ∑ drag coefficient is then C = ( D/q ) /S , based on the aircraft wetted area S = S D AC AC wet AC AC wet and excluding drag terms not associated with skin friction (specifically landing gear, rotor hub, and contingency drag).

7–10 Performance Metrics The following performance metrics are calculated for the aircraft. The aircraft hover figure of merit √ is M = W W/ 2 ρA /P . The aircraft effective drag is D = P/V , hence the effective lift-to-drag ratio ref e is L/D = W V /P . For these metrics, the aircraft power is the sum of the engine group power, jet group e propulsive power, and charge group power: P = P + V T + P . The aircraft power loading is req jet chrg W/P (lb/hp or kg/kW). Isolated rotor performance metrics are described in Chapter 11.

7–11 Weights The design gross weight W is a principal parameter defining the aircraft, usually determined D by the sizing task for the design conditions and missions. The aircraft weight statement defines the empty weight, fixed useful load, and operating weight for the design configuration. The aircraft weight statement is the sum of the weight statements for all the aircraft components, the component weight determined by input or by parametric calculations with technology factors. The definitions of the weight terms are as follows: gross weight W = W + W = W + W + W G E U L O pay fuel operating weight W = W + W O E F U L useful load W = W + W + W U L F U L pay fuel where W is the weight empty; W the fixed useful load; W the payload weight; and W the E F U L pay fuel usable fuel weight. Aircraft weight definitions are given in SAWE RP7D (ref. 3), including: Payload is any item which is being transported and is directly related to the purpose of the flight as opposed to items that are necessary for the flight operation. Payload can include, but is not limited to, passengers, cargo, passenger baggage, ammo, internal and external stores, and fuel which is to be delivered to another aircraft or site. Payload may or may not be expended in flight.

Operating weight is the sum of aircraft weight empty and operating items. Operating weight is equivalent to takeoff gross weight less usable fuel, payload, and any item to be expended in flight.

Aircraft 77 Weight empty is an engineering term which is defined as the weight of the complete aircraft as defined in the aircraft specifications, dry, clean, and empty except for fluids in closed systems such as a hydraulic system.

The weight empty consists of structure, propulsion group, systems and equipment, vibration, and contingency weights. If the weight empty is input, then the contingency weight is adjusted so W equals E the required value. If the design gross weight is input, then the payload or fuel weight must be fallout.

The structural design gross weight W and maximum takeoff weight W can be input, or SD M T O specified as an increment d plus a fraction f of a weight W : { d + f W SDGW SDGW D W = d + f W = SD SDGW SDGW d + f ( W − W + f W ) SDGW SDGW D fuel fuel fuel − cap { d + f W W M T O W M T O D W = d + f W = M T O W M T O W M T O d + f ( W − W + W ) W M T O W M T O D fuel fuel − cap This convention allows the weights to be input directly ( f = 0 ), or scaled with W . For W , W is D SD the design gross weight W , or W adjusted for a specified fuel state (input fraction of fuel capacity).

D D Alternatively, W can be calculated as the gross weight at a designated sizing flight condition. The SD structural design gross weight is used in the weight estimation. For W , W is the design gross M T O weight W , or W adjusted for maximum fuel capacity. Alternatively, W can be calculated as the D D M T O maximum gross weight possible at a designated sizing flight condition. The maximum takeoff weight is used in the cost model, in the scaled aircraft and hub drag, and in the weight estimation.

The design ultimate load factor n at the structural design gross weight W is specified, in z ult SD particular for use in the component weight estimates. The structural design gross weight W and SD design ultimate load factor n are used for the fuselage, rotor, and wing weight estimations. The z ult maximum takeoff weight W is used for the cost and drag (scaled aircraft and hub), and for the M T O weights (system, fuselage, landing gear, and engine group).

The gross weight W is specified for each flight condition and mission, perhaps in terms of the G design gross weight W . For a each flight state, the fixed useful load may be different from the design D configuration because of changes in auxiliary fuel tank weight, or kit weights, or increments in crew or equipment weights. Thus the fixed useful load weight is calculated for the flight state; and from it the useful load weight and operating weight are calculated. The gross weight, payload weight, and usable fuel weight (in standard and auxiliary tanks) complete the weight information for the flight state.

For each weight group, fixed (input) weights can be specified; or weight increments dW added to the results of the parametric weight model. The parametric weight model includes technology factors χ . Thus typically a component or element weight is obtained from W = χW + dW . Weight of model individual elements in a group can be fixed by using dW and setting the corresponding technology factor χ = 0 . With χ  = 0 , the increment dW can account for something not included in the parametric model.

For scaled weights of all components, the AFDD weight models are implemented. The user can incorporate custom weight models as well.

The operating weight is composed of scaled and fixed weights, so the design gross weight can be written W = W + W + W = W + W + W + W . The growth factor is the change D O pay fuel O fixed O scaled pay fuel 78 Aircraft in gross weight due to a change in payload: ( ) ∂W ∂W ∂W ∂W ∂W ∂W D O scaled fuel O scaled fuel D = 1 + + = 1 + + ∂W ∂W ∂W ∂W ∂W ∂W pay pay pay D D pay ( ) W W ∂W 1 O scaled fuel D ∼ = 1 + + = W W ∂W 1 − φ − φ D D pay O scaled fuel in terms of the weight fractions φ = W/W .

D The aircraft operating weight can be divided into core vehicle weight and military load. The core vehicle weight is the weight in minimum airworthy state, with the aircraft capable of normal flight throughout the envelope, but not mission capable. Military load is the sum of fixed useful load and military features in weight empty. Thus weight empty = core vehicle weight + military features military load = fixed useful load + military features in weight empty operating weight = W + W = core vehicle weight + military load E F U L In terms of the weight breakdown used here, military features in weight empty consist of folding weight (wing, rotor, tail, fuselage terms), crashworthiness weight (fuselage, landing gear terms), marinization weight (fuselage), rotor brake (drive system), avionics group (mission equipment), armament group, furnishings and equipment group, anti-icing group (including electrical group term), and load and handling group.

7–12 Weight Statement Aircraft weight information is stored in a data structure that follows SAWE RP8A Group Weight Statement format (ref. 4), as outlined in figure 7-3. The asterisks designate extensions of RP8A for the purposes of this analysis. Typically only the lowest elements of the hierarchy are specified; higher elements are obtained by summation. Fixed (input) weight elements are identified in the data structure.

A weight statement data structure exists for each component. The aircraft weight statement is the sum of the structures from all components.

7–13 References 1) McCormick, B.W. Aerodynamics, Aeronautics, and Flight Mechanics. New York: John Wiley & Sons, Second Edition, 1995.

2) Kuethe, A.M., and Chow, C.-Y. Foundations of Aerodynamics. New York: John Wiley & Sons, Fifth Edition, 1998.

3) “Mass Properties Management and Control for Military Aircraft, Revision D.” Society of Allied Weight Engineers, Recommended Practice Number 7, May 2004.

4) “Weight and Balance Data Reporting Forms for Aircraft (including Rotorcraft), Revision A.” Society of Allied Weight Engineers, Recommended Practice Number 8, June 1997.

Aircraft 79 WEIGHT EMPTY STRUCTURE wing group basic structure secondary structure fairings (*), fittings (*), fold/tilt (*) control surfaces rotor group blade assembly hub & hinge basic (*), fairing/spinner (*), blade fold (*), shaft (*) rotor support structure (*), duct (*) empennage group horizontal tail (*) basic (*), fold (*) vertical tail (*) basic (*), fold (*) tail rotor (*) blades, hub & hinge, rotor supports, rotor/fan duct fuselage group basic (*) wing & rotor fold/retraction (*) tail fold/tilt (*) marinization (*) pressurization (*) crashworthiness (*) alighting gear group basic (*), retraction (*), crashworthiness (*) engine section or nacelle group engine support (*), engine cowling (*), pylon support (*) air induction group PROPULSION GROUP engine system engine exhaust system accessories (*) propeller/fan installation blades (*), hub & hinge (*), rotor supports (*), rotor/fan duct (*) fuel system tanks and support plumbing drive system gear boxes transmission drive rotor shaft rotor brake (*) clutch (*) gas drive Figure 7-3a. Weight statement (* indicates extension of RP8A).

80 Aircraft SYSTEMS AND EQUIPMENT flight controls group cockpit controls automatic flight control system system controls fixed wing systems non-boosted (*), boost mechanisms (*) rotary wing systems non-boosted (*), boost mechanisms (*), boosted (*) conversion systems non-boosted (*), boost mechanisms (*) auxiliary power group instruments group hydraulic group fixed wing (*), rotary wing (*), conversion (*) equipment (*) pneumatic group electrical group aircraft (*), anti-icing (*) avionics group (mission equipment) armament group armament provisions (*), armor (*) furnishings & equipment group environmental control group anti-icing group load & handling group VIBRATION (*) CONTINGENCY FIXED USEFUL LOAD crew fluids (oil, unusable fuel) (*) auxiliary fuel tanks other fixed useful load (*) equipment increment (*) folding kit (*) wing extension kit (*) wing kit (*) other kit (*) PAYLOAD USABLE FUEL standard tanks (*) auxiliary tanks (*) OPERATING WEIGHT = weight empty + fixed useful load USEFUL LOAD = fixed useful load + payload + usable fuel GROSS WEIGHT = weight empty + useful load GROSS WEIGHT = operating weight + payload + usable fuel Figure 7-3b. Weight statement (* indicates extension of RP8A).

Chapter 8 Systems The systems component contains weight information (fixed useful load, vibration, contingency, and systems and equipment).

8–1 Weights The weight empty consists of structure, propulsion group, systems and equipment, vibration, and contingency weights. The vibration control weight can be input, or specified as a fraction of weight empty: W = f W . The contingency weight can be input, or specified as a fraction of weight empty: vib vib E W = f W . However, if the weight empty is input, then the contingency weight is adjusted so cont cont E W equals the required value. The weights of all components are evaluated and summed, producing the E aircraft weight empty less vibration and contingency weight, W . Then: X a) Fixed weight empty: W input or W = f W , W = W − ( W + W ) .

vib vib vib E cont E X vib b) Both fractional: W = W / (1 − f − f ), W = f W , W = f W .

E X vib cont vib vib E cont cont E c) Only vibration weight fractional: W input, W = ( W + W ) / (1 − f ) , cont E X cont vib W = f W .

vib vib E d) Only contingency weight fractional: W input, W = ( W + W ) / (1 − f ) , vib E X vib cont W = f W .

cont cont E e) Both input: W and W input, W = W + W + W .

vib cont E X vib cont Finally, the operating weight W = W + W is recalculated.

O E F U L Systems and equipment includes the following fixed (input) weights: auxiliary power group, in- struments group, pneumatic group, electrical group (aircraft), avionics group (mission equipment), armament group (armor and armament provisions), furnishings and equipment group, environmental control group, and load and handling group. Systems and equipment includes the following scaled weights: flight controls group, hydraulic group, electrical group (anti-icing), and anti-icing group.

Flight controls group includes the following fixed (input) weights: cockpit controls and automatic flight control system. Flight controls group includes the following scaled weights: fixed wing systems, rotary wing systems, and conversion or thrust vectoring systems. Rotary wing flight control weights can be calculated for the entire aircraft (using rotor parameters such as chord and tip speed for a designated rotor), an approach that is consistent with parametric weight equations developed for conventional two- rotor configurations. Alternatively, rotary wing flight control weights can be calculated separately for each rotor and then summed. The fixed wing flight controls and the conversion controls can be absent.

The fixed useful load W consists of crew ( W ), trapped fluids (oil and unusable fuel, W ), F U L crew trap auxiliary fuel tanks ( W ), equipment increment, kits (folding, wing, wing extension, other), and auxtank other fixed useful load ( W ). W , W , and W are input. For a each flight state, the F U L other crew trap F U L other 82 Systems fixed useful load may be different from the design configuration because of changes in auxiliary fuel tank weight, kit weight, and crew or equpment weight increments.

Folding weights can be calculated in several weight groups, including wing, rotor, empennage, and fuselage. These weights are the total weights for folding and the impact of folding on the group. A fraction f of these weights can be in a kit, hence optionally removable. Thus, of the total folding foldkit weight, the fraction f is a kit weight in the fixed useful load of the weight statement, while the foldkit remainder is kept in the component group weight.

8–2 Detailed Weight Definition Figure 8-1 presents a more detailed description of the weights for the systems and equipment, and the useful load. The input information for a group can be just the total weight ( Δ W in the following equations), or can include all the terms.

The electrical group consists of power supply, power conversion, power distribution and controls, lights and signal devices, and equipment supports; plus a term associated with the anti-icing group. The total weight is W = W + W electrical elect aircraft DI elect W = W + W + W + W + W + Δ W elect aircraft supply conv distrib lights support electrical The avionics group consists of equipment and installation; here the equipment weights include installa- tion. The equipment items are communications, navigation, identification, control and display, aircraft survivability, and mission system equipment; plus armament electronics. The total weight is W = W + W + W + W + W + W + Δ W + W MEQ com nav ident display survive mission MEQ arm elect The armament group consists of armament provisions (gun provisions, turret systems, expendable weapons provisions) and armor. Armament electronics weight W (such as targeting, sights, arm elect radar) is part of the avionics group.

W = W + W + W + Δ W arm prov gun turret expend arm prov W = U S + U S + U N + Δ W armor armor floor cabin floor armor wall cabin wall armor crew crew seat armor Here U , U are the armor weights per surface area; S and S are the armor floor armor wall cabin floor cabin wall cabin floor and wall areas; U is the armor weight per crew; and N is number of crew armor crew crew seat seats.

The furnishings and equipment group consists of accommodation for personnel, miscellaneous equipment, furnishings, and emergency equipment. Accommodation for personnel consists of seats, miscellaneous accommodation (including galleys, toilets), and the oxygen system. Miscellaneous equip- ment includes cockpit displays. Furnishings includes floor covering, trim, partitions, crash padding, and acoustic and thermal insulation; but excludes vibration absorbers. Emergency equipment consists of fire detection and extinguishing, and other emergency equipment (including first aid, survival kit, and life raft). The total weight is W = W + W + W + W + Δ W furnish accom misc furn emerg furnish W = ( U + U + U ) N accom seat crew accom crew ox crew crew seat + ( U + U + U ) N seat pass accom pass ox pass pass seat W = W + U S furn trim insulation cabin W = W + W emerg fire other Systems 83 Here U , U are the weights per crew and passenger; N , N are the number of zz crew zz pass crew seat pass seat crew and passenger seats; U is the acoustic and thermal insulation weight per area; and S is insulation cabin the total cabin surface area. The load and handling group consists of aircraft handling and load handling (cargo handling, hoist, external load provisions). The total weight is W = W + W + Δ W load aircraft cargo load W = U S + W + W cargo handling cabin floor hoist ext prov Here U is the cargo handling weight per cabin floor area, and S is the cabin floor area.

handling cabin floor The crew weight is W = U N + Δ W crew crew crew crew Here U is the weight per crew, and N is the number of crew. A crew weight increment for a crew crew flight condition or mission is given by δN and δW . Other fixed useful load consists of various crew crew categories, such as baggage, gun installations, weapons provisions, aircraft survivability equipment (chaff, flares), survival kits, life rafts, and oxygen.

An equipment increment can be defined for a flight condition or mission, in terms of δN , crew seat δN , and δW : pass seat equip W = U δN + U δN + δW equip inc crew seat inc crew seat pass seat inc pass seat equip The default weights per crew and passenger seats are U = U + U + U + crew seat inc seat crew accom crew ox crew U and U = U + U + U .

armor crew pass seat inc seat pass accom pass ox pass The payload consists of passengers or troops, cargo (internal and external), ammunition, and weapons: W = U N + W + W + W + W + Δ W payload pass pass cargo ext load ammo weapons payload Here U is the weight per passenger, and N is the number of passengers. A payload increment is pass pass defined by Δ W . For fallout payload, the value of Δ W is adjusted.

payload payload 84 Systems WEIGHT EMPTY SYSTEMS AND EQUIPMENT electrical group aircraft power supply power conversion power distribution and controls lights and signal devices equipment supports anti-icing avionics group (mission equipment) equipment installation armament group armament provisions armor furnishings & equipment group accommodation for personnel seats miscellaneous accommodation oxygen system miscellaneous equipment furnishings emergency equipment fire detection and extinguishing other emergency equipment load & handling group aircraft handling load handling USEFUL LOAD FIXED USEFUL LOAD crew other fixed useful load various categories equipment increment PAYLOAD passengers/troops cargo ammunition weapons Figure 8-1. Details of weight descriptions (based on RP8A).

Chapter 9 Fuselage There is one fuselage component for the aircraft.

9–1 Geometry The fuselage length can be input or calculated. The calculated length depends on the longitudinal fus positions of all components. Let x and x be the maximum (forward) and minimum (aft) position max min of all rotors, wings, and tails. Then the calculated fuselage length is = + ( x − x ) + fus nose max min aft The nose length (distance forward of hub) and aft length (distance aft of hub) are input, or nose aft calculated as = f L and = f L . Typically f = 0 or negative for the main-rotor and nose nose aft aft aft tail-rotor configuration, and f = 0 . 75 for the coaxial configuration. The fuselage width w is input.

aft fus The reference length L is the rotor radius or wing span of a designated component, or the input fuselage length.

The fuselage wetted area S (reference area for drag coefficients) and projected area S (ref- wet proj erence area for vertical drag) are input (excluding or including the tail boom terms); or calculated from the nose length: S = f (2 h + 2 w + 2 h w ) + C L wet wet nose fus nose fus fus fus boom S = f ( w ) + w L proj proj nose fus boom using input fuselage height h , and factors f and f ; or calculated from the fuselage length: fus wet proj S = f (2 h + 2 w + 2 h w ) + C L wet wet fus fus fus fus fus fus boom S = f ( w ) + w L proj proj fus fus boom Using the nose length and the tail boom area is probably best for a single-main-rotor and tail-rotor helicopter. Here C is the effective tail boom circumference (boom wetted area divided by reference boom length), and w is the effective tail boom width (boom vertical area divided by reference length).

boom Cabin areas are required for weight estimates: total cabin surface area S for acoustic and cabin thermal insulation weight (furnishings and equipment group); cabin floor area S for armor and cabin floor cargo handling weights; and cabin wall area S for armor weight. These areas are input, or cabin wall calculated from the fuselage dimensions: S = f (2 h + 2 w ) cabin cabin fus fus fus fus S = f ( w ) cabin floor floor fus fus S = f (2 h ) cabin wall wall fus fus ∼ with typically f = 0 . 6 .

86 Fuselage The fuselage contribution to the aircraft operating length is x + f (forward) and x − (1 − fus ref fus fus f ) (aft). Here f is the position of the fuselage aerodynamic reference location aft of the nose, as ref fus ref a fraction of the fuselage length. If the fuselage length is input, then f is input; if the fuselage length ref is calculated, then f = ( x + − x ) / .

ref max nose fus fus 9–2 Control and Loads F The fuselage has a position z , where the aerodynamic forces act; and the component axes are BF aligned with the aircraft axes, C = I . The fuselage has no control variables.

9–3 Aerodynamics The aerodynamic velocity of the fuselage relative to the air, including interference, is calculated in B BA component axes, v . The angle of attack α , sideslip angle β (hence C ), and dynamic pressure q fus fus B are calculated from v . The reference area for the fuselage forward flight drag is the fuselage wetted area S , which is input or calculated as described previously. The reference area for the fuselage vertical wet drag is the fuselage projected area S , which is input or calculated as described previously.

proj 9-3.1 Drag The drag area or drag coefficient is defined for forward flight, vertical flight, and sideward flight.

In addition, the forward flight drag area or drag coefficient is defined for fixtures and fittings, and for rotor-body interference. The effective angle of attack is α = α − α , where α is the angle e fus D min D min of minimum drag; in reverse flow ( | α | > 90 ), α ← α − 180 sign α . For angles of attack less than a e e e e transition angle α , the drag coefficient equals the forward flight (minimum) drag C , plus an angle of t D 0 attack term. Thus if | α | ≤ α e t X d C = C (1 + K | α | ) D D 0 d e and otherwise X d C = C (1 + K | α | ) Dt D 0 d t ( ) ( ) S π | α | − α proj e t C = C + C − C sin D Dt DV Dt S 2 π/ 2 − α wet t and similarly for the transition of payload drag ( D/q ) and contingency drag ( D/q ) . Optionally pay cont there might be no angle-of-attack variation at low angles ( K = 0 ), or quadratic variation ( X = 2 ). With d d an input transition angle, there will be a jump in the slope of the drag coefficient at α . For a smooth t transition, the transition angle that matches slopes as well as coefficients is found by solving ( ) 2 X ( S /S ) C − C d proj wet DV D 0 X X − 1 d d − 1 α − X α + = 0 d t t π K C d D 0 This calculation of the transition angle is only implemented with quadratic variation, for which ⎛ ⎞ √ 1 ( S /S ) C − C proj wet DV D 0 ⎝ ⎠ α = 1 + 1 − a t a K C d D 0 B with a = (4 /π ) − 1 ; α is however required to be between 15 and 45 deg. For sideward flight ( v = 0 ) the t x − 1 B B drag is obtained using φ = tan ( − v /v ) to interpolate between sideward and vertical coefficients: v z y S proj 2 2 C = C cos φ + C sin φ D DS v DV v S wet Fuselage 87 Then the drag force is ( ) ( ) ∑ D = qS C + C + C + q ( D/q ) + ( D/q ) wet D D fit D rb pay cont including drag coefficient for fixtures and fittings C and rotor-body interference C (summed over D fit D rb all rotors); drag area of the payload (specified for flight state); and contingency drag area.

9-3.2 Lift and Pitch Moment The fuselage lift and pitch moment are defined in fixed form ( L/q and M/q ), or scaled form ( C L and C , based on the fuselage wetted area and fuselage length). The effective angle of attack is M α = α − α , where α is the angle of zero lift; in reverse flow ( | α | > 90 ), α ← α − 180 sign α . Let e fus zl zl e e e e α be the angle-of-attack increment (above or below zero lift angle) for maximum lift. If | α | ≤ α max e max C = C α L Lα e C = C + C α M M 0 M α e and otherwise ( ) π/ 2 − | α | e C = C α sign α L Lα max e π/ 2 − | α | max ( ) π/ 2 − | α | e C = ( C + C α sign α ) M M 0 M α max e π/ 2 − | α | max for zero lift and moment at 90 deg angle of attack. In sideward flight, these coefficients are zero. Then L = qS C and M = qS C are the lift and pitch moment.

wet L wet fus M 9-3.3 Side Force and Yaw Moment The fuselage side force and yaw moment are defined in fixed form ( Y /q and N/q ), or scaled form ( C and C , based on the fuselage wetted area and fuselage length). The effective sideslip angle is Y N β = β − β , where β is the angle of zero side force; in reverse flow ( | β | > 90 ), β ← β − 180 sign β .

e fus zy zy e e e e Let β be the sideslip angle increment (above or below zero side force angle) for maximum side force.

max If | β | ≤ β e max C = C β Y Y β e C = C + C β N N 0 N β e and otherwise ( ) π/ 2 − | β | e C = C β sign β Y Y β max e π/ 2 − | β | max ( ) π/ 2 − | β | e C = ( C + C β sign β ) N N 0 N β max e π/ 2 − | β | max for zero side force and yaw moment at 90 deg sideslip angle. Then Y = qS C and N = qS C wet Y wet fus N are the side force and yaw moment. The roll moment is zero.

9–4 Weights The fuselage group consists of the basic structure; wing and rotor fold/retraction; tail fold/tilt; and marinization, pressurization, and crashworthiness structure.

88 Fuselage

Chapter 10 Landing Gear There is one landing gear component for the aircraft. The landing gear can be located on the body or on the wing. The landing gear can be fixed or retractable; a gear retraction speed is specified (CAS), or the landing gear state can be specified in the flight state.

10–1 Geometry F The landing gear has a position z , where the aerodynamic forces act. The component axes are BF aligned with the aircraft axes, C = I . The landing gear has no control variables. The height of the bottom of the landing gear above ground level, h , is specified in the flight state. The landing gear LG F position z is a distance d above the bottom of the gear.

LG 10-1.1 Drag The drag area is specified for landing gear extended, ( D/q ) . The velocity relative to the air at LG F F F F 2 z gives the drag direction e = − v / | v | and dynamic pressure q = / ρ | v | (no interference). Then d F F = e q ( D/q ) d LG is the total drag force.

10–2 Weights The alighting gear group consists of basic structure, retraction, and crashworthiness structure.

90 Landing Gear

Chapter 11 Rotor The aircraft can have one or more rotors, or no rotors. In addition to main-rotors, the rotor component can model tail-rotors, propellers, proprotors, ducted fans, thrust vectoring rotors, and auxiliary-thrust rotors. The principal configuration designation (main-rotor, tail-rotor, or propeller) is identified for each rotor component, and in particular determines where the weights are put in the weight statement (summarized in table 11-1). Each configuration can possibly have a separate performance or weight model, which is separately specified. Antitorque rotors and auxiliary-thrust rotors can be identified, for special sizing options. Other configuration features are variable diameter and ducted fan, and reaction drive.

Multi-rotor systems (such as coaxial or tandem configuration) are modeled as a set of separate rotors, in order to accommodate the description of the position, orientation, controls, and loads. Optionally the location of the center of the rotor system can be specified and the rotor locations calculated based on input separation parameters. The performance calculation for twin rotor systems can include the mutual influence of the induced velocity on the power.

The main-rotor size is defined by the radius R or disk loading W/A , thrust-weighted solidity σ , hover tip speed V , and blade loading C /σ = W/ρAV σ . With more than one main-rotor, the disk tip W tip loading and blade loading are obtained from an input fraction of design gross weight, W = f W . The W D air density ρ for C /σ is obtained from a specified takeoff condition. If the rotor radius is fixed for W the sizing task, three of ( R or W/A ), C /σ , V , and σ are input, and the other parameters are derived.

W tip Optionally the radius can be calculated from a specified ratio to the radius of another rotor. If the sizing task determines the rotor radius ( R and W/A ), then two of C /σ , V , and σ are input, and the other W tip parameter is derived. The radius can be sized for just a subset of the rotors, with fixed radius for the others.

For antitorque and auxiliary-thrust rotors, three of ( R or W/A ), C /σ , V , and σ are input, and W tip the other parameters are derived. Optionally the radius can be calculated from a specified ratio to the radius of another rotor. The disk loading and blade loading are based on f T , where f is an input factor and T is the maximum thrust from designated design conditions. Optionally the tail-rotor radius can be scaled with the main-rotor radius: R = f R (0 . 1348 + 0 . 0071 W/A ) , where f is an input factor and the mr units of disk loading W/A are lb/ft . Figure 11-1 shows the basis for this scaling.

Table 11-1. Principal configuration designation.

configuration weight statement weight model performance model main-rotor rotor group rotor rotor tail-rotor empennage group tail-rotor rotor propeller propulsion group rotor, aux thrust rotor 92 Rotor 0.26 aircraft equation 0.24 0.22 0.20 mr /R tr R 0.18 0.16 0.14 0.12 0. 2. 4. 6. 8. 10. 12. 14. 16.

2) disk loading (lb/ft Figure 11-1. Tail-rotor radius scaling.

11–1 Drive System The drive system defines gear ratios for all the components it connects. The gear ratio is the ratio of the component rotational speed to that of the primary rotor. There is one primary rotor per propulsion group (for which the reference tip speed is specified); other components are dependent (for which a gear ratio is specified). There can be more than one drive system state, in order to model a multiple-speed or variable-speed transmission. Each drive system state corresponds to a set of gear ratios.

For the primary rotor, a reference tip speed V is defined for each drive system state. By tip − ref convention, the “hover tip speed” refers to the reference tip speed for drive state #1. If the sizing task changes the hover tip speed, then the ratios of the reference tip speeds at different engine states are kept constant. By convention, the gear ratio of the primary rotor is r = 1 . For dependent rotors, either the gear ratio is specified (for each drive system state) or a tip speed is specified and the gear ratio is calculated ( r = Ω / Ω , Ω = V /R ). For the engine group, either the gear ratio is dep prim tip − ref specified (for each drive system state) or the gear ratio is calculated from the specification engine turbine speed Ω = (2 π/ 60) N and the reference tip speed of the primary rotor ( r = Ω / Ω , spec spec spec prim Ω = V /R ). The latter option means the specification engine turbine speed N corresponds prim tip − ref spec to V for all drive system states. To determine the gear ratios, the reference tip speed and radius are tip − ref used, corresponding to hover.

Rotor 93 The flight state specifies the tip speed of the primary rotor and the drive system state, for each propulsion group. The drive system state defines the gear ratio for dependent rotors and the engine groups. From the rotor radius, the rotational speed of the primary rotor is obtained ( Ω = V /R ); prim tip from the gear ratios, the rotational speed of dependent rotors ( Ω = r Ω ) and the engine groups dep prim ( N = (60 / 2 π ) r Ω ) are obtained; and from the rotor radius, the tip speed of the dependent rotor eng prim ( V = Ω R ) is obtained. The flight state specification of the tip speed can be an input value, the tip dep reference tip speed, a function of flight speed or a conversion schedule, or one of several default values.

These relationships between tip speed and rotational speed use the actual radius of the rotors in the flight state, which for a variable-diameter rotor may not be the same as the reference, hover radius.

A designated drive system state can have a variable speed (variable gear ratio) transmission, by introducing a factor f on the gear ratio when the speeds of the dependent rotors and engines are gear evaluated. The factor f is a component control, which can be connected to an aircraft control and gear thus set for each flight state.

An optional conversion schedule is defined in terms of two speeds: hover and helicopter mode for speeds below V , cruise mode for speeds above V , and conversion mode for speeds between C hover C cruise V and V . The tip speed is V in helicopter and conversion mode, and V in C hover C cruise tip − hover tip − cruise airplane mode. Drive system states are defined for helicopter, cruise, and conversion mode flight. The flight state specifies the nacelle tilt angle, tip speeds, control state, and drive system state, including the option to obtain any or all of these quantities from the conversion schedule.

Several default values of the tip speed are defined for use by the flight state, including cruise, maneuver, one-engine inoperative, drive system limit conditions, and a function of flight speed (piecewise linear input). Optionally these default values can be input as a fraction of the hover tip speed. Optionally √ 2 2 the tip speed can be calculated from μ = V /V , so V = V /μ ; or from M = M (1 + μ ) + μ , so tip tip at tip z √ 2 2 V = ( c M ) − V − V . Optionally the tip speed can be the minimum of the input value or that for tip s at z M .

at The sizing task might change the hover tip speed (reference tip speed for drive system state #1), the reference tip speed of a dependent rotor, a rotor radius, or the specification engine turbine speed N . In such cases the gear ratios and other parameters are recalculated. Note that it is not consistent spec to change the reference tip speed of a dependent rotor if the gear ratio is a fixed input.

An increment on the primary rotor rotational speed (or primary engine group, if there are no rotors) is a control variable of the propulsion group.

11–2 Geometry The rotor rotation direction is described by the parameter r : r = 1 for counter-clockwise rotation and r = − 1 for clockwise rotation (as viewed from the positive thrust side of the rotor).

The rotor solidity and blade mean chord are related by σ = N c/πR ; usually thrust-weighted values are used, but geometric values are also required by the analysis. The mean chord is the average of the chord over the rotor blade span, from root cutout to tip. The thrust-weighted chord is the average of the chord over the rotor blade span r , from root cutout to tip, weighted by r . A general blade chord distribution is specified as c ( r ) = c ˆ c ( r ) , where c is the thrust-weighted chord. Linear taper is ref ref specified in terms of a taper ratio t = c /c , or in terms of the ratio of thrust-weighted and geometric tip root chords, f = σ /σ = c /c .

t g . 75 R . 50 R 94 Rotor F The rotor hub is at position z . Optionally, a component of the position can be calculated, hub superseding the location input. The calculated geometry depends on the configuration. For a coaxial T SF F F rotor, the rotor separation is s = | k C ( z − z ) / (2 R ) | (fraction rotor diameter), or the hub hub1 hub2 locations are calculated from the input separation s , and the input location midway between the hubs: ⎛ ⎞ F F F S ⎝ ⎠ z = z ± C 0 hub center sR For a tandem rotor, the rotor longitudinal overlap is o = Δ / (2 R ) = 1 − / (2 R ) (fraction rotor diameter), or the hub locations are calculated from the input overlap o , and the input location midway between the hubs: x = x ± R (1 − o ) hub center For a tail-rotor, the longitudinal position can be calculated from the main-rotor radius R , tail-rotor radius R , and tail-rotor/main-rotor clearance d : tr tr x = x − ( R + d + R ) hub tr hub mr mr tr tr For a tiltrotor, the lateral position can be calculated from the rotor radius R (cruise value for variable- diameter rotor), fuselage/rotor clearance d , and fuselage width w : fus fus y = ± ( f R + d + / w ) hub fus fus with the pivot, pylon, and nacelle center-of-gravity lateral positions adjusted to keep the same relative position to the hub. The calculated clearance between the rotor and fuselage is d = | y |− ( R + / w ) .

fus hub 2 fus Alternatively, for a tiltrotor the lateral position can be calculated from the wing span, y = ± b/ 2 , so hub the rotors are at the wing tips, or from a designated wing panel edge, y = ± η ( b/ 2) .

hub p For twin rotors (tandem, side-by-side, or coaxial), the overlap is o = Δ / (2 R ) = 1 − / (2 R ) (fraction of diameter; 0 for no overlap and 1 for coaxial), where the hub-to-hub separation is = 2 2 1 / 2 [( x − x ) + ( y − y ) ] ( = 2 R for no overlap and = 0 for coaxial). The overlap area hub1 hub2 hub1 hub2 is mA , with A the area of one rotor disk and [ ] √ − 1 m = cos ( / 2 R ) − ( / 2 R ) 1 − ( / 2 R ) π The vertical separation is s = | z − z | / (2 R ) .

hub1 hub2 The reference areas for the component drag coefficients are the rotor disk area A = πR (for hub drag), pylon wetted area S , duct wetted area 2 S , and spinner wetted area S . The pylon wetted pylon duct spin area is input, or calculated from the drive system (gear box and rotor shaft) weight, or from the drive system plus engine system (engine, exhaust, and accessories) weight: ( ) 2 / 3 S = k w/N pylon rotor ∑ 2 2 / 3 2 2 / 3 where w = W or w = W + W and the units of k are ft /lb or m /kg . The pylon area gbrs gbrs ES is included in the aircraft wetted area if the pylon drag coefficient is nonzero. The duct wetted area is twice the duct area S . The duct area is input, or calculated from the rotor circumference and the duct duct length = kR : duct S = (2 πR ) duct duct Rotor 95 The spinner wetted area is input, or calculated from the spinner frontal area: S = k ( πR ) spin spin where R is the spinner radius, which is specified as a fraction of the rotor radius.

spin The rotor contribution to the aircraft operating length and width is calculated from the locus of F S T the rotor disk: z = z + RC (cos ψ sin ψ 0) . The longitudinal distance from the hub position is disk hub √ 2 2 Δ x = R ( a cos ψ + b sin ψ ) , so the maximum distance is Δ x = ± R a + b . The lateral distance from the √ 2 2 hub position is Δ y = R ( c cos ψ + d sin ψ ) , so the maximum distance is Δ y = ± R c + d .

11–3 Control and Loads The rotor controls consist of collective, lateral cyclic, longitudinal cyclic, and perhaps shaft inci- dence (tilt) and cant angles. Rotor cyclic control can be defined in terms of tip-path plane or no-feathering plane command. The collective control variable is the rotor thrust amplitude or the collective pitch angle.

N 2 2 The relationship between tip-path plane tilt and hub moment is M = I Ω ( ν − 1) β = K β , b hub where N is the number of blades, Ω the rotor speed, and ν the dimensionless fundamental flap frequency.

The flap moment of inertia I is obtained from the Lock number: γ = ρacR /I , for seal level standard b b (SLS) density ρ , lift-curve slope a = 5 . 7 , and thrust-weighted chord (or from the blade weight, or from an autorotation index). The flap frequency and Lock number are specified for hover radius and rotational speed. The flap frequency and hub stiffness are required for the radius and rotational speed of the flight 2 2 state. For a hingeless rotor, the blade flap spring is K = I Ω ( ν − 1) , obtained from the hover flap b N quantities; then K = K and hub flap K flap ν = 1 + I Ω b 2 2 For an articulated rotor, the hinge offset is e = Rx/ (1 + x ) , x = ( ν − 1) from the hover quantities; then 3 e/R ν = 1 + 2 1 − e/R N 2 2 and K = I Ω ( ν − 1) , using I from γ (and scaled with R for a variable diameter rotor) and Ω for hub b b the flight state.

Optionally the rotor can have a variable diameter. The rotor diameter is treated as a control, allowing it to be connected to an aircraft control and thus set for each flight state. The basic variation can be specified based on the conversion schedule, or input as a function of flight speed (piecewise linear input).

For the conversion schedule, the rotor radius is R for speeds below V , R = f R for hover C hover cruise hover speeds above V , and linear with flight speed in conversion mode. During the diameter change, the C cruise chord, chord radial distribution, and blade weight are assumed fixed; hence solidity scales as σ ∼ 1 /R , 2 2 blade flap moment of inertia as I ∼ R , and Lock number as γ ∼ R .

b 11-3.1 Control Variables The collective control variable is direct command of rotor thrust magnitude T or C /σ (in shaft T axes), from which the collective pitch angle can be calculated; or rotor collective pitch angle θ , from 0 . 75 which the thrust and inflow can be calculated.

Shaft tilt control variables are incidence (tilt) and cant angles, acting at a pivot location.

96 Rotor Tip-path plane command is direct control of the tip-path plane tilt, hence tilt of the thrust vector.

This control mode requires calculation of rotor cyclic pitch angles from the flapping. The control variables are longitudinal tilt β (positive forward) and lateral tilt β (positive toward retreating side).

c s Alternatively, the cyclic control can be specified in terms of hub moment or lift offset, if the blade flap frequency is greater than 1/rev. The relationship between tip-path plane tilt and hub moment is M = K β , and between moment and lift offset is M = o ( T R ) . Thus the flapping is hub ( ) ( ) ( ) 1 T R β rM o s x x = = β − M − o c K y K y hub hub for hub moment command or lift offset command, respectively.

No-feathering plane command is control of rotor cyclic pitch angles, usually producing tilt of the thrust vector. This control mode requires calculation of rotor tip-path plane tilt from the cyclic control, including the influence of inflow. The control variables are longitudinal cyclic pitch angle θ (positive s aft) and lateral cyclic pitch angle θ (positive toward retreating side).

c 11-3.2 Aircraft Controls Each control can be connected to the aircraft controls c : c = c + ST c , with c zero, constant, or AC 0 AC 0 a function of flight speed (piecewise linear input). The factor S can be introduced to automatically scale the collective matrix: S = a/ 6 = 1 / 60 if the collective control variable is C /σ ; S = ρV A ( a/ 6) T blade tip if the collective control variable is rotor thrust T ; S = 1 if the collective control variable is pitch angle θ . For cyclic matrices, S = 1 with no-feathering plane command, and S = − 1 for tip-path plane 0 . 75 command.

