Effects of control laws and relaxed static stability on vertical ride quality of flexible aircraft

Page Appendix Table Gust Specifications for the Study Vehicles . . . . . 70 A.1 A.2 B-52H Bare Airframe. Coupled.Ei genvalues 71

A.3 B-52P [A,". M*]. Matrix (I Ox 13) . . . . . . . . . 72

4 . 71 A.4 B-52H Control Matrix, B* (1.0x1) . . . . . . . .

A.5 B . . . 71 -52H White Noise Matrix, G* (3x1) .

A. 6 B-522 Gust . Matrix, A (30) . . 71 g . . ... . . . . .

r .

A.7 B-52H Mode Shapes , . . . . . . . . . ._ . . 73 .

A.8 B-1 Bare Airframe Coupled Eigenvalues 74 . . . . . . . . . 75 A.9 B-1 [A*:M*] Matrix (10x13) .

B-1 . . 74

A.10 Contra] Matrix, B* .(10x,1) . . . .

B-1 White Noise Matrix, G* (3x1) . . . . . . . . . . 74 A.11 . . .

. . . . . 76 Gust Matrix, A g (3x3) . . . . . .

A.12 B-1

. . . . . . . . . . . . . . . . 77

A.13 B-1 Mode Shapes . . .

.

B-52H Case Control List . . . . . . . . . . . . . . . . . 78

B.1

. . . . . . 80

B.2 B-52H Pull State Control Gains . . . .

. . . . . ... . . . . . . . 81

B.3 B-1. Case Control List . . . . .

State Control Gains . . . . . . . . . . . . . . . .

r , B:4 B-1 Full I ^ V7 2.3 B-1 Unaugmented Vehicle Load Factors . . . 17 . . . . . . . . .

2.4 B-52H Pitch Rate Feedback RQI . . . . . . 19 . . . . . . . . .

B-52H 5 RQI Discrimination . . . . . . . . . . . . . . . . . . 20 2.6 SAS Costs for Various Rides . . . . . . . . . . . . . . . 23 .

4.1 Short Period Frequency Requirements . . . . . . . . . . . . 36 4.2 B-52H, Effect of Increased Natural Frequency . . . . . . . . 41 4.3 B-52H Full State Elastic Damping Changes . . . . . . . . . . 44 5.1 Longitudinal Stability Diagram; Major Parameters . . . . . . 48 5.2 B-52H RSS, Dynamics . .

Rigid Body Only, No Elastic . . . . 53 5.3 B-52H Unaugmented Vehicle . . . . . .

.. . . . . . . . . . . 55 5.4 B-1 Unaugmented Vehicle Load Factors . . . . . . . . . . . . 56 B-52H RSS, . . . . . . . . . . .

5.5 Restored Handling, Case #33 57 5.6 B-1 Rigid Body Load Factor Comparison . . . . . . . . . . . 59 Appendix Figure

A.1 Vehicle Stability Axis Sign Convention . . . . . . . . . . . 67

RIDEQ Load .

C.1 Factors Algorithm . . . . . . . . . . . . . . . 85 C.2 B-52H Load Factors, Case . . . . . . . . . . . . . . . 89 #5 .

C.3 B-52H Load Factors, Case #6 . 90 vii

Appendix Page Figure C.4 B-52H Load Factors, Case #12 . . . . . . . . . . . . . . . . . 91 C.5 B-52H Load Factors, Case #14 . . . . . . . . . . . . . . . . . 92 C.6 B-52H Load Factors, Case #37 . . . . . . . . . . . . . . . . . 93 C.7 B-1 Load Factors, Case #4 . . . . . . . . . . . . . . . . . . 94 C.S B-1 Lo yd Factors, Case #18 . . . . . . . . . . . . . . . . . . 95 C.9 B-1 Load Factors, Case #32 . . . . . . . . . . . . . . . . . . 96 viii LIST or SYMBOLS Abbreviations CCV Control configured vehicle CPU Central processing unit NASA National. Aeronautics and Space Administration RMS Root mean square RQ Ride quality RQI Ride quality index RSS Relaxed static stability SMCS Structural mode control system USAF United States Air Force

Referenti al s

A Matrix of physical state coefficients in the vehicle equations of motion ail Matrix element located in the ith row and jth column az Acceleration in the positive z direction B M7,trix of control variable coefficients in the vehicle equations of motion b Wing spas n C* Blended pitch rate and vertical acceleration control law, also called: % star" cg C^-„ter of gravity D Derivative operator ix ^ di Open loop characteristic equation coefficients E{.} Expected value E i Total area under the load factor curve for the ith control case i ei Closed loop characteristic equation coefficients f j eMc State transition matrix t F Sum of all applied forces; Generalized force ft Feet.

G Scalar white noise coefficient vector in the vehicle equations of motion g local gravitational acceleration (9.80 m/sec t (32.17 ftisec2)) H Angular momentum vector In Inches K Gain (scalar or matrix) value(s) which multiplies the control feedback variables k Gain matrix representing the matched coefficient i differences between the closed and open loop characteristic equations L Gust scale length Distance from center of gravity along the fuselage centerline, positive forward, m (ft) Ni Sum of all applied moments Mq Change in moment due to pitch rate ( l } sec M Change in moment due to elevator deflection se 1---^) ( _ F a sec a I ) M Change in moment due to vertical velocity, ? ( ft m-sec -sec w % Change in moment due to acceleration along the z axis I (ft) m m Number of elastic modes included in the load factor expression