11-3.3 Rotor Axes and Shaft Tilt F The rotor hub is at position z , where the rotor forces and moments act; the orientation of the rotor hub SF F shaft axes relative to the aircraft axes is given by the rotation matrix C . The pivot is at position z .

pivot The hub or shaft axes S have origin at the hub node; the z -axis is the shaft, positive in the positive thrust direction; and the x -axis downstream or up. The rotor orientation is specified by selecting a nominal direction in body axes (positive or negative x -, y -, or z -axis) for the positive thrust direction; the other two axes are then the axes of control. For a main-rotor, the nominal direction would be the negative z -axis; for a tail-rotor, it would be the lateral axis ( ry -axis, depending on the direction of rotation of the main-rotor); and for a propeller, the nominal direction would be the positive x -axis. This selection defines a rotation matrix W from F to S axes. The hub and pivot axes have a fixed orientation relative to the body axes: HF hub incidence and cant: C = U V θ φ h h P F pivot dihedral, pitch, and sweep: C = X Y Z φ θ ψ p p p where U and V depend on the nominal direction, as described in table 11-2. The shaft control consists of incidence and cant about the pivot axes, from reference angles i and c : ref ref C = U V cont i − i c − c ref ref For a tiltrotor aircraft, one of the aircraft controls is the nacelle angle, with the convention α = 0 for tilt cruise, and α = 90 deg for helicopter mode. The rotor shaft incidence angle is then connected to tilt α by defining the matrix T appropriately. For the locations and orientation input in helicopter mode, tilt i i = 90 . Thus the orientation of the shaft axes relative to the body axes is: ref SF HF F P P F C = W C C C C cont Rotor 97 SF HF F or just C = W C with no shaft control. From the pivot location z and the hub location for the pivot F reference shaft control z , the hub location in general is hub0 F F F P P F T F F z = z + ( C C C ) ( z − z ) cont hub pivot hub0 pivot Similarly, the pylon location and nacelle center-of-gravity location can be calculated for given shaft control. The shift in the aircraft center of gravity produced by nacelle tilt is F F F F F P T P F F F W ( z − z ) = W ( z − z ) = W ( C C C − I ) ( z − z ) move move cg cg 0 nac nac0 cont nac0 pivot where W is the gross weight and W the weight moved. Table 11-2 summarizes the geometry options.

move Table 11-2. Rotor shaft axes.

nominal thrust incidence cant S S z -axis x -axis W + for T + for T U V θ φ h h F F main-rotor − z up − x aft Y aft right Y X 180 θ φ F F other z down − x aft Z aft right Y X 180 − θ − φ F F propeller x forward − z up Y up right Y Z 90 θ φ F F other − x aft − z up Z Y up right Y Z 180 − 90 − θ − φ F F tail-rotor ( r = 1 ) y right − x aft Z X aft up Z X 180 − 90 θ − φ F F tail-rotor ( r = − 1 ) − y left − x aft Z X aft up Z X 180 90 − θ φ ⎡ ⎤ ⎡ ⎤ − 1 0 0 0 0 − 1 ⎣ ⎦ ⎣ ⎦ Y = 0 1 0 Z Y = 0 − 1 0 180 180 − 90 0 0 − 1 − 1 0 0 ⎡ ⎤ ⎡ ⎤ − 1 0 0 − 1 0 0 ⎣ ⎦ ⎣ ⎦ Z = 0 − 1 0 Z X = 0 0 1 180 180 − 90 0 0 1 0 1 0 ⎡ ⎤ ⎡ ⎤ 0 0 − 1 − 1 0 0 ⎣ ⎦ ⎣ ⎦ Y = 0 1 0 Z X = 0 0 − 1 90 180 90 1 0 0 0 − 1 0 11-3.4 Hub Loads The rotor controls give the thrust magnitude and the tip-path plane tilt angles β and β , either c s directly or from the collective and cyclic pitch. The forces acting on the hub are the thrust T , drag H , and side force Y (positive in z -, x -, y -axis directions, respectively). The hub pitch and roll moments are proportional to the flap angles. The hub torque is obtained from the shaft power P and rotor speed shaft Ω . The force and moment acting on the hub, in shaft axes, are then: ⎛ ⎞ ⎛ ⎞ H 0 S ⎝ ⎠ ⎝ ⎠ F = Y + 0 T − f T B ⎛ ⎞ ⎛ ⎞ M K ( rβ ) x hub s S ⎝ ⎠ ⎝ ⎠ M = M = K ( − β ) y hub c − rQ − rP / Ω shaft 98 Rotor The force includes a term proportional to the rotor thrust and an input blockage factor f = Δ T /T ≥ 0 .

B This term accounts for blockage or download, as an alternative to including the drag of the fuselage or a lifting surface in the aircraft trim. For example, f can model the tail-rotor blockage caused by B operation near the vertical tail. The rotor loads in aircraft axes acting at the center of gravity are then: F F S S F = C F F F S S ˜ F F M = C M + Δ z F F F F where Δ z = z − z .

cg hub F The wind axis lift L and drag X are calculated from the net rotor hub force F and the rotor velocity F F F v . The velocity relative to the air gives the propulsive force direction e = v / | v | (no interference) p F T F and the velocity magnitude V = | v | . The drag and lift components of the force are X = − e F and p T F F T F 2 F 2 2 L = | ( I − e e ) F | , respectively. Thus XV = − ( v ) F and L = | F | − | X | . The rotor contribution p p T IF F to vertical force is the z -axis component of the force in inertial axes, F = − k C F .

V 11–4 Aerodynamics F F F F The rotor velocity relative to the air is v = v + ˜ ω Δ z in aircraft axes. The velocities in shaft AC AC axes are ⎛ ⎞ ⎛ ⎞ − μ r ˙ α x x S SF F S SF F ⎝ ⎠ ⎝ ⎠ v = C v = Ω R rμ ω = C ω = Ω ˙ α y y AC μ r ˙ α z z where Ω R is the rotor tip speed. The advance ratio μ , inflow ratio λ , and shaft angle of attack α are defined as √ 2 2 μ = μ + μ x y λ = λ + μ i z − 1 α = tan ( μ /μ ) z The blade velocity relative to the air has the maximum amplitude (advancing tip velocity) of μ = at √ 2 2 (1 + μ ) + μ , from which the advancing tip Mach number is M = M μ , using the tip Mach at tip at z number M = (Ω R ) /c . The rotor thrust coefficient is defined as C = T /ρA (Ω R ) . The dimensionless tip s T ideal induced velocity λ is calculated from μ , μ , and C ; then the dimensional velocity is v = Ω R λ .

i z T i i The ideal induced power is then P = T v . Note that for these inflow velocities, the subscript “i” ideal i denotes “ideal.” 11-4.1 Ideal Inflow The ideal wake-induced velocity is obtained from Glauert’s momentum theory: C sλ T h λ = √ = √ i 2 2 2 2 2 λ + μ λ + μ where λ = λ + μ , λ = | C | / 2 ( λ is always positive), and s = sign C . This expression is generalized i z T h T h to λ = λ s F ( μ/λ , sμ /λ ) i h h z h If μ is zero, the equation for λ can be solved analytically. Otherwise, for non-axial flow, the equation i is written as follows: sλ h √ λ = + μ z 2 2 λ + μ Rotor 99 Using λ instead of λ as the independent variable simplifies implementation of the ducted fan model. A i Newton–Raphson solution for λ gives: sλ h ̂ √ λ = in 2 2 λ + μ n ̂ λ − μ − λ n z in λ = λ − f n +1 n 2 2 ̂ 1 + λ λ / ( λ + μ ) in n n A relaxation factor of f = 0 . 5 is used to improve convergence. Three or four iterations are usually sufficient, using sλ h ∼ √ λ = + μ z 2 2 ( sλ + μ ) + μ h z to start the solution. To eliminate the singularity of the momentum theory result at ideal autorotation, the expression [ ] 2 2 0 . 373 μ + 0 . 598 μ z λ = μ − 0 . 991 z λ h is used when 2 2 2 1 . 5 μ + (2 sμ + 3 λ ) < λ z h h 2 2 2 2 The equation λ = μ ( aμ − bλ + cμ ) /λ is an approximation for the induced power measured in the z z h h turbulent-wake and vortex-ring states. Matching this equation to the axial-flow momentum theory result √ √ at μ = − 2 λ and μ = − λ gives a = 5 / 6 = 0 . 3726780 and b = (4 5 − 3) / 6 = 0 . 9907120 . Then z h z h matching to the forward-flight momentum theory result at ( μ = λ , μ = − 1 . 5 λ ) gives c = 0 . 5980197 .

h z h For axial flow ( μ = 0 ) the solution is: √ ⎧ ( ) μ μ z z ⎪ ⎪ + s + λ − λ < sμ ⎪ h z h ⎪ 2 2 ⎪ ⎪ ⎪ [ ] ⎨ 0 . 373 μ z λ = μ − 0 . 991 − 2 λ < sμ < − λ z h z h ⎪ λ ⎪ h ⎪ ⎪ √ ⎪ ( ) ⎪ 2 ⎪ μ μ z z ⎩ 2 − s − λ sμ < − 2 λ z h h 2 2 Note that λ and v are the ideal induced velocities; additional factors are required for the wake-induced i i velocity or induced power calculations.

11-4.1.1 Ducted Fan Rotor momentum theory can be extended to the case of a ducted fan. Consider a rotor system with disk area A , operating at speed V , with an angle α between V and the disk plane. The induced velocity at the rotor disk is v , and in the far wake w = f v . The far wake area is A = A/f . The axial velocity W ∞ A at the fan is f V , with f accounting for acceleration or deceleration through the duct. The edgewise V z z V z velocity at the fan is f V , with f = 1 . 0 for wing-like behavior, or f = 0 for tube-like behavior of V x x V x V x the flow. The total thrust (rotor plus duct) is T , and the rotor thrust is T = f T . For this model, the rotor T duct aerodynamics are defined by the thrust ratio f or far wake area ratio f , plus the fan velocity ratio T A f . The mass flux through the rotor disk is ˙ m = ρAU = ρA U , where U and U are respectively the V ∞ ∞ ∞ total velocity magnitudes at the fan and in the far wake: 2 2 2 U = ( f V cos α ) + ( f V sin α + v ) V x V z 2 2 2 U = ( V cos α ) + ( V sin α + w ) ∞ 100 Rotor Mass conservation ( f = A/A = U /U ) relates f and f . Momentum and energy conservation give A ∞ ∞ A W T = ˙ mw = ρAU w/f = ρAU f v ∞ A W ( ) 1 w P = ˙ mw (2 V sin α + w ) = T V sin α + 2 2 With these expressions, the span of the lifting system in forward flight is assumed equal to the rotor diameter 2 R . Next it is required that the power equals the rotor induced and parasite loss: P = T ( f V sin α + v ) = T f ( f V sin α + v ) rotor V z T V z In axial flow, this result can be derived from Bernoulli’s equation for the pressure in the wake. In forward flight, any induced drag on the duct is being neglected. From these two expressions for power, V + f v/ 2 = f ( f V + v ) is obtained, relating f and f . With no duct ( f = f = f = 1 ), the z W T V z z T W T V x V z far wake velocity is always w = 2 v , hence f = 2 . With an ideal duct ( f = f = f = 1) , the far W A V x V z √ wake velocity is f = 1 . In hover (with or without a duct), f = f = 2 f , and v = 2 /f v . The W W A T W h rotor ideal induced power is P = T w/ 2 = f T v , introducing the duct factor f = f / 2 .

ideal D D W For a ducted fan, the thrust C is calculated from the total load (rotor plus duct). To define the duct T effectiveness, either the thrust ratio f = T /T or the far wake area ratio f = A/A is specified (and T rotor A ∞ the fan velocity ratio f ). The wake-induced velocity is obtained from the momentum theory result for V √ 2 2 a ducted fan: λ = ( f λ / 2) ( f μ ) + ( f μ + λ ) . If the thrust ratio f is specified, this can be W i V x V z z i T h written sλ /f μ T z h √ f μ + λ = + V z z i 2 2 f ( f μ + λ ) + ( f μ ) T V z z i V x In this form, λ can be determined using the free-rotor expressions given previously: replacing λ , μ , i z h μ , and λ with λ /f , μ /f , f μ , and f μ + λ , respectively. Then from λ the velocity and area T z T V x V z z i i h ratios are obtained: ( ) μ z f = 2 f − (1 − f f ) W T T V z λ i √ 2 2 μ + ( μ + f λ ) z W i f = A 2 2 ( f μ ) + ( f μ + λ ) V x V z z i If instead the area ratio f is specified, it is simplest to first solve for the far wake velocity f λ : A W i sλ 2 f A h μ + f λ = √ + μ z W i z 2 2 ( μ + f λ ) + μ z W i In this form, f λ can be determined using the free-rotor expressions given previously: replacing λ W i h and λ with λ 2 f and μ + f λ , respectively. The induced velocity is A z W i h [ ] 2 2 2 2 ( f μ + λ ) = μ + ( μ + f λ ) − ( f μ ) V z z i z W i V x f A The velocity ratio is f = ( f λ ) /λ , and W W i i μ + f λ / 2 z W i f = T f μ + λ V z z i is the thrust ratio. However, physical problems and convergence difficulties are encountered with this approach in descent, if an arbitrary value of f is permitted. From the expression for f , f should T T T Rotor 101 approach 1 /f at high rates of climb or descent. To avoid problems with an arbitrary value of f , it is V z T assumed that the input value of f defines the velocity ratio f = 2 f in descent. So in descent μ is T W T z not replaced by μ /f .

z T 11-4.1.2 Ground Effect The wake-induced velocity is reduced when the rotor disk is in the proximity of the ground plane.

√ 3 / 2 Ground effect in hover can be described in terms of the figure of merit M = ( T / 2 ρA ) /P as a function of scaled rotor height above the ground, z /D = z / 2 R . Usually the test data are given as the ratio of g g 2 / 3 the thrust to OGE thrust, for constant power: T /T = ( M/M ) = κ ≥ 1 . The effect on power at ∞ ∞ g − 3 / 2 constant thrust is then P = P f , where f = κ ≤ 1 . Ground effect is generally negligible at heights ∞ g g g above z /D = 1 . 5 and at forward speeds above μ = 3 λ .

g h The ground plane is assumed to be perpendicular to the inertial frame z -axis. The ground normal F F I S SF F (directed downward) is k = C k in airframe axes, or k = C k in rotor shaft axes. The height g g g of the landing gear above ground level, h , is specified in the flight state. The height of the rotor hub LG above ground level is then F T F F z = h − ( k ) ( z − z ) + d g LG LG g hub LG F where z is the position of the landing gear in the airframe, and d is the distance from the bottom of LG LG the gear to the location z . From the velocity LG ⎛ ⎞ μ x S ⎝ ⎠ v = − rμ y − λ S T S S the angle  between the ground normal and the rotor wake is evaluated: cos  = ( k ) v / | v | (  = 0 for g √ 2 2 hover,  = 90 deg in forward flight). Note that if the rotor shaft is vertical, then cos  = λ/ μ + λ (see ref. 1). The expressions for ground effect in hover are generalized to forward flight by using ( z / cos  ) g in place of z . No ground effect correction is applied if the wake is directed upward ( cos  ≤ 0 ), or if g − 3 / 2 z / cos  > 1 . 5 D . From z /D cos  , the ground effect factor f = κ is calculated. Then g g g g ( λ ) = f ( λ ) i IGE g i OGE is the effective ideal induced velocity.

Several empirical ground effect models are implemented: from Cheeseman and Bennett (ref. 1, basic model and using blade-element (BE) theory to incorporate influence of thrust), from Law (ref. 2), from Hayden (ref. 3), and a curve fit of the interpolation from Zbrozek (ref. 4): ⎧ [ ] 3 / 2 ⎪ 1 ⎪ ⎪ 1 − Cheeseman and Bennett ⎪ ⎪ (4 z /R ) ⎪ g ⎪ ⎪ ⎪ ⎪ [ ] − 3 / 2 ⎪ ⎪ σaλ 1 ⎪ i ⎪ 1 + 1 . 5 Cheeseman and Bennett (BE) ⎪ ⎪ 2 ⎪ 4 C (4 z /R ) T g ⎪ ⎪ ⎪ ⎪ [ ] ⎪ 3 / 2 ⎨ 1 . 0991 − 0 . 1042 / ( z /D ) g Law f = g 1 + ( C /σ )(0 . 2894 − 0 . 3913 / ( z /D )) ⎪ T g ⎪ ⎪ ⎪ ⎪ [ ] − 1 ⎪ ⎪ 0 . 03794 ⎪ ⎪ 0 . 9926 + Hayden ⎪ ⎪ 2 ⎪ ( z / 2 R ) g ⎪ ⎪ ⎪ ⎪ [ ] ⎪ − 3 / 2 ⎪ ⎪ ⎪ 0 . 0544 ⎪ ⎪ 0 . 9122 + √ Zbrozek ⎩ ( z /R ) C /σ g T 102 Rotor Hayden Cheeseman & Bennett Rabbott Cheeseman & Bennett (BE) Cerbe Law 1.3 Cheeseman Hayden Zbrozek Zbrozek 1.2 ∞ 1.1 T/T 1.0 0.9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 z/R Figure 11-2. Ground effect models (hover).

These equations break down at small height above the ground, and so are restricted to z /D ≥ 0 . 15 ; g however, the database for ground effect extends only to about z/D = 0 . 3 . Also, f ≤ 1 is required.

g − 2 / 3 Figure 11-2 shows T /T = κ = f as a function of z/R for these models ( C /σ = 0 . 05 , 0 . 10 , 0 . 15 ), ∞ g g T compared with test data from several sources.

The influence of the ground on tiltrotor power required is stronger than for an isolated rotor. This further reduction of power is probably due to a reduction of wing and fuselage download when operating near the ground. For quad tiltrotors, an upload on the airframe has been measured at low heights. These effects can be modeled by using an effective distance above the ground: z = C z , with typically e g g C = 0 . 5 for a tiltrotor.

g 11-4.1.3 Inflow Gradient As a simple approximation to nonuniform induced velocity distribution, a linear variation over the disk is used: Δ λ = λ r cos ψ + λ r sin ψ . There are contributions to Δ λ from forward flight and from x y hub moments, which influence the relationship between flapping and cyclic. The linear inflow variation caused by forward flight is Δ λ = λ ( κ r cos ψ + κ r sin ψ ) , where λ is the mean inflow. Typically κ f i x y i x is positive, and roughly 1 at high speed, and κ is smaller in magnitude and negative. Both κ and κ y x y Rotor 103 must be zero in hover. Based on references 5–8, the following models are considered: 15 π 15 π μ √ Coleman and Feingold: κ = f tan χ/ 2 = f x 0 x x 2 2 32 32 μ + λ + | λ | κ = − f 2 μ y 0 y √ √ μ √ White and Blake: κ = f 2 sin χ = f 2 x 0 x x 2 2 μ + λ κ = − 2 f μ y 0 y where tan χ = | λ | /μ is the wake angle. Extending these results to include sideward velocity gives κ = ( κ μ + κ μ ) /μ and κ = ( − κ μ + κ μ ) /μ . For flexibility, the empirical factors f and f x x 0 x y 0 y y x 0 y y 0 x x y have been introduced (values of 1.0 give the baseline model). There is also an inflow variation produced by any net aerodynamic moment on the rotor disk, which can be evaluated using a differential form of momentum theory: f m Δ λ = √ ( − 2 C r cos ψ + 2 C r sin ψ ) = λ r cos ψ + λ r sin ψ m M y M x xm ym 2 2 μ + λ including empirical factor f . Note that the denominator of the hub moment term is zero for a hovering m rotor at zero thrust; so this inflow contribution should not be used for cases of low speed and low thrust.

11-4.2 Rotor Forces When direct control of the rotor thrust magnitude is used, the rotor collective pitch angle θ must 0 . 75 be calculated from the thrust C /σ . If the commanded variable is the collective pitch angle, then it is T necessary to calculate the rotor thrust, resulting in more computation, particularly since all quantities depending on the thrust (inflow, induced power factor, and mean drag coefficient) are also unknown.

There may be flight states where the commanded thrust can not be produced by the rotor, even with stall neglected in the section aerodynamics. This condition manifests as an inability to solve for the collective pitch given the thrust. In this circumstance the trim method can be changed so the required or specified thrust is an achievable value, or commanded collective pitch control can be used.

Cyclic control consists of tip-path plane command, requiring calculation of the rotor cyclic pitch angles from the flapping; or no-feathering plane command, requiring calculation of the tip-path plane tilt from the cyclic control angles. The longitudinal tip-path plane tilt is β (positive forward) and the lateral c tilt is β (positive toward retreating side). The longitudinal cyclic pitch angle is θ (positive aft), and the s s lateral cyclic pitch angle is θ (positive toward retreating side). Tip-path plane command is appropriate c for main-rotors. For rotors with no cyclic pitch, no-feathering plane command must be used.

The forces acting on the hub are the thrust T , drag H , and side force Y (positive in z -, x -, y - axis directions, respectively). The aerodynamic analysis is conducted for a clockwise rotating rotor, with appropriate sign changes for lateral velocity, flapping, and force. The analysis is conducted in dimensionless form, based on the actual radius and rotational speed of the flight state. The inplane hub forces are produced by tilt of the thrust vector with the tip-path plane, plus forces in the tip-path plane, and profile terms (produced by the blade drag coefficient). The orientation of the tip-path axes relative P S to the shaft axes is then C = X Y . Then rβ s − β c ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ C 0 C C H H tpp Ho SP ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ C = C 0 + rC + rC Y Y tpp Y o SP C C /C 0 0 T T 104 Rotor The inplane forces relative to the tip-path plane can be neglected, or calculated by blade element theory.

Note that with thrust and tip-path plane command and C and C neglected, it is not necessary to H tpp Y tpp solve for the rotor collective and cyclic pitch angles. In general the inplane forces relative to the tip-path plane are not zero, and may be significant, as for a rotor with large flap stiffness. Figures 11-3a and b show, respectively, the tip-path plane tilt and thrust vector tilt with cyclic pitch control (no-feathering plane tilt), as functions of flap stiffness (frequency), for several rotor thrust values. The difference between tip-path plane tilt (fig. 11-3a) and thrust vector tilt (fig. 11-3b) is caused by tilt of the thrust vector relative to the tip-path plane.

The profile inplane forces can be obtained from simplified equations, or calculated by blade element theory. The simplified method uses: ( ) ( ) σ C μ /μ Ho x = c F d mean H C − μ /μ Y o y where the mean drag coefficient c is from the profile power calculation. The function F ac- d mean H counts for the increase of the blade section velocity with rotor edgewise and axial speed: C = Ho ∫ ∫ 1 1 2 2 2 1 / 2 σc U ( r sin ψ + μ ) dr = σc ( u + u + u ) ( r sin ψ + μ ) dr ; so (from ref. 9) d d 2 2 T R P ∫ ∫ 2 π 1 ( ) 1 / 2 2 2 2 F = 4 ( r + μ sin ψ ) + ( μ cos ψ ) + μ ( r sin ψ + μ ) dr dψ H z 2 π 0 0 [ ] √ ( ) ( ) √ 2 1 V − 1 3 1 + V + 1 3 2 3 ∼ 1 + V 3 μ + μ + μμ + μ ln = z 2 2 4 (1 + V ) 4 V 2 2 2 with V = μ + μ .

z 11-4.3 Blade Element Theory Blade element theory is the basis for the solution for the collective and cyclic pitch angles (or thrust and flap angles) and evaluation of the rotor inplane hub forces. The section aerodynamics are described by lift varying linearly with angle of attack, c = c α (no stall), and a constant mean drag coefficient  α c (from the profile power calculation). The analysis is conducted in dimensionless form (based on d mean density ρ , rotor rotational speed Ω , and blade radius R of the flight state). So in the following σ , ν , and γ are for the actual ρ , Ω , and R ; and a = 5 . 7 is the lift-curve slope used in the Lock number γ . The blade section aerodynamic environment is described by the three components of velocity, from which the yaw and inflow angles are obtained, and then the angle of attack: 2 2 2 U = u + u T P √ u = r + μ sin ψ + μ cos ψ T x y 2 2 2 cos Λ = U/ u + u + u T P R u = μ cos ψ − μ sin ψ R x y − 1 φ = tan u /u P T ˙ u = λ + r ( β + ˙ α sin ψ − ˙ α cos ψ ) + u β P x y R α = θ − φ In reverse flow ( | α | > 90 ), α ← α − 180 sign α , and then c = c α still (airfoil tables are not used). The  α blade pitch consists of collective, cyclic, twist, and pitch-flap coupling terms. The flap motion is rigid rotation about a hinge with no offset, and only coning and once-per-revolution terms are considered: θ = θ + θ + θ cos ψ + θ sin ψ − K β 0 . 75 tw c s P β = β + β cos ψ + β sin ψ 0 c s Rotor 105 1.1 C / σ = 0.14 1.0 T C / σ = 0.10 0.9 T C / σ = 0.06 T 0.8 C / σ = 0.02 T 0.7 0.6 0.5 0.4 0.3 TPP tilt / cyclic magnitude 0.2 0.1 0.0 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 flap frequency ν (per-rev) Figure 11-3a. Tip-path plane tilt with cyclic pitch.

1.1 1.0 0.9 0.8 0.7 0.6 0.5 C / σ = 0.02 T 0.4 C / σ = 0.06 T 0.3 C / σ = 0.10 thrust tilt / cyclic magnitude T 0.2 C / σ = 0.14 T 0.1 0.0 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 flap frequency ν (per-rev) Figure 11-3b. Thrust vector tilt with cyclic pitch.

106 Rotor where K = tan δ . The twist is measured relative to 0 . 75 R ; θ = θ ( r − 0 . 75) for linear twist. The P 3 tw L mean inflow is λ = κλ , using the induced velocity factor κ from the induced power model. The inflow 0 i includes gradients caused by edgewise flight and hub moments: λ = μ + λ (1 + κ r cos ψ + κ r sin ψ ) + Δ λ z 0 x y m f m = μ + λ (1 + κ r cos ψ + κ r sin ψ ) + √ ( − 2 C r cos ψ + 2 C r sin ψ ) z 0 x y M y M x 2 2 μ + λ From the hub moments ( ) ( ) σa ν − 1 − C β M y c = C β 2 γ M x s the inflow gradient is ( ) 2 2 f σa ν − 1 ν − 1 m Δ λ = √ ( rβ cos ψ + rβ sin ψ ) = K ( rβ cos ψ + rβ sin ψ ) m c s m c s 2 2 8 γ/ 8 γ/ 8 μ + λ The constant K is associated with a lift-deficiency function: m 1 1 C = = √ ( ) 2 2 1 + K m 1 + f σa/ 8 μ + λ m The blade chord is c ( r ) = c ˆ c ( r ) , where c is the thrust-weighted chord (chord at 0 . 75 R for linear ref ref taper). Yawed flow effects increase the section drag coefficient, hence c = c / cos Λ . The section d d mean forces in velocity axes and shaft axes are 1 1 L = ρU cc F = L cos φ − D sin φ = ρU c ( c u − c u )  z  T d P 2 2 1 1 D = ρU cc F = L sin φ + D cos φ = ρU c ( c u + c u ) d x  P d T 2 2 1 1 R = ρU cc = D tan Λ F = − βF + R = − βF + ρU cc u r r z z d R 2 2 These equations for the section environment and section forces are applicable to high inflow (large μ ), z sideward flight ( μ ), and reverse flow ( u < 0 ). The total forces on the rotor hub are y T ∫ T = N F dr z ∫ H = N F sin ψ + F cos ψ dr x r ∫ Y = N − F cos ψ + F sin ψ dr x r with an average over the rotor azimuth implied, along with the integration over the radius. Lift forces are integrated from the root cutout r to the tip loss factor B . Drag forces are integrated from the root root cutout to the tip.

In coefficient form (forces divided by ρAV ) the rotor thrust and inplane forces are: tip ∫̂ C = σ F dr ̂ T z F = ˆ cU ( c u − c u ) z  T d P ∫̂ ̂ ̂ C = σ F sin ψ + F cos ψ dr F = ˆ cU ( c u + c u ) H x r x  P d T ∫ ̂ ̂ ̂ ̂ F = − β F + ˆ cU c u C = σ − F cos ψ + F sin ψ dr r z d R Y x r Rotor 107 ̂ ∼ ̂ ˙ ̂ ̂ (and the sign of C is changed for a clockwise rotating rotor). The terms Δ F = F β and Δ F = − F β Y x z r z produce tilt of the thrust vector with the tip-path plane ( C = − C β and C = − C β ), which H T c Y T s are accounted for directly. The section drag coefficient c produces the profile inplane forces. The d ∼ approximation u = μ is consistent with the simplified method (using the function F ), hence P z H ∫̂ ∫ σ ̂ ̂ C = σ F sin ψ + F cos ψ dr = ˆ cU c ( r sin ψ + μ ) dr F = ˆ cU c u Ho xo ro 0 d x xo 0 d T ∫ ∫ σ ̂ ̂ ̂ F = ˆ cU c u C = σ − F cos ψ + F sin ψ dr = − ˆ cU c ( r cos ψ + μ ) dr ro 0 d R Y o xo ro 0 d y 2 2 2 2 2 where U = u + μ , and c = c / cos Λ . Using blade element theory to evaluate C and C d d mean Ho Y o 0 T z accounts for the planform ( ˆ c ) and root cutout. Using the function F implies a rectangular blade and no H root cutout (plus at most a 1% error approximating the exact integration). The remaining terms in the section forces produce the inplane loads relative to the tip-path plane: 1 1 ̂ ̂ ̂ ˙ ̂ ˙ ˙ F = F − F β − F = ˆ cU c ( u − u β ) + ˆ cU c ((1 − U /U ) u + u β ) xi x z xo  P T d 0 T P 2 2 ̂ ̂ ̂ ̂ F = F + F β − F = ˆ cU c (1 − U /U ) u ri r z ro d 0 R ∫̂ ̂ C = σ F sin ψ + F cos ψ dr H tpp xi ri ∫ ̂ ̂ C = σ − F cos ψ + F sin ψ dr Y tpp xi ri (including small profile terms from U  = U ).

Evaluating these inplane forces requires the collective and cyclic pitch angles and the flapping motion. The thrust equation must be solved for the rotor collective pitch or the rotor thrust. The relationship between cyclic pitch and flapping is defined by the rotor flap dynamics. The flap motion is rigid rotation about a central hinge, with a flap frequency ν > 1 for articulated or hingeless rotors. The flapping equation of motion is ∫̂ γ 2 2 ¨ β + ν β + 2 ˙ α sin ψ + 2 ˙ α cos ψ = F r dr + ( ν − 1) β y x z p a including precone angle β ; the Lock number γ = ρac R /I . This equation is solved for the mean p ref b (coning) and 1/rev (tip-path plane tilt) flap motion: ∫̂ γ 2 2 ν β = F r dr + ( ν − 1) β 0 z p a ( ) ( ) ( ) ∫̂ γ β 2 cos ψ 2 ˙ α 2 c x ( ν − 1) = F r dr + z β 2 sin ψ 2 ˙ α a s y with an average over the rotor azimuth implied. The solution for the coning is largely decoupled by introducing the thrust: ∫̂ γ 6 C γ T 2 2 ν β = + ( ν − 1) β + F ( r − 3 / 4) dr 0 p z 0 0 8 σa a A separate flap frequency ν is used for coning, in order to model teetering and gimballed rotors. For an articulated rotor, β = 0 should be used.

p 108 Rotor The thrust and flapping equations of motion that must be solved are: ∫̂ 6 6 C T E = F dr − t z a σa ( ) ∫̂ ( ) ( ) ( ) 8 ν − 1 16 E 2 cos ψ β ˙ α c c x = F r dr − + z E a 2 sin ψ γ/ 8 β γ ˙ α s s y T The solution v such that E ( v ) = 0 is required. The variables are v = ( θ θ θ ) for thrust and tip-path 0 . 75 c s T T plane command; v = ( θ β β ) for thrust and no-feathering plane command; v = ( C /σ θ θ ) for 0 . 75 c s T c s T collective pitch and tip-path plane command; v = ( C /σ β β ) for collective pitch and no-feathering T c s plane command. Note that since c = c α is used (no stall), these equations are linear in θ . However,  α if ∂T /∂θ is small, the solution may not produce a reasonable collective for commanded thrust. A 0 . 75 ∼ Newton–Raphson solution method is used: from E ( v ) E ( v ) + ( dE/dv )( v − v ) = 0 , the = n +1 n n +1 n iterative solution is v = v − C E ( v ) n +1 n n − 1 where C = f ( dE/dv ) , including the relaxation factor f . The derivative matrix dE/dv is obtained by numerical perturbation. Convergence of the Newton–Raphson iteration is tested in terms of | E | <  for each equation, where  is an input tolerance.

11–5 Power The rotor power consists of induced, profile, interference, and parasite terms: P = P + P + P + P .

i o t p The parasite power (including climb/descent power for the aircraft) is obtained from the wind axis drag F T F force: P = − XV = ( v ) F .

p The interference power can be produced by interactions from the wing. The component of the F F wing interference velocity v parallel to the rotor force vector F (velocity roughly normal to the rotor ind ∼ T disk) produces a power change P v F . The component of the interference velocity perpendicular = t ind to the rotor force vector (velocity roughly in the plane of the rotor disk) produces interference through the swirl, hence P ∝ ( V / Ω R ) | v || F | . Thus the interference power is calculated from t ind √ V T 2 T 2 P = − K v F + K ( | v || F | ) − ( v F ) t int n int p ind ind ind Ω R Separate interference factors K are used for the two terms. K is negative for favorable interference.

int int p The induced power is calculated from the ideal power: P = κP = κf T v . The empirical i ideal D ideal factor κ accounts for the effects of nonuniform inflow, non-ideal span loading, tip losses, swirl, blockage, and other phenomena that increase the induced power losses ( κ > 1 ). For a ducted fan, f = f / 2 is D W introduced. The induced power at zero thrust is zero in this model (or accounted for as a profile power increment). If κ is deduced from an independent calculation of induced power, nonzero P at low thrust i will be reflected in large κ values.

The profile power is calculated from a mean blade drag coefficient: P = ρA (Ω R ) C , C = o P o P o ( σ/ 8) c F . The function F ( μ, μ ) accounts for the increase of the blade section velocity with rotor d mean P P z Rotor 109 ∫ ∫ 1 3 1 2 2 2 3 / 2 edgewise and axial speed: C = σc U dr = σc ( u + u + u ) dr ; so (from ref. 9) P o d d 2 2 T R P ∫ ∫ 2 π 1 ( ) 3 / 2 2 2 2 F = 4 ( r + μ sin ψ ) + ( μ cos ψ ) + μ dr dψ P z 2 π 0 0 ( ) √ 2 4 4 5 3 4 + 7 V + 4 V 9 μ 2 2 ∼ 2 = 1 + V 1 + V + μ − 2 2 2 2 8 (1 + V ) 16 1 + V [ ] √ ( ) 3 3 9 1 + V + 1 4 2 2 4 + μ + μ μ + μ ln z z 2 2 16 V 2 2 2 3 with V = μ + μ . This expression is exact when μ = 0 , and f ∼ 4 V for large V .

P z Two performance methods are implemented, the energy method and the table method. The induced power factor and mean blade drag coefficient are obtained from equations with the energy method, or from tables with the table method. Optionally κ and c can be specified for each flight state, d mean superseding the performance method values.

11-5.1 Energy Performance Method 11-5.1.1 Induced Power The induced power is calculated from the ideal power: P = κP = κf T v . Reference values i ideal D ideal of κ are specified for hover, axial cruise (propeller), and edgewise cruise (helicopter): κ , κ , and hover prop κ . Two models are implemented: constant model and standard model. The constant model uses edge κ = κ if μ = μ = 0 ; or κ = κ if | μ | < 0 . 1 | μ | ; or κ = κ otherwise.

hover z prop z edge The standard model calculates an axial flow factor κ from κ , κ , and κ . Let Δ = axial hover climb prop C /σ − ( C /σ ) . For hover and low-speed axial climb, including a variation with thrust, the inflow T T ind factor is [ ] X X − 1 axial h 2 κ = κ + k Δ + k | Δ | + ( κ − κ ) tan (( | μ | /λ ) /M ) h hover h 1 h h 2 h climb hover z h axial π where | μ | /λ = M is the midpoint of the transition between hover and climb, and X is large z h axial axial for a fast transition. Figure 11-4 illustrates κ in hover (with a minimum value). Figure 11-5 shows the behavior of this function for a helicopter in climb ( X = 0 . 65 ). A polynomial describes the variation axial with axial velocity, scaled so κ = κ at μ = 0 and κ = κ at μ = μ . Including variations with h z p z z prop thrust and shaft angle: X 2 X p 2 pα κ = κ + k Δ + k | Δ | + k μ | μ | p prop p 1 p p 2 p pα z 2 X a κ = κ + k | μ | + S ( k μ + k | μ | ) axial h a 1 z a 2 a 3 z z 2 X a where S = ( κ − ( κ + k μ )) / ( k μ + k | μ | ) ; S = 0 if k = k = 0 (not scaled); and p h a 1 z prop a 2 a 3 z prop a 2 a 3 z prop κ = κ if μ = 0 . A polynomial describes the variation with edgewise advance ratio, scaled so axial h z prop κ = κ at μ = 0 and κ = f f κ at μ = μ . Thus the induced power factor is axial α off edge edge 2 X e κ = κ + k μ + S ( k μ + k μ ) axial e 1 e 2 e 3 2 X e where S = ( f f κ − ( κ + k μ )) / ( k μ + k | μ | ) ; S = 0 if k = k = 0 (not α off edge axial e 1 edge e 2 e 3 edge e 2 e 3 edge scaled); and κ = κ if μ = 0 . The function f = 1 − k μ accounts for the influence of angle of axial edge α eα z − k o o 2 x attack ( μ /μ ) or rotor drag ( C ). The function f = 1 − k (1 − e ) accounts for the influence of z X off o 1 110 Rotor 1.30 1.25 1.20 κ 1.15 1.10 1.05 1.00 0.00 0.03 0.06 0.09 0.12 0.15 0.18 C / σ T Figure 11-4. Induced power factor for rotor in hover.

1.14 1.12 1.10 1.08 κ 1.06 M = 1.176 axial 1.04 M = 0.5 axial M = 2.0 1.02 axial 1.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 μ / λ z h Figure 11-5. Induced power factor for rotor in axial flight.

lift offset, o = rM /T R = ( K /T R ) β . Figure 11-6 illustrates κ in edgewise flight. Minimum and x x hub s maximum values of the induced power factor, κ and κ , are also specified.

min max 11-5.1.2 Profile Power The profile power is calculated from a mean blade drag coefficient: C = ( σ/ 8) c F . Since the P o d mean P ∼ blade mean lift coefficient is c C /σ , the drag coefficient is estimated as a function of blade loading = 6  T C /σ (using thrust-weighted solidity). With separate estimates of the basic, stall, and compressibility T drag, the mean drag coefficient is: c = χS ( c + c + c ) d mean d basic d stall d comp 0 . 2 where χ is a technology factor. The factor S = ( Re /Re ) accounts for Reynolds number effects on ref Rotor 111 4.50 C / σ = 0.08 T 4.00 C / σ = 0.14 T 3.50 ( ∝ , κ ) edge edge 3.00 κ 2.50 2.00 1.50 1.00 0.00 0.10 0.20 0.30 0.40 0.50 μ Figure 11-6. Induced power factor for rotor in edgewise flight.

the drag coefficient; Re is based on the thrust-weighted chord, 0 . 75 V , and the flight state; and Re tip ref corresponds to the input c information. The following models are implemented for the basic drag: d a) Array model: The basic drag c is input as a function of C /σ ; the array is linearly interpolated.

d basic T b) Equation model: The basic drag c is a quadratic function of C /σ , plus an additional term d basic T allowing faster growth at high (sub-stall) angles of attack. Let Δ = | C /σ − ( C /σ ) | , where T T D min ( C /σ ) corresponds to the minimum drag and Δ = | C /σ | − ( C /σ ) . Values of the basic T D min sep T T sep drag equation are specified for helicopter (hover and edgewise) and propeller (axial climb and cruise) operation: 2 X sep c = d + d Δ + d Δ + d Δ dh 0hel 1hel 2hel sep sep 2 X 2 X sep pα c = d + d Δ + d Δ + d Δ + d μ | μ | dp 0prop 1prop 2prop sep sep pα z The separation term is present only if Δ > 0 . The helicopter and propeller values are interpolated as sep a function of μ : z − 1 X f c = c + ( c − c ) tan ( | μ | /λ ) + d μ + d μ d basic dh dp dh z h f 1 f 2 π so | μ | /λ = 1 is the midpoint of the transition. The last terms are the effect of edgewise flow.

z h The stall drag increment represents the rise of profile power caused by the occurrence of significant stall on the rotor disk. Let Δ = | C /σ | − ( f /f f )( C /σ ) ( f is an input factor). The function s T s α off T s s f = 1 − d μ accounts for the influence of angle of attack ( μ /μ ) or rotor drag ( C ). The function α sα z z X − d o o 2 x f = 1 − d (1 − e ) accounts for the influence of lift offset, o = rM /T R = ( K /T R ) β . Then off o 1 x x hub s X X s 1 s 2 c = d Δ + d Δ (zero if Δ ≤ 0 ). The blade loading at which the stall affects the entire rotor d stall s 1 s 2 s s s √ 2 2 power, ( C /σ ) , is an input function of the velocity ratio V = μ + μ .

T s z The compressibility drag increment depends on the advancing tip Mach number M , and the tip at airfoil thickness-to-chord ratio τ . The following models are implemented: a) Drag divergence model: Let Δ M = M − M , where M is the drag divergence Mach number of at dd dd the tip section. Then the compressibility increment in the mean drag coefficient is X m c = d Δ M + d Δ M d comp m 1 m 2 112 Rotor transient limit steady limit 0.20 high stall low stall 0.16 s 0.12 ) σ / T (C 0.08 0.04 0.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 μ Figure 11-7. Stall function.

(ref. 10). M is a function of the advancing tip lift coefficient, c . The advancing tip lift is estimated dd  (1 , 90) ∼ from α = ( θ + 0 . 25 θ + θ − ( λ − β ) / (1 + μ )) . 6(1 − 2 . 97 μ + 2 . 21 μ )(6 C /σa ) + 0 . 25 θ (zero = 1 (1 , 90) . 75 L s c T tw above μ = 0 . 6 ). Then the Korn expression (ref. 11) gives M for small lift coefficient: dd M = κ − κ | c | − τ = M − κ | c | dd A  dd 0  where M is the drag divergence Mach number at zero lift, and typically κ = 0 . 16 .

dd 0 b) Similarity model: From transonic small-disturbance theory (refs. 12–13), the scaled wave drag must 2 2 2 / 3 be a function only of K = ( M − 1) / [ M τ (1 + γ )] . An approximation for the wave drag increment at at is 5 / 3 5 / 3 τ τ 5 / 2 Δ c = D ( K ) = 1 . 774( K + 1 . 674) d 1 1 2 2 1 / 3 1 / 3 [ M (1 + γ )] [ M (1 + γ )] at at (constant for K > − 0 . 2 ). Integration of Δ c over the rotor disk gives the compressibility increment in 1 d the profile power. Following Harris, the resulting compressibility increment in the mean drag coefficient is approximately: 2 5 / 2 1 / 2 c = 1 . 52 f ( K + 1) [(1 + μ ) τ ] (1 + γ ) d comp 1 including the input correction factor f ; c is zero for K < − 1 , and constant for K > − 0 . 2 .

d comp 1 1 Figure 11-7 shows typical stall functions ( C /σ ) for two rotors with different stall characteristics, T s designated high-stall and low-stall, resulting from design features such as different airfoils, Figure 11-7 also shows, for reference, typical helicopter rotor steady and transient load limits. Figure 11-8 illustrates the mean drag coefficient in hover, showing c with and without the separation term, and the total for dh the high-stall and low-stall cases. Figure 11-9 illustrates the mean drag coefficient in forward flight, showing the compressibility term c , and the growth in profile power with C /σ and μ as the stall d comp T drag increment increases.

Rotor 113 0.0350 c (low stall) d mean 0.0300 c (high stall) d mean c (with separation) dh 0.0250 c (quadratic) dh 0.0200 d mean 0.0150 c 0.0100 0.0050 0.0000 0.00 0.03 0.06 0.09 0.12 0.15 0.18 C / σ T Figure 11-8. Mean drag coefficient for rotor in hover.

11-5.1.3 Twin Rotors For twin rotors, the induced power is determined by the induced velocity of the rotor system, not the individual rotors. The induced power is still obtained using P = κP = κf T v for each rotor, i ideal D ideal but the ideal induced velocity is calculated for an equivalent thrust C based on the thrust and geometry T e of both rotors. The profile power calculation is not changed for twin rotors.

√ In hover, the twin rotor induced velocity is v = κ T / 2 ρA , from the total thrust T and the i twin p projected disk area A = (2 − m ) A . The overlap fraction m is calculated from the rotor hub separation p . A correction factor for the twin rotor ideal power is also included. For a coaxial rotor, typically ∼ κ = 0 . 90 . So the ideal inflow is calculated for C = ( C + C ) / (2 − m ) .

twin T e T 1 T 2 In forward flight, the induced velocity of a coaxial rotor is v = κ T / (2 ρAV ) , from the total i twin ∼ thrust T and a span of 2 R . The correction factor for ideal induced power (biplane effect) is κ = 0 . 88 twin to 0 . 81 for rotor separations of 0 . 06 D to 0 . 12 D . The ideal inflow is thus calculated for C = C + C .

T e T 1 T 2 The induced velocity of side-by-side rotors is v = κ T / (2 ρA V ) , from the total thrust T and a span of i twin e 2 2 2 R + , hence A = A (1 + / 2 R ) . The ideal inflow is thus calculated for C = ( C + C ) / (1 + / 2 R ) .

e T e T 1 T 2 The induced velocity of tandem rotors is v = κ ( T / (2 ρAV ) + x T / (2 ρAV )) for the front rotor and F twin F R R ∼ ∼ v = κ ( T / (2 ρAV ) + x T / (2 ρAV )) for the rear rotor. For large separation, x and x ; = 0 = 2 R twin R F F R F for the coaxial limit x = x = 1 is appropriate. Here x = m and x = 2 − m is used.

R F R F To summarize, the model for twin rotor ideal induced velocity uses C = x C + x C and the T e 1 T 1 2 T 2 correction factor κ . In hover, x = 1 / (2 − m ) ; in forward flight of coaxial and tandem rotors, x = 1 twin h f for this rotor and x = m or x = 2 − m for the other rotor; in forward flight of side-by-side rotors, f f x = 1 / (1 + / 2 R ) ( x = 1 / 2 if there is no overlap, / 2 R > 1 ). The transition between hover and forward f flight is accomplished using 2 2 x μ + x Cλ f h h x = 2 2 μ + Cλ h with typically C = 1 to 4 . This transition is applied to x for both rotors, and to κ .

twin 114 Rotor C / σ = 0.14 T 0.0350 C / σ = 0.12 T 0.0300 C / σ = 0.10 T C / σ = 0.08 T 0.0250 c d comp 0.0200 d mean 0.0150 c 0.0100 0.0050 0.0000 0.00 0.10 0.20 0.30 0.40 0.50 μ Figure 11-9a. Mean drag coefficient for rotor in forward flight, high-stall.

C / σ = 0.14 T C / σ = 0.12 T C / σ = 0.10 T C / σ = 0.08 T 0.0350 c d comp 0.0300 0.0250 0.0200 d mean 0.0150 c 0.0100 0.0050 0.0000 0.00 0.10 0.20 0.30 0.40 0.50 μ Figure 11-9b. Mean drag coefficient for rotor in forward flight, low-stall.

With a coaxial rotor in hover, the lower rotor acts in the contracted wake of the upper rotor.

Momentum theory gives the ideal induced power for coaxial rotors with large vertical separation (ref. 14): √ 2 2 P = T v , v = T / 2 ρA for the upper rotor; and P = (¯ αs/ τ ) T v , v = T / 2 ρA for the lower rotor.

u u u u     u  Here τ = T /T ; ¯ α is the average of the disk loading weighted by the induced velocity, hence a measure  u of nonuniform loading on the lower rotor ( ¯ α = 1 . 05 to 1 . 10 typically); and the momentum theory solution Rotor 115 is (√ ) ¯ αs 1 √ = 1 + 4(1 + τ ) ¯ ατ − 1 3 / 2 τ 2 τ ∼ The optimum solution for equal power of the upper and lower rotors is ¯ αsτ = 1 , giving τ = T /T = 2 / 3 .

 u Hence for the coaxial rotor in hover the ideal induced velocity is calculated from C = C for the T e T u √ upper rotor and from C = (¯ αs/ τ ) C for the lower rotor, with κ = 1 . Thus x = 1 / (2 − m ) = 1 / 2 T e T  twin h and the input hover κ is not used, unless the coaxial rotor is modeled as a tandem rotor with zero twin longitudinal separation.

11-5.2 Table Performance Method The induced power is calculated from the ideal power: P = κP = κf T v . The induced i ideal D ideal power factor κ is obtained from an input table (linearly interpolated) that can be a function of up to three independent variables. Optionally κ from the equations can be retained as an increment to the table value.

The profile power is calculated from a mean blade drag coefficient: P = ρA (Ω R ) C = o P o σ ρA (Ω R ) c F . The mean drag coefficient c , or alternatively c F = 8 C /σ , is ob- d mean P d mean d mean P P o tained from an input table (linearly interpolated) that can be a function of up to three independent variables. Optionally c from the equations can be retained as an increment to the table value.

d mean The independent variables can be edgewise advance ratio μ (hub plane or tip-path plane), axial − 1 velocity ratio μ (hub plane or tip-path plane), shaft angle of attack α = tan ( μ /μ ) (hub plane or z z tip-path plane), blade loading C /σ , lift offset M /T R , or advancing tip Mach number M .

T x at 11–6 Power Required and Reaction Drive The total rotor power required is P = P + P + P + P . In most helicopter designs the power req i o t p is delivered to the rotor by a mechanical drive, through the rotor shaft torque. Such designs require a transmission and a means for balancing the main-rotor torque. The shaft power is P = P , which shaft req contributes to the propulsion group power required, P , and produces a torque on the aircraft.

reqP G An alternative is to supply the power by a jet reaction drive of the rotor, using cold or hot air ejected out of the blade tips or trailing edges. Helicopters have also been designed with ram jets on the blade tips, or with jet flaps on the blade trailing edges that use compressed air generated in the fuselage.

Since there is no torque reaction between the helicopter and rotor (except for the small bearing friction), no transmission or antitorque device is required, resulting in a considerable weight saving. With a jet reaction drive, the propulsion system is potentially lighter and simpler, although the aerodynamic and thermal efficiency are lower. The helicopter must still have a mechanism for yaw control. With reaction drive the shaft power is P = P − P , where the reaction power P contributes to the engine shaft req react react group or jet group power required. The reaction drive produces a force F on the rotor blades at react effective radial station r , so P = Ω r F . Momentum balance gives the total force react react react react ∑ [ ( ) ] F = ˙ m V − Ω r e − ˙ mV e react react ψ x where e and e are unit vectors perpendicular to the blade and in the free stream direction; and the ψ x sum is over all blades. The average force in the nonrotating frame is the drag of the inlet momentum ( ˙ m V ), which is accounted for in the engine group or jet group model. The mean in the rotating react frame gives the total jet force required ( ) F = ˙ m V − Ω r react react react react 116 Rotor The engine group or jet group performance includes the blade duct and nozzle, perhaps even with tip burning. Optionally the reaction power can be set equal to the rotor power required ( P = P ), so react req P = 0 . If the reaction drive is turned off ( P = 0 and F = 0 ), then the rotor must be trimmed shaft react react such that P = 0 .

req 11–7 Performance Metrics Several performance metrics are calculated for each rotor. The induced power factor is κ = P /P .

i ideal The rotor mean drag coefficient is c = (8 C /σ ) /F , using the function F ( μ, μ ) given previously. The d P o P z rotor effective lift-to-drag ratio is a measure of the induced and profile power: L/D = V L/ ( P + P ) .

e i o The hover figure of merit is M = T f v/P . The propeller propulsive efficiency is η = T V /P . These two D metrics can be combined as a momentum efficiency: η = T ( V + w/ 2) /P , where w/ 2 = f v/ 2 = f v .

mom W D 11–8 Interference The rotor can produce aerodynamic interference velocities at the other components (fuselage, wings, and tails). The induced velocity at the rotor disk is κv , acting opposite the thrust ( z -axis of tip-path i P P F F P P plane axes). So v = − k κv , and v = C v . The total velocity of the rotor disk relative to the i ind ind ind F F F air consists of the aircraft velocity and the induced velocity from this rotor: v = v − v . The total ind P P F F F P P direction of the wake axis is thus e = − C v / | v | (for zero total velocity, e = − k is used).

w w total total − 1 P T P The angle of the wake axis from the thrust axis is χ = cos | ( k ) e | .

w F F The interference velocity v at each component is proportional to the induced velocity v (hence int ind is in the same direction), with factors accounting for the stage of wake development and the position of the component relative to the rotor wake. The far wake velocity is w = f v , and the contracted wake W i area is A = πR = A/f . The solution for the ideal inflow gives f and f . For an open rotor, f = 2 .

c A W A W c For a ducted rotor, the inflow and wake depend on the wake area ratio f , or on the ratio of the rotor A thrust to total thrust: f = T /T . The corresponding velocity and area ratios at an arbitrary point on T rotor the wake axis are f and f , related by w a √ 2 2 μ + ( μ + f λ ) z w i f = a 2 2 ( f μ ) + ( f μ + λ ) V x V z z i Vortex theory for hover gives the variation of the induced velocity with distance z below the rotor disk: ( ) z/R v = v (0) 1 + √ 1 + ( z/R ) With this equation the velocity varies from zero far above the disk to v = 2 v (0) far below the disk. To use this expression in edgewise flow and for ducted rotors, the distance z/R is replaced by ζ /tR , where w ζ is the distance along the wake axis, and the parameter t is introduced to adjust the rate of change ( t w small for faster transition to far wake limit). Hence the velocity inside the wake is f v , where w i ⎧ ζ /tR ⎪ w ⎪ ⎪ 1 + √ ζ < 0 w ⎪ ⎨ 2 1 + ( ζ /tR ) w f = f f = w W z ⎪ ⎪ ζ /tR w ⎪ ⎪ √ 1 + ( f − 1) ζ > 0 W w ⎩ 1 + ( ζ /tR ) w √ and the contracted radius is R = R/ f .

c a Rotor 117 P The wake is a skewed cylinder, starting at the rotor disk and with the axis oriented by e . The w F interference velocity is required at the position z on a component. Whether this point is inside or B outside the wake cylinder is determined by finding its distance from the wake axis, in a plane parallel to P P F F F the rotor disk. The position relative to the rotor hub is ξ = C ( z − z ) ; the corresponding point on B B hub P P P P the wake axis is ξ = e ζ . Requiring ξ and ξ have the same z value in the tip-path plane axes gives w A w B A P T P F F F ( k ) C ( z − z ) B hub ζ = w P T P ( k ) e w from which f , f , f , and R are evaluated. The distance r from the wake axis is then z w a c ( ) ( ) 2 2 2 P T P P P T P P r = ( i ) ( ξ − ξ ) + ( j ) ( ξ − ξ ) B A B A The transition from full velocity inside the wake to zero velocity outside the wake is accomplished in the distance sR , using c ⎧ 1 r ≤ R ⎨ c f = 1 − ( r − R ) / ( sR ) c c r ⎩ 0 r ≥ (1 + s ) R c ( s = 0 for an abrupt transition, s large for always in wake).

F F The interference velocity at the component (at z ) is calculated from the induced velocity v , B ind the factors f f accounting for axial development of the wake velocity, the factor f accounting for W z r immersion in the wake, and an input empirical factor K : int F F v = K f f f f v int W z r t int ind An additional factor f for twin rotors is included. Optionally the development along the wake axis t can be a step function ( f f = 0 , 1 , f above the rotor, on the rotor disk, and below the rotor disk, W z W respectively); nominal ( t = 1 ); or use an input rate parameter t . Optionally the wake immersion can use the contracted radius R or the uncontracted radius R ; can be a step function ( s = 0 , so f = 1 and 0 c r inside and outside the wake boundary); can be always immersed ( s = ∞ so f = 1 always); or can use r an input transition distance s . Optionally the interference factor K can be reduced from an input value int at low speed to zero at high speed, with linear variation over a specified speed range.

To account for the extent of the wing or tail area immersed in the rotor wake, the interference velocity is calculated at several points along the span and averaged. The increment in position is F F B T Δ z = C (0 Δ y 0) , Δ y = ( b/ 2)( − 1 + (2 i − 1) /N ) for i = 1 to N ; where b is the wing span. The B average interference is calculated separately for each wing panel (left and right), by interpolating the interference velocity at N points along the wing to N/ 2 points along the panel span.

For twin main-rotors (tandem, side-by-side, or coaxial), the performance may be calculated for the rotor system, but the interference velocity is still calculated separately for each rotor, based on its disk loading. At the component, the velocities from all rotors are summed, and the total used to calculate the angle of attack and dynamic pressure. This sum must give the interference velocity of the twin rotor system, which requires the correction factor f . Consider differential momentum theory t to estimate the induced velocity of twin rotors in hover. For the first rotor, the thrust and area in the √ non-overlap region are (1 − m ) T and (1 − m ) A , hence the induced velocity is v = κ T / 2 ρA ; similarly 1 1 1 √ v = κ T / 2 ρA . In the overlap region the thrust and area are mT + mT and mA , hence the induced 2 2 1 2 118 Rotor √ velocity is v = κ ( T + T ) / 2 ρA . So for equal thrust, the velocity in the overlap region (everywhere m 1 2 √ for the coaxial configuration) is 2 larger. The factor K is introduced to adjust the overlap velocity: T √ √ v = κ ( K / 2) ( T + T ) / 2 ρA . The interference velocities are calculated separately for the two m T 1 2 √ √ rotors, with the correction factor f : v = f κ T / 2 ρA and v = f κ T / 2 ρA . The sum v + v t int1 t 1 int2 t 2 int1 int2 must take the required value. Below the non-overlap region, the component is in the wake of only one of the rotors, so the interference velocity from the other rotor is zero, and thus f = 1 . Below the overlap t region, the component is in the wake of both rotors, and the sum of the interference velocities equals v m if √ K / 2 T f = √ √ th τ + τ 1 2 √ where τ = T / ( T + T ) is the thrust ratio. For equal thrusts, f = K / 2 ; or f = 1 / 2 for the n n 1 2 th T th 2 2 nominal velocity. The expression f = f cos χ + sin χ gives the required correction factor, with f = 1 t th t √ in edgewise flight. Optionally the correction for twin rotors can be omitted ( f = 1 ); nominal ( K = 2 ); t T or use an input velocity factor in the overlap region ( K ).

T 11–9 Drag F The rotor component includes drag forces acting on the hub and spinner (at z ) and on the pylon hub F (at z ). The component drag contributions must be consistent. In particular, a rotor with a spinner pylon (such as on a tiltrotor aircraft) would likely not have hub drag. The pylon is the rotor support and the nacelle is the engine support. The drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag), since the pylon is connected to the rotor shaft axes; with non-tilting engines it would use the nacelle drag as well.

The body axes for the drag analysis are rotated about the y -axis relative to the rotor shaft axes: BF BS SF BS C = C C , where C = Y . The pitch angle θ can be input, or the rotation appropriate for − θ ref ref a helicopter rotor or a propeller can be specified.

a) Consider a helicopter rotor, with the shaft axes oriented z -axis up and x -axis downstream. It is appropriate that the angle of attack is α = 0 for forward flight and α = − 90 deg for hover, meaning that the body axes are oriented z -axis down and x -axis forward. Hence θ = 180 deg.

ref b) Consider a propeller or tiltrotor, with the shaft axes oriented z -axis forward and x -axis up. It is appropriate that the angle of attack is α = 0 in cruise and α = 90 deg for helicopter mode (with a tilting pylon), meaning that the body axes are oriented z -axis down and x -axis forward. Hence θ = 90 deg.

ref B The aerodynamic velocity relative to the air is calculated in component axes, v . The angle of attack B α and dynamic pressure q are calculated from v . The reference areas for the drag coefficients are the rotor disk area A = πR (for hub drag), pylon wetted area S , ducted wetted area 2 S , and spinner pylon duct wetted area S ; these areas are input or calculated as described previously.

spin The hub drag can be fixed, specified as a drag area D/q ; or the drag can be scaled, specified as a drag coefficient C based on the rotor disk area A = πR ; or the drag can be estimated based on the gross D weight, using a squared-cubed relationship or a square-root relationship. Based on historical data, the drag coefficient C = 0 . 004 for typical hubs, C = 0 . 0024 for current low-drag hubs, and C = 0 . 0015 for D D D 2 / 3 faired hubs. For the squared-cubed relationship: ( D/q ) = k (( W /N ) / 1000) ( W /N hub M T O rotor M T O rotor 2 2 / 3 2 2 / 3 is the maximum takeoff gross weight per lifting rotor; units of k are ft /k-lb or m /Mg ). Based on Rotor 119 historical data, k = 1 . 4 for typical hubs, k = 0 . 8 for current low-drag hubs, and k = 0 . 5 for faired hubs √ (English units). For the square-root relationship: ( D/q ) = k W /N ( W /N is the hub M T O rotor M T O rotor 2 1 / 2 2 1 / 2 maximum takeoff gross weight per lifting rotor; units of k are ft /lb or m /kg ); based on historical data (ref. 15), k = 0 . 074 for single rotor helicopters, k = 0 . 049 for tandem rotor helicopters (probably a blade number effect), k = 0 . 038 for hingeless rotors, and k = 0 . 027 for faired hubs (English units).

To handle multi-rotor aircraft, the scaling weight w = W /N is calculated as for disk loading: M T O rotor w = f W for main-rotors or w = f T for antitorque and auxiliary-thrust rotors.

W M T O The hub vertical drag can be fixed, specified as a drag area D/q ; or the drag can be scaled, specified as a drag coefficient C based on the rotor disk area A = πR .

D The pylon forward flight drag and vertical drag are specified as drag area or drag coefficient, based on the pylon wetted area. The duct forward flight drag and vertical drag are specified as drag area or drag coefficient, based on the duct wetted area. The spinner drag is specified as drag area or drag coefficient, based on the spinner wetted area.

The drag coefficient for the hub or pylon or duct at angle of attack α is X d C = C + ( C − C ) | sin α | D D 0 DV D 0 Optionally the variation can be quadratic ( X = 2 ). For sideward flight, C = C for the hub, d D hub D 0 C = C for the pylon, and C = C for the duct. Then the total component drag force is D pylon DV D duct DV D = qAC + qS C + qS C + qS C D hub pylon D pylon duct D duct spin D spin The dynamic pressure and position of the hub are used for the duct and spinner. The force and moment produced by the drag are ∑ F F = e D d ∑ F F ˜ F M = Δ z F F F F where Δ z = z − z (separate locations are defined for the rotor hub and for the pylon), and e is the d cg F F drag direction. The velocity relative to the air gives e = − v / | v | (no interference).

d 11–10 Weights The rotor configuration determines where the weights occur in the weight statement, as summarized in table 11-3. The rotor group consists of blade assembly, hub and hinge, fairing/spinner, blade fold structure, inter-rotor shaft, rotor support, and duct. The tail-rotor (in empennage group) or the propeller/ fan installation (in propulsion group) consists of blade assembly, hub and hinge, rotor support, and duct.

There are separate weight models for main-rotors, tail-rotors, and auxiliary-thrust systems (pro- pellers). The tail-rotor model requires a torque calculated from the drive system rated power and main-rotor rotational speed: Q = P / Ω . The auxiliary-thrust model requires the design max- DS limit mr imum thrust of the propeller. The engine section or nacelle group includes the engine support weight and pylon support weight; these must be consistent with the use of the rotor support structural weight.

Table 11-3. Principal configuration designation.

configuration weight statement weight model performance model main-rotor rotor group rotor rotor tail-rotor empennage group tail-rotor rotor propeller propulsion group rotor, aux thrust rotor 120 Rotor The flap moment of inertia I and the Lock number γ = ρacR /I are required for the blade motion b b solution. Several options are implemented to calculate I . The Lock number can be specified, and b then I = ρacR /γ used, independent of the blade weight; this is the only option for the tail-rotor and b auxiliary-thrust weight models, which do not give separate blade and hub weight estimates. The moment of inertia I can be calculated from the blade weight and the weight distribution. The Lock number b can be specified, hence I = ρacR /γ , and then mass added to the blade to achieve this value. An b 1 2 autorotation index AI = KE/P = N I Ω /P can be specified, hence the required I , and then mass b b added to the blade to achieve this value. Reference 16 describes this and other autorotation indices; AI = KE/P ≥ 3 sec gives good autorotation characteristics for small helicopters.

In order to increase the moment of inertia, a tip weight W can be added to each blade at radial t station r . Thus the total blade weight is W = χw + dW + (1 + f ) W N (lb or kg); where w is the blade t b b b t b weight estimate, χ the technology factor, dW a specified weight increment; and the factor f accounts for b the blade weight increase required by the centrifugal force due to W . The mass per blade is M = W /N t b b (slug or kg), or M without the tip weight. The blade moment of inertia is b 0 2 2 2 2 2 2 I = R ( r ( M + f M ) + r M ) = I + R ( r + f r ) M b b 0 t t b 0 t 2 t t 2 √ ∼ where r is the radius of gyration of the distributed mass. Typically r . 6 ; r = 1 / 3 = 0 . 577 = 0 2 2 2 for uniform mass distribution. If the required moment of inertia I is greater than I , the tip mass b b 0 M is needed. Additional mass is required inboard to react the centrifugal force increase due to M .

t t This additional mass is less effective than M at increasing I . Assume a fraction a of the blade mass t b ∫ reacts the centrifugal force F , so Δ M/M = a Δ F/F . The reference values are M = R m dr and b 0 0 b 0 ∫ 2 2 2 1 ∼ F = Ω R rm dr = Ω Rr M , where r = . Then 0 1 b 0 1 ( ) ( ) M 1 r ar /r b 0 t t 1 2 2 Δ M = a Δ F = a Ω Rr Δ M + Ω Rr M = a Δ M + M = M = f M 1 t t t t t F Ω Rr r 1 − a 0 1 1 1 2 2 2 ∼ ∼ With a = , f = 1 . The tip mass required to produce Δ I = I − I is M = Δ I / ( R ( r + f r )) , and b b b 0 t b t 2 the total blade weight increment is Δ W = (1 + f ) M N (lb or kg).

b t 11–11 References 1) Cheeseman, I.C., and Bennett, W.E. “The Effect of the Ground on a Helicopter Rotor in Forward Flight.” ARC R&M 3021, September 1955.

2) Law, H.Y.H. “Two Methods of Prediction of Hovering Performance.” USAAVSCOM TR 72-4, February 1972.

3) Hayden, J.S. “The Effect of the Ground on Helicopter Hovering Power Required.” American Helicopter Society 32nd Annual National V/STOL Forum, Washington, D.C., May 1976.

4) Zbrozek, J. “Ground Effect on the Lifting Rotor.” ARC R&M 2347, July 1947.

5) Coleman, R.P.; Feingold, A.M.; and Stempin, C.W. “Evaluation of the Induced-Velocity Field of an Idealized Helicopter Rotor.” NACA ARR L5E10, June 1945.

6) Mangler, K.W., and Squire, H.B. “The Induced Velocity Field of a Rotor.” ARC R & M 2642, May 1950.

Rotor 121 7) Drees, J.M. “A Theory of Airflow Through Rotors and Its Application to Some Helicopter Problems.” Journal of the Helicopter Association of Great Britain, Vol. 3, No. 2, July–September 1949.

8) White, T., and Blake, B.B. “Improved Method of Predicting Helicopter Control Response and Gust Sensitivity.” Annual National Forum of the American Helicopter Society, May 1979.

9) Harris, F.B. “Rotor Performance at High Advance Ratio; Theory versus Test.” NASA CR 2008- 215370, October 2008.

10) Gessow, A., and Crim, A.D. “A Theoretical Estimate of the Effects of Compressibility on the Performance of a Helicopter Rotor in Various Flight Conditions.” NACA TN 3798, October 1956.

11) Mason, W.H. “Analytic Models for Technology Integration in Aircraft Design.” AIAA Paper No.

90-3262, September 1990.

12) Ashley, H., and Landahl, M. Aerodynamics of Wings and Bodies. Reading, Massachusetts: Addison- Wesley Publishing Company, Inc., 1965.

13) Spreiter, J.R., and Alksne, A.Y. “Thin Airfoil Theory Based on Approximate Solution of the Transonic Flow Equation.” NACA Report 1359, 1958.

14) Johnson, W. “Influence of Lift Offset on Rotorcraft Performance.” NASA TP 2009-215404, November 2009.

15) Keys, C.N., and Rosenstein, H.J. “Summary of Rotor Hub Drag Data.” NASA CR 152080, March 1978.

16) Wood, T.L. “High Energy Rotor System.” American Helicopter Society 32nd Annual National V/STOL Forum, Washington, D.C., May 1976.

122 Rotor

Chapter 12 Wing The aircraft can have one or more wings, or no wings.

12–1 Geometry The wing is described by planform area S , span b , mean chord c = S/b , and aspect ratio A R = b /S .

These parameters are for the entire wing. The geometry is specified in terms of two of the following parameters: S or wing loading W/S , b (perhaps calculated from other geometry), c , A R = b /S . With more than one wing, the wing loading is obtained from an input fraction of design gross weight, W = f W . Optionally the span can be calculated from a specified ratio to the span of another wing; or W D the span can be calculated from a specified ratio to the radius of a designated rotor, b = 2 f R . Optionally the wing span can be calculated from an appropriate specification of all wing panel widths.

Optionally for the tiltrotor configuration, the wing span can be calculated from the fuselage and rotor geometry: b = 2( f R + d ) + w , where R is the rotor radius (cruise value for variable-diameter rotor), fus fus d the rotor-fuselage clearance, and w the fuselage width. Note that the corresponding option for the fus fus rotor hub position is y = ± ( f R + d + / w ) . Optionally the wing span can be calculated from the hub fus fus rotor hub position: b = 2 | y | (regardless of how the rotor position is determined). As implemented, hub symmetry is not assumed; rather the radius or hub position of the outermost designated rotors is used.

F The wing is at position z , where the aerodynamic forces act. The component axes are the aircraft BF body axes, C = I .

The wing planform is defined in terms of one or more wing panels (fig. 12-1). Symmetry of the wing is assumed. The number of panels is P , with the panel index p = 1 to P . The wing span station η is scaled with the semi-span: y = η ( b/ 2) , η = 0 to 1 . Each panel is a trapezoid, with a straight aerodynamic center and linear taper. The aerodynamic center locus (in wing axes) is defined by sweep Λ ; dihedral δ ; p p and offsets ( x , z ) at the inboard edge relative to the aerodynamic center of the previous panel. The Ip Ip F wing position z is the mean aerodynamic center. The offset ( ¯ x , ¯ z ) of the mean aerodynamic center A A from the root chord aerodynamic center is calculated (so the wing planform can be drawn; typically the aerodynamic center is drawn as the quarter-chord). Outboard panel edges are at η (input or calculated).

Ep A panel is characterized by span b (each side), mean chord c , and area S = 2 b c (both sides). The p p p p p taper is defined by inboard and outboard chord ratios, λ = c/c (where c is a panel or wing reference ref ref chord, depending on the options for describing the geometry).

The span for each panel (if there are more than two panels) can be a fixed input; a fixed ratio of the wing span, b = f ( b/ 2) ; or free. The panel outboard edge (except at the wing tip) can be at a fixed input p bp position y ; at a fixed station η , y = η ( b/ 2) ; calculated from the rotor radius, y = f R ; calculated Ep Ep p Ep p from the fuselage and rotor geometry, y = f R + d + / w (for a designated rotor); calculated from p fus 2 fus 124 Wing centerline panel p (x , z ) A A aero center locus mean aero center (wing location) inboard outboard edge edge η outboard panel edge, η = y /( b /2) Ep η η wing station Ip Op chord ratio, λ = c / c λ λ ref Ip Op chord c c Ip Op Λ sweep (+ aft) p δ dihedral (+ up) p aero center offset (inboard, + aft) x Ip aero center offset (inboard, + up) z Ip span (each side) b p c mean chord p S area = 2 b c p p p Figure 12-1. Wing geometry (symmetric, only right half-wing shown).

the hub position, y = | y | (for a designated rotor); or adjusted. An adjusted station is obtained from p hub the last station and the span of this panel, y = y + b or y = y + f ( b/ 2) ; or from the next station p p − 1 p p p − 1 bp and the span of the next panel, y = y − b or y = y − f ( b/ 2) . The specification of panel p p +1 p +1 p p +1 b ( p +1) spans and panel edges must be consistent, and sufficient to determine the wing geometry. Determining the panel edges requires the following steps: a) Calculate the panel edges that are either at fixed values (input, or from width, or from hub position) or at fixed stations; root and tip edges are known.

b) Working from root to tip, calculate the adjusted panel edge y if panel span b or ratio f p p bp is fixed, and if previous edge y is known.

p − 1 c) Working from tip to root, calculate the adjusted panel edge y (if not yet known) if panel p span b or ratio f is fixed, and if next edge y is known.

p +1 b ( p +1) p +1 At the end of this process, all edges must be known and the positions y must be unique and sequential.

p If this geometry is being determined for a known span, then there must not be a fixed panel span or span Wing 125 ratio that has not been used. Alternatively, if the wing span is being calculated from the specification of all panel widths, then the process must leave one and only one fixed panel span or span ratio that has not been used. Since the wing span is to be calculated, each panel edge is known in the form y = c + c b/ 2 .

p 0 1 Then the unused fixed panel span gives the equation ( c + c b/ 2) − ( c + c b/ 2) = b (subscript O 0 1 O 0 1 I p denotes outboard edge, subscript I denotes inboard edge), or the unused fixed panel span ratio gives the equation ( c + c b/ 2) − ( c + c b/ 2) = f b/ 2 , which can be solved for the semispan b/ 2 .

0 1 O 0 1 I p To complete the definition of the geometry, one of the following quantities is specified for each panel: panel area S ; ratio of panel area to wing area, f = S /S ; panel mean chord c ; ratio of panel p s p p mean chord to wing mean chord, f = c /c ; chord ratios λ = c /c and λ = c /c (taper); or free.

c p I I ref O O ref The total wing area equals the sum of all panel areas: ∑ ∑ ∑ ∑ ∑ S = S + S f + 2 b c + 2 c b f + 2 c b ( λ + λ ) p s p p p c ref p I O If there is one or more taper specification (and no free), then c is calculated from this equation for S , ref 1 1 and the mean chord is c = ( c + c ) = c ( λ + λ ) , S = 2 b c . If there is one (and only one) free p I O ref I O p p p 2 2 specification, then S is calculated from this equation for S , and the mean chord is c = S / (2 b ) , with p p p p c = 2 c / (1 + λ /λ ) , c = 2 c − c .

I p O I O p I Since the panels have linear taper ( c = c λ ), the mean aerodynamic chord is ref ∫ ∫ b/ 2 1 2 2 2 S ¯ c = c dy = b c λ dη A ref − b/ 2 0 ∑ ∑ 1 1 2 2 2 2 2 = b c ( λ + λ λ + λ ) Δ η = ( c + c c + c ) 2 b I O p I O p ref I O I O 3 3 ∫ ∫ b/ 2 1 S = c dy = b c λ dη ref − b/ 2 0 ∑ ∑ 1 1 = b c ( λ + λ ) Δ η = ( c + c ) 2 b ref I O p I O p 2 2 These expressions are evaluated from panel c and c , as calculated using λ and λ , or using the ratio I O I O λ /λ ( c may not be the same for all panels).

O I ref The mean aerodynamic center is the point where there is zero moment due to lift: ¯ x C S = A L ∫ ∫ ¯ x c c dy = xc c dy , with cc = ( y ) the spanwise lift distribution. Thus A    ∫ ( η )(¯ x − x ( η )) dη = 0 A AC The locus of section aerodynamic centers x is described by the panel sweep Λ and the offset x at AC p Ip the inboard end of the panel. These offsets can be a fixed input, a fraction of the root chord, or a fraction √ of the panel inboard chord. Assuming elliptical loading ( = 1 − η ) gives ∫ ∫ 1 (̂ ) ∑ √ π b ¯ x = ( η ) x dη = 1 − η x + tan Λ η dη A AC Ip p 4 2 [̂ ] η ( ) O ∑ √ 1 b 1 − 1 2 3 / 2 = x η 1 − η + sin η − tan Λ (1 − η ) Ip p 2 2 3 η I ( ) ∑ p where ̂ x = x + ( b/ 2) tan Λ ( η − η ) − ( b/ 2) tan Λ η . The vertical position of Ip Iq q − 1 O ( q − 1) I ( q − 1) p Ip q =2 the mean aerodynamic center is obtained in a similar fashion, from panel dihedral δ and offset z at p Ip the inboard edge of the panel. Assuming uniform loading ( = 1 ) gives [̂ ] ∫ ∫ η 1 (̂ ) O ∑ ∑ b b 1 ¯ z = z dη = z + tan δ η dη = z η + tan δ η A AC Ip p Ip p 2 2 2 0 η I 126 Wing Then ( ¯ x , ¯ z ) is the offset of the mean aerodynamic center from the root chord aerodynamic center.

A A Finally, ( ∑ ) b p − 1 Λ = tan tan Λ p b/ 2 ( ∑ ) b p − 1 δ = tan tan δ p b/ 2 2 c λ = − 1 c root are the wing overall sweep, dihedral, and taper.

The wing contribution to the aircraft operating length is x + (0 . 25 c ) cos i (forward), x − wing wing (0 . 75 c ) cos i (aft), and y ± b/ 2 (lateral).

wing 12–2 Control and Loads The control variables are flap δ , flaperon δ , aileron δ , and incidence i . The flaperon deflection F f a can be specified as a fraction of flap deflection, or as an increment relative to the flap deflection, or the flaperon can be independent of the flap. The flaperon and aileron are the same surface, generating symmetric and antisymmetric loads, respectively, hence with different connections to pilot controls.

With more than one wing panel, each panel can have control variables: flap δ , flaperon δ , aileron F p f p δ , and incidence i . The outboard panel ( p ≥ 2 ) control or incidence can be specified independently, ap p or in terms of the root panel ( p = 1 ) control or incidence (either fraction or increment).

Each control is described by the ratio of the control surface chord to the wing panel chord, = c /c ; f f p and by the ratio of the control surface span to wing panel span, f = b /b , such that the control surface b f p area is obtained from the panel area by S = f S .

f f b p 12–3 Aerodynamics The aerodynamic velocity of the wing relative to the air, including interference, is calculated in B BA component axes, v . The angle of attack α (hence C ) and dynamic pressure q are calculated from wing B v . The reference area for the wing aerodynamic coefficients is the planform area, S . The wetted area contribution is twice the exposed area: S = 2( S − cw ) , where w is the fuselage width.

wet fus fus The wing vertical drag can be fixed, specified as a drag area ( D/q ) ; or the drag can be scaled, V specified as a drag coefficient C based on the wing area; or calculated from an airfoil section drag DV coefficient (for − 90 deg angle of attack) and the wing area immersed in the rotor wake: ( ) C = c S − S − f b c (1 − cos δ ) − f b c (1 − cos δ ) DV d 90 center d 90 F F F d 90 f f f S The term S = c ( w + 2 d ) (where w is the fuselage width and d the rotor-fuselage clearance) center fus fus fus fus is the area not immersed in the rotor wake, and is used only for tiltrotors. The last two terms account for the change in wing area due to flap and flaperon deflection, with an effectiveness factor f .

d 90 From the control surface deflection and geometry, the lift coefficient, maximum lift angle, moment coefficient, and drag coefficient increments are evaluated: Δ C , Δ α , Δ C , and Δ C . These Lf max f M f Df increments are the sum of contributions from flap and flaperon deflection, hence weighted by the control surface area. The drag coefficient increment includes the contribution from aileron deflection.

Wing 127 12-3.1 Lift The wing lift is defined in terms of lift-curve slope C and maximum lift coefficient C (based Lα L max on wing planform area). The three-dimensional lift-curve slope is input directly or calculated from the two-dimensional lift-curve slope: c α C = Lα 1 + c (1 + τ ) / ( πA R ) α where τ accounts for non-elliptical loading. The effective angle of attack is α = α + i − α , where e wing zl α is the angle of zero lift; in reverse flow ( | α | > 90 ), α ← α − 180 sign α . Let α = C /C be zl e e e e max L max Lα the angle-of-attack increment (above or below zero lift angle) for maximum lift. Including the change of maximum lift angle caused by control deflection, A = α + Δ α and A = − α + Δ α .

max max max f min max max f Then ⎧ C α + Δ C A ≤ α ≤ A Lα e Lf min e max ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ f π/ 2 − | α | e ⎨ ( C A + Δ C ) max 0 , α > A Lα max Lf e max C = f π/ 2 − | A | L max ⎪ ⎪ ( ) ⎪ ⎪ f π/ 2 − | α | ⎪ e ⎪ ⎩ ( C A + Δ C ) max 0 , α < A Lα min Lf e min f π/ 2 − | A | min (for zero lift at f 90 deg angle of attack, f = . 9 ). Note that C A + Δ C = C α + Δ C . In Lα max Lf Lα max L max f sideward flight, C = 0 . Finally, L = qSC is the lift force.

L L 12-3.2 Pitch Moment The wing pitch moment coefficient is C = C +Δ C . Then M = qScC is the pitch moment.

M M ac M f M 12-3.3 Roll Moment The only wing roll moment considered is that produced by aileron control. Typically the flaperon and aileron are the same surface, but they are treated separately in this model. The aileron geometry is specified as for the flaperon and flap, hence includes both sides of the wing. The lift coefficient increment Δ C is evaluated as for the flaperon, so one-half of this lift acts up (on the right side) and one-half La acts down. The roll moment is then M = 2(Δ L / 2) y , where y is the lateral position of the aileron x a aerodynamic center, measured from the wing centerline (defined as a fraction of the wing semi-span).

y The roll moment coefficient is C = − Δ C . Then M = qSbC is the roll moment.

 La x  b/ 2 2 12-3.4 Drag The drag area or drag coefficient is defined for forward flight and vertical flight. The effective angle of attack is α = α + i − α , where α is the angle of minimum drag; in reverse flow e wing D min D min ( | α | > 90 ), α ← α − 180 sign α . For angles of attack less than a transition angle α , the drag coefficient e e e e t equals the forward flight (minimum) drag C , plus an angle of attack term and the control increment.