x

m Meters K Vertical load factor normalized with local gravity (9's/ft/sec or g's/.3m/sec) n j Humber of s Oates in the vector x i ;i p Linaar momentum vector P W Deterministic row vector which is a function of fuselage station; generates the instantaneous load factor when multiplied times the state vector Perturbation pitch angle rate of change (same as e) q due to variations in vertical t properties along the centerline of the vehicle Md/sec) rad Radian(s) sec Second(s) T Similarity transform matrix which produces the phase variable canonical form matrix t Time (sec) t i Similarity transform matrix, ith column vector U Steady state vehicle velocity u Matrix of control elements W vertical velocity, positive down along Perturbation z axis x Column vector of states associated with the physical output states of the aircraft Column vector of gust state variables x ]ZI Determinant of a matrix, Z Change in vertical force due to vertical velocity z (sec) Z d Change in vertical force due to elevator deflection, rad-sec rad-sec xi Greek a Pe urbation angle of attack (rad) rt Perturbation angle of attack change due to a vertical ^g gust (rad) Non-physical dummy state for the time domain 9i turbulence representation Y Perturbation flight path angle (rad) s e Elevator deflection (rad) Damping Scalar unit white noise, ti,fle domain representation TI e Perturbation pitch angle (rad) Generalized coordinate of the ith elastic mode 9 i Summation operator Q Power spectra function ¢ i (^) Orthogonal elastic mode shape, ith elastic mode value at fuselage position t.

w Frequency (rad/sec) Subscripts Average (mean) value avg BL. Base line value, used with the ride quality metric g Gust related value or variable Maximum value max Minimum value min Natural frequency n Row 5, Row 6 and so on of a particular matrix Rs, Rs,...

Rook mean square value or variable rms SP Short period X11 i Superscripts Q' Transpose of a matrix; the matrix Q is used here only as an example Q Modified matrix Q Derivative with respect to time of Q Q -1 Matrix inverse Q* Augmented matrix; implies the addition of more states or control variables xiii 1.1 Control Configured Vehicles The advent of Control Configured Vehicle (CCV) technology has changed the design process for modern aerospace vehicles and promises important improvements in future advanced control capabilities. CCV technology includes important performance areas such as flutter mode control, relaxed static stability, ride control, fatigue reduction, gust load alleviation, and maneuver load control. This research endeavor will focus on Ride Quality (RQ) and its design sensitivity to various types or states of control philosophies. For this study RQ is defined as the RMS (root mean square) normal acceleration level which the vehi- cle manifests when subjected to cruise condition unit turbulence inten- sities.

2 'd a 's 1. Ride Qu 17ty History and Import Systematic review of flight vehicle design history reveals that payload, range, cost, speed, maneuverability, handling characteristics,' and economic factors were the primary guidelines which structured a final production vehicle. Until the mid-1960's the vehicle RQ .^as determined by structural fatigue constraints, the right kind of passen- ger seat padding, and whatever pilot handling characteristics were required for the vehicle. In retrospect RQ was handled after the vehi- cle was produced by reducing passenger awareness of vibrations or dis- comforts. This was accomplished through various mental or physical activities (including alcoholic beverages) designed as diversions.

Design history is about to undergo another quantum jump in capability.

The availability of mini-computers, fly-by--wire, and active control technologies are necessary for this new capability in design. The melding of optimal solution techniques with traditional design groups like structures, aerodynamics, controls, and propulsion is providing iterative design capabilities which promise great economy, efficiency, and marketability for future aerospace vehicles. More specifically it will not only be feasible to design vehicles to certain RQ specifica- tions, but the physics and economics of design will force RQ to be a design constraint.

From the economic point of vies;, future aerospace designs must provide a range of RQ to the commercial vehicle consumers. The consumer companies or government contractors will then have a management decision capability in determining what the market (or mission) will support or need. The times when the consumers were forced to accept whatever RQ they could get are now past.

The physics of design will force RQ into the picture. Each new design vehicle seems to show one important trend. As total gross take- off weight spirals upward, empty weight remains at least the same as the last generation vehicles or in some cases is dramatically lowered. This phenomena is due to composite materials, better structural design, and high lift technology. It ultimately results in considerably more elastic effects on RQ. This effect is epitomized by the 8-1 bomber used in this study.

These elastic contributions are exhibiting a tendency toward lower undamped natural frequencies. Thus large flexible vehicles of the future will probably be subject to rigid body/elastic mode interactions.

This problem must be investigated before such difficulties are physical- ly encountered. When the short period frequency and lower frequency elastic modes begin to interface, these interactions will certainly affect the pilot's assessment of the handling qualities. Therefore, some lcnical study of these elastic effects and revised design standards must be inaugurated for large flexible vehicles. To date no design criteria exist for RQ in terms of control system specifications. It is hoped that this thesis will be a basic step in that direction.

1.3 RQ Jest Parameters The deficiency and usual omission of RQ design constraints demon-

strates the

critical need for research in this area. Some of the most recent work includes the Boeing B-52E 11 F CCU studies performed under con- tract with the USAF

Flight Dynamics

Laboratory, reference 13. These studies and test flights demonstrated the feasibility of ride smoothing (control of RQ) and snowed that it was compatible with other CCU perform- ance areas.

In another research effort I.D. •?acobsen and others at the Univer- sity of Virginia are gathering evidence for integrated RQ expressions

utilizing

three dimensional acceleration

information. Through statisti-

cal compar i sons of actual turbulent conditions with passenger reactions, their correlation studies are showing trends toward certain theoretical

ride comfort

expressions (reference 11).

More recently Rockwell International introduced the B-1 which has a Structural Mode Control System (SMCS) consisting of two symmetrically opposite vanes on the forward fuselage. The SMCS serves basically as a ride control system. This system is required because of the highly elastic properties exhibited by the basic vehicle. The question arises then, could an active control with appropriate programming utilize normal control surfaces and achieve the same effects? If so, what control philosophy would be best? And last, but maybe most important, what are the cost benefits of this . capability?

To provide a systematic approach toward RQ design, Dr. Robert Swaim proposed and received funding from NASA Dryden Research Center, Edwards AFB, CA., for the study of: 1.

RQ Sensitivity to Type of Control Philosophy 2.

RQ Effects Under Relaxed Static Stability Implementations 3.

Effects of Dynamic Elasticity on Handling Qualities and Pilot Rating 4.