D 0 If the angle of attack is greater than a separation angle α < α , there is an additional drag increase.

s t Thus if | α | ≤ α , the profile drag is e t X X d s C = C (1 + K | α | + K ( | α | − α ) ) + Δ C Dp D 0 d e s e s Df where the separation ( K ) term is present only for | α | > α ; and otherwise s e s X X d s C = C (1 + K | α | + K ( | α | − α ) ) + Δ C Dt D 0 d t s t s Df ( ) π | α | − α e t C = C + ( C − C ) sin Dp Dt DV Dt 2 π/ 2 − α t 128 Wing Optionally there might be no angle of attack variation at low angles ( K = 0 and/or K = 0 ), or d s quadratic variation ( X = 2 ), or cubic variation for the separation term ( X = 3 ). For sideward flight d s B − 1 B B ( v = 0 ) the drag is obtained using φ = tan ( − v /v ) to interpolate the vertical coefficient: C = v D x z y 2 2 C cos φ + C sin φ . The induced drag is obtained from the lift coefficient, aspect ratio, and D 0 v DV v Oswald efficiency e : ( C − C ) L L 0 C = Di πeA R Conventionally the Oswald efficiency e represents the wing parasite drag variation with lift, as well as the induced drag (hence the use of C ). If C varies with angle of attack, then e is just the span efficiency L 0 Dp factor for the induced power (and C should be zero). The wing-body interference is specified as a L 0 drag area, or a drag coefficient based on the wing area. Then ( ) D = qSC = qS C + C + C D Dp Di Dwb is the drag force. The other forces and moments are zero.

12-3.5 Wing Panels The wing panels can have separate controls, different incidence angles, and different interference from the rotors. Thus the lift, drag, and moment coefficients are evaluated separately for each panel, based on the panel area S and mean chord c . The coefficient increments due to control surface deflection p p are calculated using the ratio of the control surface area to panel area, S /S = f . The lateral position f p f b of the aileron aerodynamic center is η b from the panel inboard edge, so y/ ( b/ 2) = η + η b / ( b/ 2) a p E ( p − 1) a p from the wing centerline. Then the total wing loads are: ∑ L = q S C p p Lp ∑ M = q S c C p p p M p ∑ M = q S bC x p p p ∑ D = q S C + 〈 qS 〉 ( C + C ) p p Dpp Di Dwb ∑ The sums are over all panels (left and right). The reference area is S = S , accounting for possible p ∑ absence of wing extensions. The total wing coefficients are based on 〈 qS 〉 = q S . The three- p p dimensional lift-curve slope C is calculated for the entire wing and used for each panel. The induced Lα BF drag is calculated for the entire wing, from the total C . Since C = I , L ⎛ ⎞ − C − C − C Dpp Di Dwb ∑ F F B BA A BA ⎝ ⎠ F = C C F = C q S 0 p p p − C L is the wing aerodynamic force.

12-3.6 Interference With more than one wing, the interference velocity at other wings is proportional to the induced F F velocity of the wing producing the interference: v = K v . The induced velocity is obtained int int ind B B from the induced drag, assumed to act in the k direction: α = v / | v | = C /C = C / ( πeA R ) , ind ind Di L L F F B B B v = C k | v | α . For tandem wings, typically K = 2 for the interference of the front wing on ind int ind the aft wing, and K = 0 for the interference of the aft wing on the front wing. For biplane wings, the int Wing 129 mutual interference is typically K = 0 . 7 (upper on lower, and lower on upper). The induced drag is int then ∑ ( C − C ) L L 0 C = + C α Di L int πeA R other wings ( ) C L α = K α = K int int ind int πeA R other wing The induced velocity from the rotors is included in the angle of attack of the wing. The rotor interference must also be accounted for in the wing induced power: ⎡ ⎤ ∑ ∑ ( C − C ) L L 0 ⎣ ⎦ C = + C K α + C α Di L int ind int ind πeA R rotors other wings The angle α = v /V is obtained from the rotor induced velocity λ . If the interference is wing-like, ind ind i v = Ω Rκλ (so v ∝ L/ρb V ∝ T / 2 ρAV ). If the interference is propeller-like, v = V κλ (so ind i ind ind i v ∝ Γ /b ∝ T / 2 ρA Ω R ). The interference factor can be evaluated from wing induced drag calculations: ind C = Δ C / ( C κλ /μ ) or C = Δ C / ( C κλ ) . Typically for tiltrotors the interference is wing-like, int Di L i int Di L i ∼ with C − 0 . 06 .

= int The wing interference at the tail produces an angle-of-attack change  = E ( C /C ) , where L Lα B E = d/dα is an input factor determined by the aircraft geometry. Then from the velocity v of the wing, ⎛ ⎞ B − v z F F B ⎝ ⎠ v = C 0 int B v x is the interference velocity at the tail.

The wing interference at the rotor can produce interference power. The induced velocity at the rotor F F F F B B B disk is v = K v , with v = C k | v | α again. Separate interference factors K are used int ind int int ind ind for the components of the interference velocity parallel to and perpendicular to the rotor force vector (roughly normal to and in the plane of the rotor disk).

12–4 Wing Extensions The wing can have extensions, defined as wing portions of span b at each wing tip. For the X tiltrotor configuration in particular, the wing weight depends on the distribution of wing area outboard (the extension) and inboard of the rotor and nacelle location. Wing extensions are defined as a set of ∑ wing panels at the tip. The extension span and area are the sum of the panel quantities, b = b X p ext ∑ and S = S . The inboard span and area are then b = b − 2 b , S = S − S . Optionally X p I X I X ext the wing extensions can be considered a kit, hence the extensions can be absent for designated flight conditions or missions. As a kit, the wing extension weight is considered fixed useful load. With wing extensions removed, the aerodynamic analysis considers only the remaining wing panels. The total wing coefficients are then based on the area without the extensions. For the induced drag and interference, the effective aspect ratio is then reduced by the factor ( b /b ) , since the lift and drag coefficients are still I based on total wing area S .

12–5 Wing Kit The wing can be a kit, the kit weight an input fraction of the total wing weight. The wing kit weight can be part of the wing group, or considered fixed useful load. With the kit removed, there are 130 Wing no aerodynamic loads or aerodynamic interference generated by the wing, and the wing kit weight is omitted.

12–6 Weights The wing group consists of: basic structure (primary structure, consisting of torque box and spars, plus extensions); fairings (leading edge and trailing edge); fittings (non-structural); fold/tilt structure; and control surfaces (flaps, ailerons, flaperons, and spoilers). There are separate models for a tiltrotor or tiltwing configuration and for other configurations (including a compound helicopter).

The AFDD wing weight models are based on parameters for the basic wing plus the wing tip extensions (not the total wing and extensions). The tiltrotor wing model requires the weight on the wing tips (both sides), consisting of: rotor group, engine system, drive system (except drive shaft), engine section or nacelle group, air induction group, rotary wing and conversion flight controls, hydraulic group, trapped fluids, and wing extensions. An adjustment of this calculated weight can be used; a negative increment is required when the engine and transmission are not at the tip location with the rotor.

Chapter 13 Empennage The aircraft can have one or more tail surfaces, or no tail surface. Each tail is designated as horizontal or vertical, affecting some parameter definitions.

13–1 Geometry The tail is described by planform area S , span b , chord c = S/b , and aspect ratio A R = b /S . The tail volume can be referenced to rotor radius and disk area, V = S /RA ; to wing area and chord for horizontal tails, V = S /S c ; or to wing area and span for vertical tails, V = S /S b . Here the tail w w w w length is = | x − x | or = | x − x | for horizontal tail or vertical tail, respectively. The geometry ht cg vt cg is specified in terms of S or V ; and b , or A R , or c . The elevator or rudder is described by the ratio of control surface chord to tail chord, c /c ; and the ratio of control surface span to tail span, b /b .

f f The tail contribution to the aircraft operating length is x + 0 . 25 c (forward), x − 0 . 75 c (aft), and tail tail y ± ( b/ 2) C (lateral), where C = cos φ for a horizontal tail and C = cos( φ − 90) for a vertical tail.

tail 13–2 Control and Loads F The tail is at position z , where the aerodynamic forces act. The scaled input for tail position can be referenced to the fuselage length, or to the rotor radius.

The horizontal tail can have a cant angle φ (positive tilt to left, becomes vertical tail for φ = 90 deg).

BF Thus the component axes are given by C = X . The control variables are elevator δ and incidence − φ e i .

The convention for nominal orientation of the vertical tail is positive lift to the left, so aircraft sideslip (positive to right) generates positive tail angle of attack and positive tail lift. The vertical tail can have a cant angle φ (positive tilt to right, becomes horizontal tail for φ = 90 ), so the component axes BF are given by C = X . The control variables are rudder δ and incidence i .

− 90+ φ r 13–3 Aerodynamics The aerodynamic velocity of the tail relative to the air, including interference, is calculated in B BA component axes, v . The angle of attack α (hence C ) and dynamic pressure q are calculated from tail B v . The reference area for the tail aerodynamic coefficients is the planform area, S . The wetted area contribution is S = 2 S . From the elevator or rudder deflection and geometry, the lift coefficient, wet maximum lift angle, and drag coefficient increments are evaluated: Δ C , Δ α , and Δ C .

Lf max f Df 132 Empennage 13-3.1 Lift The tail lift is defined in terms of lift-curve slope C and maximum lift coefficient C (based Lα L max on tail planform area). The three-dimensional lift-curve slope is input directly or calculated from the two-dimensional lift-curve slope: c α C = Lα 1 + c (1 + τ ) / ( πA R ) α where τ accounts for non-elliptical loading. The effective angle of attack is α = α + i − α , where e tail zl α is the angle of zero lift; in reverse flow ( | α | > 90 ), α ← α − 180 sign α . Let α = C /C be zl e e e e max L max Lα the angle-of-attack increment (above or below zero lift angle) for maximum lift. Including the change of maximum lift angle caused by control deflection, A = α + Δ α and A = − α + Δ α .

max max max f min max max f Then ⎧ C α + Δ C A ≤ α ≤ A Lα e Lf min e max ⎪ ⎪ ⎪ ( ) ⎪ ⎪ ⎪ f π/ 2 − | α | e ⎨ ( C A + Δ C ) max 0 , α > A Lα max Lf e max C = f π/ 2 − | A | L max ⎪ ⎪ ( ) ⎪ ⎪ f π/ 2 − | α | ⎪ e ⎪ ⎩ ( C A + Δ C ) max 0 , α < A Lα min Lf e min f π/ 2 − | A | min (for zero lift at f 90 deg angle of attack, f = . 9 ). Note that C A + Δ C = C α + Δ C . In Lα max Lf Lα max L max f B 2 B 2 B 2 sideward flight (defined by ( v ) + ( v ) < (0 . 05 | v | ) ), C = 0 . Finally, L = qSC is the lift force.

L L x z 13-3.2 Drag The drag area or drag coefficient is defined for forward flight and vertical flight. The effective angle of attack is α = α + i − α , where α is the angle of minimum drag; in reverse flow ( | α | > 90 ), e tail D min D min e α ← α − 180 sign α . For angles of attack less than a transition angle α , the drag coefficient equals e e e t the forward flight (minimum) drag C , plus an angle-of-attack term and the control increment. Thus D 0 if | α | ≤ α , the profile drag is e t X d C = C (1 + K | α | ) + Δ C Dp D 0 d e Df and otherwise X d C = C (1 + K | α | ) + Δ C Dt D 0 d t Df ( ) π | α | − α e t C = C + ( C − C ) sin Dp Dt DV Dt 2 π/ 2 − α t Optionally there might be no angle-of-attack variation at low angles ( K = 0 ), or quadratic variation d B 2 B 2 B 2 ( X = 2 ). In sideward flight (defined by ( v ) + ( v ) < (0 . 05 | v | ) ), the drag is obtained using d x z − 1 B B 2 φ = tan ( − v /v ) to interpolate the vertical coefficient: C = C cos φ + C sin φ . The v Dp D 0 v DV v z y induced drag is obtained from the lift coefficient, aspect ratio, and Oswald efficiency e : ( C − C ) L L 0 C = Di πeA R Conventionally the Oswald efficiency e can represent the tail parasite drag variation with lift, as well as the induced drag (hence the use of C ). Then L 0 ( ) D = qSC = qS C + C D Dp Di is the drag force. The other forces and moments are zero.

Empennage 133 13–4 V-Tail A V-tail is modeled as a pair of horizontal and vertical tails, each sized by area or tail volume. The aerodynamic loads are calculated separately for each tail. The weight is calculated only for the second tail, using the V-tail area and aspect ratio. The V-tail area and span are the sum of the two tails: S = S + S V ht vt b = b + b V ht vt √ 2 − 1 and the aspect ratio is A R = b /S . The V-tail dihedral angle is δ = tan S /S , and the cant angle V vt ht V φ = 0 for both. The location (which should be the same for the two tails) is the midpoint of the V-tail, vertically and laterally. An upward V-tail and a downward V-tail are identical in this model.

13–5 Weights The empennage group consists of the horizontal tail, vertical tail, and tail-rotor. The tail plane weight consists of the basic structure and fold structure. The tail weight (empennage group) model depends on the configuration: helicopters and compounds, or tiltrotors and tiltwings. Separate weight models are available for horizontal and vertical tails.

The AFDD tail weight model depends on the design dive speed at sea level (input or calculated).

The calculated dive speed is V = 1 . 25 V , from the maximum speed at the design gross weight and dive max sea level standard conditions.

134 Empennage

Chapter 14 Propulsion System The aircraft propulsion system can be constructed from a number of components: propulsion groups, engine groups, jet groups, charge groups, and fuel tank systems. Figure 14-1 illustrates the configuration possibilities.

The aircraft can have one or more propulsion groups, or none. Each propulsion group is a set of components (rotors) and engine groups, connected by a drive system. The components define the power required, and the engine groups define the power available. There are one or more drive states, with a set of gear ratios for each state. The power required equals the sum of component power, transmission losses, and accessory losses.

An engine group consists of one or more engines of a specific type. An engine group transfers power by shaft torque, so it is associated with a propulsion group. For each engine type an engine model is defined. The engine model describes a particular engine, used in one or more engine groups.

The models include turboshaft engines (perhaps convertible, for turbojet operation or reaction drive), reciprocating engines, compressors, electric motors (perhaps with fuel cells), electric generators, and generator-motors.

The aircraft can have one or more jet groups, or none. A jet group produces a force on the aircraft.

A jet model describes a particular jet, used in one or more jet groups. The models include turbojet and turbofan engines (perhaps convertible, for reaction drive), reaction drive, and a simple force. A reaction drive supplies a blade force that provides the rotor power required.

The aircraft can have one or more charge groups, or none. A charge group generates energy for the aircraft. A charge model describes a particular charger, used in one or more charge groups. The models include fuel cells and solar cells.

There are one or more fuel tank systems for the aircraft. Fuel tank systems are associated with the engine groups, jet groups, and charge groups. Fuel quantity is measured as either weight or energy.

Fuels considered include jet fuel, gasoline, diesel, and hydrogen. Storage systems considered include batteries, capacitors, and flywheels.

14–1 Referred Performance Referred performance parameters are used for propulsion system components that operate with air.

The operating condition and atmosphere give the standard conditions (temperature T and pressure std p ) for a specified pressure altitude; the sea-level standard conditions (temperature T and pressure p ); std 0 0 ◦ ◦ and the operating temperature T and pressure p . Here the temperatures are R or K. The performance characteristics depend on the temperature ratio θ = T /T and pressure ratio δ = p/p . The flight Mach 0 0 √ number M = V /c = V /c θ is obtained from the aircraft speed V .

s s 0 136 Propulsion System power flow fuel flow each fuel tank can serve one or more components fuel tank engine group charge group fuel tank turboshaft energy jet fuel solar cell engine propulsion reciprocating fuel fuel cell gasoline group engine rotors energy compressor drive train compressor rotor react drive motor + fuel drive fuel cell train electric energy motor jet group fuel tank generator- energy turbojet motor jet fuel turbofan drive generator energy simple weight or train force energy reaction rotor fuel drive Figure 14-1. Propulsion system components.

The referred or corrected performance parameters are: P ˙ w power √ fuel flow √ δ θ δ θ ˙ m F mass flow √ force δ δ/ θ P/ ˙ m N specific power rotational speed √ θ θ where P is power, ˙ m is mass flow, ˙ w is fuel flow, F is a force, and N is a rotational speed. The performance at sea-level-standard static conditions is indicated by subscript 0 .

Propulsion System 137 14–2 Engine Ratings Engine performance depends on the engine rating. Each engine rating has specific operating limitations, most importantly an operating time limit intended to avoid damage to the engine.

The power available from a turboshaft engine depends on the engine rating. Typical engine ratings are given in table 14-1. Engine power is generally specified in terms of sea level standard (SLS) static MCP. Takeoff typically uses MRP. CRP or ERP is restricted to use in one-engine inoperative (OEI) emergencies.

Table 14-1. Typical turboshaft engine ratings.

rating description time limit MCP maximum continuous power ∞ IRP intermediate rated power 30 min MRP maximum rated power 10 min CRP contingency rated power 2.5 min ERP emergency rated power 1.0 min The thrust available from a turbojet or turbofan engine depends on the engine rating. Typical engine ratings are given in table 14-2.

Table 14-2. Typical jet engine ratings.

rating description time limit MCT maximum continuous thrust ∞ MTO maximum takeoff thrust 5 min 14–3 Efficiency from Equivalent Circuit The efficiency of an electrical device can be expressed in terms of its power by considering an equivalent circuit, defined by internal resistance R and current I . The total voltage is V = V + IR and 0 x the total current is I = I + I . Then the useful power is x 0 P = V I = ( V − IR ) I = V I − I R − V I = P − P x x x x 0 total loss and the efficiency is P P P η = = = 2 2 P P + P P + P R/V + P total loss 0 In terms of a reference power, let ( ) 1 1 R/V = − 1 P η ref ref P = cP 0 ref Then ( ) 1 P 1 P ref = 1 + P ( R/V ) + P /P = 1 + − 1 + c η P η P ref ref − 1 So η = (1 /η + c ) at P = P . The efficiency decreases with P because of the internal resistance, ref ref but is zero at P = 0 because of the internal current term.

138 Propulsion System 14–4 Control and Loads Geometry and control are defined for engine groups, jet groups, and charge groups. The group amplitude A and mode B are control variables: A = A + T c 0 A AC B = B + T c 0 B AC with A and B zero, constant, or a function of flight speed (piecewise linear input). The amplitude can 0 0 be power (engine group), thrust (jet group), or power (charge group). The mode can be mass flow (for convertible engines), or power flow (for generator-motor).

The group orientation is specified by selecting a nominal direction e in body axes (positive or f 0 negative x -, y -, or z -axis; usually thrust forward, hence positive x -axis); then applying a yaw angle ψ ; and then an incidence or tilt angle i (table 14-3). The yaw and incidence angles can be connected to the aircraft controls c : AC ψ = ψ + T c 0 ψ AC i = i + T c 0 i AC with ψ and i zero, constant, or a function of flight speed (piecewise linear input). Hence the incidence 0 0 and yaw angles can be fixed orientation or can be control variables. Optionally the lateral position of the group can be set equal to that of a designated rotor (useful for tiltrotors when the rotor hub lateral position is calculated from the clearance or wing geometry).

The group produces a force T , acting in the direction of the group; and a drag D , acting in the wind F direction. The group is at location z . The force and moment acting on the aircraft in body axes are thus: F F = e T + e D f d F F ˜ F M = Δ z F F F F where Δ z = z − z , e is the force direction, and e is the drag direction. The velocity relative to the cg f d F F BF air gives e = − v / | v | (no interference). The group axes are C = U V , where U and V depend on d i ψ F B the nominal direction, as described in table 14-3. The force direction is e = C e .

f f 0 For a tiltrotor aircraft, one of the aircraft controls is the nacelle angle, with the convention α = 0 tilt for cruise and α = 90 deg for helicopter mode. The incidence angle is then connected to α by tilt tilt defining the matrix T appropriately. If the nominal direction is defined for airplane mode ( + x ), then i i = α should be used; if the nominal direction is defined for helicopter mode ( − z ), then i = α − 90 tilt tilt should be used.

Table 14-3. Group orientation.

BF nominal (F axes) e incidence, + for force yaw, + for force C = U V f 0 i ψ x forward i up right Y Z i ψ − x aft − i up right Y Z − i − ψ y right j aft up Z X i − ψ − y left − j aft up Z X − i ψ z down k aft right Y X − i − ψ − z up − k aft right Y X i ψ Propulsion System 139 14–5 Nacelle Drag An aerodynamic model is defined for engine groups, jet groups, and charge groups. The group includes a nacelle, which contributes to the aircraft drag. The component drag contributions must be consistent. The pylon is the rotor support and the nacelle is the engine support. The drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag), since the pylon is connected to the rotor shaft axes; with non-tilting engines it would use the nacelle drag as well.

F BF The nacelle drag acts at the group location z . The nacelle axes are the group axes, hence C is calculated as described previously (see table 14-1). For the nominal direction forward ( + x -axis), the nacelle z -axis is downward and the x -axis is forward; zero incidence angle corresponds to zero angle of attack; and 90 deg incidence angle corresponds to 90 deg angle of attack (vertical drag). The velocity, angle of attack, and dynamic pressure are calculated at the nacelle (without interference). The reference area for the nacelle drag coefficient is the nacelle wetted area. The wetted area is input, or calculated from the weight w : ( ) 2 / 3 S = k w/N wet 2 2 / 3 2 2 / 3 where N is the number of engines, jets, or chargers; and the units of k are ft /lb or m /kg . The reference area is then S = N S . The nacelle area is included in the aircraft wetted area if the drag nac wet coefficient is nonzero. The drag area or drag coefficient is defined for forward flight and for vertical flight. The drag coefficient at angle of attack α is X d C = C + ( C − C ) | sin α | D D 0 DV D 0 typically using X = 2 . In sideward flight, C = C is used. The nacelle drag is D = qS C .

d D D 0 nac nac D 140 Propulsion System

Chapter 15 Fuel Tank The fuel quantity stored and burned can be measured in weight or energy. Each component (engine group, jet group, or charge group) that uses or generates fuel is associated with a fuel tank system of the appropriate type. The unit of fuel energy is Mega-Joules (MJ). For reference, 1 British Thermal Unit (BTU) = 1055.056 Joule and 1 kW-hr = 3.6 MJ.

15–1 Fuel Tank System Store and Burn Weight For fuel use measured by weight, the fuel properties are density ρ (weight per volume, lb/gal or fuel kg/liter) and specific energy e (MJ/kg). Table 15-1 gives the properties of a number of aviation fuels, fuel based on military and industry specifications (refs. 1–3). Fuels considered include jet fuel, gasoline, diesel, and hydrogen. From the fuel weight W , the energy is E = e W (MJ) and the volume fuel fuel fuel fuel is V = W /ρ (gallons or liters). A motive device has a fuel flow ˙ w (lb/hour or kg/hour), and its fuel fuel fuel specific fuel consumption is sfc = ˙ w/P or sfc = ˙ w/T .

15-1.1 Fuel Capacity The fuel tank capacity W (maximum usable fuel weight) is determined from designated fuel − cap sizing missions. The maximum mission fuel required, W (excluding reserves and any fuel in fuel − miss auxiliary tanks), gives W = max( f W , W + W ) fuel − cap fuel − cap fuel − miss fuel − miss reserve where f ≥ 1 is an input factor. Alternatively, the fuel tank capacity W can be input. The fuel − cap fuel − cap corresponding volumetric fuel tank capacity is V = W /ρ .

fuel − cap fuel − cap fuel For missions that are not used to size the fuel tank, the fuel weight may be fallout, or the fuel weight may be specified (with or without auxiliary tanks). The fuel weight for a flight condition or the start of a mission can be specified as an increment d , plus a fraction f of the fuel tank capacity, plus auxiliary tanks: ∑ W = min( d + f W , W ) + N W fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap where W is the capacity of each auxiliary fuel tank. The fuel capacity of the wing can be estimated aux − cap from ∑ W = ρ f c t b fuel − wing fuel tb w w where c is the torque box chord, t the wing thickness, and b the wing span; and f is the input fraction tb w w of the wing torque box that is filled by primary fuel tanks, for each wing. This calculation is performed in order to judge whether fuel tanks outside the wing are needed.

142 Fuel Tank if W > W fuel fuel − max for designated auxiliary tank N = N + n auxtank auxtank if fixed total weight: Δ W = − nf W fuel auxtank aux − cap Δ W = nW fuel − max aux − cap repeat if W > W fuel fuel − max if fixed total weight and W ≤ W − nW fuel fuel − max aux − cap N = N − n auxtank auxtank Δ W = − nW fuel − max aux − cap W = W (capped) fuel fuel − max else if W < W fuel fuel − max for designated auxiliary tank (then for last nonzero N ) auxtank N = N − n auxtank auxtank if fixed total weight: Δ W = nf W fuel auxtank aux − cap Δ W = − nW fuel − max aux − cap repeat if W < W fuel fuel − max undo last increment N = N + n auxtank auxtank if fixed total weight: Δ W = − nf W fuel auxtank aux − cap Δ W = nW fuel − max aux − cap Figure 15-1. Outline of N calculation.

auxtank 15-1.2 Fuel Reserves Mission fuel reserves can be specified in several ways for each mission. Fuel reserves can be defined in terms of specific mission segments, for example 200 miles plus 20 minutes at V . Fuel be reserves can be an input fraction of the fuel burned by all (except reserve) mission segments, so W = fuel (1 + f ) W . Fuel reserves can be an input fraction of the fuel capacity, so W = W + res fuel − miss fuel miss − seg f W . If more than one criterion for reserve fuel is specified, the maximum reserve is used.

res fuel − cap 15-1.3 Auxiliary Fuel Tank Auxiliary fuel tanks are defined in one or more sizes. The capacity of each auxiliary fuel tank, W , is an input parameter. The number of auxiliary fuel tanks on the aircraft, N for each aux − cap auxtank size, can be specified for the flight condition or mission segment. Alternatively (if the mission is not used to size the fuel tank), the number of auxiliary fuel tanks at the start of the mission can be determined from the mission fuel.

Figure 15-1 describes the process for determining N from the required fuel weight W auxtank fuel ∑ and the aircraft maximum fuel capacity W = W + N W . The fuel weight fuel − max fuel − cap auxtank aux − cap adjustment Δ W is made if fuel weight is fallout from fixed gross weight and payload, accounting fuel for the operating weight update when N changes. If the auxiliary tank weight is greater than auxtank the increment in fuel weight needed, then the fallout fuel weight W = W − W − W can not be fuel G O pay achieved; in such a case, the fuel weight is capped at the maximum fuel capacity and the payload weight adjusted instead. The tanks changed can be the first size, the first size already used, or a designated size.

The tanks can be added or dropped in groups of n ( n = 2 for pairs).

Fuel Tank 143 The weight and drag of N tanks are included in the performance calculation. Optionally the auxtank number of auxiliary tanks required can be calculated at the beginning of designated mission segments (based on the aircraft fuel weight at that point), and tanks dropped if no longer needed. The weight of the ∑ auxiliary fuel tanks is an input fraction of the tank capacity: W = f N W .

auxtank auxtank auxtank aux − cap 15-1.4 Auxiliary Fuel Tank Drag F The auxiliary fuel tanks are located at position z . The drag area for one auxiliary tank is specified, F F ( D/q ) . The velocity relative to the air gives the drag direction e = − v / | v | and dynamic auxtank d F 2 pressure q = / ρ | v | (no interference). Then F F = e q N ( D/q ) d auxtank auxtank is the total drag force, calculated for each auxiliary tank size.

15-1.5 Weights The fuel system consists of the tanks (including support) and the plumbing. For the fractional model, the fuel tank weight is W = χ f W .

tank tank tank fuel − cap The weight of the auxiliary fuel tanks is part of the fixed useful load; it is an input fraction of the ∑ tank capacity: W = f N W .

auxtank auxtank auxtank aux − cap The AFDD weight model for the plumbing requires the fuel flow rate (for all engines), calculated for the takeoff rating and conditions.

15–2 Fuel Tank System Store and Burn Energy For fuel use and storage measured by energy, there is no weight change as energy is used. The energy storage (tank) is characterized by specific energy e (MJ/kg) and energy density ρ (MJ/liter). Table tank tank 15-2 gives the properties of a number of systems. The tank weight and volume are obtained from the ˙ fuel energy E (MJ). The fuel weight W is zero. A motive device has an energy flow E , and its fuel − cap fuel ˙ ˙ specific fuel consumption is sfc = E/P (inverse of efficiency). An equivalent fuel flow is ˙ w = E/e , eq ref based on the specific energy e (MJ/kg) of the first fuel tank that burns weight (or GP-4, e = 42 . 8 ref ref MJ/kg). The corresponding equivalent specific fuel consumption is sfc = ˙ w /P . The specific power eq eq is π (kW/kg).

tank Storage systems considered include batteries, capacitors, and flywheels. A battery (or capacitor) stores charge (A-hr), so the capacity is expressed as energy for a nominal voltage. Variation of the voltage with operation affects the efficiency of the relation between useful power and the rate of change of the energy stored. Each fuel tank system that stores and burns energy has a battery model for computation of the charge/discharge efficiency. The components associated with a fuel tank system define the total ˙ energy flow E (charge or discharge). Accounting for battery capacity, efficiency, and losses gives comp ˙ the effective energy flow E .

eff 15-2.1 Fuel Capacity The fuel tank capacity E (maximum usable fuel energy) is determined from designated fuel − cap sizing missions. The maximum mission fuel required, E (excluding reserves and any fuel in fuel − miss auxiliary tanks), gives E = max( f E , E + E ) fuel − cap fuel − cap fuel − miss fuel − miss reserve 144 Fuel Tank if E > E fuel fuel − max for designated auxiliary tank N = N + n auxtank auxtank Δ E = nE fuel − max aux − cap repeat if E > E fuel fuel − max else if E < E fuel fuel − max for designated auxiliary tank (then for last nonzero N ) auxtank N = N − n auxtank auxtank Δ E = − nE fuel − max aux − cap repeat if E < E fuel fuel − max undo last increment N = N + n auxtank auxtank Δ E = nE fuel − max aux − cap Figure 15-2. Outline of N calculation.

auxtank where f ≥ 1 is an input factor. Alternatively, the fuel tank capacity E can be in- fuel − cap fuel − cap put. The corresponding fuel tank weight is W = χ E /e (lb or kg) and the fuel tank tank fuel − cap tank tank volume is V = E /ρ (gallons or liters). The corresponding power capacity is tank fuel − cap tank P = ( π /e ) E (kW and MJ). Optionally the maximum mission battery discharge power cap tank tank fuel − cap gives P .

cap For missions that are not used to size the fuel tank, the fuel energy may be fallout, or the fuel energy may be specified (with or without auxiliary tanks). The fuel energy for a flight condition or the start of a mission can be specified as an increment d , plus a fraction f of the fuel tank capacity, plus auxiliary tanks: ∑ E = min( d + f E , E ) + N E fuel fuel fuel fuel − cap fuel − cap auxtank aux − cap where E is the capacity of each auxiliary fuel tank.

aux − cap 15-2.2 Fuel Reserves Mission fuel reserves can be specified in several ways for each mission. Fuel reserves can be defined in terms of specific mission segments, for example 200 miles plus 20 minutes at V . Fuel reserves can be be an input fraction of the fuel burned by all (except reserve) mission segments, so E = (1+ f ) E .

fuel res fuel − miss Fuel reserves can be an input fraction of the fuel capacity, so E = E + f E . If more fuel miss − seg res fuel − cap than one criterion for reserve fuel is specified, the maximum reserve is used.

15-2.3 Auxiliary Fuel Tank Auxiliary fuel tanks are defined in one or more sizes. The capacity of each auxiliary fuel tank, E , is an input parameter. The number of auxiliary fuel tanks on the aircraft, N for each aux − cap auxtank size, can be specified for the flight condition or mission segment. Alternatively (if the mission is not used to size the fuel tank), the number of auxiliary fuel tanks at the start of the mission can be determined from the mission fuel.

Figure 15-2 describes the process for determining N from the required fuel energy E and auxtank fuel ∑ the aircraft maximum fuel capacity E = E + N E . The tanks changed fuel − max fuel − cap auxtank aux − cap Fuel Tank 145 can be the first size, the first size already used, or a designated size. The tanks can be added or dropped in groups of n ( n = 2 for pairs).

The drag of N tanks is included in the performance calculation. Optionally the number of auxtank auxiliary tanks required can be calculated at the beginning of designated mission segments (based on the aircraft fuel energy at that point), and tanks dropped if no longer needed. The weight of the auxiliary fuel ∑ tanks is obtained from e (MJ/kg) and the tank capacity: W = N E /e .

auxtank auxtank auxtank aux − cap auxtank 15-2.4 Weights The fuel system consists of the tanks (including support) and the plumbing. The plumbing weight here means the power distribution (wiring). The fuel tank weight is E /e + W (lb or kg) fuel − cap tank BM S and the fuel tank volume is V = E /ρ (gallons or liters). The battery management fuel − cap fuel − cap tank system (BMS) weight is a fraction of the basic tank weight: W = f ( E /e ) . The BM S BM S fuel − cap tank wiring weight is a fraction of the basic tank weight: f ( E /e ) . Alternatively, the wiring wire fuel − cap tank weight can be input as part of the electrical group weight.

The weight of the auxiliary fuel tanks is part of the fixed useful load; it is obtained from e auxtank ∑ (MJ/kg) and the tank capacity: W = N E /e .

auxtank auxtank aux − cap auxtank 15–3 References 1) Department of Defense Military Specification. “Glossary of Definitions, Ground Rules, and Mission Profiles to Define Air Vehicle Performance Capability.” MIL-STD-3013A, September 2008.

2) Department of Defense Detail Specification. “Turbine Fuel, Aviation, Grades JP-4 and JP-5.” MIL- DTL-5624U, September 1998.

3) Department of Defense Detail Specification. “Turbine Fuel, Aviation, Kerosene Type, JP-8 (NATO F-34), NATO F-35, and JP-8+100 (NATO F-37).” MIL-DTL-83133H, October 2011.

146 Fuel Tank Table 15-1. Fuel properties.

fuel specification density specific energy energy dens.

lb/gal kg/L MJ/kg BTU/lb lb/hp-hr MJ/L gasoline MIL-STD-3013A 6.0* 0.719 43.50 18700* 0.136 31.3 diesel 7.0 0.839 43.03 18500 0.138 36.1 6.84– 0.820– 43.0 18487 0.138 35.8 7.05 0.845 Jet A/A-1 MIL-STD-3013A 6.7* 0.803 42.80 18400* 0.138 34.4 6.84/ 0.820/ 42.8 18401 0.138 34.8 6.71 0.804 JP-4 6.5 0.779 42.80 18400 0.138 33.3 MIL-DTL-5624U 6.23– 0.751*– 42.8* 18401 0.138 32.2 6.69 0.802* JP-5 MIL-STD-3013A 6.6* 0.791 42.57 18300* 0.139 33.7 alternate design 6.8* 0.815 42.91 18450* 0.138 35.0 MIL-DTL-5624U 6.58– 0.788*– 42.6* 18315 0.139 34.8 7.05 0.845* JP-8 MIL-STD-3013A 6.5* 0.779 42.80 18400* 0.138 33.3 alternate design 6.8* 0.815 43.19 18570* 0.137 35.2 MIL-DTL-83133H 6.45– 0.775*– 42.8* 18401 0.138 34.6 7.01 0.840* hydrogen (700 bar) 0.328 0.03930 120. 51591 0.0493 4.72 hydrogen (liquid) 0.592 0.07099 120. 51591 0.0493 8.52 *specification value Table 15-2. Energy storage properties.

tank specific energy tank energy density efficiency power MJ/kg kW-hr/kg MJ/L kW-hr/m kW/kg lead-acid battery 0.11–0.14 0.03–0.04 0.22–0.27 60–75 70–90% 0.18 nickel-cadmium battery 0.14–0.20 0.04–0.06 0.18–0.54 50–150 70–90% 0.15 lithium-ion state-of-art 0.54–0.90 0.15–0.25 0.90–1.30 250–360 ∼ 99% 1.80 +5 years 1.26 0.35 1.80 500 +15 years 2.34 0.65 2.25 625 ultracapacitor 0.01–0.11 0.004-0.03 0.02–0.16 6–45 1.00 flywheel steel 0.11 0.03 ∼ 90% graphite 0.90 0.25

Chapter 16 Propulsion Group The propulsion group is a set of components and engine groups, connected by a drive system. The engine model describes a particular engine, used in one or more engine group. The components (rotors) define the power required. The engine groups define the power available. Figure 16-1 illustrates the power flow.

16–1 Drive System The drive system defines gear ratios for all the components it connects. The gear ratio is the ratio of the component rotational speed to that of the primary rotor. There is one primary rotor per propulsion group (for which the reference tip speed is specified); other components are dependent (for which a gear ratio is specified). There can be more than one drive system state, in order to model a multiple-speed or variable-speed transmission. Each drive system state corresponds to a set of gear ratios.

For the primary rotor, a reference tip speed V is defined for each drive system state. By tip − ref convention, the “hover tip speed” refers to the reference tip speed for drive state #1. If the sizing task changes the hover tip speed, then the ratios of the reference tip speeds at different engine states are kept constant. By convention, the gear ratio of the primary rotor is r = 1 . For dependent rotors, either the gear ratio is specified (for each drive system state) or a tip speed is specified and the gear ratio is calculated ( r = Ω / Ω , Ω = V /R ). For the engine group, either the gear ratio is dep prim tip − ref specified (for each drive system state) or the gear ratio is calculated from the specification engine turbine speed Ω = (2 π/ 60) N and the reference tip speed of the primary rotor ( r = Ω / Ω , spec spec spec prim Ω = V /R ). The latter option means the specification engine turbine speed N corresponds prim tip − ref spec to V for all drive system states. To determine the gear ratios, the reference tip speed and radius are tip − ref used, corresponding to hover.

The flight state specifies the tip speed of the primary rotor and the drive system state, for each propulsion group. The drive system state defines the gear ratio for dependent rotors and the engine groups. From the rotor radius, the rotational speed of the primary rotor is obtained ( Ω = V /R ); prim tip from the gear ratios, the rotational speed of dependent rotors ( Ω = r Ω ) and the engine groups dep prim ( N = (60 / 2 π ) r Ω ) are obtained; and from the rotor radius, the tip speed of the dependent rotor eng prim ( V = Ω R ) is obtained. The flight state specification of the tip speed can be an input value, the tip dep reference tip speed, a function of flight speed or a conversion schedule, or one of several default values.

These relationships between tip speed and rotational speed use the actual radius of the rotors in the flight state, which for a variable-diameter rotor may not be the same as the reference, hover radius.

A designated drive system state can have a variable speed (variable gear ratio) transmission, by introducing a factor f on the gear ratio when the speeds of the dependent rotors and engines are gear 148 Propulsion Group POWER POWER AVAILABLE REQUIRED ENGINE P P q a UNINSTALLED installation losses P loss mechanical ( N / N ) P spec mech power limit ENGINE SHAFT P P av req INSTALLED sum distribute all operable engines flight state f p power fraction engine ( Ω / Ω ) P shaft prim ref ES limit limit ENGINE GROUP P P avEG reqEG sum all engine groups distribute drive ( Ω / Ω ) P system prim ref DS limit limit PROPULSION P P avPG reqPG GROUP transmission losses P xmsn accessory power P acc distribute all components sum rotor ( Ω / Ω ) P shaft prim ref RS limit limit P = P + P + P P req comp i o p av comp ROTORS Max Gross Weight Options Max Effort or Trim Options P = fP + d P = P reqPG avPG reqPG avPG Q ≤ Q Q ≤ Q req limit req limit Figure 16-1. Power flow.

Propulsion Group 149 evaluated. The factor f is a component control, which can be connected to an aircraft control and gear thus set for each flight state.

An optional conversion schedule is defined in terms of two speeds: hover and helicopter mode for speeds below V , cruise mode for speeds above V , and conversion mode for speeds between C hover C cruise V and V . The tip speed is V in helicopter and conversion mode, and V in C hover C cruise tip − hover tip − cruise airplane mode. Drive system states are defined for helicopter, cruise, and conversion mode flight. The flight state specifies the nacelle tilt angle, tip speeds, control state, and drive system state, including the option to obtain any or all of these quantities from the conversion schedule.

Several default values of the tip speed are defined for use by the flight state, including cruise, maneuver, one-engine inoperative, drive system limit conditions, and a function of flight speed (piecewise linear input). Optionally these default values can be input as a fraction of the hover tip speed. Optionally √ 2 2 the tip speed can be calculated from μ = V /V , so V = V /μ ; or from M = M (1 + μ ) + μ , so tip tip at tip z √ 2 2 V = ( c M ) − V − V . Optionally the tip speed can be the minimum of the input value or that for tip s at z M .

at The sizing task might change the hover tip speed (reference tip speed for drive system state #1), the reference tip speed of a dependent rotor, a rotor radius, or the specification engine turbine speed N . In such cases the gear ratios and other parameters are recalculated. Note that it is not consistent spec to change the reference tip speed of a dependent rotor if the gear ratio is a fixed input.

An increment on the primary rotor rotational speed (or primary engine group, if there are no rotors) is a control variable of the propulsion group.

16–2 Power Required The component power required P is evaluated for a specified flight condition, as the sum of comp the power required by all the components of the propulsion group. The total power required for the propulsion group is obtained by adding the transmission losses and accessory power: P = P + P + P reqP G comp xmsn acc The transmission losses are calculated as an input fraction of the component power, plus windage xmsn loss: P = f | P | + P (Ω / Ω ) xmsn xmsn xmsn comp windage prim ref The factor f can equal 1, or can include a function of the drive shaft limit (increasing the losses at xmsn low power): ⎧ 1 1 ⎪ P Q < X limit ⎨ 2 4 ( ) 7 4 1 f | P | = − Q | P | < Q < 1 xmsn comp comp 3 3 4 ⎪ ⎩ | P | 1 < Q comp where Q = | P | /P , P = rP , and r = N/N = Ω / Ω . Accessory losses are comp X limit X limit DS limit spec prim ref calculated as the sum of an input constant; terms that scale with air density and rotor speed; a fraction of power required (such as environmental control unit (ECU) losses); infared suppressor fan loss (if IRS system is on); deice power loss (if deice system is on); and an increment specified for each flight state: P = P + P σ + P σ (Ω / Ω ) + ( | P | + P ) + σN P + P + dP acc acc0 acc d acc n prim ref acc comp xmsn IRfan eng eng acc i acc where σ = ρ/ρ is the density ratio.

150 Propulsion Group The power required for the propulsion group must be distributed to the engine groups. With only one engine group, P = P . An engine group power can be fixed at P = ( N − N ) A , where reqEG reqP G reqEG eng inop A is the input power amplitude; or fraction A of engine power available, P = ( N − N ) AP ; reqEG eng inop av or fraction A of engine rated power P = ( N − N ) AP . The power required for the remaining reqEG eng inop eng (perhaps all) engine groups is distributed proportional to the engine rated power: ∑ ( ) ( N − N ) P eng inop eng ∑ P = P − P reqEG reqP G reqEG ( N − N ) P eng inop eng notfixed fixed omitting engine groups that do not supply shaft power. If the sum of the fixed P exceeds the reqEG propulsion group power required, or if the power is fixed for all engine groups, then each is scaled by ∑ the ratio P / P . The fuel flow of the propulsion group is obtained from the sum over reqP G reqEG fixed ∑ the engine groups: ˙ w = ˙ w .

reqP G reqEG 16–3 Geometry The length of the drive system can be input or calculated. The calculated length is the sum of DS the longitudinal, lateral, and vertical distances from the primary rotor hub to the other hub locations, for ∑ all rotors in the propulsion group: = f ( | Δ x | + | Δ y | + | Δ z | ) , where f is an input factor.

DS 16–4 Drive System limit The drive system limit is defined as a power limit, P . The limit is properly a torque limit, DS limit Q = P / Ω , but is expressed as a power limit for clarity. The drive system limit can be DS limit DS limit ref specified as follows (with f an input factor): limit a) Input P .

DS limit ∑ b) From the engine takeoff power limit, P = f N P (summed over DS limit limit eng eng all engine groups).

c) From the power available at the transmission sizing conditions and missions, ∑ P = f (Ω / Ω ) N P (largest of all conditions and segments).

DS limit limit ref prim eng av d) From the power required at the transmission sizing conditions and missions, ∑ P = f (Ω / Ω ) N P (largest of all conditions and segments).

DS limit limit ref prim eng req The drive system limit is a limit on the entire propulsion system. To account for differences in the distribution of power through the drive system, limits are also used for the torque of each rotor shaft ( P ) and of each engine group ( P ). The engine shaft limit is calculated as for the drive system RS limit ES limit limit, without the sum over engine groups. The rotor shaft limit is either input or calculated from the rotor power required at the transmission sizing flight conditions. The power limit is associated with a reference rotational speed, and when applied the limit is scaled with the rotational speed of the flight state. The rotation speed for the drive system limit P is the hover speed of the primary rotor of DS limit the propulsion group (for the first drive state). The rotation speed for the engine shaft limit P ES limit is the corresponding engine turbine speed. The rotation speed for the rotor shaft limit P is the RS limit corresponding speed of that rotor.

The drive system limits can be specified for several levels, analogous to engine ratings. The limit P is associated with the maximum continuous rating (MCQ or MCP). An alternate rating changes DS limit the torque limit by the factor x . Typically x > 1 for ratings associated with short duration operation.

Propulsion Group 151 The torque limit is calculated from Q = Q/x for the flight condition or mission segment. The torque limit limit is applied as Q = xQ limit 16–5 Weights The drive system consists of gear boxes and rotor shafts, drive shafts, rotor brakes, clutches, and gas drive. The drive system weight depends on the rotor and engine rotational speeds, evaluated for the propulsion group primary rotor and a specified engine group, at a specified drive system state (gear ratio).

The AFDD drive system weight model depends on f , the second (main or tail) rotor rated torque Q as a fraction of the total drive system rated torque; and on f , the second (main or tail) rotor rated power P as a fraction of the total drive system rated power. These parameters are related by the rotational speeds of the two rotors: f = f Ω / Ω . Typically f = f = 0 . 6 for twin rotors (tandem, coaxial, and P Q other main P Q tiltrotor configurations). For the single-main-rotor and tail-rotor configuration, typically f = 0 . 03 and Q f = 0 . 15 ( 0 . 18 with a two-bladed teetering main-rotor).

P 152 Propulsion Group

Chapter 17 Engine Group The engine group consists of one or more engines of a specific type. An engine group transfers power by shaft torque, so it is associated with a propulsion group. For each engine type an engine model is defined. The engine model describes a particular engine, used in one or more engine groups.

The models include turboshaft engines (perhaps convertible, for turbojet operation or reaction drive), reciprocating engines, compressors, electric motors (perhaps with fuel cells), electric generators, and generator-motors.

17–1 Engine Group Performance The engine size is described by the power P , which is the sea-level static power available per eng engine at a specified takeoff rating. The number of engines N is specified for each engine group.

eng If the sizing task determines the engine power for a propulsion group, the power P of at least one eng ∑ engine group is found (including the first engine group). The total power required is P = r N P , P G eng eng ∑ where r = max( P /P ) . The sized power is P = P − N P . Then the sized reqP G avP G sized P G eng eng fixed ∑ engine power is P = f P /N for the n -th engine group (with f = f for the first eng n sized eng 1 n n  =1 , sized group). If an engine group does not consume power (compressor or generator) or does not contribute to shaft power (converted), the size is scaled with r = max( P /P ) .

reqEG avEG The propulsion group power available is obtained from the sum over the engine groups: P = avP G ∑ P .

avEG The propulsion group component power P includes compressor power, generator power re- comp quired, and generator-motor power when it is producing energy.

The flight condition information includes the altitude, temperature, flight speed, and primary rotor speed; a power fraction f ; and the states of the engine, drive system, and infrared suppressor (IRS). The P engine turbine speed is N = (60 / 2 π ) r Ω , where Ω is the current rotor speed and r is the gear eng prim prim eng ratio (depending on the drive system state, including a factor f for a variable speed transmisson).

gear If the reference primary rotor speed Ω corresponds to the specification turbine speed N , then prim spec r = Ω / Ω ; alternatively, the engine gear ratio can be a fixed input.

eng spec prim The drive system limit at the flight condition is rxP , where r = Ω / Ω and x is the rating DS limit prim ref factor. Optionally this limit is applied to the propulsion group power: P = min( P , rxP ) .

avP G avP G DS limit Similarly the engine shaft limit at the flight condition is optionally applied to the engine group power: P = min( P , rxP ) .

avEG avEG ES limit 154 Engine Group 17–2 Turboshaft Engine Turboshaft engine performance is obtained from the Referred Parameter Turboshaft Engine Model (RPTEM).

17-2.1 Power Available Given the flight condition and engine rating, the power available P is calculated from the specific a power SP = P / ˙ m and mass flow ˙ m : a a a a SP a = SP g ( θ, M, n ) 0 sp θ ˙ m a √ = ˙ m g ( θ, M, n ) 0 m δ/ θ P a √ = P g ( θ, M, n ) 0 p δ θ as functions of temperature ratio θ = T /T , Mach number M , and referred engine turbine speed n = √ N/ θ .

In the engine model, installation losses P are subtracted from P ( P = P − P ), and then the loss a av a loss mechanical limit is applied: P = min( P , rP ) , r = N/N . The mechanical limit is properly a av av mech R spec torque limit, Q = P /N , but is expressed as a power limit for clarity.

mech mech spec The engine model gives the performance of a single engine. The power available of the engine group is obtained by multiplying the single engine power by the number of engines operational (total number of engines less inoperable engines): P = f ( N − N ) P avEG P eng inop av including a specified power fraction f .

P 17-2.2 Performance at Power Required The engine performance (mass flow, fuel flow, and gross jet thrust) is calculated for a specified power required P and flight condition: q ˙ m req √ = ˙ m g ( q, θ, M, n ) 0 C m δ/ θ ˙ w req √ = ˙ w g ( q, θ, M, n ) 0 C w δ θ F g = F g ( q, θ, M, n ) g 0 C f δ √ as functions of q = P / ( P δ θ ) , temperature ratio θ = T /T , Mach number M , and referred engine q 0 C 0 √ turbine speed n = N/ ( N θ ) .

spec The engine model deals with a single engine. The power required of a single engine is obtained by dividing the engine group power by the number of engines operational (total number of engines less inoperable engines): P = P / ( N − N ) req reqEG eng inop In the engine model, installation losses P are added to P : P = P + P .

loss req q req loss Engine Group 155 The engine model gives the performance of a single engine. The performance of the engine group is obtained by multiplying the single engine characteristics by the number of engines operational (total number of engines less inoperable engines): ˙ m = ( N − N ) ˙ m reqEG eng inop req ˙ w = ( N − N ) ˙ w K reqEG eng inop req f f d F = ( N − N ) F N EG eng inop N D = ( N − N ) D aux EG eng inop aux The fuel flow has also been multiplied by a factor K accounting for deterioration of the engine f f d efficiency.

17-2.3 Installation The difference between installed and uninstalled power is the inlet and exhaust losses P : P = loss av P − P and P = P − P . The inlet ram recovery efficiency η is included in the engine model a loss req q loss d calculations. The inlet and exhaust losses are modeled as fractions of power available or power required: P = ( + ) P or P = ( + ) P . The installed gross jet thrust is F = K F , where K loss in ex a loss in ex q G f gr g f gr accounts for exhaust effects. The net jet thrust is F = F − ˙ m V . The momentum drag of the N G req auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency. Exhaust losses ( ) and auxiliary air flow parameters ( η , aux ex aux f ) are defined for IR suppressor on and off. Inlet particle separator loss is added to the inlet losses aux ( ).

in 17-2.4 Convertible Engine: Turbojet/Turbofan The engine mode B is the mass flow fraction diverted for a convertible engine: B = 0 for all mass flow to the power turbine (turboshaft operation), and B = 1 for all mass flow to the jet exhaust or a fan (turbojet/turbofan operation).

A separate engine model defines the performance for turbojet/turbofan operation ( B = 1 ). The engine group power P is prescribed, as a measure of the jet thrust, and this engine does not reqEG contribute to the propulsion group shaft power available. The turbojet/turbofan thrust is the engine group net jet thrust, F = F − ˙ m V .

N G req 17-2.5 Convertible Engine: Reaction Drive The engine mode B is the mass flow fraction diverted for a convertible engine: B = 0 for all mass flow to the power turbine (turboshaft operation), and B = 1 for all mass flow to the rotor (reaction jet operation).

A separate engine model defines the engine performance for reaction jet operation ( B = 1 ). The engine group power is fixed by amplitude input, or obtained from the rotor power ( P = P ), reqEG react and this engine does not contribute to the propulsion group shaft power available. The gross jet thrust is zero, so the net thrust is the momentum drag, F = − ˙ m V .

N req 156 Engine Group 17–3 Turboshaft Engine Tabular Model A simple tabular model of the engine performance is implemented, suitable for use with data from a limited exercise of an engine deck. The tables are for power available P , fuel flow ˙ w , and net jet a thrust F as a function of altitude h , flight speed V , and rating R . The altitude influence is for a selected N atmosphere, hence for a temperature variation with altitude. The ratings reflect engine throttle setting, including partial power. Power turbine speed variation is not considered. The tables are used with linear interpolation.

Given the altitude, speed, and rating, the power table T ( h, V, R ) is interpolated. Then the power p available is P = K T . For fixed altitude and speed, the table T ( h, V, R ) implies a variation of power a p p p with rating; and the tables for fuel flow and jet thrust can be interpreted as T ( h, V, P ) and T ( h, V, P ) .

w q f q So given altitude, speed, and power required, the tables are interpolated; then the fuel flow is ˙ w = K T w w and the net jet thrust is F = K T . The input factors K , K and K can account for technology level.

N f f p w f The tables are for an installed engine, including losses. Hence P = 0 ( = = 0 ) with this loss in ex model, and P = P , P = P . Mechanical limits are included in the power available data. Engine av a req q mass flow is not considered, so D = 0 , and the table is for net jet thrust. The fuel flow is multiplied aux by the factor K , accounting for deterioration of the engine efficiency. The engine is not scaled. The f f d engine weight W is fixed.

one eng 17–4 Reciprocating Engine Reciprocating engine performance is described in reference 1. The work per cycle is W = P n /R , c where R is the rotational speed (rev/sec), n is the revolutions per cycle ( n = 2 for a 4-stroke engine, c c n = 1 for a 2-stroke engine), and R/n is the cycles per second. The engine displacement is V (volume).

c c d The mean effective pressure (mep) is defined as W P n Qn 2 π c c mep = = = V V R V d d d from the power P = N Q = 2 πRQ ; so mep is the specific torque. The engine output is the brake horsepower (BHP), which equals indicated power less friction power: BHP = IHP − FHP = IHP − (MHP + PHP + CHP + AHP − THP) The friction power is composed of losses due to mechanical friction (MHP), pumping (PHP, the work of the piston during inlet and exhaust strokes), compressor or supercharger (CHP), auxiliary or accessories (AHP, such as oil pump, water pump, cooling fan, generator), and exhaust turbine (THP, treated as negative friction). The mechanical efficiency is η = BHP / IHP . The sum of airflow ( ˙ m ) and fuel flow ( ˙ w ) is the charge flow: ˙ w = ˙ w + ˙ w . The fuel-air ratio is F = ˙ w / ˙ w . The mass flow is c f a f a ˙ m = e ˙ m = e ρV ( R/n ) ; so mep ∼ ( P/ ˙ m ) ρ . The indicated power is the product of the fuel flow, v ideal v d c fuel specific energy ( e = JQ , from the heat of combustion Q and Joule’s constant J relating work and fuel c c heat), and thermal efficiency: P = e η ˙ w . These equations have constant factors when conventional i fuel th units are used.

2 2 The maximum break mean effective pressure is typically 125–250 lb/in (850–1700 kN/m ). Typ- ical break specific fuel consumption is bsfc = ˙ w/P = 0 . 38 to 0 . 45 lb/hp-hr. Typically the volumetric efficiency e = ˙ m/ ˙ m = 0 . 8 to 0 . 9 .

v ideal Engine Group 157 Reciprocating engine performance is obtained from the Referred Parameter Turboshaft Engine Model (RPTEM).

17-4.1 Power Available The power available P is calculated from the specific power SP = P / ˙ m and mass flow ˙ m : a a a a a SP mep a = SP g ∼ 0 sp θ δ ˙ m N/N a spec ∼ √ = ˙ m g m √ = ˙ 0 m 0 δ/ θ θ Installation losses P are subtracted from P ( P = P − P ), and then the mechanical limit is loss a av a loss applied: P = min( P , rP ) , r = N/N . The mechanical limit is properly a torque limit, av av mech R spec Q = P /N , but is expressed as a power limit for clarity. The power available of the engine mech mech spec group is obtained by multiplying the single engine power by the number of engines operational (total number of engines less inoperable engines): P = f ( N − N ) P , including a specified power avEG P eng inop av fraction f .

P 17-4.2 Performance at Power Required The engine performance for a specified power required P is: q ˙ w req ∼ √ = ˙ w g = ˙ w q 0 C w 0 C δ θ ˙ m N/N req spec ∼ √ = ˙ m g m √ = ˙ 0 C m 0 C δ/ θ θ F g ∼ = F g = 0 g 0 C f δ √ where q = P / ( P δ θ ) . Installation losses P are added to P : P = P + P , P = q 0 C loss req q req loss req P / ( N − N ) . The performance of the engine group is obtained by multiplying the single reqEG eng inop engine characteristics by the number of engines operational (total number of engines less inoperable engines): ˙ m = ( N − N ) ˙ m reqEG eng inop req ˙ w = ( N − N ) ˙ w K reqEG eng inop req f f d F = ( N − N ) F N EG eng inop N D = ( N − N ) D aux EG eng inop aux The fuel flow has also been multiplied by a factor K accounting for deterioration of the engine f f d efficiency.

17–5 Compressor A compressor converts input shaft power to a jet velocity and thrust. The shaft power contributes to the propulsion group power required. The compressor does not use fuel.

17-5.1 Power Available Given the flight condition and engine rating, the power available P is calculated from the specific a power SP = P / ˙ m and mass flow ˙ m : a a a a 158 Engine Group SP a = SP g ( θ, M, n ) 0 sp θ ˙ m a √ = ˙ m g ( θ, M, n ) 0 m δ/ θ P a √ = P g ( θ, M, n ) 0 p δ θ √ as functions of temperature ratio θ = T /T , Mach number M , and referred compressor speed n = N/ θ .

Installation losses P are subtracted from P ( P = P − P ), and then the mechanical limit loss a av a loss is applied: P = min( P , rP ) , r = N/N . The mechanical limit is properly a torque limit, av av mech R spec Q = P /N , but is expressed as a power limit for clarity.

mech mech spec The compressor model gives the performance of a single compressor. The power available of the engine group is obtained by multiplying the single compressor power by the number of engines operational (total number of engines less inoperable engines): P = f ( N − N ) P avEG P eng inop av including a specified power fraction f .

P 17-5.2 Performance at Power Required The compressor performance (mass flow and gross jet thrust) is calculated for a specified power required P and flight condition: q ˙ m req √ = ˙ m g ( q, θ, M, n ) 0 C m δ/ θ ST req √ = ST g ( q, θ, M, n ) 0 C st θ F g = F g ( q, θ, M, n ) g 0 C f δ √ as functions of q = P / ( P δ θ ) , temperature ratio θ = T /T , Mach number M , and referred compressor q 0 C 0 √ speed n = N/ θ ) . The specific thrust gives the gross thrust, F = ( ST ) ˙ m .

g The power required of a single compressor is obtained by dividing the engine group power by the number of engines operational (total number of engines less inoperable engines): P = P / ( N − N ) req reqEG eng inop Accounting for installation losses gives the uninstalled power required P = P + P .

q req loss The compressor model gives the performance of a single compressor. The performance of the engine group is obtained by multiplying the single compressor characteristics by the number of engines operational: ˙ m = ( N − N ) ˙ m reqEG eng inop req F = ( N − N ) F N EG eng inop N D = ( N − N ) D aux EG eng inop aux P = ( N − N ) P K comp eng inop q f f d Engine Group 159 The component power is the product of the uninstalled power required and the number of operational engines, and a factor K accounting for deterioration of the engine efficiency.

f f d 17-5.3 Installation The difference between installed and uninstalled power is the inlet and exhaust losses P : P = loss av P − P and P = P − P . The inlet and exhaust losses are modeled as fractions of power available a loss req q loss or power required: P = ( + ) P or P = ( + ) P . The installed gross jet thrust is loss in ex a loss in ex q F = K F , where K accounts for exhaust effects. The net jet thrust is F = F − ˙ m V . The G f gr g f gr N G req momentum drag of the auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency.

aux 17-5.4 Compressor for Reaction Drive If the compressor supplies the jet force for rotor reaction drive, then the engine group power required is fixed by ampitude input, or obtained from the rotor power ( P = P ). The gross jet thrust is reqEG react zero, so the net thrust is the momentum drag, F = − ˙ m V .

N req 17–6 Electric Motor or Generator A motor converts electrical energy (fuel) to shaft power. A generator converts input shaft power to electrical energy, and the shaft power contributes to the propulsion group power required.

The power available is related to the size P . Given the flight condition and motor rating, the eng uninstalled power available P is calculated, including a torque limit. The motor model gives the av performance of a single engine. The power available of the engine group is obtained by multiplying the single engine power by the number of engines operational: P = f ( N − N ) P avEG P eng inop av including a specified power fraction f .

P From the engine group power required P , the power required of a single engine is reqEG P = P / ( N − N ) req reqEG eng inop 17-6.1 Motor The motor power required determines the energy flow from the fuel tank. The energy flow is calculated for P and a specified flight condition: req ˙ ˙ E = E g ( q, n ) req 0 C e as a function of q = P /P and engine speed n = N/N . The motor model gives the performance req eng spec of a single engine. The performance of the engine group is obtained by multiplying the single engine characteristics by the number of engines operational: ˙ ˙ E = ( N − N ) E K reqEG eng inop req f f d 160 Engine Group The energy flow has also been multiplied by a factor K accounting for deterioration of the engine f f d efficiency.

17-6.2 Generator The generator energy flow to the fuel tank defines the power required. The energy flow is calculated for P and a specified flight condition: req ˙ ˙ E = E g ( q, n ) req 0 C e as a function of q = P /P and engine speed n = N/N . The motor model gives the performance req eng spec of a single engine. The performance of the engine group is obtained by multiplying the single engine characteristics by the number of engines operational: ˙ ˙ E = ( N − N ) E reqEG eng inop req P = ( N − N ) P K comp eng inop req f f d The component power is the product of the power required and the number of operational engines, and a factor K accounting for deterioration of the engine efficiency.

f f d 17-6.3 Generator-Motor The engine mode B is the direction of power flow for a generator-motor: B positive for motor operation, and B negative for generator operation. Separate motor models are used for the two modes.

17-6.4 Motor and Fuel Cell A motor with a fuel cell burns a fuel (typically hydrogen) to produce electrical energy, which is converted to shaft power. The device can also be modelled as separate motor and fuel cell components, with a battery (fuel tank) to transfer the electrical energy.

The engine performance (mass flow and fuel flow) is calculated for P and a specified flight req condition: ˙ m = ˙ m g ( q, n ) = K ˙ w req 0 C m mf req ˙ w = ˙ w g ( q, n ) = sfc P req 0 C w req as a function of q = P /P and engine speed n = N/N . Inlet and exhaust losses are included in req eng spec the specific fuel consumption. The net thrust is the inlet momentum drag, F = − ˙ m V . The motor N req model gives the performance of a single engine. The performance of the engine group is obtained by multiplying the single engine characteristics by the number of engines operational: ˙ m = ( N − N ) ˙ m reqEG eng inop req ˙ w = ( N − N ) ˙ w K reqEG eng inop req f f d F = ( N − N ) F N EG eng inop N D = ( N − N ) D aux EG eng inop aux The fuel flow has also been multiplied by a factor K accounting for deterioration of the engine f f d efficiency. The momentum drag of the auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency.

aux Engine Group 161 17–7 Control and Loads The engine power amplitude A and mode B are control variables. The mode can be mass flow (for convertible engines), or power flow (for generator-motor). In distributing the propulsion group power required to the engine groups, an engine group power can be fixed at P = ( N − N ) A ; or reqEG eng inop fraction A of engine power available, P = ( N − N ) AP ; or fraction A of engine rated power reqEG eng inop av P = ( N − N ) AP .

reqEG eng inop eng The engine orientation is specified by selecting a nominal direction e in body axes (positive or f 0 negative x -, y -, or z -axis; usually thrust forward, hence positive x -axis); then applying a yaw angle ψ ; and then an incidence or tilt angle i . The yaw and incidence angles can be control variables.

The engine group produces a jet thrust F , acting in the direction of the engine (actually, F = N N F − ˙ mV , with the gross thrust in the direction of the engine and the momentum thrust in the wind G direction); a momentum drag D (if the component has mass flow), acting in the wind direction; and aux F a nacelle drag D , acting in the wind direction. The engine group is at location z . The force and nac moment acting on the aircraft in body axes are thus: F F = ( e F + e ˙ mV ) + e D + e D f G d d aux d nac F F ˜ F M = Δ z F F F F where Δ z = z − z , e is the engine thrust direction and e is the drag direction.

f d cg 17–8 Nacelle Drag The engine group includes a nacelle, which contributes to the aircraft drag. The component drag contributions must be consistent. The pylon is the rotor support and the nacelle is the engine support.

The drag model for a tiltrotor aircraft with tilting engines would use the pylon drag (and no nacelle drag), since the pylon is connected to the rotor shaft axes; with non-tilting engines it would use the nacelle drag as well.

The reference area for the nacelle drag coefficient is the nacelle wetted area. The wetted area per engine is input, or calculated either from the engine system (engine, exhaust, and accessories) weight or from the engine system plus drive system weight: ( ) 2 / 3 S = k w/N wet eng 2 2 / 3 2 2 / 3 where w = W or w = W + W /N , and the units of k are ft /lb or m /kg . The reference ES ES gbrs EG area is then S = N S .

nac eng wet 17–9 Weights The component weight consists of engine system, engine section or nacelle group, and air induction group. The engine system consists of engine, exhaust system, and accessories. The engine section or nacelle group consists of engine support, engine cowling, and pylon support. These weights are for the engine group, consisting of N engines. The engine system weight W = W + W + W is used eng ES eng exh acc for the engine nacelle wetted area, the rotor pylon wetted area, and the rotor moving weight. The rotor group includes the rotor support structural weight; this must be consistent with the use of the engine support weight and pylon support weight.

162 Engine Group 17–10 References 1) Taylor, C.F. The Internal-Combustion Engine in Theory and Practice. Volume 1: Thermodynamics, Fluid Flow, Performance. Second edition. Cambridge, MA: MIT Press, 1985.

Chapter 18 Jet Group A jet group produces a force on the aircraft, possibly used for lift, propulsion, or control. A jet model describes a particular jet, used in one or more jet groups. The models include turbojet and turbofan engines (perhaps convertible, for reaction drive), reaction drive, and a simple force. A reaction drive supplies a blade force that provides the rotor power required.

18–1 Jet Group Performance The jet size is described by the thrust T , which is the sea-level static thrust available per jet at a jet specified takeoff rating. The number of jets N is specified for each jet group.

jet If the sizing task determines the jet thrust, the total thrust required is T = rN T , where JG jet jet r = max( T /T ) .

reqJG avJG The flight condition information includes the altitude, temperature, and flight speed; and a thrust fraction f .

T 18–2 Turbojet or Turbofan The thrust of a turbojet is ( ) T = ˙ m (1 + f ) V − V + ( p − p ) A e e atm e where ˙ m is the mass flow; f = ˙ w/ ˙ m is the fuel-air ratio; and V , p , and A are the velocity, pressure, e e e and area at the exit. The pressure term is zero or small, and the fuel-air ratio is small, so the net thrust is approximately T = ˙ m ( V − V ) e from the gross thrust T = ˙ mV and the inlet-momentum or ram drag ˙ mV .

G e The thrust of a turbofan is ( ) ( ) ( T = ˙ m (1 + f ) V − V + ˙ m V − V = ˙ m (1 + f ) V + βV ) − ˙ m (1 + β ) V e fan e fan e e fan with bypass ratio β = ˙ m / ˙ m .

fan Turbojet or turbofan performance is obtained from the Referred Parameter Jet Engine Model (RP- JEM), based on references 1 and 2. The referred thrust, specific thrust ST = T / ˙ m , and specific fuel con- 164 Jet Group √ sumption sfc = ˙ w/T are functions of the flight Mach number M and compressor speed n = N/ ( N θ ) : T = G ( M, n ) t δ T / ˙ m √ = G ( M, n ) st θ ˙ w/T 1 √ = G ( M, n ) sfc η θ b The independent variable can be the compressor speed, or the turbine inlet temperature, or the fuel flow.

The combustion efficiency η depends on the atmosphere (altitude and temperature), hence the specific b fuel consumption is not a function of just M and n .

18-2.1 Thrust Available Given the flight condition and engine rating, the thrust available T is calculated from the specific a thrust ST = T / ˙ m and mass flow ˙ m : a a a a ST a √ = ST g ( θ, M ) 0 st θ ˙ m a √ = ˙ m g ( θ, M ) 0 m δ/ θ T a = T g ( θ, M ) 0 t δ as functions of temperature ratio θ = T /T and Mach number M .

In the jet model, installation losses T are subtracted from T ( T = T − T ), and then the loss a av a loss mechanical limit is applied: T = min( T , T ) .

av av mech R The jet model gives the performance of a single jet. The thrust available of the jet group is obtained by multiplying the single jet thrust by the number of jets operational (total number of jets less inoperable jets): T = f ( N − N ) T avJG T jet inop av including a specified thrust fraction f .

T 18-2.2 Performance at Thrust Required The thrust required of a single jet is obtained by dividing the jet group thrust by the number of jets operational (total number of jets less inoperable jets): T = T / ( N − N ) req reqJG jet inop In the jet model, installation losses T are added to T ( T = T + T ).

loss req q req loss The jet performance (mass flow and fuel flow) is calculated for a specified thrust required T and q flight condition: ˙ m req √ = ˙ m g ( t, θ, M ) 0 C m δ/ θ ˙ w req √ = ˙ w g ( t, θ, M ) 0 C w δ θ Jet Group 165 as functions of t = T / ( T δ ) (or referred gross thrust), temperature ratio θ = T /T , and Mach number q 0 C 0 M .

The jet model gives the performance of a single jet. The performance of the jet group is obtained by multiplying the single jet characteristics by the number of jets operational: ˙ m = ( N − N ) ˙ m reqJG jet inop req ˙ w = ( N − N ) ˙ w K reqJG jet inop req f f d D = ( N − N ) D aux JG jet inop aux The fuel flow has also been multiplied by a factor K accounting for deterioration of the jet efficiency.

f f d 18-2.3 Installation The difference between installed and uninstalled thrust is the inlet and exhaust losses T : T = loss av T − T and T = T − T . The inlet and exhaust losses are modeled as fractions of thrust available a loss req q loss or thrust required: T = ( + ) T or T = ( + ) T . The momentum drag of the auxiliary loss in ex a loss in ex q air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency. Exhaust losses ( ) and auxiliary air flow parameters ( η , aux ex aux f ) are defined for IR suppressor on and off.

aux 18-2.4 Convertible Engine: Reaction Drive The jet mode B is the mass flow fraction diverted for a convertible engine: B = 0 for all mass flow to the exhaust (turbojet operation), and B = 1 for all mass flow to the rotor (reaction jet operation).

A separate jet model defines the engine performance for B = 1 . The jet thrust required is fixed by amplitude input, or obtained from the rotor power ( T = F = P / Ω r ). The mass flow and reqJG react req react fuel flow follow. The jet group net thrust is the inlet momentum drag, F = − ˙ m V .

N req Thrust, specific thrust, and mass flow are defined by the jet model for turbojet operation. The specific fuel consumption of the jet model for reaction jet operation is scaled proportional to the sfc scaling for turbojet operation.

18–3 Reaction Drive Rotor power can be supplied by a reaction drive, using cold or hot air ejected out of the blade tips or trailing edges. Helicopters have also been designed with ram jets on the blade tips, or with jet flaps on the blade trailing edges that use compressed air generated in the fuselage. The jet group performance includes the blade duct and nozzle, perhaps even with tip burning.

The net jet thrust required is fixed by amplitude input, or obtained from the rotor power. The rotor power required P gives the required force on the rotor blade req ( ) F = P / Ω r = ˙ m V − Ω r react req react react react react at effective radial station r . The net jet group thrust required is react T = F = T − ˙ m Ω r reqJG react G reqJG react 166 Jet Group T is the gross thrust. From T , the jet performance (mass flow and fuel flow) is calculated. The jet G reqJG group net thrust is the inlet momentum drag, F = − ˙ m V .

N req 18–4 Simple Force For the simple force model, the design maximum thrust is T (per jet). The thrust available is max thus T = T . The force generation can use fuel as weight or as energy.

a max If the component burns fuel weight, the fuel flow is calculated from an input thrust-specific fuel consumption: ˙ w = T (sfc) = ˙ w q , where q = T /T . Units of sfc are pound/hour/pound or req q 0 C q max kilogram/hour/Newton. Then ˙ w = ( N − N ) ˙ w K reqJG jet inop req f f d The fuel flow has also been multiplied by a factor K accounting for deterioration of the jet efficiency.

f f d If the component uses fuel energy, the energy flow is calculated from an input thrust-specific ˙ ˙ fuel consumption: E = T (sfc) = E q , where q = T /T . Units of sfc are MJ/hour/pound or req q 0 C q max MJ/hour/Newton. Then ˙ ˙ E = ( N − N ) E K reqJG jet inop req f f d The energy flow has also been multiplied by a factor K accounting for deterioration of the jet efficiency.

f f d The simple force weight is calculated from specific weight S plus a fixed increment: W = ST + max Δ W . This weight is identified as either engine system or propeller/fan installation weight, both of the propulsion group; or tail-rotor, of empennage group.

18–5 Control and Loads The jet thrust amplitude A and mode B are control variables. The mode can be mass flow (for convertible engines). The jet group thrust can be fixed at T = ( N − N ) A ; or fraction A of jet reqJG jet inop thrust available, T = ( N − N ) AT ; or fraction A of jet rated thrust T = ( N − N ) AT .

reqJG jet inop av reqJG jet inop jet The jet orientation is specified by selecting a nominal direction e in body axes (positive or negative f 0 x -, y -, or z -axis; usually thrust forward, hence positive x -axis); then applying a yaw angle ψ ; and then an incidence or tilt angle i . The yaw and incidence angles can be control variables.

The jet group produces a jet thrust T , acting in the direction of the engine (actually, the gross thrust in the direction of the engine and the momentum thrust in the wind direction); a momentum drag D , aux acting in the wind direction; and a nacelle drag D , acting in the wind direction. The jet group is at nac F location z . The force and moment acting on the aircraft in body axes are thus: F F = ( e T + e T ) + e D + e D f G d mom d aux d nac F F ˜ F M = Δ z F F F F where Δ z = z − z , e is the jet thrust direction and e is the drag direction.

f d cg 18–6 Nacelle Drag The jet group includes a nacelle, which contributes to the aircraft drag. The reference area for the nacelle drag coefficient is the nacelle wetted area. The wetted area per jet is input, or calculated from Jet Group 167 the engine system (engine, exhaust, and accessories) weight: ( ) 2 / 3 S = k w/N wet jet 2 2 / 3 2 2 / 3 where w = W , and the units of k are ft /lb or m /kg . The reference area is then S = N S .

ES nac jet wet 18–7 Weights The component weight consists of engine system, engine section or nacelle group, and air induction group. The engine system consists of engine, exhaust system, and accessories. The engine section or nacelle group consists of engine support, engine cowling, and pylon support. These weights are for the jet group, consisting of N engines. The engine system weight W = W + W + W is used for jet ES eng exh acc the engine nacelle wetted area.

18–8 References 1) Sanders, N.D. “Performance Parameters for Jet-Propulsion Engines.” NACA TN 1106, July 1946.

2) Hill, P.G., and Peterson, C.R. Mechanics and Thermodynamics of Propulsion. Reading, MA: Addison-Wesley Publishing Company, Inc., 1965.

168 Jet Group

Chapter 19 Charge Group A charge group generates energy for the aircraft. A charger model describes a particular charger, used in one or more charge groups. The models include fuel cells and solar cells.

19–1 Charge Group Performance The charger size is described by the power P , which is the sea-level static power available per chrg charger at a specified takeoff rating. The number of chargers N is specified for each charge group.

chrg If the sizing task determines the charger power, the charger power required is rP , where r = chrg max( P /P ) .

req total av total The flight condition information includes the altitude, temperature, and flight speed; and a power fraction f .

P ˙ Given the flight condition and the charger rating, the cell power available is E = P . The total a cell 0 cell power available is obtained by multiplying the single charger power by the number of chargers operational (total number of chargers less inoperable chargers): ˙ P = f ( N − N ) E av total P chrg inop a cell including a specified power fraction f .

P ˙ The energy flow to the fuel tank gives the charge group power required: P = E . The reqCG reqCG energy flow required of a single charger is ˙ ˙ E = E K / ( N − N ) req reqCG f f d chrg inop The energy flow has also been multiplied by a factor K accounting for deterioration of the charger f f d ˙ efficiency. The cell energy flow required E is obtained from the energy flow. Then q cell ˙ P = ( N − N ) E req total chrg inop q cell is the total cell power required.

19–2 Fuel Cell A fuel cell burns a fuel (typically hydrogen) and generates electrical energy. The cell energy flow ˙ is calculated for E and a specified flight condition: req ˙ ˙ E = E g ( q ) q cell 0 C e 170 Charge Group ˙ ˙ where q = E /P . The fuel cell performance (mass flow and fuel flow) is obtained from E , or req 0 q cell ˙ m = ˙ m g ( q ) req 0 C m ˙ w = ˙ w g ( q ) req 0 C w Installation losses are included in the specific fuel consumption. The net thrust is the inlet momentum drag, F = − ˙ m V . The charger model gives the performance of a single charger. The performance of N req the charge group is obtained by multiplying the single charger characteristic by the number of chargers operational: ˙ m = ( N − N ) ˙ m reqCG chrg inop req ˙ w = ( N − N ) ˙ w reqCG chrg inop req F = ( N − N ) F N CG chrg inop N D = ( N − N ) D aux CG chrg inop aux The momentum drag of the auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency.

aux 19–3 Solar Cell The power available from solar radiation is approximately 1.36 kW/m , reduced by atmospheric effects (absorption, reflection, and scattering) to about 1.00 kW/m . The average solar radiation in the continental United States is 3.5–7.0 (kW/m )(hour/day), hence approximately 15 to 25% of the available power. Typical efficiencies of solar cells are 10–35%, with some sensitivity to temperature. The solar 2 2 cell is characterized by power density (W/m ) and weight density (kg/m ).

˙ The cell energy flow is calculated for E and a specified flight condition: req ˙ ˙ E = E g ( q ) q cell 0 C e ˙ where q = E /P .

req 0 19–4 Control and Loads The charger power amplitude A and mode B are control variables. The charge group power can ˙ ˙ be fixed at E = P = ( N − N ) A ; or fraction A of charger rated power E = P = reqCG chrg inop reqCG ( N − N ) AP .

chrg inop chrg The charger orientation is specified by selecting a nominal direction e in body axes (positive or f 0 negative x -, y -, or z -axis); then applying a yaw angle ψ ; and then an incidence or tilt angle i . The yaw and incidence angles can be control variables.

The charge group produces a jet thrust F (actually, F = F − ˙ mV , with the gross thrust in the N N G direction of the engine and the momentum thrust in the wind direction), acting in the direction of the charger; a momentum drag D (if the component has mass flow), acting in the wind direction; and aux F a nacelle drag D , acting in the wind direction. The charge group is at location z . The force and nac moment acting on the aircraft in body axes are thus: F F = ( e F + e ˙ mV ) + e D + e D f G d d aux d nac F F ˜ F M = Δ z F Charge Group 171 F F F where Δ z = z − z , e is the charger thrust direction and e is the drag direction.

f d cg 19–5 Nacelle Drag The charge group includes a nacelle, which contributes to the aircraft drag. The reference area for the nacelle drag coefficient is the nacelle wetted area. The wetted area per charger is input, or calculated from the engine system (engine, exhaust, and accessories) weight: ( ) 2 / 3 S = k w/N wet chrg 2 2 / 3 2 2 / 3 where w = W , and the units of k are ft /lb or m /kg . The reference area is then S = N S .

ES nac chrg wet 19–6 Weights The component weight consists of the engine system. The engine system consists of the charger weight. These weights are for the charge group, consisting of N chargers. The engine system weight chrg W = N W is used for the charger nacelle wetted area.

ES chrg one chrg 172 Charge Group

Chapter 20 Referred Parameter Turboshaft Engine Model Aircraft gas turbine engine performance capabilities are formally specified by computer programs known as engine decks, which are created by engine manufacturers in an industry-standard format.

Engine decks are typically based on thermodynamic cycle analysis using real engine component per- formance maps. The most important performance maps for turboshaft engines are compressor, gas generator turbine, and power turbine. These component performance maps are critical to obtaining realistic off-design engine performance. Design and analysis codes calculate aircraft performance for a very wide range of operating conditions. Thus engine performance must be realistic even far from the engine design point. A simple thermodynamic cycle analysis that assumes design point component efficiencies everywhere is not realistic for such an application. Rather than developing models for com- ponent performance, the approach taken is to use a model for the total engine performance. The engine is not being designed.