Sensitivity of SAS Designs to Uncertainties in the Mathematical Models of Elastic Modes.

The first two parts of Dr. Swaim°s proposal were used as research topics for this thesis.

1.4 Relaxed Static Stab Control philosophies such as rate feedback are familiar topics with the modern engineer. Relaxed static stability is rather new when used as a "variable" in the preliminary design process. Therefore it deserves a brief introduction.

Relaxed Static Stability (RSS) is herein defined as the reduction or elimination of inherent aerodynamic static and dynamic vehicle stabil- ity requirements. An active control system is used to restore or main- tain desired stability and handling characteristics. This CCV perform- ance area is destined to be the first flight-critical concept used in the next generation of advanced cargo or large commercial vehicles.

Since the supersonic transport studies were completed, various payoff and trade studies on RSS have shown great promise in specific fuel consumption parameters. The RSS concept is currently incorporated in the design of the new F-16 fighter that the USAF selected as its next generation lightweight combat airplane. The trends in fossil fuel availability and price indicate that RSS will necessarily be an integral part of advanced vehicle design.

1.5 Objectives The overall objective of this work is a clear statement of the effects on vertical RQ when control philosophy or RSS changes are in- corporated on the study vehicles. It is hoped that this study will lead to specific RQ design standards that are realizable and practical for future aerospace vehicles. It was necessary that some comparative para- meter be found that would quantify good and bad RQ. Hence, a secondary but fundamental objective was finding a metric that would judge RQ according to some predetermined philosophy.

1.6 Organization This thesis is divided into four investigative chapters. The next chapter is designed as a review of ride quality and suggests the metric which will discriminate between various vertical acceleration curves representing ride quality. Chapter 3 is devoted to a parametric study i `f of control law effects on RQ. Chapter 4 delves into handling quality specifications and their effects. Chapter 5 reviews the static stabil- ity effect and traces its impact on the vertical load factors of the R-1 !

and 8-52H. The summary, results, and conclusions as well as further research recommendations are included in Chapter 6. The three appen- dices provide basic vehicle flight condition data, a complete listing of the study cases, and a section on interesting computational aspects.

' J

-

2.1 Definition and Model.

Since general RQ in terms of passenger comfort has successfully eluded analytical representation to this date, RQ will be defined in this thesis as the RMS load factor curve along the fuselage centerline of the study vehicles. Equation 2.1 represent's the vertical load factor at station Z of the centerline.

N 1 1 Z 1 [U ^(t) + Z e(t) -^ ^ i i (t)?

t) = (Z) ^ (2.1) zWl where; N the local gravity normalized load factor Z the distance from the cg, posit ve forward t time U steady state forward velocity y the perturbation flight path angle e the perturbation pitch angle the ith orthogonal elastic mode shape value at station Z m the number of elastic modes included the generalized coordinate of the ith mode 9 i g the local gravitational acceleration This simplified load factor expression assumes trimmed cruise conditions with no lateral coupling.

Throughout this paper reference will be made to "rigid body only" load factor contributions to RQ. Unless otherwise noted this situation implies the omission of the summation term in equation 2.1. It should be pointed out that the rigid body physical output variables remaining i (33 are still coupled to the elastic degrees of freedom in the vehicle equa- tions of motion. Hence, the vehicle response dynamics are still elastic.

The study vehicle equations of motion are arranged with physical variables as the states shown; in Appendix A. Recalling equations A.8 and A.9: Bu } Gn (A.8)

x = Ax +

^ II (A.9) X ' = 93 C4 n 9 k 1 ^I ^ 2 12 13 14 a g 1 ag qg_.f where: the ith mode generalized coordinate ^i CL the perturbation angle of attack e the perturbation pitch angle the dummy gust state a9t • g the change in angle of attack due to a vertical gust q g the change in pitch rate due to a vertical gust, penetration effect the state vector (13 x 1) u the elevator control (se) TI scalar unit white noise A state coefficient matrix (13 x 13) B control coefficient matrix (13 x 1) G gust forcing coefficient matrix (13 x l) The sign convention used throughout this work is the standard right-handed stability axis system with origin at the vehicle center of gravity and the x axis positive forward along the centerline. A drawing of the stability axis system is shown in Figure 2.1.a. The y axis is positive toward the right wing tip and the z axis is positive down. The load factor will be positive in the positive z direction. The sign (onvention for the generalized coordinates and mode shapes of symmetric fuselage modes is shown in figure 2.1.b, y=ly a — Horizon I j 2.2 RMS load Factors and

Solution Technique

To determine RQ the RMS load factor is used throughout this work.

a response to unit turbulent excitation for mission-

We are seeking

or vehicle-critical flight conditions. For this thesis the B-52H and B-1 flight conditions satisfy the critical condition requirement. To

generate the RMS load factor several matrix and operator manipulations

are required. The load factor equation 2.1 can be reconstituted in a

state variable

format 2.2:

N(.Z,t) = P(t) x(t)

(2.2)

The row vector P(Z) is (1xn) where n is the number of states in the gust augmented x vector. From the

state vector equation A.8 we can expand

slightly to see that: x = Ax + Bu + Gn (A.B) The fifth equation: a = AR5 X + BR U + 0 • n (2.3) The sixth equation: 6 = AR6 x + B,R6 u + 0•n (2.4) The i th generalized coordinate equation: (2.5) i i = AR(6+j) x + BR(6+,j)u + 0• The pitch rate state is the 6th column of A: a = 000001 000000OIX (2.6) For a given control (u

-Kx) and a specified gain K on that

control, the row vector P is deterministic. Only the states are now

subject to statistical uncertainty in that they are forced by random

turbulence modeled as shown in Appendix A, equation A.6.

g The turbulence model used for this study is the Dryden modal in a state vector format. The state vector model is due to Heath and is fully discussed in reference 8. Power spectral density representations of the vertical and pitch gust statistics for clear air turbulence are - utilized in transfer function format to generate first order differen tial equations representing the appropriate aerodynamic force changes.

Appendix A gives the power spectral density forms and the resulting differential equation set.

Squaring equation 2.2, rearranging the terms, and using the linear expected value operator produces equations 2.7 which represent the mean square and RMS load factor:

i"{N 2 } = Pi;{xx' 1P' (2.7a)

(2.7b) Nrms =R^ If we can calculate the covariance matrix of the states, then the RMS load factor value is the square root of 2.7a.

Returning to the state equation A.8, we assume that the gain and control is now specified. Then A.8 becomes 2.8: (2.8) x = A*x + Gn where A* denotes the matrix augmented with the specified control values Constructing the transpose of 2.8 and pre-multiplying by x yields 2.9a.

P,73t-multiplying by x' gives 2.9b: xx' - xx'A*' + XnG' (2.9a) (2.9b)

Xx' = A*'xx' + Gn x'

Adding: D{xx'l - A*xx' + xx'A*' + xnG' + Gn x' (2.10) Under the assumptions that this process is statistically stationary with a zero mean, the derivative of E;xx'} is identically zero. Equation 2,10 can be rewritten: 0 = A*E{xx'I + E{xx',IA 4k ' + E{xn}G' + GE{nx'} (2.11) ~ Bryson and Ho have shown in reference l that E{xn} = 2 and that E{nx'l = G' for a unit Gaussian white noise process. Thus equation 2.11 reduces to a linear covariance equation of the form 2.12: A*E{xx'} + E{xx'}A*' + GG' = 0 (2.12) The solution for a 16x16 system can be obtained on the Purdue University CDC 6500 in approximately 30 seconds. The algorithm has been tested and used up to 16 x 16 matrix sizes. The numerical technique used for this solution is suggested by Gelb and others in reference 2.

The technique in actual use is an unpublished modification suggested by Dr. David Schmidt, Purdue 'University, School of Aeronautics and Astronautics.

The stopping condition used in this algorithm deserves some explana- tion. Since the main diagonal terms are dominant in the equation, the maximum diagonal element ca n the right hand side in every trial solution was discerned and saved. At the next trial solution this value was compared to the last trial value for convergence -tendencies and was assigned a percentage convergence value based on the corresponding element value in the covariance matrix. When the trial solution maximum error was less than 5%, the solution was considered complete. An out- line of this method appears in Appendix C.

Now that the covariance matrix E{xx'l is known, the load factor problem is completed by matrix multiplications as shown in equation 2.7.

As each individual sequence of this solutirti technique was proven, test cases were run on the CDC 6500 to verity computational feasibility and utility. The load factor curves were compared with references 13 ^# and 14 to insure the results were reasonable.

2.3 The Study Vehicles The B-52H and B-1 were chosen for this study because they exemplify the trend toward more elast i c structures for future large vehicles. The B-52, and commercial derivatives thereof, was a member of the first generation of elastic vehicles. Since that era, -improved structural design techniques and composite materials have made possible advanced vehicles like the highly elastic B-1.

The flight conditions were chosen because they represent cruise condit=ions which are mission essential, and because turbulence encoun- ters at low altitudes must be included in design considerations.

The B-52H is used by the U.S. Air Force as a long range bomber. It is 47.55 meters (156 ft) long and has a wing span of 56.4 meters (185 ft). Originally designed as a high altitude bomber, it must now cope with penetration problems by combined high/low altitude profiles. Table 2.1 descri bes the flight condition for the B-52H.

TABLE 2.1, B-52H Flight Condition Mass = 158,757 kilograms (350,000 lbs.j Mach = .55

Velocity = 185,56

meters/sec (608.8 fps)

(856 cg at fuselage station 21.74 meters inches) AItitude = 609.6 meters (2000 ft) The B-1 is currently being test flown in a major pre-production effort by Rockwell International and the USAF. It is designed as the replacement vehicle for the aging B--52 fleet. The advanced structures and integrated technology make this vehicle an outstanding example for load factor contributicns due to elasticity. The overall I-,tgth of the B-1 is 46 meters (151 ft). The reference wing span utilized at the flight condition in Table 2.2 is 41.8 meters (136.7 ft).

.3 TABLE 2.2: B-1 Fli ght Condition Mass = 103,315 kilograms (227,770 lbs) Mach = .85 Velocity = 289.4 meters/sec (949.45 fps) cg at fuselage station 40.67 meters (1061.2 inches) Altitude = 30.48 meters {i0O feet) 2.4 A Ride Quality Index Same metric is now required to compare the resulting load factor curves. The latest work in this area by Rustenburg (reference 9) relates pilot tracking error to vibration levels. The result is a suggested specification for pilot experienced vibrations. Jacobsen (reference 111 is investigating mathematical relationships between sub- jective comfort statements and environmental variables on commercial passenger flights. However, no standardized specification exists for RQ in today's design guides such as MIL.-F--8785B, "Military Specification - Flying Qualities of Piloted Airplanes", reference 16. As a minimum, this thesis is predicated on the ability to compute and compare the vertical RQ for the B-52H and B-1. It is also hoped that a more general usage for the suggested RQI (Ride Quality Index) can be justified.

As a first level requirement, the RQI must be discriminating for the vertical and lateral decoupled cases. Application to the coupled cases would then logically follow the weighting suggested by Jacobsen.

With this requirement, an examination of a typical vertical load factor curve (Figure 2.2) for the B-52H reveals these preliminary observations about the ride in terms of RMS load fac}ors.

ir Observation 1: The area under the load factor curve is a represen- tation of the energy dissipated by the vehicle when disturbed by unit intensity turbulence.

Observation 2: The maximum and minimum Ioad factors indicate the dispersion along the structure of better or worse acceleration conditions.

Observation 3: The mean load factor value indicates some average level of acceleration experience.