The Referred Parameter Turboshaft Engine Model (RPTEM) is based on curve-fits of performance data for existing or projected engines over a range of operating conditions. The curve-fits are typically obtained by exercising an engine deck. The use of referred parameters tends to collapse the data, and provides a basis for scaling the engine. The operating condition is described by pressure altitude, ambient air temperature, flight Mach number, power turbine speed, exhaust nozzle area, and either engine rating or engine power required. These curve-fits, typically based on real engines, are scaled to the required size and adjusted to the appropriate technology level to represent a notional engine. Engine size is represented by mass flow. Engine technology is represented by specific power available and specific fuel consumption at maximum continuous power (MCP), sea level/standard day (SLS), static (zero airspeed) conditions. Engine installation effects (inlet and exhaust losses) are also modeled.

The use of referred parameters to curve-fit engine performance data was suggested by David Woodley from Boeing during the JVX program (1983). The RPTEM was developed and documented by Michael P. Scully and Henry Lee of ASRAO, U.S. Army Aeroflightdynamics Directorate (AFDD), with a subsequent implementation written by Sam Ferguson (1995).

20–1 Operating Environment The operating condition and atmosphere give the standard conditions (temperature T and pressure std p ) for a specified pressure altitude; the sea-level standard conditions (temperature T and pressure std 0 ◦ ◦ p ); and the operating temperature T and pressure p . Here the temperatures are R or K. The engine characteristics depend on the temperature ratio θ = T /T and pressure ratio δ = p/p .

0 0 √ The flight Mach number M = V /c = V /c θ is obtained from the aircraft speed V .

s s 0 174 Referred Parameter Turboshaft Engine Model The inlet ram air temperature ratio and pressure ratio are obtained then from M and the inlet ram recovery efficiency η : d ( ) ( ) γ − 1 2 2 θ = 1 + M = 1 + 0 . 2 M M γ ( ) γ − 1 ( ) γ − 1 3 . 5 2 2 δ = 1 + η M = 1 + 0 . 2 η M M d d where the ratio of specific heats γ = 1 . 4 .

20–2 Performance Characteristics The engine performance is described by: the power available P , at each engine rating and the a specification engine turbine speed N ; the mass flow ˙ m and fuel flow ˙ w required to produce power spec required P at engine turbine speed N ; and the gross jet thrust F at a given power required P . Then the q q specific power is SP = P/ ˙ m , and the specific fuel consumption is sfc = ˙ w/P .

The reference performance is at sea-level-standard static conditions (subscript 0 ), and MCP (sub- script C ). For each rating R, the performance is characterized by the following quantities for sea-level- standard static conditions: power P , specific power SP , and mechanical power limit P . The 0 R 0 R mech R mass flow is then ˙ m = P /SP . The gross jet thrust F is given at MCP. These characteristics 0 R 0 R 0 R g 0 C are at the specification turbine speed N .

spec The installed power required P and power available P > P are measured at the engine output req av req shaft. In addition to shaft power, the engine exhaust produces a net jet thrust F , from mass flow that N goes through the engine core. The fuel flow and mass flow are the total required to produce the shaft power and jet thrust. The forces produced by mass flow that does not go through the engine core (such as infrared suppressor or cooling air) are treated as momentum drag D .

aux The difference between net and gross jet thrust is the momentum drag: F = F − ˙ m V = n g req ˙ m ( V − V ) , where V is the engine jet exhaust velocity. Note that traditional units for mass flow are req j j pound/sec (pps), while this equation requires slug/sec ( ˙ m /g replaces ˙ m ).

req req The uninstalled power required is P , the power available P , the gross jet thrust F , and net jet thrust q a g F . The engine model calculates P as a function of flight condition and engine rating; or calculates n a engine mass flow, fuel flow, and jet thrust at P .

q 20–3 Installation The difference between installed and uninstalled power is the inlet and exhaust losses P : loss P = P − P av a loss P = P − P req q loss The inlet ram recovery efficiency η (through δ ) is included in the engine model calculations. The inlet d M and exhaust losses are modeled as fractions of power available or power required: P = ( + ) P loss in ex a or P = ( + ) P . So loss in ex q P = P (1 − − ) av a in ex P = P / (1 − − ) q req in ex Referred Parameter Turboshaft Engine Model 175 The engine model gives uninstalled power and the gross thrust F for a nominal exhaust nozzle area. The g gross jet thrust F and exhaust power loss ( ) are both functions of the exhaust nozzle area. Smaller G ex exhaust nozzle areas increase exhaust losses and increase gross thrust. Thus the ratio of installed to uninstalled thrust is approximated by a function of the exhaust power loss: 2 3 F /F = K = K + K + K + K G g f gr f gr 0 f gr 1 ex f gr 2 f gr 3 ex ex so the net installed jet thrust is F = K F − ˙ m V N f gr g req The momentum drag of the auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency. Exhaust losses ( ) and auxiliary air flow parameters ( η , aux ex aux f ) are defined for IR suppressor on and off. Inlet particle separator loss is added to the inlet losses aux ( ).

in 20–4 Power Turbine Speed The shaft power available is a function of the gas power available P and the power turbine efficiency G η : P = η P . Generally the power turbine speed N has a significant effect on η , but almost no effect t a t G t X ∼ N η on P . The model used for the efficiency variation is η = 1 − | ( N/N ) − 1 | , where N is the G t opt opt speed for peak efficiency, hence X N η P ( N ) η ( N ) 1 − | ( N/N ) − 1 | t opt = = X N η P ( N ) η ( N ) 1 − | ( N /N ) − 1 | spec t spec spec opt Two approximations for the optimum turbine speed are used. The first is a linear expression in p = √ P/ ( P δ θ ) : 0 R ( ) √ √ K N opt B N = N θ K θ + p opt spec N opt A M δ M √ and the second is a cubic function of p = P/ ( P δ θ ) : 0 C √ ( ) X 2 3 N opt N = N θ K + K p + K p + K p [ θ ] opt opt0 C N opt0 N opt1 N opt2 N opt3 M The second expression is based on a larger data sample. For power available calculations, P = P ( N ) ; a spec for power required calculations, P = P ( N ) .

q 20–5 Power Available Given the flight condition and engine rating, the power available P is calculated as follows. The a specific power and referred mass flow (at N , relative to SP and ˙ m for this rating) are approximated spec 0 0 by functions of the ambient temperature ratio θ and inlet ram air ratios: [ ] √ X spa SP ( N ) = SP θ K δ θ a spec 0 spa M M ( ) [ ] √ √ X mf a K mf a ˙ m ( N ) = ˙ m δ/ θ e δ θ a spec 0 M M 176 Referred Parameter Turboshaft Engine Model where the static lapse rate ( K , K ) and ram air exponents ( X , X ) are piecewise linear spa mf a spa mf a functions of θ . The power available is then ( ) [ ] √ √ X + X spa mf a SP ( N ) ˙ m ( N ) a spec a spec K mf a P ( N ) = P = P δ θ K e δ θ a spec 0 0 spa M M SP ˙ m 0 0 This expression for ˙ m is used only to calculate P ; elsewhere the ˙ m expression below (for performance a a q at power required) is used to obtain the mass flow at a power P . Finally q X N η 1 − | ( N/N ) − 1 | opt P ( N ) = P ( N ) a a spec X N η 1 − | ( N /N ) − 1 | spec opt is the power available at turbine speed N . Installation losses P are subtracted from P ( P = loss a av P − P ), and then the mechanical limit is applied: P = min( P , rP ) , r = N/N . The a loss av av mech R spec mechanical limit is properly a torque limit, Q = P /N , but is expressed as a power limit for mech mech spec clarity.

20-5.1 Piecewise Linear Parameters Several parameters K are input as piecewise linear functions of the temperature ratio θ . One format of the input is a set of I regions, with the function in the i -th region given by K = K + K θ . The break 0 i 1 i point between the i and i − 1 regions is K − K 0 i 0( i − 1) θ = − bi K − K 1 i 1( i − 1) K = K + K θ bi 0 i 1 i b for i = 2 to I . Another format is a set of I + 1 break points, with the values at the i -th point ( θ , K ).

bi bi Then the coefficients between i and i + 1 are K θ − K θ bi bi b ( i +1) b ( i +1) K = 0 i θ − θ b ( i +1) bi − K + K bi b ( i +1) K = 1 i θ − θ b ( i +1) bi for i = 1 to I . The interpolation is performed using K = K + K θ for θ in the range θ to θ .

0 i 1 i bi b ( i +1) Outside the defined regions, the linear expression is continued; hence θ is used only for i = 2 to I .

bi 20–6 Performance at Power Required The engine performance (mass flow, fuel flow, and gross jet thrust) is calculated for a specified power required P (which might equal the power available), flight condition, and engine rating. Installation q losses P are added to P ( P = P + P ). The referred quantities (relative to SLS static MCP loss req q req loss √ quantities) are approximated by cubic functions of q = P ( N ) / ( P δ θ ) : q spec 0 C ( ) √ ( ) − X 2 3 f f q ˙ w = ˙ w δ θ K + K q + K q + K q [ θ ] req 0 C f f q 0 f f q 1 f f q 2 f f q 3 M ( ) √ ( ) X 2 3 mf q ˙ m = ˙ m δ/ θ K + K q + K q + K q [ θ ] req 0 C mf q 0 mf q 1 mf q 2 mf q 3 M ( ) X 2 3 f gq F = F ( δ ) K + K q + K q + K q [ θ ] g g 0 C f gq 0 f gq 1 f gq 2 f gq 3 M Referred Parameter Turboshaft Engine Model 177 at N , with ˙ w = sfc P . The mass flow and fuel flow are primarily functions of the gas power P , spec 0 C 0 C 0 C G and are assumed to be independent of η , hence independent of turbine speed. However, these equations t are functions of P ( N ) , obtained from P ( N ) using q spec q X N η 1 − | ( N /N ) − 1 | spec opt P ( N ) = P ( N ) q spec q X N η 1 − | ( N/N ) − 1 | opt Then the installed net jet thrust F and momentum drag D are calculated.

N aux 20–7 Scaling The parameters of the engine model can be defined for a specific engine, but it is also necessary to scale the parameters as part of the aircraft sizing task, in order to define an engine for a specified power.

In addition, advanced technology must be represented in the model. Scaling and advanced technology are handled in terms of specific power and specific fuel consumption (at SLS static conditions, MCP, and N ). Figures 20-1 through 20-3 present historical data for engine specific fuel consumption, weight, spec and specific power; the variation at a given power reflects technology insertion with time.

The engine model includes reference values of the engine performance parameters: P , SP , 0 R 0 R P , sfc , SF , N , and N . Mass flow and fuel flow are obtained from ˙ m = P /SP and mech R 0 C 0 C spec opt0 C 0 R 0 R 0 R ˙ w = sfc P . The reference power at each engine rating R defines a ratio to MCP: r = P /P .

0 C 0 C 0 C p 0 R 0 R 0 C Similarly for specific power and mechanical limits: r = SP /SP and r = P /P . These s 0 R 0 R 0 C m 0 R mech R 0 C ratios are kept fixed when the engine is scaled.

The engine size is specified as takeoff power P = P , which is the power at rating R, for SLS to eng static conditions and specification turbine speed N . Hence the MCP is P = P /r , and the spec 0 C to p 0 R power at all other ratings follows. If P is not equal to the reference value of the engine model, then 0 C the engine is scaled. To reflect advanced technology, the specific power, specific fuel consumption, and specification turbine speed can be specified: SP = SP , sfc = sfc , and N = N 0 C tech 0 C tech spec tech (replacing the engine model reference values). The default values are the reference values of the engine model. In the following paragraph, the subscript “ tech ” refers to these quantities; the subscript “ ref ” means the engine model reference values.

The engine technology parameters SP and sfc are assumed to vary linearly with mass flow ˙ m 0 C 0 C 0 C up to a limit ˙ m , and constant thereafter at SP and sfc . The mass flow at the technology condition lim lim lim is ˙ m = P /SP , with the technology values SP and sfc . The intercept values are projected tech ref tech tech tech from the technology values: K = SP − K ˙ m , K = ( SP − SP ) / ( ˙ m − ˙ m ) ; and sp 0 tech sp 1 tech sp 1 lim tech lim tech similarly for sfc . Then for ˙ m < ˙ m 0 C lim SP = K + K ˙ m 0 C sp 0 sp 1 0 C sfc = K + K ˙ m 0 C sf c 0 sf c 1 0 C and for ˙ m ≥ ˙ m 0 C lim SP = SP 0 C lim sfc = sfc 0 C lim These equations are used if ˙ m > ˙ m . Otherwise the model sets K = K = 0 , so there is no lim tech sp 1 sf c 1 variation with scale: SP = SP and sfc = sfc . Usually the effect of size gives K ≥ 0 and 0 C tech 0 C tech sp 2 K ≤ 0 . The power at the limit is P = SP ˙ m . Figure 20-4 illustrates the scaling of SP , sfc , sf c 2 lim lim lim and SW .

178 Referred Parameter Turboshaft Engine Model Using ˙ m = P /SP , the specific power equation can be solved for the mass flow given the 0 C 0 C 0 C power: ⎧ P /K K = 0 0 C sp 0 sp 1 ⎪ ⎪ ⎪ ⎨ P /SP P ≥ P 0 C lim 0 C lim ˙ m = 0 C ⎪ √ ⎪ ⎪ K P K ⎩ 0 2 0 C 0 ( ) + − otherwise 2 K K 2 K 1 1 1 From this mass flow, SP and sfc are calculated, hence the fuel flow ˙ w = sfc P . The specific 0 C 0 C 0 C 0 C 0 C thrust available at MCP is assumed to be constant, and the specification power turbine speed decreases with the mass flow: F = SF ˙ m g 0 C 0 C 0 C ( ) √ √ √ N = N − K / ˙ m + K / ˙ m = K + K / ˙ m spec spec N s 2 0 C N s 2 0 C N s 1 N s 2 0 C tech ( N ) opt0 C ref N = N = K N opt0 C spec N o spec N tech Then the power and specific power at all ratings R are obtained from the ratios: P = r P , 0 R p 0 R 0 C SP = r SP , and P = r P .

0 R s 0 R 0 C mech R m 0 R 0 C The actual (perhaps scaled) values of the performance parameters are available for the engine group: P , SP , P , sfc , F , N , and N .

0 R 0 R mech R 0 C g 0 C spec opt0 C 20–8 Engine Speed The model as described in the previous sections may not adequately account for variation of engine performance with engine speed, so it is also possible to define the parameters corresponding to a set of engine speed ratios r = N/N . Then the engine performance and power available quantities are spec linearly interpolated to obtain the values at the required engine speed N . If this option is used, then the correction based on P ( N ) /P ( N ) = η ( N ) /η ( N ) is not applied.

spec t t spec 20–9 Weight The engine weight can be a fixed input value, calculated as a function of power, or scaled with engine mass flow. As a function of power, the weight of one engine is: X eng W = K + K P + K P one eng 0eng 1eng 2eng where P is the installed takeoff power (SLS static, specified rating) per engine. A constant weight per power W/P is given by using only K . Alternatively, the specific weight SW = P/W can be scaled 1eng with the mass flow ˙ m . The scaling is determined from the specific weight SW at the mass flow 0 C ref ˙ m , and the limit SW at ˙ m . Then tech lim lim { K + K ˙ m ˙ m < ˙ m sw 0 sw 1 0 C 0 C lim SW = SW ˙ m ≥ ˙ m lim 0 C lim (if ˙ m < ˙ m , then K = 0 , so SW = SW ) and W = P/SW .

lim tech sw 1 ref one eng 20–10 Reciprocating Engine Reciprocating engine performance is obtained from the Referred Parameter Turboshaft Engine Model. For the power available, ( ) √ N/N spec √ ˙ m ( N ) = ˙ m δ/ θ a 0 θ Referred Parameter Turboshaft Engine Model 179 assuming constant volumetric efficiency; with no other variation of P with engine speed. For the a performance at power required, ( ) √ ˙ w = ˙ w δ θ q req 0 C ( ) √ N/N spec ˙ m = ˙ m δ/ θ √ req 0 C θ ∼ F = 0 g √ with q = P / ( P δ θ ) , ˙ w = sfc P , and no other variation with engine speed. Typical weight q 0 C 0 C 0 C 0 C X eng models are W = K P , or SW ∼ mep (so W ∼ ˙ m ).

one eng 2eng one eng 20–11 Units In this engine model, only the reference values and scaling constants are dimensional. Conventional English units and SI units are shown in table 20-1. Units of specific power and specific fuel consumption follow from these conventions.

Table 20-1. Conventional units.

power P mass flow ˙ m fuel flow ˙ w force F turbine speed N English: horsepower pound/sec pound/hour pound rpm SI: kiloWatt kilogram/sec kilogram/hour Newton rpm 20–12 Typical Parameters Typical values of the principal parameters describing the engine and its performance are given in table 20-2 for several generic engine sizes. These values represent good current technology. Advanced technology can be introduced by reducing the specific fuel consumption and weight, and increasing the specific power. Typical ratios of the power, specific power, and mass flow to the values at MCP are given in table 20-3 for several ratings. Figure 20-5 shows typical performance characteristics: fuel flow ˙ w , mass flow ˙ m , and net jet thrust F variation with power P and speed (referred quantities, normalized, g at N ). Figure 20-6 shows typical variation of the power available with engine turbine speed. Figures spec 20-7 to 20-12 show typical power available characteristics: specific power SP , mass flow ˙ m , and power P variation with temperature ratio θ , for static and 200 knots conditions and several engine ratings (referred quantities, normalized, at N ).

spec 180 Referred Parameter Turboshaft Engine Model 0.8 0.7 0.6 0.5 0.4 specific fuel consumption (lb/hp-hr) 0.3 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp) Figure 20-1. Historical data for turboshaft engine specific fuel consumption.

Table 20-2. Typical engine performance parameters.

power (MCP) P 500 1000 2000 4000 8000 16000 hp 0 C specific power SP 116 125 134 143 153 164 hp/lb/sec 0 C mechanical limit P 750 1500 3000 6000 12000 24000 hp mech specific fuel cons. sfc 0.54 0.48 0.44 0.40 0.38 0.35 lb/hp-hr 0 C specific jet thrust SF 5.9 7.3 8.9 11.0 13.6 16.7 lb/lb/sec 0 C gross jet thrust F 25 58 134 308 707 1625 lb g 0 C mass flow ˙ m 4.3 8.0 15.0 27.9 52.1 97.3 lb/sec 0 C turbine speed N 35600 26100 19100 14000 10200 7500 rpm spec weight W/P 0.34 0.23 0.18 0.16 0.16 0.15 lb/hp to Table 20-3. Typical parameter ratios for various ratings (percent).

IRP MRP CRP P mech power P /P 120 127 133 150 0 0 C specific power SP /SP 117 123 128 0 0 C mass flow ˙ m / ˙ m 102.6 103.2 103.9 0 0 C Referred Parameter Turboshaft Engine Model 181 0.7 0.6 0.5 0.4 0.3 weight (lb/hp) 0.2 0.1 0.0 0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp) Figure 20-2. Historical data for turboshaft engine weight.

250.

200.

150.

100.

SP (hp/lb/sec) 50.

0.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000.

takeoff power (hp) Figure 20-3. Historical data for turboshaft engine specific power.

182 Referred Parameter Turboshaft Engine Model . . . .

m m m m ≤ > lim tech lim tech lim lim SP SP 0C 0C tech tech tech lim lim tech . .

m m 0C 0C sfc sfc tech tech 0C 0C lim lim tech lim lim tech . .

m m 0C 0C lim lim SW SW ref ref tech lim lim tech . .

m m 0C 0C Figure 20-4. Turboshaft engine scaling; SP = P/ ˙ m , sfc = ˙ w/P , and SW = P/W .

Referred Parameter Turboshaft Engine Model 183 2.0 1.5 1.0 · · · · w / w static w / w 200 knots 0C 0C · · · · 0.5 m / m static m / m 200 knots 0C 0C F /F static F /F 200 knots g g0C g g0C 0.0 0.0 0.5 1.0 1.5 2.0 P(N )/P spec 0C Figure 20-5. Fuel flow, mass flow, and net jet thrust variation with power.

1.2 1.1 1.0 ) spec 0.9 P(N)/P(N 0.8 0.7 0.6 0.4 0.6 0.8 1.0 1.2 N/N spec Figure 20-6. Power variation with turbine speed.

184 Referred Parameter Turboshaft Engine Model 1.8 MPC IRP 1.6 MRP CRP 1.4 1.3 1.1 SP/SP 0.9 0.8 0.6 0.4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-7. Specific power variation with temperature ratio, static.

1.8 MPC IRP 1.6 MRP CRP 1.4 1.2 SP/SP 1.0 0.8 0.6 0.4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-8. Specific power variation with temperature ratio, 200 knots.

Referred Parameter Turboshaft Engine Model 185 1.20 MPC IRP 1.15 MRP CRP 1.10 · / m 1.05 · m 1.00 0.95 0.90 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-9. Mass flow variation with temperature ratio, static.

1.20 MPC IRP 1.15 MRP CRP 1.10 · / m 1.05 · m 1.00 0.95 0.90 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-10. Mass flow variation with temperature ratio, 200 knots.

186 Referred Parameter Turboshaft Engine Model 1.8 MPC IRP 1.6 MRP CRP 1.4 1.2 P/P 1.0 0.8 0.6 0.4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-11. Power variation with temperature ratio, static.

1.8 MPC IRP 1.6 MRP CRP 1.4 1.2 P/P 1.0 0.8 0.6 0.4 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 temperature ratio θ Figure 20-12. Power variation with temperature ratio, 200 knots.

Chapter 21 Compressor Model A compressor converts input shaft power to a jet velocity and thrust. The shaft power contributes to the propulsion group power required. The compressor does not use fuel.

The operating condition is described by pressure altitude, ambient air temperature, flight Mach number, and either compressor rating or power required. The parametric model is scaled to the required size and adjusted to the appropriate technology level to represent a notional compressor. Compressor size is represented by mass flow. Technology is represented by specific power available at maximum continuous power (MCP), sea level/standard day (SLS), static (zero airspeed) conditions. Installation effects are also modeled.

21–1 Operating Environment The operating condition and atmosphere give the standard conditions (temperature T and pressure std p ) for a specified pressure altitude; the sea-level standard conditions (temperature T and pressure p ); std 0 0 ◦ ◦ and the operating temperature T and pressure p . Here the temperatures are R or K. The characteristics depend on the temperature ratio θ = T /T and pressure ratio δ = p/p .

0 0 √ The flight Mach number M = V /c = V /c θ is obtained from the aircraft speed V .

s s 0 The inlet ram air temperature ratio and pressure ratio are obtained then from M and the inlet ram recovery efficiency η : d ( ) ( ) γ − 1 2 2 θ = 1 + M = 1 + 0 . 2 M M γ ( ) γ − 1 ( ) γ − 1 3 . 5 2 2 δ = 1 + η M = 1 + 0 . 2 η M M d d where the ratio of specific heats γ = 1 . 4 .

21–2 Performance Characteristics The uninstalled power required is P , the power available P , the gross jet thrust F , and net jet q a G thrust F . The compressor model calculates P as a function of flight condition and rating; or calculates N a jet thrust and mass flow at P . The specific power is SP = P/ ˙ m , the specific thrust is ST = F / ˙ m . The q G forces produced by mass flow that does not go through the core are treated as momentum drag D .

aux The reference performance is at sea-level-standard static conditions (subscript 0 ), and MCP (sub- script C ). For each rating R, the performance is characterized by the following quantities for sea-level- standard static conditions: Power P , specific power SP , and mechanical power limit P . The 0 R 0 R mech R mass flow is then ˙ m = P /SP .

0 R 0 R 0 R 188 Compressor Model The difference between net and gross jet thrust is the momentum drag: F = F − ˙ m V . Note N G req that traditional units for mass flow are pound/sec (pps), while this equation requires slug/sec ( ˙ m /g req replaces ˙ m ).

req 21–3 Installation The difference between installed and uninstalled power is the inlet and exhaust losses P : loss P = P − P av a loss P = P − P req q loss The inlet and exhaust losses are modeled as fractions of power available or power required: P = loss ( + ) P or P = ( + ) P . So in ex a loss in ex q P = P (1 − − ) av a in ex P = P / (1 − − ) q req in ex The momentum drag of the auxiliary air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency.

aux 21–4 Power Available Given the flight condition and engine rating, the power available P is calculated as follows. The a specific power and referred mass flow (relative to SP and ˙ m for this rating) are approximated by 0 0 functions of the ambient temperature ratio θ , here just: [ ] √ X spa SP = SP θ δ θ a 0 M M ( ) [ ] √ √ X mf a ˙ m = ˙ m δ/ θ δ θ a 0 M M The power available is then ( ) [ ] √ √ X + X spa mf a P = SP ˙ m = P δ θ δ θ a a a 0 M M This expression for ˙ m is used only to calculate P ; elsewhere the ˙ m expression below (for performance a a q at power required) is used to obtain the mass flow at a power P . The influence of compressor rotational q speed is not considered.

21–5 Performance at Power Required The compressor performance (mass flow and gross jet thrust) is calculated for a specified power required P , flight condition, and rating. Installation losses P are added to P ( P = P + P ).

q loss req q req loss The referred quantities (relative to SLS static MCP quantities) are approximated by functions of q = √ P / ( P δ θ ) : q 0 C ( ) √ ( ) X 2 3 mf q ˙ m = ˙ m δ/ θ K + K q + K q + K q [ θ ] req 0 C mf q 0 mf q 1 mf q 2 mf q 3 M √ X stq ST = ST θ [ θ ] req 0 C M Compressor Model 189 The gross jet thrust is then ( ) X + X 2 3 stq mf q F = ST ˙ m = F δ K + K q + K q + K q [ θ ] G req req g 0 C mf q 0 mf q 1 mf q 2 mf q 3 M Then the installed net jet thrust F and momentum drag D are calculated. The influence of compressor N aux rotational speed is not considered.

21–6 Scaling The parameters of the compressor model can be defined for a specific compressor, but it is also necessary to scale the parameters as part of the aircraft sizing task, in order to define a compressor for a specified power. In addition, advanced technology must be represented in the model. Scaling and advanced technology are handled in terms of specific power and specific thrust (at SLS static conditions and MCP).

The compressor model includes reference values of the performance parameters: P , SP , and 0 R 0 R P . Mass flow is obtained from ˙ m = P /SP . The reference power at each rating R defines a mech R 0 R 0 R 0 R ratio to MCP: r = P /P . Similarly for specific power and mechanical limits: r = SP /SP p 0 R 0 R 0 C s 0 R 0 R 0 C and r = P /P . These ratios are kept fixed when the compressor is scaled.

m 0 R mech R 0 C The compressor size is specified as power P , which is the power at takeoff rating R, for SLS eng static conditions. Hence the MCP power is P = P /r , and the power at all other ratings follows.

0 C to p 0 R If P is not equal to the reference value of the compressor model, then the compressor is scaled.

0 C 21–7 Weight The compressor weight can be a fixed input value, or calculated as a function of power. As a function of power, the weight of one compressor is: X comp W = K + K P + K P one eng 0comp 1comp 2comp where P is the installed power (SLS static, specified rating) per compressor. A constant weight per power W/P is given by using only K .

1comp 190 Compressor Model

Chapter 22 Motor Model A motor converts electrical energy (fuel) to shaft power. A generator converts input shaft power to electrical energy, and the shaft power contributes to the propulsion group power required. The model follows references 1 to 4.

22–1 Motor Characteristics The motor or generator size is defined by the maximum power available, P = P , and a peak max eng torque Q . For clarity, the torque limit is expressed as a power limit Q = P /N . The ratio peak peak peak spec of maximum power and peak torque is the base rotational speed: P /Q = N ; at N , the max peak base base power and torque limits coincide. Figure 22-1 shows the maximum power and peak torque for a number of motors. The base rotational speed is typically N = 700–7000 rpm, lower for high-torque motors.

base The ratio of maximum power to continuous power is typically MRP/MCP = 1.25–3 (fig. 22-2).

Motor weight depends primarily on the peak torque. Figure 22-3 shows the weight for a number of motors. For a given size, there is a wide range of Q/W values. Here high torque-to-weight is defined as Q/W > 3 . 5 ft-lb/lb. Motor electrical controller weight is typically 10–30% of the basic motor weight, up to 70% for small motors. The density is typically 100–250 lb/ft (fig. 22-4).

The engine performance is described by the power available P at each engine rating, and the av ˙ ˙ energy flow E required to produce the power required P . The specific fuel consumption is sfc = E/P req (inverse of efficiency). For each rating R, the maximum power is P , and the torque limit is P .

0 R peak R 22–2 Power Available and Performance at Power Required Given the flight condition and engine rating R, the power available is ( ) P = min P , rP av 0 R peak R where r = N/N , and N is the motor rotational speed.

spec The performance is calculated for a specified power required (which might equal the power avail- able), flight condition, and engine rating. The motor power required determines the energy flow: ˙ ˙ E = E g ( q, n ) = P /η req 0 C e req motor where q = P /P , n = N/N , and η is the motor efficiency. The generator energy flow to the req eng spec motor fuel tank is related to the power required: ˙ ˙ E = E g ( q, n ) = P η req 0 C e req motor where η is the generator efficiency.

motor 192 Motor Model For a motor and fuel cell, the performance (mass flow and fuel flow) is calculated for a specified power required: ˙ w = ˙ w ( q/η ) = sfc ( P /η ) req 0 C 0 C req ˙ m = ˙ m ( q/η ) = K ˙ w req 0 C mf req Inlet and exhaust installation losses are included in the specific fuel consumption. The efficiency η = η η includes both motor and fuel cell losses. The fuel cell efficiency as a function of power is cell motor ( ) 1 P 1 P eng = 1 + − 1 + c η P η P cell eng ref from η and c for the fuel cell. The ratio of mass flow and fuel flow ( K ) follows from the chemistry ref mf of the reaction.

22–3 Efficiency Motor loss sources include copper (internal resistance, proportional to current-squared hence torque-squared), iron core (eddy current and hysteresis, proportional to rotational speed), and me- chanical (friction, proportional to speed, and windage, proportional to speed-cubed). Equivalent circuit and efficiency map models are implemented. The efficiency can also be a fixed value.

22-3.1 Equivalent Circuit The motor or generator efficiency as a function of power is estimated considering an equivalent circuit, defined by internal resistance R and current I . The voltage is V = V − IR . The efficiency is 0 o ∼ η = P/ ( P + P ) , from the power loss P = I R + P . For small loss, I P/V . In terms of = motor loss loss 0 o P , let eng ( ) 1 1 R/V = − 1 o P η eng ref P = cP 0 eng Then ( ) 1 P 1 P eng = 1 + P ( R/V ) + P /P = 1 + − 1 + c o η P η P motor eng ref − 1 So η = (1 /η + c ) at P = P . The efficiency decreases with P because of the internal resistance, motor ref eng but is zero at P = 0 because of the internal current term.

22-3.2 Efficiency Map The motor or generator power loss is described as a polynomial in the motor torque and rotational speed: 3 3 ∑ ∑ i j P = P f C t n loss eng loss ij i =0 j =0 where q = P /P , n = N/N , and t = q/n . The factor f allows adjustment of the peak efficiency.

req eng spec loss Controller losses, including power conversion and conditioning, are represented by an efficiency η .

cont Then P q req η = η = η motor cont cont P + P q + P /P req loss loss eng is the motor or generator efficiency. Constant efficiency implies just C = 1 /η − 1 . The copper, iron, ( ) 2 3 2 3 and windage losses imply P = K Q + K N + K N + K = P C t + C n + C n + C .

loss c i w 0 eng 20 01 03 00 Motor Model 193 Consider an efficiency map with a peak η at t = Q /Q and n = N /N . Taking the 0 0 0 eng 0 0 spec ( ) 2 3 derivatives of C t + C n + C n + C = nt (1 − η ) /η with respect to t and n , and evaluating at t 20 01 03 00 0 and n so ∂η/∂t = ∂η/∂n = 0 , gives t 1 − η 3 C 0 0 00 C = − 4 η 2 n 0 0 t 1 − η C 0 0 00 C = + 2 3 4 n η 2 n 0 0 n 1 − η 0 0 C = 2 t η 0 0 (ref. 2). Note that C > 0 requires C < ( n t / 6)(1 − η ) /η .

01 00 0 0 0 0 In general, expanding the efficiency in t and n about the peak gives ( ) 1 P P 1 1 loss loss 2 2 ∼ ˜ ˜ = + 1 = + 1 1 + a ( t − 1) + b ( t − 1)( ˜ n − 1) + c ( ˜ n − 1) = η QN P tn η eng 0 ˜ where t = t/t and ˜ n = n/n . Hence 0 0 ( ) [ ] ( ) P 1 1 loss 2 2 2 2 ˜ ˜ ˜ = tn − 1 = tn 1 + a ( t − 1) + b ( t − 1)( ˜ n − 1) + c ( ˜ n − 1) − 1 + d ( t − 1) ( ˜ n − 1) P η η eng 0 The last term is introduced so the efficiency is zero at zero torque or zero speed.

Alternatively, consider an equivalent circuit defined by internal resistance R , no load current I , motor constant K , and friction K . So V = V + IR (back emf V = N/K ), I = I + I ( I = v f m m v m 0 m Q N/V = K Q ), Q = Q − Q (output torque, Q = K N ). The input power is V I = ( N/K + IR ) I , m m v m m f f f v with I = K Q + I = K ( Q + K N ) + I . The shaft power is QN . Then the power loss is v m 0 v f 0 ( ) P = V I − QN = N + ( I/K ) K R ( I/K ) − QN loss v v v ( ) ( ) = N Q + K N + I /K + Q + K N + I /K K R − QN f 0 v f 0 v v ( ) ( ) = N K N + I /K + Q + K N + I /K K R f 0 v f 0 v v This expression does not show a peak efficiency as a function of torque and speed.

22–4 Weight The motor or generator weight can be a fixed input value, or calculated as a function of power or torque. As a function of power, the weight of one engine is: X motor W = K + K P + K P one eng 0motor 1motor 2motor where P = P .

eng Motor weight depends primarily on the peak torque. For the NASA15 model, the weight is: 0 . 8129 W = 0 . 5382 f Q one eng design where Q = P /N (ft-lb), for takeoff rating R. The structural design factor f = 1 . 0 for high peak R spec design torque-to-weight motors ( Q/W > 3 . 5 ft-lb/lb), and f = 2 . 5606 for others. Based on 64 motors, the design 194 Motor Model average error is 27.5% (fig. 22-5); 25% for high torque-to-weight and 29% for others. Including either maximum rotational speed or maximum power does not reduce the weight estimation error significantly.

Considering only the high torque-to-weight motors, the weight is 0 . 8587 W = 0 . 3928 Q one eng Based on 25 motors, the average error is 21.8% (fig. 22-5). This sensitivity of motor weight to output torque capability is somewhat greater than the drive system trend.

22–5 Scaling The parameters of the motor model can be defined for a specific motor, but it is also necessary to scale the parameters as part of the aircraft sizing task, in order to define a motor for a specified power.

In addition, advanced technology must be represented in the model.

The motor model includes reference values of the performance parameters: P , P . The 0 R peak R reference power at each rating R defines a ratio to MCP: r = P /P . Similarly for torque limits: p 0 R 0 R 0 C r = P /P . These ratios are kept fixed when the motor is scaled. The specification motor m 0 R peak R 0 C speed is scaled with the power: ( ) K N s P ref N = N spec spec − ref P eng The speed ratio N/N influences the efficiency map and the torque limit. If the base rotational speed spec N = P /Q scales with power as K instead of K , then the scaled values of P require base max peak N b N s peak R K − K N b N s a factor of ( P /P ) as well as r . Depending on the design approach, K ranges from eng ref m 0 R N b about 0 . 8 to zero (constant N ) to about − 1 . 2 ( N increase with power).

spec spec The motor size is specified as power P , which is the power at takeoff rating R, for sea level eng standard (SLS) static conditions. Hence the MCP power is P = P /r , and the power at all other 0 C to p 0 R ratings follows. If P is not equal to the reference value of the motor model, then the motor is scaled.

0 C 22–6 References 1) McDonald, R.A. “Electric Motor Modeling for Conceptual Aircraft Design.” AIAA Paper No.

2013-0941, January 2013.

2) McDonald, R.A. “Electric Propulsion Modeling for Conceptual Aircraft Design.” AIAA Paper No.

2014-0536, January 2014.

3) Sinsay, J.D.; Alonso, J.J.; Kontinos, D.A.; Melton, J.E.; and Grabbe, S. “Air Vehicle Design and Technology Considerations for an Electric VTOL Metro-Regiional Public Transportation System.” AIAA Paper No. 2012-5404, September 2012.

4) Datta, A., and Johnson, W. “Requirements for a Hydrogen Powered All-Electric Manned Helicopter.” AIAA Paper No. 2012-5405, September 2012.

Motor Model 195 100000.

high Q/W other base rpm = 100 10000.

1000.

10000 peak torque (ft-lb) 100.

10.

1. 10. 100. 1000.

maximum power (hp) Figure 22-1. Motor maximum power and peak torque.

5.

high Q/W other 4.

3.

2.

1.

maximum power / continuous power 0.

1. 10. 100. 1000.

maximum power (hp) Figure 22-2. Motor maximum power and continuous power (MRP/MCP).

196 Motor Model 10000.

high Q/W other 1000.

100.

weight (lb) 10.

1.

10. 100. 1000. 10000. 100000.

peak torque (ft-lb) Figure 22-3. Motor weight.

350.

high Q/W 300.

other 250.

) 200.

150.

density (lb/ft 100.

50.

0.

10. 100. 1000. 10000. 100000.

peak torque (ft-lb) Figure 22-4. Motor density.

Motor Model 197 high Q/W other 80.

only high Q/W 60.

40.

20.

0.

error (%) -20.

-40.

-60.

-80.

1. 10. 100. 1000. 10000.

actual weight Figure 22-5. Motor weight (NASA15).

198 Motor Model

Chapter 23 Referred Parameter Jet Engine Model Design and analysis codes calculate aircraft performance for a very wide range of operating con- ditions. Thus the jet performance model must be realistic even far from the design point. A simple thermodynamic cycle analysis that assumes design point component efficiencies everywhere is not real- istic for such an application. Rather than developing models for component performance, the approach taken is to use a model for the total turbojet or turbofan performance. The jet is not being designed.

The Referred Parameter Jet Engine Model (RPJEM) is based on curve-fits of performance data for existing or projected jets over a range of operating conditions. The use of referred parameters tends to collapse the data and provides a basis for scaling the jet. The operating condition is described by pressure altitude, ambient air temperature, flight Mach number, and either jet rating or jet thrust required.

The parametric model is scaled to the required size and adjusted to the appropriate technology level to represent a notional jet. Jet size is represented by mass flow. Jet technology is represented by specific thrust available and specific fuel consumption at maximum continuous thrust (MCT), sea level/standard day (SLS), static (zero airspeed) conditions. Jet installation effects are also modeled.

23–1 Operating Environment The operating condition and atmosphere give the standard conditions (temperature T and pressure std p ) for a specified pressure altitude; the sea-level standard conditions (temperature T and pressure p ); std 0 0 ◦ ◦ and the operating temperature T and pressure p . Here the temperatures are R or K. The characteristics depend on the temperature ratio θ = T /T and pressure ratio δ = p/p .

0 0 √ The flight Mach number M = V /c = V /c θ is obtained from the aircraft speed V .

s s 0 The inlet ram air temperature ratio and pressure ratio are obtained then from M and the inlet ram recovery efficiency η : d ( ) ( ) γ − 1 2 2 θ = 1 + M = 1 + 0 . 2 M M γ ( ) γ − 1 ( ) γ − 1 3 . 5 2 2 δ = 1 + η M = 1 + 0 . 2 η M M d d where the ratio of specific heats γ = 1 . 4 .

23–2 Performance Characteristics The uninstalled thrust required is T , the thrust available T . The jet model calculates T as a q a a function of flight condition and engine rating; or calculates mass flow and fuel flow at T . The specific q thrust is ST = T / ˙ m , and the specific fuel consumption is sfc = ˙ w/T . A turbofan engine has a bypass 200 Referred Parameter Jet Engine Model ratio β = ˙ m / ˙ m . The forces produced by mass flow that does not go through the core or fan are treated fan as momentum drag D .

aux The reference performance is at sea-level-standard static conditions (subscript 0 ), and MCT (sub- script C ). For each rating R, the performance is characterized by the following quantities for sea-level- standard static conditions: thrust T , specific thrust ST , and mechanical thrust limit T . The 0 R 0 R mech R mass flow is then ˙ m = T /ST .