G G FT/SEC M1SEC .©1300 ED ,0600 .0400 I CC iy Only to .0200

+5 (METERS)

.0000 vam. ( INCHES) 600. 900. 1200. 1500.

0. 300.

BODY STATION Figure 2.2 Q-52H Unaugmented Vehicle Load Factor Curve.

Mach .55, Altitude 610 m (2000 ft).

AEI I i In addition to these observations some reasonable assumptions can be ='F invoked to decrease RQI complexity.

Assumption 1: The structure is at least quasi-continuous along the fuselage centerline.

Assumption 2: control law complexity is directly proportional to control implementation cost and exhibits discrete cost jumps relative to the type of control policy used.

Assumption 3: The probability for acceptance of an RQI is inversely proportional to its complexity.

Assuming that a merit value can be assigned to various rides, the most important assumption mentioned above is that simplicity encourages acceptance and use. Following this reasoning I propose the ride quality metric as equation 2.13: Nmax + Nmin. + N avg l .

Ei + (2.I3)

RQI = E N N

BL max BL minBL avgBt The terms in 2.13 are defined as follows: total area under the load factor curve for the ith E i control case maximum load factor value for the ith control case N maxi minimum load factor value for the ith control case Nmai = mean load factor value for the ith control case i N a^,g E-^_ j The subscript BL represents the baseline value used for the comparison, N. the load factor value at the j th fuselage lumped mass point, and x j^the number of fuselage mass points.

The baseline values take on special meaning for preliminary design purposes. For example, after the production decision model is well developed, the baseline values could be changed to indicate a marketing value of RQ to the consumer. A ':wief scenario of the marketing aspect will be included at the end of this chapter.

2.5 Testing and Justification of the RQI The success of this metric for preliminary design use must certainly be tied to its ability to discriminate and inform design decision managers about the RQ of a current design iteration. The metric is not intended to be an absolute scaling function. However it must inform the manager about RQ in relation to all the other design tradeoffs.

As an example Figure 2.3 shows the load factor curve for the un- augmented (no stability augmentation system) B-1. This model does not include the structural mode control system which the B-1 utilizes in its present configuration.

These load factor curves were generated as described in equation 2.7. The variables EBL, are assigned to Nmax BL ' 'NminBL$ and Na49BL equation 2.13 from the curve with all four modes. A quick calculation of the RQI for this initial (i = 0) case reveals the RQI is unity. With unit' as the comparative point it follows that the RQI equal to zero would represent a "perfect" ride and an RQI greater than one implies a degraded ride in comparison to the baseline ride.

The first good quality evident about the index is that its com- plexity is independent of the number of degrees of freedom used in the model.. As a matter of fact the computation time is the same for 25 degrees of freedom as for 2 degrees of freedom. Yet the RQI will show freedom required to generate a

parametrically the tradeoff in degrees of

meaningful load factor curve and will identify major contributors to the load factor curve.

Utilizing this concept, the RQI for the B-1 unaugmented vehicle was computed to separate the contributions of each elastic mode and the rigid body with the fully flexible vehicle as the baseline. Figure 2.3 shows the B-1 load factor curve with additional modes added into the load factor expression. Modes 1 and 3 are the major contributors to the total ride. A check in the mode shape data shows these two modes are primarily fuselage bending modes. Referring to Table 2.3 we can see the RQI associated with each mode. Clearly the index discriminates between major contributing modes and the less important ones. Note that a G G

FTISEC MISFC

.2500 --1 .2000 IED .1500

U

cr- o Rigid body L...

X Plus mode 1 Q + Plus mode 2 cr- A Plus mode 3 Q n Plus mode 4 _' .1000 U) fkf .0500 0 20 25 30 35 40 45 (METERS) 0.0000-f- 0. 300. 600.

900. 1200. 1500.

1600. (INCHES)

favorable effect has been gained by including the second elastic mode.

It is barely recognizable on Figure 2.3 but is immediately evident with the RQI. This cancellation phenomenon will be discussed in chapters 5 and 6.

RQI TABLE 2.3 B-1 Unaugmented Vehicle Load Factor Curve Includes: RQI Rigid body 4 elastic modes 1.0000 plus Rigid body plus 3 elastic modes 0.9997 Rigid body plus 2 elastic modes 0.5534 Rigid body plus I elastic mode 0.5643 Rigid body only 0.3406 Two more cases will be demonstrated concerning the RQI discrimina- f tion capability. The first case involves an obviously better ride.

I Figure 2.4 shows the B--52H load factor curve for pitch rate feedback with a gain c F -.2 as compared to the unaugmented vehicle. Both curves modes in the dynamic equations of and include 4 symmetric elastic motion the load factor expressions. The destabilizing gain should, and actuaI- i ly does produce higher load factors.

The major discriminator in the RQI for this case is the area under the curve, Ei. The average, minimum, and maximum value discriminators all promote the lower curve but not with the degree of change seen by example will show the metric's utility the area variable, Ei. The next with intersecting curves.

In the cases where the areas under the load factor curves are nearly equal, some philosophy about RQ must be expressed by the manager.

Throughout this work we will assume that uniform load factor values are highly sloped load factor curves with large minimum/ better rides than Consider Figure 2.5 for the B-52H maximum load Factor differences.

relaxed static stability case. The handling qualities have been restored by a SAS. Which load factor curve is a better ride? Under the assumption at the beginning of this paragraph, the RSS ride looks worse.

.4000

1 .2

Co .3000 w Cr_ U- M CC O 1000 r

.3

5 10 is 25 30 35

40 45 (METERS)

I a i I d I - .0000-1 I I I 600.

0. 300. 900. 1200. 1500. 1800.

f INCHES)

BODY STATION

Figure 2.4 B-521 Pitch Rate Feedback R41. Mach .55, Altitude 610 m (2000 ft).