0 R 0 R 0 R The difference between net and gross thrust is the momentum drag: T = T − ˙ m ( ST ) , N G mom where ( ST ) = (1 + β ) V for turbojet or turbofan, or ( ST ) = Ω r for reaction drive. Note mom mom react that traditional units for mass flow are pound/sec (pps), while this equation requires slug/sec ( ˙ m /g req replaces ˙ m ).

req 23–3 Installation The difference between installed and uninstalled thrust is the inlet and exhaust losses T : T = loss av T − T and T = T − T . The inlet and exhaust losses are modeled as fractions of thrust available a loss req q loss or thrust required: T = ( + ) T or T = ( + ) T . The momentum drag of the auxiliary loss in ex a loss in ex q air flow is a function of the mass flow ˙ m = f ˙ m : aux aux req D = (1 − η ) ˙ m V = (1 − η ) f ˙ m V aux aux aux aux aux req where η is the ram recovery efficiency. Exhaust losses ( ) and auxiliary air flow parameters ( η , aux ex aux f ) are defined for infrared suppressor on and off.

aux 23–4 Thrust Available Given the flight condition and jet rating, the thrust available T is calculated as follows. The gross a specific thrust and referred mass flow (relative to ST and ˙ m for this rating) are approximated by 0 0 functions of the ambient temperature ratio θ , here just: [ ] √ √ X sta ST = ST θ δ θ a 0 M M ( ) [ ] √ √ X mf a ˙ m = ˙ m δ/ θ δ θ a 0 M M The thrust available is then [ ] √ X + X sta mf a T = ST ˙ m − ˙ m ( ST ) = T δ δ θ − ˙ m ( ST ) a a a a mom 0 M M a mom This expression for ˙ m is used only to calculate T ; elsewhere the ˙ m expression below (for performance a a q at thrust required) is used to obtain the mass flow at a thrust T .

q 23–5 Performance at Thrust Required The jet performance (mass flow and fuel flow) is calculated for a specified thrust required T , flight q condition, and jet rating. The referred quantities (relative to SLS static MCT quantities) are approximated by functions of referred gross thrust t = ( T + ˙ m ( ST ) ) / ( T δ ) : q mom 0 C ( ) √ ( ) − X 2 f f q ˙ w = ˙ w δ θ K + K t + K t [ θ ] req 0 C f f q 0 f f q 1 f f q 2 M ( ) √ X K mf q mf q ˙ m = ˙ m δ/ θ t [ θ ] req 0 C M Referred Parameter Jet Engine Model 201 The mass flow solution is ⎧ 1 K = 0 mf q ⎨ ˙ m req b/ (1 − a ) K = 1 √ = mf q √ X mf q ⎩ ˙ m ( δ/ θ ) [ θ ] 2 0 C M − ( a/ 2) + ( a/ 2) + b K = 1 / 2 mf q √ X mf q where a = ( ST ) ˙ m [ θ ] /T θ and b = T /T δ .

mom 0 C M 0 C q 0 C 23–6 Scaling The parameters of the jet model can be defined for a specific turbojet or turbofan, but it is also necessary to scale the parameters as part of the aircraft sizing task, in order to define a jet for a specified thrust. In addition, advanced technology must be represented in the model. Scaling and advanced technology are handled in terms of specific thrust and specific fuel consumption (at SLS static conditions and MCT). Figures 23-1 through 23-3 present historical data for jet specific fuel consumption, weight, and specific thrust.

The jet model includes reference values of the performance parameters: T , ST , T , and 0 R 0 R mech R sfc . Mass flow and fuel flow are obtained from ˙ m = T /ST and ˙ w = sfc T . The reference 0 C 0 R 0 R 0 R 0 C 0 C 0 C thrust at each rating R defines a ratio to MCT: r = T /T . Similarly for specific thrust and t 0 R 0 R 0 C mechanical limits: r = ST /ST and r = T /T . These ratios are kept fixed when the s 0 R 0 R 0 C m 0 R mech R 0 C jet is scaled.

The jet size is specified as takeoff thrust T = T , which is the thrust at rating R, for SLS static to eng conditions. Hence the MCT is T = T /r , and the thrust at all other ratings follows. If T is not 0 C to t 0 R 0 C equal to the reference value of the jet model, then the jet is scaled. To reflect advanced technology, the specific thrust and specific fuel consumption can be specified: ST = ST and sfc = sfc 0 C tech 0 C tech (replacing the jet model reference values). The default values are the reference values of the jet model.

In the following paragraph, the subscript “ tech ” refers to these quantities; the subscript “ ref ” means the jet model reference values.

The jet technology parameters ST and sfc are assumed to vary linearly with mass flow ˙ m up 0 C 0 C 0 C to a limit ˙ m , and constant thereafter at ST and sfc . The mass flow at the technology condition lim lim lim is ˙ m = T /ST , with the technology values ST and sfc . The intercept values are projected tech ref tech tech tech from the technology values: K = ST − K ˙ m , K = ( ST − ST ) / ( ˙ m − ˙ m ) ; and st 0 tech st 1 tech st 1 lim tech lim tech similarly for sfc . Then for ˙ m < ˙ m 0 C lim ST = K + K ˙ m 0 C st 0 st 1 0 C sfc = K + K ˙ m 0 C sf c 0 sf c 1 0 C and for ˙ m ≥ ˙ m 0 C lim ST = ST 0 C lim sfc = sfc 0 C lim These equations are used if ˙ m > ˙ m . Otherwise the model sets K = K = 0 , so there is no lim tech st 1 sf c 1 variation with scale: ST = ST and sfc = sfc . Usually the effect of size gives K ≥ 0 and 0 C tech 0 C tech st 2 K ≤ 0 . The thrust at the limit is T = ST ˙ m . Figure 23-4 illustrates the scaling of SP , sfc , and sf c 2 lim lim lim SW .

202 Referred Parameter Jet Engine Model 1.4 1.2 1.0 0.8 0.6 0.4 specific fuel consumption (lb/lb-hr) 0.2 0.0 0. 20000. 40000. 60000. 80000. 100000.

takeoff thrust (lb) Figure 23-1. Historical data for turbojet and turbofan specific fuel consumption.

0.6 0.5 0.4 0.3 weight (lb/lb) 0.2 0.1 0.0 0. 20000. 40000. 60000. 80000. 100000.

takeoff thrust (lb) Figure 23-2. Historical data for turbojet and turbofan weight.

Referred Parameter Jet Engine Model 203 80.

60.

40.

ST (lb/lb/sec) 20.

0.

0. 20000. 40000. 60000. 80000. 100000.

takeoff thrust (lb) Figure 23-3. Historical data for turbojet and turbofan specific thrust.

Using ˙ m = T /ST , the specific thrust equation can be solved for the mass flow given the 0 C 0 C 0 C thrust: ⎧ T /K K = 0 0 C st 0 st 1 ⎪ ⎪ ⎪ ⎨ T /ST T ≥ T 0 C lim 0 C lim ˙ m = 0 C ⎪ √ ⎪ ⎪ K T K ⎩ 0 2 0 C 0 ( ) + − otherwise 2 K K 2 K 1 1 1 From this mass flow, ST and sfc are calculated, hence the fuel flow ˙ w = sfc T . Then the thrust 0 C 0 C 0 C 0 C 0 C and specific thrust at all ratings R are obtained from the ratios: T = r T , ST = r ST , and 0 R t 0 R 0 C 0 R s 0 R 0 C T = r T .

mech R m 0 R 0 C The actual (perhaps scaled) values of the performance parameters are available for the jet group: T , ST , T , and sfc .

0 R 0 R mech R 0 C 23–7 Weight The jet weight can be a fixed input value, or calculated as a function of thrust. As a function of thrust, the weight of one jet is: X jet W = K + K T + K T one jet 0jet 1jet 2jet where T is the installed takeoff thrust (SLS static, specified rating) per jet. A constant weight per thrust W/T is given by using only K .

1jet 204 Referred Parameter Jet Engine Model . .

. .

m m ≤ m m > lim tech lim tech lim lim ST ST 0C 0C tech tech lim tech tech lim .

.

m m 0C 0C sfc sfc tech tech 0C 0C lim lim tech lim lim tech . .

m m 0C 0C lim lim SW SW ref ref tech lim lim tech . .

m m 0C 0C Figure 23-4. Turbojet or turbofan scaling; ST = T / ˙ m , sfc = ˙ w/T , and SW = T /W .

23–8 Units In this jet model, only the reference values and scaling constants are dimensional. Conventional English units and SI units are shown in table 23-1. Units of specific thrust and specific fuel consumption follow from these conventions.

Table 23-1. Conventional units.

thrust T mass flow ˙ m fuel flow ˙ w English: pound pound/sec pound/hour SI: Newton kilogram/sec kilogram/hour

Chapter 24 Fuel Cell Model A fuel cell burns a fuel (typically hydrogen) and generates electrical energy, which is stored in a fuel tank system or used directly by a motor. The energy flow defines the power required. The power available is related to the size P . The model follows reference 1.

chrg ˙ Given the flight condition and the charger rating, the cell power available is E = P . The power a cell 0 ˙ required E is defined by the charge group energy flow. The cell power required is req ˙ ˙ ˙ E = E g ( q ) = E /η q cell 0 C e req chrg ˙ where q = E /P , and η is the fuel cell efficiency. The fuel cell performance (mass flow and fuel req 0 chrg flow) is calculated from the cell power required: ˙ ˙ w = sfc E = ˙ w q/η req 0 C q cell 0 C chrg ˙ m = K ˙ w = ˙ m q/η req mf req 0 C chrg ˙ where q = E /P , ˙ w = sfc P , and ˙ m = K ˙ w . Inlet and exhaust installation losses are req 0 0 C 0 C chrg 0 C mf 0 C included in the specific fuel consumption. The specific fuel consumption is given by the fuel specific energy and the fuel cell thermal efficiency: sfc = e /η . The ratio of mass flow and fuel flow follows fuel th from the chemistry of the reaction. For hydrogen and air λ m λ A A A K = = 68 . 59 mf λ x m λ H O H H The molar masses of hydrogen and air are m = 2 . 016 and m = 28 . 97 g/mole; x = 0 . 2095 is the molar H A O ∼ fraction of oxygen in air. The supply ratio λ /λ = 1 / 2 from stoichiometry, and typically λ /λ . 25 = 1 A H A H in practice.

The fuel cell efficiency as a function of power is estimated considering an equivalent circuit, defined by internal resistance R and current I . The voltage is V = V − IR . The efficiency is η = P/ ( P + P ) , 0 o chrg loss 2 ∼ from the power loss P = I R + P . For small loss, I P/V . In terms of P , let = loss 0 o chrg ( ) 1 1 R/V = − 1 o P η chrg ref P = cP 0 chrg Then ( ) 1 P 1 P chrg = 1 + P ( R/V ) + P /P = 1 + − 1 + c o η P η P chrg chrg ref − 1 So η = (1 /η + c ) at P = P . The efficiency decreases with P because of the internal resistance, chrg ref chrg but is zero at P = 0 because of the internal current term. Alternatively, the efficiency can be a fixed value.

206 Fuel Cell Model The fuel cell weight can be a fixed input value or calculated as a function of power. The weight of one charger is: X cell W = K + K P + K P one chrg 0cell 1cell 2cell where P = P .

chrg 24–1 References 1) Datta, A., and Johnson, W. “Requirements for a Hydrogen Powered All-Electric Manned Helicopter.” AIAA Paper No. 2012-5405, September 2012.

Chapter 25 Solar Cell Model A solar cell generates electrical energy, which is stored in a fuel tank system. The energy flow defines the power required. The power available is related to the size P . The solar cell is characterized chrg 2 2 by power density e (W/m ) and weight density σ (kg/m ). From the size P , the area is solar solar chrg A = P /e ; and then the weight is W = W = A σ .

solar chrg solar one chrg solar solar solar ˙ Given the flight condition and the charger rating, the cell power available is E = P . The power a cell 0 ˙ required E is defined by the charge group energy flow. The cell power required is req ˙ ˙ ˙ E = E g ( q ) = E /η q cell 0 C e req chrg ˙ where q = E /P , and η is the solar cell efficiency.

req 0 chrg The solar cell efficiency as a function of power is estimated considering an equivalent circuit, defined by internal resistance R and current I . The voltage is V = V − IR . The efficiency is 0 o ∼ η = P/ ( P + P ) , from the power loss P = I R + P . For small loss, I P/V . In terms of P , = chrg loss loss 0 o chrg let ( ) 1 1 R/V = − 1 o P η chrg ref P = cP 0 chrg Then ( ) 1 P 1 P chrg = 1 + P ( R/V ) + P /P = 1 + − 1 + c o η P η P chrg chrg ref − 1 So η = (1 /η + c ) at P = P . The efficiency decreases with P because of the internal resistance, chrg ref chrg but is zero at P = 0 because of the internal current term. Alternatively, the efficiency can be a fixed value.

208 Solar Cell Model

Chapter 26 Battery Model A battery is a fuel tank system for which the fuel quantity stored and burned is measured in energy.

The unit of fuel energy is Mega-Joules (MJ). For reference, 1 kW-hr = 3.6 MJ. The operating state affects the efficiency of the relation between useful power and the rate of change of the energy stored. The battery model produces the charge/discharge efficiency. The battery model can be used for capacitors and flywheels as well. References 1–6 provide background for the model.

The components associated with a fuel tank system (motor, generator, fuel cell, and solar cell) ˙ define the total energy flow E (charge or discharge). Accounting for efficiency and losses gives the comp ˙ ˙ battery energy flow E . Accounting for battery capacity then gives the effective energy flow E . The batt eff ˙ ˙ change in stored energy is calculated from E and time. The convention is that energy flow E > 0 for eff ˙ discharge, and E < 0 for charge. Power and current are positive for both discharge and charge.

The battery capacity is E (maximum usable fuel energy). The battery is characterized fuel − cap by specific energy e (MJ/kg) and energy density ρ (MJ/liter), so the tank weight and volume tank tank are obtained from the capacity. The current amount of energy stored is E . The state-of-charge is fuel s = E /E ; the depth-of-discharge is d = 1 − s .

fuel fuel − cap The charge capacity is C (A-hr). C is the usable capacity at low current and a reference temperature.

Typically 20–30% of the stored charge is not usable at nominal conditions, which is accounted for in the capacity through the specific energy and energy density values. The corresponding energy capacity is obtained for a reference voltage, E = CV (W-hr, expresssed in MJ). Then the specific energy ref (conventional units W-hr/kg) gives the weight, and the energy density (conventional units W-hr/liter) gives the volume. The current is measured in terms of the charge capacity: I = xC (A), corresponding to a discharge time of 1 /x hours. The units of x are 1/hr. The maximum burst discharge current ( mbd ) is x .

mbd The usable capacity decreases with time and duty cycles, depending on the temperature and current.

This factor is assumed to be constant during a mission or flight condition. Capacity fade is accounted for using a factor f < 1 on the available capacity or on the energy flow. Thus the effective energy fade flow is obtained by dividing the actual energy flow by f .

fade k The capacity depends on the discharge current. The Peukert model assumes I T = constant, where T is the discharge time for current I = xC . The Peukert coefficient k = 1 . 2 to 1 . 3 for lead-acid batteries, ref k − 1 and k = 1 . 01 to 1 . 05 for lithium-ion batteries (weak dependence). Thus the capacity C = IT ∼ 1 /x .

An increase in current by a factor of 10 means a 2–11% reduction of capacity for lithium-ion batteries.

For a larger current, the battery reaches a specified discharge voltage sooner, hence effectively has a reduced capacity.

210 Battery Model The specific power is π (kW/kg). The power capacity is obtained at the maximum burst discharge batt current: P = I V = x CV = x E cap mbd ref mbd ref mbd fuel − cap (for P in W and E in W-hr). Hence given x , π = x e / 3 . 6 (kW/kg from MJ/kg).

cap fuel − cap mbd batt mbd tank Typically x is smaller for large capacity, implying a trade between high specific power and high mbd specific energy.

26–1 Equivalent Circuit Model The discharge or charge efficiency as a function of power is estimated considering an equivalent circuit, defined by internal resistance R and current I . The voltage is V = V − IR . The efficiency is 0 o ∼ η = P/ ( P + P ) , from the power loss P = I R + P . For small loss, I = P/V . In terms of a batt loss loss 0 o ˙ reference power P = P , and battery power P = | E | , let ref cap comp ( ) 1 1 R/V = − 1 o P η ref ref P = cP 0 ref Then ( ) 1 P 1 P ref = 1 + P ( R/V ) + P /P = 1 + − 1 + c o η P η P batt ref ref − 1 So η = (1 /η + c ) at P = P . The efficiency decreases with P because of the internal resistance, batt ref ref but is zero at P = 0 because of the internal current term. Alternatively, the efficiency can be a fixed value.

∼ ∼ ˙ The current is x = I/C P/V C x P/P = x | E | /P . Given the maximum current x , = = 0 mbd ref mbd comp cap max the maximum battery power is P = ( x /x ) P .

max max mbd cap ˙ ˙ ˙ For discharge, the battery energy flow is E = E /η = E + P . Then the effective batt comp batt comp loss ˙ ˙ ˙ energy flow is E = E /f . For charge (negative energy flow), the battery energy flow is E = eff batt fade batt ˙ ˙ ˙ ˙ E η = E + P . Then the effective energy flow is E = E /f . The battery power comp batt comp loss eff batt fade ˙ margin is P − | E | .

max batt 26–2 Lithium-Ion Battery Model The capacity and power of currently available lithium-ion batteries are shown in figure 26-1. The power shown corresponds to the maximum burst discharge current, P = x E . The cell cap mbd fuel − cap reference voltage is typically V = 4 . 2 V. The maximum continous discharge current is in the range ref x = 1–30 (fig. 26-2). The maximum burst discharge current (which equals the power divided by mcd ∼ capacity) is in the range x = 10–50 (fig. 26-3), with x for large capacity. Values of x = 10 mbd mbd mbd above 100 are achievable, but such high-power/short-duration capability will not have much application to aircraft. The maximum charge current is typically x = 1–5 (fig. 26-4). Figure 26-5 shows CC max the corresponding specific energy e and specific power π = x e . The specific energy is tank batt mbd tank typically e = 50–200 W-hr/kg = 0.2–0.7 MJ/kg. Figure 26-6 shows the specific energy e (W- tank tank hr/kg) and the energy density ρ (W-hr/liter). The battery density is about 0.6 kg/liter for cylindrical tank batteries, and about 2.0 kg/liter for other configurations (pouch, prismatic).

The battery temperature is likely controlled, for safety and efficiency. The operating temperature depends on the environment ambient temperature, internal heat production, and active heating or cooling.

Internal heat production includes that due to internal resistance ( I R ), and radiation and convection Battery Model 211 (roughly proportional to the difference between cell and ambient temperatures). The cell temperature T is defined for a flight condition, or at the start of a mission. Without active temperature control, c during a mission the cell temperature changes: dT /dt is proportional to the internal heat production.

c With temperature control, there is a power loss associated with the control system, depending on the cell temperature relative to ambient, hence on the internal heat production. As a simple model, it is assumed that this loss is a fraction f of the total power.

T C 26-2.1 Discharge Characteristics The lithium-ion battery voltage during discharge depends on the discharge capacity (depth-of- discharge d ), current ( I = xC ), and cell temperature ( T ). Figure 26-7 shows typical variation of the c voltage V at constant current and constant temperature. The nominal energy capacity E = CV is ref obtained from the nominal charge capacity C and a reference voltage V . The open-circuit voltage V ref d is the voltage at zero current, a reference temperature T , and 100% charge. Typically the cell voltage ref is V = 4.2 V, and V is used as the reference V . The actual battery capacity is determined by the d d ref voltage reaching a critical value V (typically 2–3 V). Usually V decreases gradually with d , so the crit voltage provides a measure of depth-of-discharge. The decrease of V with current is primarily due to internal resistance. The voltage also decreases as temperature decreases (fig. 26-7). There is a maximum discharge current (continuous or burst), which gives the rated power: P = I V = x CV . The cap mbd ref mbd ref actual battery power supplied is P = IV .

The voltage variation with current is primarily due to the internal resistance: V = V − IR . The o open circuit voltage decreases with depth-of-discharge: V = V F ( d ) , such that F = 1 at d = 0 o d V V and F = f = V /V at d = 1 ; and V = f V . For large discharge ( d approaching 1), there V crit crit d d d ref is a secondary influence of current, which can be modelled by adjusting the discharge by a factor k .

Then the capacity is given by F ( kd ) = f . Variations of the voltage with current are scaled using V crit I = xC = ( x C )( x/x ) . The influence of temperature on the open circuit voltage V is reasonably mbd mbd o accounted for by an increment proportional to the cube of the temperature difference Δ T = T − T .

c ref The model for the discharge voltage is x CR mbd 3 3 V = V F ( kd ) + k Δ T − IR = V F ( kd ) + k Δ T − V ( x/x ) d V V T d V V T ref mbd V ref with k = 1 + k I − k Δ T = 1 + ( k x C ) ( x/x ) − k Δ T dI dT dI mbd mbd dT The open circuit voltage function F ( d ) is derived from V ( d ) as a function of I and T : F = ( V − V c V k Δ T + IR ) /V at kd . The model parameters R , k , k , and k are adjusted for a good fit to the V T d V T dI dT measured discharge characateristics. Figure 26-8 shows F ( d ) obtained for a number of lithium-ion V ∼ batteries. Typically f . 6 . That d  = 1 at F = f reflects the definition of nominal capacity for = 0 crit V crit the battery characteristics. Table 26-1 gives a typical F ( d ) function. The function extends below f V crit since high temperature can increase the voltage. Figure 26-9 shows the current parameters x CR/V mbd ref and k x C for a number of batteries. Generally x CR/V = 0.05–0.20 and k x C = 0–0.25.

dI mbd mbd ref dI mbd Figure 26-10 shows the temperature parameters k and k for a number of batteries.

V T dT The battery power available to the components is ( ) 3 2 ˙ ˙ E = P = IV = I V F ( kd ) + k Δ T − I R − P = E − P comp d V V T T C batt loss 212 Battery Model including power loss for temperature control. Hence ( ) ˙ E = P ξ f F ( kd ) + ( k /V )Δ T batt cap d V V T ref ̂ ˙ P = P Rξ + f | E | loss cap T C comp ̂ k = 1 + k ξ − k Δ T dI dT ̂ ̂ where ξ = x/x , R = x CR/V , and k = k x C . The battery capacity for a given current mbd mbd ref dI dI mbd and temperature is obtained by finding kd such that crit V = V F ( kd ) + k Δ T − IR = V f d V crit V T d crit or ̂ F ( kd ) = f + ( Rξ − ( k /V )Δ T ) /f V crit crit V T ref d Then d = ( kd ) /k , and the effective capacity is d f E . Rather than change the fuel tank crit crit crit fade fuel − cap ˙ ˙ capacity, an effective fuel flow is used: E = E / ( d f ) .

eff batt crit fade ˙ For a flight condition or mission segment, the total energy flow required is E . The temperature comp T is specified. The depth-of-discharge is d = 1 − s = 1 − E /E . The discharge current c fuel fuel − cap x = x ξ is calculated from the required energy flow: mbd ˙ ( E + P ) /P comp loss cap ξ = f F ( kd ) + ( k /V )Δ T d V V T ref Successive-substitution solution for ξ , with f as the minimum for F , converges in about 10 iterations.

crit V Since F is a monotonically decreasing function of d , d is obtained from x by interpolating d ( F ) .

V crit V ˙ ˙ ˙ ˙ Then E = E + P , and the battery efficiency is η = E / E . The effective energy flow batt comp loss batt comp batt ˙ ˙ ˙ is E = E / ( d f ) . The maximum battery discharge power is P = E evaluated at x , eff batt crit fade max batt mbd ˙ or at a specified discharge current. The battery power margin is P − | E | .

max batt Table 26-1. Lithium-ion battery open circuit voltage.

depth-of-charge d F depth-of-charge d F V V 0.00 1.000 0.92 0.842 0.10 0.970 0.93 0.835 0.20 0.950 0.94 0.826 0.30 0.930 0.95 0.815 0.40 0.915 0.96 0.800 0.50 0.900 0.97 0.780 0.60 0.890 0.98 0.750 0.70 0.880 0.99 0.700 0.80 0.870 1.00 0.600 0.90 0.850 1.01 0.400 0.91 0.847 1.02 0.000 26-2.2 Charge Characteristics Lithium-ion battery charging is controlled for safety, with limits on minimum and maximum voltage, maximum current, and minimum and maximum temperature. Standard charging procedure consists of Battery Model 213 constant current (CC) followed by constant voltage (CV). Figure 26-11 shows the voltage, current, and capacity variation with charging time. The charging current is I = x C ; the maximum charging CC CC current is typically x = 1–5 (fig. 26-4). The charging voltage is V = f V ; typically V = V = 4 . 2 CC max c c ref c ref V. The state-of-charge is s .

During the CC phase, the voltage starts at a fraction k (typically 0.05–0.20) below V and increases cV c with time, reaching V at time t = σ/x . The charging model assumes linear variation of the voltage.

c c CC During the CV phase, the voltage is fixed at V , while the current decreases rapidly. The charging model c assumes the current varies inversely with the square of the time increment since t . The charge is then c ∫ t the integral of the current, Q = I dt ; and the state-of-charge s = Q/C . The charging power is P = IV .

Table 26-2 gives the charging model as a function of normalized time τ = t/t − 1 = tx /σ − 1 .

c CC Table 26-2. Lithium-ion battery charging model.

CC phase, t < t , τ = − 1 to 0 , s < σ CV phase, t > t , τ > 0 , s > σ c c x C CC current I = xC = x C I = xC = CC (1 + k τ ) cI voltage V = V (1 + k τ ) V = V c cV c 1 + ( k + 1) τ cI charge Q = x Ct = σC (1 + τ ) Q = σC CC 1 + k τ cI 1 + ( k + 1) τ cI state-of-charge s = x t = σ (1 + τ ) s = σ CC 1 + k τ cI power P = xCV = x CV (1 + k τ ) P = xCV = x CV CC c cV c CC c (1 + k τ ) cI s − 1 s σ time τ = − 1 τ = s σ 1 − k ( − 1) cI σ At t = t ( τ = 0 ), the current and voltage have the CC/CV values ( I = x C , V = V ), the state-of-charge c CC c s = σ , and the charging power is at the peak ( P = x CV = P ( x /x ) f ). For s = 1 at large CC c cap CC mbd c time, k must be related to σ : 1 /σ = 1 + 1 /k ; the measured CV capacity is matched better with smaller cI cI values for k . Figure 26-12 shows the charging parameters k and k for a number of batteries.

cI cV cI For small charging current x , the voltage V is reached at the nominal charging time t = 1 /x .

CC c c CC So the CC phase is the entire process, and s = x t . For higher current, the voltage V is reached sooner, CC c at t = σ/x with σ < 1 . Then the CV phase is required to complete the charging. For a given battery, c CC σ = 1 at low charge current, and σ decreases with increasing x . Figure 26-13 shows 1 /σ and 1 + 1 /k CC cI for a number of batteries. An approximation for the variation of σ is 1 1 = 1 + = 1 + k ( x − 0 . 2) σ CC σ k cI above x = 0 . 2 , with k depending on the battery.

CC σ ˙ For a flight condition or mission segment, the total energy flow delivered to the battery is E . The comp temperature T is specified. The state-of-charge is s = E /E . The charge current x = x ξ c fuel fuel − cap mbd 214 Battery Model ˙ is calculated from the required energy flow, | E | = P = IV = ξx CV = P ξV /V . In the CC batt mbd cap ref phase ( τ < 0 , s < σ ): ( ) ( ) ˙ ξ = | E | − P /P /f (1 + k τ ) comp loss cap c cV and x = x . In the CV phase ( τ > 0 , s > σ ): CC ( ) ( ) ˙ ξ = | E | − P /P /f comp loss cap c and x = x (1 + k τ ) . Successive-substitution solution for ξ converges in about 10 iterations. First CC cI the CC solution is found, then if τ > 0 the CV solution is found (with an upper limit on x ). Then CC ˙ ˙ ˙ ˙ E = E + P , and the battery efficiency is η = E / E . The effective energy flow is batt comp loss batt batt comp ˙ ˙ E = E /f . The maximum battery charging power is P = P evaluated at current x , or eff batt fade max CC max ˙ at a specified charge current x . The battery power margin is P − | E | .

CC max batt 26–3 References 1) Gao, L.; Liu, S.; and Dougal, R.A. “Dynamic Lithium-Ion Battery Model for System Simulation.” IEEE Transactions on Components and Packaging Technologies, 25 :3 (September 2002).

2) Ramadass, P.; Haran, B.; White, R.; and Popov, B.N. “Mathematical Modeling of the Capacity Fade of Li-Ion Cells.” Journal of Power Sources, 123 :2 (September 2003).

3) Chen, M., and Rinc´ on-Mora, G.A. “Accurate Electrical Battery Model Capable of Predicting Runtime and I-V Performance.” IEEE Transactions on Energy Conversion, 21 :2 (June 2006).

4) Erdinc, O.; Vural, B.; and Uzunoglu, M. “A Dynamic Lithium-Ion Battery Model Considering the Effects of Temperature and Capacity Fading.” Second International Conference on Clean Electrical Power, June 2009.

5) Datta, A., and Johnson, W. “Requirements for a Hydrogen Powered All-Electric Manned Helicopter.” AIAA Paper No. 2012-5405, September 2012.

6) Barth, A.; Feil, R.; Konkak, K.; and Hajek, M. “Conceptual Study for an Autonomous Rotorcraft for Extreme Altitudes.” Fortieth European Rotorcraft Forum, Southampton, UK, September 2014.

Battery Model 215 100000.

10000.

1000.

power (W) 100.

10.

1.

1. 10. 100. 1000. 10000.

capacity (W-hr) Figure 26-1. Lithium-ion battery capacity (W-hr) and power (W).

35.

30.

25.

20.

(1/hr) 15. mcd x 10.

5.

0.

0.1 1. 10. 100. 1000. 10000.

capacity (W-hr) Figure 26-2. Lithium-ion battery maximum continuous discharge current.

216 Battery Model 150.

125.

100.

75.

= power/capacity (1/hr) 50.

mbd x 25.

0.

0.1 1. 10. 100. 1000. 10000.

capacity (W-hr) Figure 26-3. Lithium-ion battery maximum burst discharge current.

16.

12.

(1/hr) 8.

CCmax x 4.

0.

0.1 1. 10. 100. 1000. 10000.

capacity (W-hr) Figure 26-4. Lithium-ion battery maximum charge current.

Battery Model 217 100000.

10000.

1000.

specific power (W/kg) 100.

10.

0. 50. 100. 150. 200. 250.

specific energy (W-hr/kg) Figure 26-5. Lithium-ion battery specific energy (W-hr/kg) and specific power (W/kg).

700.

600.

other 500.

400.

300.

200.

energy density (W-hr/liter) cylindrical 100.

0.

0. 50. 100. 150. 200. 250. 300.

specific energy (W-hr/kg) Figure 26-6. Lithium-ion battery specific energy (W-hr/kg) and energy density (W-hr/liter).

218 Battery Model ο charge current ( ) 25 C 4.2 0.3C 0.5C 4.0 1.0C 2.0C 3.0C 3.8 4.0C 3.6 voltage (V) 3.4 3.2 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 depth-of-discharge 4.2 temperature (0.3C) ο 40 C ο 4.0 25 C ο 0 C ο -20 C 3.8 ο -40 C 3.6 voltage (V) 3.4 3.2 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 depth-of-discharge Figure 26-7. Lithium-ion battery discharge characteristics.

Battery Model 219 1.1 1.0 V 0.9 0.8 0.7 open circuit voltage F 0.6 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 depth-of-discharge d Figure 26-8. Lithium-ion battery open circuit voltage function.

0.25 0.20 0.15 C mbd x dI k 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 0.25 internal resistance, x CR/V mbd ref Figure 26-9. Lithium-ion battery discharge model, current parameters.

220 Battery Model 0.010 0.008 0.006 dT k 0.004 0.002 0.000 0.0 0.3 0.6 0.9 1.2 1.5E-5 k VT Figure 26-10. Lithium-ion battery discharge model, temperature parameters.

Battery Model 221 1.2 1.05 1.0 1.00 x = 0.5 CC x = 1.0 CC 0.8 c 0.95 0.6 x = 0.5 CC 0.90 voltage V/V current x=I/C x = 1.0 0.4 CC 0.85 0.2 0.0 0.80 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 time (hr) time (hr) 1.2 1.2 1.0 1.0 0.8 0.8 c x = 0.5 CC x = 1.0 CC 0.6 0.6 power P/CV 0.4 0.4 state-of-charge s x = 0.5 CC x = 1.0 CC 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 1.0 2.0 3.0 state-of-charge s time (hr) Figure 26-11. Lithium-ion battery charge characteristics.

222 Battery Model 0.25 0.20 0.15 cV k 0.10 0.05 0.00 0. 3. 6. 9. 12. 15.

k cI Figure 26-12. Lithium-ion battery charge model parameters.

1.8 k = 0.8 σ 1.6 1 / σ 1 + 1 / k cI 1.4 k = 0.1 σ 1.2 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 charge current x CC Figure 26-13. Lithium-ion battery charge model, CV parameters.

Chapter 27 AFDD Weight Models This chapter presents the rotorcraft weight models developed by the U.S. Army Aeroflightdynamics Directorate (AFDD). For some weight groups several models are available, designated AFDDnn. The weights are estimated from parametric equations, based on the weights of existing turbine-powered helicopters and tiltrotors. The figures of this chapter compare the weights calculated from these equations with the actual weights. The results of these equations are the weight in pounds, and the units of the parameters are noted in the tables. Technology factors χ are included in the weight equations. The input usually includes a weight increment dW that can be added to the results of the weight model. Thus typically a component or element weight is obtained from W = χW + dW . Weight of individual model elements in a group can be fixed by using dW and setting the corresponding technology factor χ = 0 .

With χ  = 0 , the increment dW can account for something not included in the parametric model.

The weight models are implemented as part of the aircraft components. The weights are entered into the weight statement data structure (extended RP8A format) for each component, reflected in the organization of this chapter.

27–1 Wing Group The wing group consists of: basic structure (primary structure, consisting of torque box and spars, plus extensions); fairings (leading edge and trailing edge); fittings (non-structural); fold/tilt structure; and control surfaces (flaps, ailerons, flaperons, and spoilers). There are separate models for a tiltrotor or tiltwing configuration and for other configurations (including compound helicopter).

27-1.1 Tiltrotor or Tiltwing Wing Wing weight equations for a tiltrotor or tiltwing aircraft are based on methodology developed by Chappell and Peyran (refs. 1 and 2). The wing is sized primarily to meet torsional stiffness requirements.

The primary structure weight is calculated from torque box and spar weights: W = A ρ b /e box tb tb w tb W = C A ρ b /e spar t sp sp w sp w = ( W + W ) f prim box spar units W = χ w prim prim prim A consistent mass-length-time system is used in the equations for W and W , which therefore have box spar units of slug or kg. The primary structure weight W however has units of lb or kg, hence a conversion prim factor f = g is required for English units. The wing fairing (leading edge and trailing edge), control units 224 AFDD Weight Models surface (flaps, ailerons, flaperons, and spoilers), fittings (non-structural), and fold/tilt weights are: w = S U W = χ w fair fair fair fair fair fair w = S U W = χ w flap flap flap flap flap flap w = f T W = χ w fit fit cap fit fit fit w = f ( W + W + W + W + W ) W = χ w fold fold prim fair flap fit tip fold fold fold The control surface area S for a tiltrotor wing is the sum of the flap and flaperon areas. The fairing flap area is S = ( b − w ) c (1 − w ) − S fair w attach w tb flap The wing extension weight is: w = S U W = χ w ext ext ext ext ext ext w = f W W = χ w efold efold ext efold efold efold and these terms are added to W and W . The tiltrotor wing weight (and wing folding weight in prim fold fuselage group) depends on W , the weight on the wing tips; which is the sum of rotor group, engine tip section or nacelle group, air induction group, engine system, drive system (except drive shaft), rotary wing and conversion flight controls, hydraulic group, trapped fluids, and wing extensions. An adjustment of this calculated weight can be used; a negative increment is required when the engine and transmission are not at the tip location with the rotor. The weight on wing tip is used as the fraction f = W /W ; tip tip SD the mass on the wing tip is M (slug or kg).

tip To estimate the wing weights, the required stiffness is scaled with input frequencies (per rev) of the wing primary bending and torsion modes. First the torque box is sized to meet the torsional stiffness (frequency) requirement. Next spar cap area is added as required to meet the chord and beam bending frequency requirements. Finally spar cap area is added if necessary for a jump takeoff condition.

Wing section form factors, relating typical airfoil and torque box geometry to ideal shapes, are input or calculated from the thickness-to-chord ratio and the torque box chord to wing chord ratio: F = 0 . 073 sin(2 π ( τ − 0 . 151) / 0 . 1365) + 0 . 14598 τ B w w + 0 . 610 sin(2 π ( w + 0 . 080) / 2 . 1560) − (0 . 4126 − 1 . 6309 τ )( w − 0 . 131) + 0 . 0081 tb w tb F = 0 . 640424 w − 0 . 89717 w + 0 . 4615 τ + 0 . 655317 C tb w tb ( ) √ F = ((0 . 27 − τ ) / 0 . 12)0 . 12739 − 0 . 96 + 3 . 32 + 94 . 6788 w − ( w / 0 . 08344) T w tb tb − 2 . 7545 w + 5 . 1799 w − 0 . 2683 tb tb F = 0 . 25 sin(5 . 236 w ) + 0 . 325 V H tb for beam bending, chord bending, torsion, and spar cap vertical/horizontal bending. The ideal shape for torsional stiffness is a tube of radius t , so the torsional stiffness J = F A t / 4 . The ideal shape for w T tb w chord bending is two caps c apart, so I = F A c / 4 . The ideal shape for beam bending is two caps tb Ctb C tb tb 2 2 t apart, so I = F A t / 4 and I = F A t / 4 . The torque box cross-sectional area is obtained w Bsp V H sp Btb B tb w w from the wing torsion frequency; 1 1 2 2 GJ = ( ω Ω) ( b − w ) M r T w attach tip pylon 2 2 A = 4 GJ/ ( G F t ) tb tb T w AFDD Weight Models 225 It is assumed that between the points of attachment to the fuselage, the torque box is very stiff in torsion, 1 1 but does deflect in bending. So the effective length is ( b − w ) for torsion, and b for bending.

w attach w 2 2 The spar cap cross-sectional area (in addition to torque box material) is obtained from beam and chord bending frequencies: 1 1 2 3 EI = ( ω Ω) b M f C C tip mode w 24 2 1 1 2 3 EI = ( ω Ω) b M f B B tip mode w 24 2 EI = E F A c / 4 Ctb tb C tb tb EI = EI − EI Csp C Ctb A = EI / ( E c / 4) Csp Csp sp tb EI = E F A t / 4 Btb tb B tb w EI = E F A t / 4 V H sp V H Csp w EI = EI − EI − EI Bsp B Btb V H A = EI / ( E t / 4) Bsp Bsp sp w EI = EI + EI sp V H Bsp A = A + A sp Csp Bsp where EI and EI are replaced by zero if negative (no additional spar material required). The factor Csp Bsp f = 1 − f is a mode shape correction for fuselage motion. Next the primary structure, fairing, mode tip flap, and fitting weights are calculated as before; and the sum W = W + W + W + W .

wing prim fair flap fit Additional spar cap material for a jump takeoff condition is obtained from the ultimate applied bending moment at the wing root: ( ) M = T 0 . 75(1 − f ) − 0 . 375( /b )( W /W ) U cap w tip w w wing SD where T is the maximum thrust capability of one rotor, equal to the greater of n W /N or cap jump SD rotor ( C /σ ) ρA V (from an input C /σ at the jump takeoff condition, sea level standard, and hover rotor T b T tip speed). The bending moment capacity of the wing is M = 2 EI  /t tb Btb U w M = 2 C EI  /t sp m sp U w Then the additional cross-section area is obtained from the moment deficit: Δ M = M − ( M + M ) U tb sp Δ A = 2Δ M/ (  E t ) sp U sp w Δ W = C Δ A ρ b /e spar j sp sp w sp where Δ M is replaced by zero if negative (no additional spar material required). If Δ W is positive, it spar is added to W , and the primary structure, fairing, flap, and fitting weights are recalculated. Parameters spar are defined in table 27-1, including units as used in these equations. Here a consistent mass-length-time system is used, producing W and W in slug or kg. Typically the input uses conventional English box spar 3 2 units for density (lb/in ) and modulus (lb/in ).