Y V

FTISEC MISEC

N^y

C) .5000 .4000

't 1.2

X

.3000

.9

a

U

cr_ 42000 CC

J

U3 .11100

10 15 20 25 30 35 40 ^{5

(METERS)

' ^ ' ^ ^ ^ 1. ^ r

I5

--^

.0000 1

i 300. 600. 900. 1200. 1500.

0. 1800. (INCHES)

BODY STATION

Figure 2.5 B-52H RQI Discrimination. Mach .55, Altitude 610 m (2000 ft).

Note that there is an order of magnitude scale factor difference between the B-1 and B-52H curves.

The Mote the RQI attached to each curve computed by equation 2.13.

} in the major discriminators at work here counterbalance each other following way. The minimum value is Morse. The maximum value is better.

t The total area under the curves is nearly the same but favors the un- The augmented case. The average value favors the unaugmented case.

Certainly result is a worse ride for the R5S case according to the RQI.

visual examination infers the same, but closer cases might not be as graphically discriminated.

the obvious modification for the RQI is a weighting assigned

Finall y

If the purpose of the by the design manager for particular effects.

then designed vehicle is transportation of vibration sensitive cargo, more weighting could be applied to the area under the curve and the Alternatively, the acceleration at a maximum value discriminators.

particular station might be of interest and carefully adjusted because The possibilities are unlimited depend- of highly sensitive equipment.

This index will ing on the manager's design problem and his philosophy.

be used -'Co evaluate the B-52H and B-1 RQ in the remainder of this thesis.

Instead of summarizing this chapter's contents the last section will discuss the utility of the RQI in a marketing context.

2.5 Marketing Example RQT_ For this problem I propose an even simpler RQI than equation 2.13.

It represents Consider only the area under the load factor curve.

energy imparted to passengers, cargo, avionics, pilots, and equipment Suppose we wish to show the cost trade- along the vehicle centerline.

offs to a consumer airline board of executives for better passenger The better ride costs more because of increased control require- rides.

But it might generate favorable advertising or selling points ments.

for their customers that would offset the initial direct cost and the lifetime costs.

First the executives need to determine what level of vibration is They can easily calcu- acceptable to their particular passenger market.

late the equivalent of the RQI baseline figure.

The RQI can be computed according to formula 2.14: I _ ^i RQI (2.14) I E BL V where: Fi the total area under the load factor curve for a particular control and handling quality.

f EBL the baseline energy computed by multiplying the human rms perception level (in g's/m/sec (g's/ft/sec)) tines the fuselage length of the vehicle.

The RQI for marketing is now a weighted multiple of the human perception level accelerations. Of course any baseline could be used.

Taking discrete jump values for control law complexity and linear multipliers for handling quality requirements within that control law, the cost function might look like 2.15: (2.15) Cost($) = Control Law Complexity Value + [RQI][HQ] The control law complexity value in 2.15 is meant as an initial hardware cost for implementing the desired control philosophy. The RQI is define) in equation 2.14. The "HQ" function is envisioned as a dollar cost per RQI value. Basically this represents the cost of engineering development, interface, and testing problems which each control system must have resolved. It is-necessarily a function of the amount of ride control required on the vehicle. Thus, the marketing experience of the manufacturers would probably give a reasonable initial definition for this costing variable.

Figure 2.5 shows a fictional example of what costing information might be available to the consumer airline's board. The constant dollar lines represent the initial hardware cost for different control law complexity. For example here we will assume that rate feedback hardware will cost $150,000 per vehicle. R modified state feedback system might cost $400,000 for initial hardware.

Once a type of control law is chosen, the maximum ability to reduce the RQ1 is set. The curves in Figure 2.6 would then represent Barrier indicates LO the beginning of feasible RQ

ti

x for that contro policy +-3 W

-- Rate Feedback

_ Full Stag Feedback U w ra r- ro •r Hardware Cost \ _Hardware Cast RQI 2 3 4 5 6 7 Figure 2.6 SAS Costs For Various Rides. Fictional Cost Curves (For Example Only).

w J c the feasible RQ and its direct cost to the consumer airline. The utility of this representation depends on the board's comprehension of what rides their passengers are willing to buy.

Again, this scenario has been completely fictional. Chapter 3 will

control

show that the RQ is not significantly affected by the type of law. Chapter 4 will sho,i that RQ is very sensitive to handling quality specifications. The "HQ" function in equation 2.15 is therefore closely related to the cost of developing certain handling qualities under all the other design constraints.

Finally, the marketing index might be adapted to a particular value appropriate to avionics maximum vibration levels. The possibilities are limitless. The design engineer and consumer managers would have a tradeoff tool to judge or delineate differences for the ride quality problem.

Chapter 3 RIDE QUALITY SENSITIVITY TO DIFFERENT CONTROL LAWS 3.1 Control Law Descriptions and Study Constraints.

The achievement of a specification for RQ is dependent on a thor- ough understanding of the effects of various contemporary control policies. 'here are wide ranging effects and advantages to be gained by Increasing complexity for control laws. But the cost is not always justifiable. For an instance, rate feedback is used on many vehicles because it is easy to model, costs less to implement, and ultimately results in lower life-cycle maintenance requirements. A more complex control policy was introduced in the 1960's by D.T. Makers (reference 3) and is called %-Star" (C*). The policy is a blend of pitch rate and plunge acceleration such that handling characteristics which pilots seem to favor can be maintained over larger ranges of steady state angle of attack. The design and implementation costs for C* are higher but the handling qualities are more acceptable over a wider range of flight conditions.

Is the RQ better for more complex control laws? Examination of this question requires several constraints and assumptions. First, the handling qualities of the short period longitudinal equations will be maintained as nearly equal as possible between the various control law 1* test cases. Second the -Four elastic modes for both vehicles will be included in the parameterizations. Third, sensing of required physical output will be accomplished at the cg for cases where it is required.

Admittedly this last constraint is certainly sub-optimal but our purpose is not to optimize the ride for a specific vehicle. Fourth, it is assumed that only major control surfaces, such as elevator, aileron, and rudder, are available for control in this model. This specifically eliminates the B-1 Structural Mode Control System (SMCS) for our purposes. Again, the primary objective for this study does not include optimization of a particular aircraft's control system to provide good RQ. The results of this section must be considered in light of the above assumptions.

The control laws investigated include pitch rate, pitch rate/pitch attitude, C*, and full state feedback. Full state feedback is not currently used in aerospace vehicles as a physically realizable feedback law, but its exact pole placing capability insured "perfect" matching of controls for several cases ir, this part of the investigation.

3.2 Control Law Modeling.

.

The common starting point for all of the derivations is the state vector equation A.8. The states, matrices, and white noise are defined in the same manner as the Chapter 2 treatment of this 13th order system.

x = Ax + Bu + Gn (A.8) The control u(t) will be elevator deflections for all cases in this thesis.

itch Rate Control 3.3 P First for piton rate we set u = -Kx 6 , where x 6 In expanded form the pitch rate control looks like 3.1: .... al5 (a lb - KB 1 ) a17 all .... al,l3 .... a 25 (a26 - KB2) a21 a27 .... a2,I3 X - -KB alo,l .... a 10,5 (a 10,6 x + Gn IO ) a10,7 .... a 10,13 p . . .. . . . . . . . . . . .

(3x3)

...............

0 Ag Q...............

(3.l ) i i Selection of a gain value K results in a deterministic A* matrix which possesses certain coupled eigenvalues. Table 3.1 shoes the gains, short period damping, and short period natural frequency for each case used on the study vehicles. The frequencies are shown in rad/sec. A reference case number will be shown in all the tables. Some cases were inconclusive or irrelevant for certain parts of this investigation.

Therefore the tables will not be strictly sequenced according to case numbers. The four elastic mode free-free natural frequency and damping values remain the sage as in Appendix A and are not listed in Table 3.1.

TABLE 3.1: Pitch Rate Feedback Parsmeterizations B-52H m B-1 GATH cu 5P SP CASE nSP nSP CASE # # 1 2.806 .5157 0. 0. .4708 2.790 1 2 2.635 .4040 -.2 -.1 .2650 2.981 4 2.970 2 3 .6170 +.2 +.l .6551 3.168 +.4 3 4 3.126 .7120 +.2 .8240 2.594 3.4 Pitch Rate/Pitch Attitude Control Utilizing this same procedure the parameterization for pitch rate/ pitch attitude was computed under the control equation 3.2. The artifi-

a

cial state xo was generated by adding the integral of to the state system, making it 14th order.

u ^ -Kzxq - K2 x (3.2) where K, Pitch attitude gain Kz Pitch rate gain Table 3.2 shows the cases chosen for testing under this control law.

3.5 Blended Pitch Rate anal Acceleration (C*) This control philosophy is used mostly with fighter or maneuvering types of vehicles. However, the purpose here was to parameterize ovAr common control laws and so it is included in the investigation.

} TABLE 3.2: Pitch Rate/Pitch Attitude Parameterizations B-52H 1K1

K2-m

SP nSp CASE # 10 .25 -.2 .3590 2.754 11 .25 .5 .7230 3.272 12 .7060 3.497 .75 .6 13 .25 .1 .5280 2.982 14 .25 -.3 .2970 2.677 B-1 CASE # 2.955 15 .1 0. .4230 16 .2 .7760 3.288 .1 17 .3 .2 .6950 3.550 18 .4 .3 .8170 3.817

u = 2 0 (3.3)

- K j az -- K

The values chosen for load factor evaluations are shown in Table 3.3.

Since the cg plunge acceleration is approximately equal to U(e -- a}, the form of equation 3.3 for implementation was actually equation 3.4: E u = - (K1 U + K2)xa (3.4) K,U x 5 where K1 Acceleration gain K2 Pitch rate gain 3.6 Full State Feedback In this procedure the roots of the desired characteristic polyno- mial were specified. Then the difference between the open and closed- These values loop characteristic polynomial coefficients were computed.

were transformed under the inverse of the phase variable canonical trans- formation matrix to find the physical state variable gain matrix, K.

'i 'i 'i TABLE 3.3: C* Parameterizations B-52H K, K2 c^ nSP SP CASE # 5 0. 2.521 -.0005 .3403 6 -.0005 .3 .5140 2.779 7 --.0001 0. .4820 2.751 8 2.8U6 .0003 -.2 .5060 9 .8560 3.402 .0004 .5 B-1 CASE # 5 -.0005 .3220 2.483 .4 6 2.504 -.0004 .3100 7 -.0004 .5230 2.685 .4 8 -.0004 .7160 2.862 .5 9 .0001 0. .6410 3.013 10 .0001 .1 .8080 3.204 .4340 2.881 11 .0003 --.3 12 .6120 3.084 .0003 -.2 13 .9005 -.4 .5840 3.160 14 .0005 --.2 .8890 3.570 The procedure is mathematically outlined below. Beginning with the original state variable system: x = Ax + Bu + Gn (A.8) Uelete the three gust state equations and rewrite the equation as 3.5: x = Ax + Bu + Agf (3.5) x where: A and B do not include the gust coefficients Agf is the aerodynamic gust force coefficients from A in A.8 x are the longitudinal gust states Now we perform a similarity transformation on x such that x = Ty or y = T-1x.

Substituting in 3.5 yields T-1 A (3.6) y= T y+ T -1 B u+ T -1 A gf x ..1 The matrix [T AT] is called the phase variable canonical farm and has the expanded form: - CT - 'AT . (3.7) . .0 0 .^.•0 1 - d 1 . . . . . . . . dn-1 -do . .

The coefficients, d i , relate to the characteristic equation of A as in 3.8: n-1 + ... + d i s + do (3.8) IsIY-A1 = s + do-l s The canonical form 3.7 provides an easy method for determination of the physical variable gain matrix, K.