226 AFDD Weight Models Table 27-1. Parameters for tiltrotor wing weight.

parameter definition units W , W wing torque box and spar weights slug or kg box spar W structural design gross weight lb SD ( C /σ ) rotor maximum thrust capability (jump takeoff) T jump n load factor at W (jump takeoff) jump SD N number of rotors rotor b wing span (length of torque box) ft or m w = b − w wing length (span less fuselage width) ft or m w w f us c wing chord ft or m w w torque box chord to wing chord ratio tb c = w c torque box chord ft or m tb tb w τ wing airfoil thickness-to-chord ratio w t = τ c wing thickness ft or m w w w r pylon radius of gyration (pitch inertia = r M ) ft or m pylon tip pylon Ω rotor speed for wing weight design condition rad/sec ω wing torsion mode frequency (fraction rotor speed) per rev T ω wing beam bending mode frequency (fraction rotor speed) per rev B ω wing chord bending mode frequency (fraction rotor speed) per rev C 3 3 ρ density of torque box material slug/ft or kg/m tb 3 3 ρ density of spar cap material slug/ft or kg/m sp 2 2 G torque box shear modulus lb/ft or N/m tb 2 2 E torque box modulus lb/ft or N/m tb 2 2 E spar modulus lb/ft or N/m sp  ultimate strain allowable (minimum of spar and torque box) U C weight correction for spar taper (equivalent stiffness) t C weight correction for spar taper (equivalent strength) j C strength correction for spar taper (equivalent stiffness) m e structural efficiency factor, torque box tb e structural efficiency factor, spars sp 2 2 U unit weight of leading and trailing-edge fairings lb/ft or kg/m fair 2 2 U unit weight of control surfaces lb/ft or kg/m flap 2 2 S area of leading and trailing-edge fairings ft or m fair 2 2 S area of control surfaces ft or m flap f wing fittings and brackets (fraction maximum thrust of one rotor) fit f wing fold/tilt (fraction total weight excluding fold, including weight on tips) fold w width of wing structural attachments to body ft or m attach 2 2 U unit weight of wing extension lb/ft or kg/m ext 2 2 S area of wing extensions (span b times mean chord c ) ft or m ext ext ext f wing extension fold/tilt (fraction extension weight) efold AFDD Weight Models 227 27-1.2 Aircraft Wing There are two models intended for the wing of a compound helicopter: area method and parametric method. For the area method (based on weight per unit area), the total wing weight excluding folding is: w = S U wing w w f = 1 − f − f − f prim fair flap fit Typically U = 5 to 9 lb/ft . For the parametric method (AFDD93), the total wing weight including w folding is: ( ) 0 . 847 W SD 0 . 39579 0 . 21754 0 . 50016 w = 5 . 66411 f n S A wing LG loc z w w 1000 cos Λ w 0 . 09359 − 0 . 14356 ((1 + λ ) /τ ) (1 − b ) w w fold f = 1 − f − f − f − f prim fair flap fit fold where f = 1 . 7247 if the landing gear is on the wing, and 1 . 0 otherwise. Based on 25 fixed-wing LG loc aircraft, the average error of the aircraft wing equation is 3.4% (fig. 27-1). Then the primary structure, secondary structure, and control surface weights are: W = χ f w prim prim prim wing W = χ f w fair fair fair wing W = χ f w flap flap flap wing W = χ f w fit fit fit wing W = χ f w fold fold fold wing The wing extension weight is: w = S U W = χ w ext ext ext ext ext ext w = f W W = χ w efold efold ext efold efold efold and these terms are added to W and W . Parameters are defined in table 27-2, including units as prim fold used in these equations.

Table 27-2. Parameters for aircraft wing weight.

parameter definition units S wing planform area (theoretical) ft w 2 2 U unit weight of wing planform lb/ft or kg/m w W structural design gross weight lb SD n design ultimate flight load factor at W g z SD Λ wing sweep angle deg w A wing aspect ratio w λ wing taper ratio (tip chord/root chord) w τ wing airfoil thickness-to-chord ratio w b fraction wing span that folds (0 to 1) fold f fairings (fraction total wing weight) fair f control surfaces (fraction total wing weight) flap f fittings (fraction total wing weight) fit f fold/tilt (fraction total wing weight) fold 2 2 U unit weight of wing extension lb/ft or kg/m ext S area of wing extensions (span b times mean chord c ) ft ext ext ext f wing extension fold/tilt (fraction extension weight) efold 228 AFDD Weight Models 27–2 Rotor Group The rotor group consists of: blades, hub and hinge, spinner, and blade fold structure. The blade and hub-hinge weights for the AFDD82 model are: 0 . 6592 1 . 3371 0 . 9959 0 . 6682 2 . 5279 w = 0 . 02606 N N R c V ν W = χ w blade rotor blade blade blade blade tip blade 0 . 2807 1 . 5377 0 . 4290 2 . 1414 0 . 5505 w = 0 . 003722 N N R V ν ( W /N ) W = χ w hub rotor blade rotor hub hub hub blade tip hub Based on 37 aircraft, the average error of the blade equation is 7.7% (fig. 27-2). Based on 35 aircraft, the average error of the hub equation is 10.2% (fig. 27-3). The blade and hub-hinge weights for the AFDD00 model are: 0 . 53479 1 . 74231 0 . 77291 0 . 87562 2 . 51048 w = 0 . 0024419 f N N R c V ν W = χ w blade tilt rotor blade blade blade blade tip blade 0 . 20373 0 . 60406 0 . 52803 1 . 00218 0 . 87127 w = 0 . 0061182 N N R V ν ( W /N ) W = χ w hub rotor blade rotor hub hub hub blade tip hub where f = 1 . 17940 for tilting rotors; 1 . 0 otherwise. Based on 51 aircraft, the average error of the blade tilt equation is 7.9% (fig. 27-4) and the average error of the hub equation is 12.2% (fig. 27-5). This equation for hub weight was developed using the actual blade weight. Using the blade weight equation instead gives 0 . 16383 0 . 19937 0 . 06171 0 . 46203 1 . 02958 w = 0 . 18370 N N R V ν ( w /N ) hub rotor blade rotor blade tip hub 1 . 02958 0 . 71443 1 . 99321 0 . 79577 0 . 96323 0 . 46203 2 . 58473 = 0 . 00037547 f N N R c V ν ν rotor tilt blade tip hub blade which really just adds chord and f to the regression parameters. Based on 51 aircraft, the average tilt error of this equation is 9.2% (fig. 27-5). For teetering and gimballed rotors, the flap frequency ν should be the coning frequency. Thus using the blade weight equation results in a lower average error, and best represents legacy rotor systems. The hub weight equation using the actual blade weight is best for advanced technology rotors with blades lighter than trend.

The fairing/spinner and blade fold weights are: W = χ 7 . 386 N D spin spin rotor spin W = χ f W fold fold fold blade The blade weight is for all blades of the rotors. If the weight is evaluated separately for each rotor, then N = 1 should be used in the equations. Typically f = 0 . 04 for manual fold and f = 0 . 28 for rotor fold fold automatic fold. Parameters are defined in table 27-3, including units as used in these equations. The rotor support and duct weights are: W = χ f W supt supt supt M T O W = χ N S U duct duct rotor duct duct AFDD Weight Models 229 Table 27-3. Parameters for rotor weight.

parameter definition units N number of rotors rotor N number of blades per rotor blade R rotor radius in hover ft c rotor mean geometric blade chord ft V rotor hover tip velocity ft/sec tip ν , ν flap natural frequency (for weight estimate) per rev blade hub D spinner diameter ft spin f blade fold weight (fraction total blade weight) fold W maximum takeoff weight lb M T O f rotor support structure weight (fraction maximum takeoff weight) supt 2 2 U unit weight of duct lb/ft or kg/m duct 2 2 S area of duct ft or m duct 27-2.1 Lift-Offset Rotor For lift-offset rotors, the blade and hub weights can be calculated based on the methodology of reference 3. The blade weight is estimated based on the beam stiffness required to maintain the clearance s when the blade is loaded by the lift offset. The blade tip deflection is proportional to δ ∝ P R /EI , where EI is the bending stiffness. The beam loading is P ∝ n W L/N . With A the blade z SD blade xs cross-section area, the moment of inertia I ∝ A t . The criterion is δ = h − s . Hence the blade weight xs 4 2 is W ∝ ρN RA ∝ ( ρ/E ) n W LR / ( t ( h − s )) with E the elastic modulus, and ρ here the blade blade xs z SD material density. The hub weight is estimated based on the structure in upper and lower hub plates required to react a tensile force F = C + M / ( x/ 2) due to combined centrifugal force and bending cent bend moment at the root; where the hub plate separation x scales with the blade thickness t . The centrifugal force C ∝ ( W /N ) V /R . The bending moment M ∝ n W R . The limit tensile stress cent blade blade bend z SD tip σ ∝ F/A gives a criterion for the total hub arm area A . The radius of the hub scales with the blade xs xs ( ) thickness t . Hence the hub weight is W ∝ ρN A ∝ ( ρ/σ ) W V t/R + Kn W RN .

hub blade xs blade z SD blade tip The distribution factor K is determined from the calibration weights. The inter-rotor shaft weight is estimated based on the structure to react the hub moment caused by lift offset. The hub moment M ∝ n W LR . The shaft diameter d scales with the blade thickness t . The shaft length scales with z SD the rotor separation h . The ultimate bending stress σ = M/ ( I/c ) gives a criterion for the area moment I/c ∝ d w , hence for the shaft wall thickness w . Then W ∝ ρ dw ∝ ( ρ/σ ) n W LRh/t is the shaft shaft z SD weight.

The blade, hub-hinge, and inter-rotor shaft weights are: 3 2 w = N 0 . 000083770 wL R / (2( h − s ) t ) blade rotor W = χ w . 2 R blade blade blade ( ) w = N 0 . 17153 w RN + 0 . 000010534( W /N ) V t /R W = χ w hub rotor blade blade rotor . 2 R hub hub hub tip W = χ w shaft shaft shaft w = N 0 . 081304 wL R 2 h/t shaft rotor . 2 R where w = n W / 1000 , and z SD ( ) 0 . 8 + 0 . 2 λ t = τ c . 2 R . 2 R 0 . 5 + 0 . 5 λ 230 AFDD Weight Models is the blade thickness at 20% R . These equations were developed for the coaxial rotor configuration ( N = 2 ), and calibrated to the XH-59A weights. The material factors ( ρ/E and ρ/σ ) are included in rotor the technology factors χ . These weight terms can be used separately, with other hub and blade weight models. Parameters are defined in table 27-4, including units as used in these equations.

Table 27-4. Parameters for lift-offset rotor weight.

parameter definition units N number of rotors rotor N number of blades per rotor blade W structural design gross weight lb SD n design ultimate flight load factor at W g z SD λ blade taper ratio (tip chord/root chord) τ blade airfoil thickness-to-chord ratio (at 20% R ) . 2 R R rotor radius ft c blade mean chord ft h coaxial rotor separation (fraction rotor diameter) s coaxial rotor tip clearance (fraction rotor diameter) L lift offset ( M /T R ) roll V rotor hover tip velocity ft/sec tip 27–3 Empennage Group The empennage group consists of: horizontal tail, vertical tail, and tail-rotor. The tail plane weight consists of the basic structure and fold structure. There are two models for tail plane basic weight: area method and parametric method. For the area method (based on weight per unit area), the weight is W = S U . The parametric weight model depends on the aircraft configuration. The helicopter tail tail tail or compound model is AFDD82. The horizontal tail weight is: 0 . 2 tiltrotor or tiltwing W = χ S (0 . 00395 S V − 0 . 4885) ht ht ht dive ht 1 . 1881 0 . 3173 helicopter or compound W = χ 0 . 7176 S A ht ht ht ht Based on 13 aircraft, the average error of the helicopter horizontal tail equation is 22.4% (fig. 27-6).

The vertical tail weight is: 0 . 2 tiltrotor or tiltwing W = χ S (0 . 00395 S V − 0 . 4885) vt vt vt dive vt 0 . 9441 0 . 5332 helicopter or compound W = χ 1 . 0460 f S A vt vt tr vt vt where f = 1 . 6311 if the tail-rotor is located on the vertical tail; 1 . 0 otherwise. Based on 12 aircraft, the tr average error of the helicopter vertical tail equation is 23.3% (fig. 27-7). V is the design dive speed, dive calculated or input; V = 1 . 25 V , where V is the maximum speed at design gross weight and sea dive max max level standard conditions. The fold weight is a fraction of the basic weight: W = χ f W , fold fold fold basic where W = W or W .

basic ht vt The tail-rotor weight is: 0 . 0897 0 . 8951 W = χ 1 . 3778 R ( P R/V ) tr tr DS limit tip tr AFDD Weight Models 231 Based on 19 aircraft, the average error of the helicopter tail-rotor equation is 16.7% (fig. 27-8). The tail-rotor weight is calculated by the rotor component model, including rotor support and duct weights.

Rotor weight models can be used for the tail-rotor. Parameters are defined in table 27-5, including units as used in these equations.

Table 27-5. Parameters for tail weight.

parameter definition units S horizontal tail planform area ft ht S vertical tail planform area ft vt A horizontal tail aspect ratio ht A vertical tail aspect ratio vt V design dive speed at sea level kts dive R tail-rotor radius ft tr P drive system power limit (MCP) hp DS limit R main-rotor radius in hover ft V main-rotor hover tip velocity ft/sec tip f fold (fraction basic weight excluding fold) fold 2 2 S horizontal or vertical tail planform area ft or m tail 2 2 U unit weight of tail planform lb/ft or kg/m tail 27–4 Fuselage Group The fuselage group consists of: basic structure; wing and rotor fold/retraction; tail fold/tilt; and marinization, pressurization, and crashworthiness structure. The AFDD84 model is a universal body weight equation, used for tiltrotor and tiltwing as well as for helicopter configurations. The AFDD82 model is a helicopter body weight equation, not used for tiltrotor or tiltwing configuration.

For the AFDD84 (UNIV) model, the basic structure weight is ( ) ( ) 0 . 4879 0 . 2075 W n W M T O z SD 0 . 1676 0 . 1512 w = 25 . 41 f f f S basic LG loc LG ret ramp body 1000 1000 W = χ w basic basic basic where f = 1 . 1627 if the landing gear is located on the fuselage, and 1 . 0 otherwise; f = 1 . 1437 LG loc LG ret if the landing gear is on the fuselage and retractable, and 1 . 0 otherwise; f = 1 . 2749 if there is a cargo ramp ramp, and 1 . 0 otherwise. Based on 35 aircraft, the average error of the body equation is 6.5% (fig. 27-9).

The tail fold, wing and rotor fold, marinization, pressurization, and crashworthiness weights are: w = f W W = χ w tfold tfold tail tfold tfold tfold w = f ( W + W ) W = χ w wfold wfold wing tip wfold wfold wfold w = f W W = χ w mar mar basic mar mar mar w = f W W = χ w press press basic press press press w = f ( W + W + W + W + W ) W = χ w cw cw basic tfold wfold mar press cw cw cw Typically f = 0 . 30 for a folding tail, and f = 0 . 06 . For wing folding the weight on the wing tip tfold cw ( W ) is required (calculated as for the wing group). Parameters are defined in table 27-6, including tip units as used in these equations.

232 AFDD Weight Models Table 27-6. Parameters for fuselage weight (AFDD84 model).

parameter definition units W maximum takeoff weight lb M T O W structural design gross weight lb SD S wetted area of body ft body R main-rotor radius ft n design ultimate flight load factor at W g z SD length of fuselage ft f tail fold weight (fraction tail weight) tfold f wing and rotor fold weight (fraction wing/tip weight) wfold W tail group weight tail W + W wing group weight plus weight on wing tip wing tip f marinization weight (fraction basic body weight) mar f pressurization (fraction basic body weight) press f crashworthiness weight (fraction fuselage weight) cw For the AFDD82 (HELO) model, the basic structure weight is ( ) 0 . 4908 W M T O 0 . 1323 0 . 2544 0 . 6100 w = 5 . 896 f n S basic ramp z body W = χ w basic basic basic where f = 1 . 3939 if there is a cargo ramp, and 1 . 0 otherwise. Based on 30 aircraft, the average error ramp of the body equation is 8.7% (fig. 27-10). The tail fold, wing and rotor fold, marinization, pressurization, and crashworthiness weights are: w = f W W = χ w tfold tfold basic tfold tfold tfold w = f ( W + W ) W = χ w wfold wfold basic tfold wfold wfold wfold w = f W W = χ w mar mar basic mar mar mar w = f W W = χ w press press basic press press press w = f ( W + W + W + W + W ) W = χ w cw cw basic tfold wfold mar press cw cw cw Typically f = 0 . 05 for a folding tail, and f = 0 . 06 . Parameters are defined in table 27-7, including tfold cw units as used in these equations.

Table 27-7. Parameters for fuselage weight (AFDD82 model).

parameter definition units W maximum takeoff weight lb M T O S wetted area of body ft body R main-rotor radius ft n design ultimate flight load factor at W g z SD length of fuselage ft f tail fold weight (fraction basic structure) tfold f wing and rotor fold weight (fraction basic structure and tail fold) wfold f marinization weight (fraction basic body weight) mar f pressurization (fraction basic body weight) press f crashworthiness weight (fraction fuselage weight) cw AFDD Weight Models 233 27–5 Alighting Gear Group The alighting gear group consists of: basic structure, retraction, and crashworthiness structure.

There are two models, parametric (AFDD82) and fractional. The basic landing gear weight is: 0 . 6662 0 . 5360 0 . 1525 parametric w = 0 . 4013 W N ( W/S ) LG M T O LG fractional w = f W LG LG M T O and W = χ w . Typically f = 0 . 0325 (fractional method). Based on 28 aircraft, the average LG LG LG LG error of the parametric equation is 8.4% (fig. 27-11). The retraction and crashworthiness weights are: w = f W W = χ w LG ret LG ret LG LG ret LG ret LG ret w = f ( W + W ) W = χ w LG cw LG cw LG LG ret LG cw LG cw LG cw Typically f = 0 . 08 , and f = 0 . 14 . Parameters are defined in table 27-8, including units as used LG ret LG cw in these equations.

Table 27-8. Parameters for landing gear weight.

parameter definition units W maximum takeoff weight lb M T O f landing gear weight (fraction maximum takeoff weight) LG W/S wing loading (1.0 for helicopter) lb/ft N number of landing gear assemblies LG f retraction weight (fraction basic weight) LG ret f crashworthiness weight (fraction basic and retraction weight) LG cw 27–6 Engine Section or Nacelle Group and Air Induction Group The engine section or nacelle group consists of: engine support structure, engine cowling, and pylon support structure. The weights (AFDD82 model) are: 1 . 1433 1 . 3762 W = χ 0 . 0412(1 − f )( W /N ) N supt supt airind eng eng eng 1 . 3476 W = χ 0 . 2315 S cowl cowl nac W = χ f W pylon pylon pylon M T O Based on 12 aircraft, the average error of the engine support equation is 11.0% (fig. 27-12). Based on 12 aircraft, the average error of the engine cowling equation is 17.9% (fig. 27-13). The air induction group weight (AFDD82 model) is: 1 . 1433 1 . 3762 W = χ 0 . 0412 f ( W /N ) N airind airind airind eng eng eng Typically f = 0 . 3 (range 0 . 1 to 0 . 6 ). Based on 12 aircraft, the average error of the air induction airind equation is 11.0% (fig. 27-14). Parameters are defined in table 27-9, including units as used in these equations.

234 AFDD Weight Models Table 27-9. Parameters for engine section, nacelle, and air induction weight.

parameter definition units W maximum takeoff weight lb M T O W weight all main engines lb eng N number of main engines eng S wetted area of nacelles and pylon (less spinner) ft nac f air induction weight (fraction nacelle plus air induction) airind f pylon support structure weight (fraction W ) pylon M T O 27–7 Propulsion Group The propulsion group consists of the engine system, fuel system, and drive system.

27-7.1 Engine System The engine system consists of the main engines, the engine exhaust system, and the engine acces- sories. The engine system weights are: engine group: jet group: W = χ N W W = χ N W eng eng eng one eng eng jet jet one jet W = χ N ( K + K P ) W = χ N ( K + K T ) exh exh eng 0exh 1exh exh exh jet 0exh 1exh 0 . 5919 0 . 7858 W = χ 2 . 0088 f ( W /N ) N acc acc lub eng eng eng where f = 1 . 4799 if the accessory weight includes the lubrication system weight, 1 . 0 if the lubrication lub system weight is in the engine weight. The exhaust system weight is per engine, including any infrared suppressor. The accessory weight equation is the AFDD82 model. Based on 16 aircraft, the average error of the accessories equation is 11.5% (fig. 27-15). The engine system weight W = W + W + W .

ES eng exh acc Parameters are defined in table 27-10, including units as used in these equations.

Table 27-10. Parameters for engine system weight.

parameter definition units N number of main engines eng N number of jets jet P installed takeoff power (SLS static, specified rating) per engine hp T installed takeoff thrust (SLS static, specified rating) per jet lb K , K engine exhaust weight vs. power or thrust, constants 0exh 1exh 27-7.2 Propeller/Fan Installation The auxiliary propulsion or propeller weight is: 1 . 04771 − 0 . 07821 AFDD82 W = χ 0 . 0809484 N T ( T /A ) at at at at at at 0 . 91996 − 0 . 48578 − 0 . 45904 0 . 15690 AFDD10 W = χ 9 . 90350 N P N Ω D f at at at m at prop prop blade AFDD Weight Models 235 where T is at maximum speed, design gross weight, and SLS conditions, calculated or input. The at material factor f = 1 for composite construction; 1.20 for wood; 1.31 for aluminum spar; and 1.44 for m aluminum construction. Based on 16 aircraft, the average error of the AFDD10 equation is 10.5%. The propeller weight is calculated by the rotor component model, including rotor support and duct weights.

Rotor weight models can be used for the propeller. Parameters are defined in table 27-11, including units as used in these equations.

Table 27-11. Parameters for propeller weight.

parameter definition units N number of auxiliary thrusters at T thrust per propeller lb at A auxiliary thruster disk area ft at P power per propeller hp at N number of blades per propeller blade Ω propeller rotation speed at P rpm prop at D propeller diameter ft prop 27-7.3 Fuel System The fuel system consists of tanks and support structure (including fuel tanks, bladders, supporting structure, filler caps, tank covers, and filler material for void and ullage), and fuel plumbing (including fuel system weight not covered by tank weight). There are two models for tank weight, parametric (AFDD82) and fractional. The fuel system weights (AFDD82 model) are: 0 . 7717 0 . 5897 1 . 9491 W = χ 0 . 4341 C N f f tank tank cw int int bt [ ] 0 . 866 W = χ K + K (0 . 01 N + 0 . 06 N )( F/N ) plumb plumb 0plumb 1plumb plumb eng eng where f = 1 . 3131 for ballistically survivable (UTTAS/AAH level) and 1 . 0 otherwise. The ballistic cw tolerance factor f = 1 . 0 to 2 . 5 . Based on 15 aircraft, the average error of the fuel tank equation is 4.6% bt (fig. 27-16). The fuel flow rate F is calculated for the takeoff power rating at static SLS conditions.

K is a crashworthiness and survivability factor; typically K = 2 . K is the sum of 1plumb 1plumb 0plumb weights for auxiliary fuel (typically up to 120 lb), in-flight refueling (up to 150 lb), pressure refueling (up to 150 lb), inerting system (up to 20 lb), etc.; typically K = 50 to 250 lb. Alternatively, for the 0plumb fractional model the fuel tank weight is W = χ f W . Parameters are defined in table tank tank tank fuel − cap 27-12, including units as used in these equations.

Table 27-12. Parameters for fuel system weight.

parameter definition units N number of internal fuel tanks int C internal fuel tank capacity gallons int f ballistic tolerance factor bt N total number of fuel tanks (internal and auxiliary) for plumbing plumb N number of main engines eng K , K plumbing weight, constants 0plumb 1plumb F fuel flow rate lb/hr f fuel tank weight (fraction fuel capacity) tank 236 AFDD Weight Models 27-7.4 Drive System The drive system consists of gear boxes and rotor shafts, drive shafts, and rotor brake. This distribution of drive system weights is based on the following functional definitions. Gearboxes are parts of the drive system that transmit power by gear trains, and the structure that encloses them. Rotor shafts are the structure (typically a shaft) that transmits power to the rotor. Drive shafts are the structure (typically a shaft) that transmits power in the propulsion system, but not directly to the rotor or by a gear train. The rotor brake weight encompasses components that can prevent the rotor from freely turning.

The gear box and rotor shaft weights for the AFDD83 model are: 0 . 8195 0 . 0680 0 . 0663 0 . 0369 0 . 6379 w = 57 . 72 P f N (Ω / 1000) / Ω gbrs DS limit Q gb eng rotor W = χ (1 − f ) w gb gb rs gbrs W = χ f w rs rs rs gbrs Based on 30 aircraft, the average error of the gear box and rotor shaft equation is 7.7% (fig. 27-17). The gear box and rotor shaft weights for the AFDD00 model are: 0 . 38553 0 . 78137 0 . 09899 0 . 80686 w = 95 . 7634 N P Ω / Ω gbrs rotor DS limit eng rotor W = χ (1 − f ) w gb gb rs gbrs W = χ f w rs rs rs gbrs Based on 52 aircraft, the average error of the gear box and rotor shaft equation is 8.6% (fig. 27-18).

Typically f = 0 . 13 (range 0 . 06 to 0 . 20 ). Parameters are defined in table 27-13, including units as used rs in these equations.

The drive shaft (AFDD82 model) and rotor brake weights are: 0 . 3828 1 . 0455 0 . 3909 0 . 2693 W = χ 1 . 166 Q x N (0 . 01 f ) ds ds P DS limit hub ds W = χ 0 . 000871 W (0 . 01 V ) rb rb blade tip where f = f Ω / Ω . Based on 28 aircraft, the average error of the drive shaft equation is 16.0% P Q other main (fig. 27-19). Based on 23 aircraft, the average error of the rotor brake equation is 25.1% (fig. 27-20).

The clutch weight in the weight statement is associated with an auxiliary power unit, and is a fixed input value. The conventional rotor drive system clutch and free wheeling device weights are included in the gear box and rotor shaft weight equations. Parameters are defined in table 27-14, including units as used in these equations.

Typically f = f = 60% for twin main-rotors (tandem, coaxial, and tiltrotor); for a single main- P Q rotor and tail-rotor, f = 3% and f = 15% ( 18% for 2-bladed rotors).

Q P Table 27-13. Parameters for drive system weight.

parameter definition units P drive system power limit (MCP) hp DS limit N number of main-rotors rotor N number of gear boxes gb Ω main-rotor rotation speed rpm rotor Ω engine output speed rpm eng f second (main or tail) rotor torque limit* % Q f rotor shaft weight (fraction gear box and rotor shaft) rs *input as fraction of total drive system torque limit AFDD Weight Models 237 Table 27-14. Parameters for drive shaft and rotor brake weight.

parameter definition units Q P / Ω (MCP) hp/rpm DS limit DS limit rotor N number of intermediate drive shafts ds x length of drive shaft between rotors ft hub f second (main or tail) rotor power limit* % P V main-rotor tip speed ft/sec tip *input as fraction of total drive system power limit 27–8 Flight Controls Group The flight controls group consists of cockpit controls, automatic flight control system, and system controls. W and W weights are fixed (input). System controls consist of fixed wing flight controls, cc af cs rotary wing flight controls, and conversion (rotor tilt) flight controls. The weight equations model separately non-boosted controls (which do not see aerodynamic surface or rotor loads), boost mechanisms (actuators), and boosted controls (which are affected by aerodynamic surface or rotor loads). The load path goes from pilot, to cockpit controls, to non-boosted controls, to boost mechanisms, to boosted controls, and finally to the component.

Fixed wing flight controls consist of non-boosted flight controls and flight control boost mechanisms.

The weights are: 0 . 6 full controls w = 0 . 91000 W M T O 0 . 64345 0 . 40952 only stabilizer controls w = 0 . 01735 W S M T O ht and then W = χ f w F W nb F W nb F W nb W = χ (1 − f ) w F W mb F W mb F W nb For a helicopter, the stabilizer control equation is used. Parameters are defined in table 27-15, including units as used in these equations.

Table 27-15. Parameters for fixed wing flight control weight.

parameter definition units S horizontal tail planform area ft ht W maximum takeoff weight lb M T O f fixed wing non-boosted weight F W nb (fraction total fixed wing flight control weight) Rotary wing flight controls consist of non-boosted flight controls, flight control boost mechanisms, and boosted flight controls. The non-boosted flight control weight (AFDD82 model) is: fraction method W = χ f (1 − f ) w RW nb RW nb RW nb RW hyd f c 0 . 3999 1 . 3855 parametric method W = χ 2 . 1785 f W N RW nb RW nb nbsv M T O rotor 238 AFDD Weight Models where f = 1 . 8984 for ballistically survivable (UTTAS/AAH level); 1 . 0 otherwise. The parametric nbsv method assumes the rotor flight controls are boosted and computes the weight of the non-boosted portion up to the control actuators. Based on 20 aircraft, the average error of the non-boosted flight controls equation is 10.4% (fig. 27-21). The flight control boost mechanism weight and boosted flight control weight (AFDD82 model) are: 0 . 6257 1 . 3286 2 . 1129 0 . 8942 w = 0 . 2873 f ( N N ) c (0 . 01 V ) f f c mbsv rotor blade tip RW red W = χ (1 − f ) w RW mb RW mb RW hyd f c 1 . 0042 0 . 1155 2 . 2296 3 . 1877 W = χ 0 . 02324 f ( N N ) N c (0 . 01 V ) RW b RW b bsv rotor blade tip rotor where f = 1 . 3029 and f = 1 . 1171 for ballistically survivable (UTTAS/AAH level); 1 . 0 otherwise; mbsv bsv and f = 1 . 0 to 3 . 0 . Typically f = 0 . 6 (range 0 . 3 to 1 . 8 ); f = 0 . 4 . Based on 21 aircraft, RW red RW nb RW hyd the average error of the boost mechanisms equation is 6.5% (fig. 27-22). Based on 20 aircraft, the average error of the boosted flight controls equation is 9.7% (fig. 27-23). Parameters are defined in table 27-16, including units as used in these equations.

Table 27-16. Parameters for rotary wing flight control weight.

parameter definition units W maximum takeoff weight lb M T O N number of main-rotors rotor N number of blades per rotor blade c rotor mean blade chord ft V rotor hover tip velocity ft/sec tip f rotary wing non-boosted weight (fraction boost mechanisms weight) RW nb f rotary wing hydraulics weight RW hyd (fraction hydraulics plus boost mechanisms weight) f flight control hydraulic system redundancy factor RW red The conversion controls consist of non-boosted tilt controls, and tilt control boost mechanisms; they are used only for tilting rotor configurations. The weights are: w = f W W = χ w CV mb CV mb M T O CV mb CV mb CV mb w = f W W = χ w CV nb CV nb CV mb CV nb CV nb CV nb Parameters are defined in table 27-17, including units as used in these equations.

Table 27-17. Parameters for conversion control weight.

parameter definition units W maximum takeoff weight lb M T O f conversion non-boosted weight (fraction boost mechanisms weight) CV nb f conversion boost mechanisms weight (fraction maximum takeoff weight) CV mb 27–9 Hydraulic Group The hydraulic group consists of hydraulics for fixed wing flight controls, rotary wing flight controls, conversion (rotor tilt) flight controls, and equipment. The hydraulic weight for equipment, W , is EQhud AFDD Weight Models 239 fixed (input). The weights (AFDD82 model) are W = χ f W F W hyd F W hyd F W hyd F W mb W = χ f w RW hyd RW hyd RW hyd f c W = χ f W CV hyd CF hyd CV hyd CV mb Typically f = 0 . 4 . Parameters are defined in table 27-18, including units as used in these equations.

RW hyd Table 27-18. Parameters for hydraulic group weight.

parameter definition units f fixed wing hydraulics weight (fraction boost mechanisms weight) F W hyd f rotary wing hydraulics weight RW hyd (fraction hydraulics plus boost mechanisms weight) f conversion hydraulics weight (fraction boost mechanisms weight) CV hyd 27–10 Anti-Icing Group The anti-icing group consists of the anti-ice system. The electrical system for anti-icing is part of the electrical group. The weights are obtained from the sum over all rotors, all wings, and all engines: ∑ W = χ k A DI elect DI elect elec blade (∑ ) ∑ ∑ ∑ W = χ k A + k + k W + k W DI sys DI sys rotor blade wing wing air eng jet jet Parameters are defined in table 27-19, including units as used in these equations.

Table 27-19. Parameters for anti-icing group weight.

parameter definition units 2 2 A total blade area of rotor, from geometric solidity ft or m blade wing length (wing span less fuselage width) ft or m wing k electrical system weight factor elect k rotor deice system weight factor rotor k wing deice system weight factor wing k engine air intake deice system weight factor air k jet air intake deice system weight factor jet 27–11 Other Systems and Equipment The following weights are fixed (input) in this model: auxiliary power group; instruments group; pneumatic group; electrical group (aircraft); avionics group (mission equipment); armament group (armament provisions and armor); furnishing and equipment group; environmental control group; and 240 AFDD Weight Models load and handling group. Typical fixed weights are given in table 27-20, based on medium to heavy helicopters and tiltrotors.

Table 27-20. Other systems and equipment weight.

group typical weight (lb) SYSTEMS AND EQUIPMENT flight controls group cockpit controls 100–125 automatic flight control system 35–200 flight control electronics, mechanical 35–100 flight control electronics, fly-by-wire 250 auxiliary power group 130–300 instruments group 150-250 hydraulic group equipment 50–300 electrical group 400–1000 avionics group (mission equipment) 400–1500 furnishings & equipment group 600–1000 crew only 100–200 environmental control group 50–250 anti-icing group 50–300 load & handling group internal 200–400 external 150–300 FIXED USEFUL LOAD crew 500–800 fluids (oil, unusable fuel) 50–150 27–12 Folding Weight Folding weights are calculated in a number of groups: wing W (including extensions), rotor fold W , tail W , fuselage W and W . These are the total weights for folding and the impact of fold fold tfold wfold folding on the group. A fraction f of these weights can be in a kit, hence optionally removable.

foldkit Thus of the total folding weight, the fraction f is a kit weight in the fixed useful load of the weight foldkit statement, while the remainder is kept in the wing, rotor, or fuselage group weight.

27–13 Parametric Weight Correlation Table 27-21 summarizes the satistics of the parametric weight estimation equations. Figure 27-24 shows the error of the calculated weight for the sum of all parametric weight, accounting on average for 42% of the empty weight. This sum is composed of the structural group (based on the AFDD00 equation for rotor blade and hub weights, and the AFDD84 equation for body weight), the propulsion group (based on the AFDD00 equation for drive system weight), and the flight controls group. Based AFDD Weight Models 241 on 42 aircraft, the average error of the sum of all parametric weight is 5.3%. The corresponding average error is 6.1% for the structural group (8.6% for the rotor group alone), 10.9% for the propulsion group, and 8.7% for the flight controls group.

Table 27-21. Statistics of parametric weight equations.

group number of aircraft average error (%) wing 25 3.4 rotor blade AFDD82 37 7.7 rotor hub AFDD82 37 10.2 rotor blade AFDD00 51 7.9 rotor hub AFDD00 51 12.2 horizontal tail 13 22.4 vertical tail 12 23.3 tail-rotor 19 16.7 fuselage AFDD82 30 8.7 fuselage AFDD84 35 6.5 alighting gear 28 8.4 engine support 12 11.0 engine cowling 12 17.9 air induction 12 11.0 accessory 16 11.5 propeller AFDD10 16 10.5 fuel tank 15 4.6 gear box + rotor shaft AFDD83 30 7.7 gear box + rotor shaft AFDD00 52 8.6 drive shaft 28 16.0 rotor brake 23 25.1 rotary wing flight controls non-boosted 20 10.4 rotary wing flight controls boost mechanisms 21 6.5 rotary wing flight controls boosted 20 9.7 27–14 References 1) Chappell, D., and Peyran, R. “Methodology for Estimating Wing Weights for Conceptual Tilt-Rotor and Tilt-Wing Aircraft.” SAWE Paper No. 2107, Category No. 23, May 1992.

2) Chappell, D.P. “Tilt-rotor Aircraft Wing Design.” ASRO-PDT-83-1, 1983.

3) “Weight Trend Estimation for the Rotor Blade Group, Rotor Hub Group, and Upper Rotor Shaft of the ABC Aircraft.” ASRO-PDT-83-2, 1983.

242 AFDD Weight Models 15.

10.

5.

0.

error (%) -5.

-10.

-15.

0. 10000. 20000. 30000. 40000. 50000. 60000.

actual weight Figure 27-1. Wing group (AFDD93).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 500. 1000. 1500. 2000. 2500. 3000. 3500.

actual weight Figure 27-2. Rotor group, blade weight (AFDD82).

.

AFDD Weight Models 243 25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 1500. 3000. 4500. 6000. 7500.

actual weight Figure 27-3. Rotor group, hub weight (AFDD82).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 1500. 3000. 4500. 6000. 7500.

actual weight Figure 27-4. Rotor group, blade weight (AFDD00).

244 AFDD Weight Models 25.

using actual blade weight 20.

using calculated blade weight 15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 1500. 3000. 4500. 6000. 7500.

actual weight Figure 27-5. Rotor group, hub weight (AFDD00).

50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 110. 120.

actual weight Figure 27-6. Empennage group, horizontal tail weight (AFDD82).

AFDD Weight Models 245 50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 110. 120.

actual weight Figure 27-7. Empennage group, vertical tail weight (AFDD82).

50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450. 500.

actual weight Figure 27-8. Empennage group, tail-rotor weight (AFDD82).

246 AFDD Weight Models 30.

rotorcraft fixed wing 20.

other RC other FW 10.

0.

error (%) -10.

-20.

-30.

0. 2500. 5000. 7500. 10000. 12500. 15000.

actual weight Figure 27-9. Fuselage group, fuselage weight (AFDD84).

50.

rotorcraft 40.

fixed wing 30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 5000. 10000. 15000. 20000. 25000. 30000.

actual weight Figure 27-10. Fuselage group, fuselage weight (AFDD82).

AFDD Weight Models 247 50.

rotorcraft fixed wing 40.

other RC 30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 2000. 4000. 6000. 8000. 10000. 12000.

actual weight Figure 27-11. Alighting gear group, landing gear weight (AFDD82).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300.

actual weight Figure 27-12. Engine section or nacelle group, engine support weight (AFDD82).

248 AFDD Weight Models 30.

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

-30.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450.

actual weight Figure 27-13. Engine section or nacelle group, cowling weight (AFDD82).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300.

actual weight Figure 27-14. Air induction group, air induction weight (AFDD82).

AFDD Weight Models 249 25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450.

actual weight Figure 27-15. Propulsion group, accessories weight (AFDD82).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 200. 400. 600. 800. 1000. 1200. 1400. 1600.

actual weight Figure 27-16. Propulsion group, fuel tank weight (AFDD82).

250 AFDD Weight Models 25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. 9000. 10000.

actual weight Figure 27-17. Propulsion group, gear box and rotor shaft weight (AFDD83).

25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 1000. 2000. 3000. 4000. 5000. 6000. 7000. 8000. 9000. 10000.

actual weight Figure 27-18. Propulsion group, gear box and rotor shaft weight (AFDD00).

AFDD Weight Models 251 50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 100. 200. 300. 400. 500. 600. 700.

actual weight Figure 27-19. Propulsion group, drive shaft weight (AFDD82).

75.

60.

45.

30.

15.

0.

error (%) -15.

-30.

-45.

-60.

-75.

0. 20. 40. 60. 80. 100. 120. 140. 160. 180. 200.

actual weight Figure 27-20. Propulsion group, rotor brake weight.

252 AFDD Weight Models 25.

20.

15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 50. 100. 150. 200. 250. 300. 350. 400. 450. 500.

actual weight Figure 27-21. Flight controls group, rotor non-boosted control weight (AFDD82).

50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 100. 200. 300. 400. 500. 600. 700.

actual weight Figure 27-22. Flight controls group, rotor boost mechanisms weight (AFDD82).

AFDD Weight Models 253 50.

40.

30.

20.

10.

0.

error (%) -10.

-20.

-30.

-40.

-50.

0. 200. 400. 600. 800. 1000. 1200. 1400. 1600. 1800. 2000. 2200.

actual weight Figure 27-23. Flight controls group, rotor boosted control weight (AFDD82).

all parametric weight structure group 25.

propulsion group 20.

flight control group 15.

10.

5.

0.

error (%) -5.

-10.

-15.

-20.

-25.

0. 5000. 10000. 15000. 20000. 25000. 30000. 35000. 40000.

weight Figure 27-24. Sum of all parametric weight.

254 AFDD Weight Models