Let the row vector be equal to the differences of the matched

Ilil

coefficients of the open and closed loop characteristic equations. Then T_ I Bk is an nxn matrix which is null except the last row: 0 . . . . . . . . . . . 0

[T"'Bkj

(3.9) E ko k, . . . . . . .

kn-i ^.

The closed Ioop characteristic equation coefficients are: + n t e n-1 s n-i f + ... e l s e p (3.10) IsI-A*1 = s Therefore the 11 i matrix is related to the open and closed loop co- efficients by: e i = d i - k i . Substitution for u = -Kx = -KTy yields: [T - 'Bk i ] = [-T -; BKT] (3.11) But, [T - 'B]' _ ^0 ... 0 1 1 which implies: k T-1 (3.12) K = L d Hence with the characteristic equation coefficient differences known, the physical state gains K can be computed. The similarity transform T is computed using an algorithm suggested in reference 4. Given the control coefficient matrix B, the state coefficient matrix A, and the open loop characteristic equation coefficients, d i , we compute T as follows: Let T = [t,:t2.....tnI where t i are column vectors. Then t o = B t i _ I = At i + d i B (i=n,n-1,...,2) (3.13) A computational problem arose with the accuracy of the T matrix. Appen- dix C, section C.4 discusses the problem and the solution method used as a result.

The gains and associated short period damping and frequencies are logged in Table M fir the B-52H and Table 3.5 for the B-1. All of the gains are reasonable in magnitude and in that sense represent physically realizable system values,. Two exceptions to this statement are the B-1 cases #21 and #22. In these two cases a large elastic damping increase was imposed for investigative purposes. The high gains re q uired were not unexpected nor were they intended as physically realizable.

3.7 Control Law Va riation Results For the cases where damping and natural frequency were nearly identical, the load factor curves were nearly identical. They were so close that numerical 4 digit load factor hardcopy had to be compared to find any differences. As a result the RQI was utilized on these cases and the results are compiled in Table 3.6.

TABLE 3.6: Control Law Parameterizations, B-52H B-52H Type w CASE # Control nSP ASP RQI

1 e 2.806 .5157

I.0000

C* 2.779 .5140 0.9813 24 Full State 2.806

.5157 0.9991

12 e/6

3.497 .7060 0.7992

Full State 3 400 .8175 0.8128

Cases 12 and 31 provide a reasonable implication that the same equivalence phzanomena exists for e/e controls. In all cases shown the RQI difference is less than two percent. Appendix C, section 5 contains the load factor plots for cases 6 and 12 so that the reader may compare them to cases 1 and 31, respectively. Cases 1 and 24 in Table 3.6 are both for the bare airframe (no SAS), but the pitch rate state equation 3.1 was used for Case 1 and the full state formulation of equations 3.6, 3.7, and 3.9 used for Case 24. These served as check cases for the two formulations on identical dynamics.

I V TABLE 3.4: B-52H Full State Feedback aarameterizations w B-52H K1 K2 K3 K4 K5 K6' K^ K 8 kg Kla SP CASE # nSp 24 2.806 .5157 0 0 0 0 0 0 0 0 0 0 2.806 25 ,3250 -.004 .06 -.01 .02 -.34 .28 -.0006 -.007 .002 -.001 26 2.806 .8176 .006 -.09 .03 -.03 .54 -.44 .001 .01 -.002 .001 27* 2.806 .5157 -.005 .59 -1.4 -.01 .19 -.02 .001 -.08 .329 -.024 28* 2.806 .3250 •-.01 .73 -1.73 -185 -.14 .25 .009 -.08 .313 -.03 29* 2.806 .8176 .004 .38 -.85 -1.0 .71 .003 .35 -.02 -.45 -.08 30 3.000 .8176 .008 -.11 .03 -.03 .34 -.52 .001 .015 -.003 .002 31 3.400 .8175 .01 -.04 -.003 .002 -.13 .04 -.14 -.68 .001 .022 *Increased damping on elastic modes.

TABU 3.5: B-1 Full State Feedback B-1 K K K K K K K K K 10 1 2 3 ^ 5 6 7 8 9 K wn ASP CASE • SP 19 2.'90 .4708 0 0 0 0 0 0 0 0 0 0 20 2.790 .4708 0 d 000 0 0 0 0 0 0 0 2.710 21* .4707 -.01 -.76 -.I9 659. -1.04 28.97 -.17 -.002 .02 -.003 22* 2.790 .4707 -.02 -_06 -.24 894.6 -1.92 -.21 -.003 .03 -.002 27.43 23 2.790 .6551 -.0008 .003 .001 -1.73 .06 -.08 -.00003 .0001 .23 -.00007 24 2.980 .6131 -.000 .003 .001 7 - 1.72 -.02 -.08 - . 00002 .0001 -.00007 .22 25 .2296 -0- -.002 .003 .001 -.OT .66 -.76 .0001 .0002 --.000036 .000096 *Increased damping on 1st and 3rd elastic modes.

5 ^.f Chapter 4 RIDE QUALITY SENSITIVITY TO HANDLING CHARACTERISTICS 4.1 Handling Quality Specifications The recognized compendium of handling quality information is reference 16, NIL-F-8785B (ASG) "Military Specification -- Flying Qualities of Piloted Airplanes". This document divides the vehicle handling quality design specifications into the following: Kind of airplane - Class Job to be done - Category of Flight Phase Degree of job accomplishment Level For the study vehicles used herein the following definitions will apply: Class III - Heavy bomber Flight Phase - Category B (CR-cruise) Level 1 - Qualities are clearly adequate for the mission flight phase.

Short period damping and natural frequency are very important to the pilot's perception of response. If the damping is too high, then _t the vehicle will be sluggish and slow to respond to the pilot inputs.

If the damping is too low, the pilot will over-control the system because it reacts too quickly and Will overshoot his desired output. This can develop into a "PIO" or pilot induced oscillation. As a result, the 5 2.0 MIL-F-8785B short period damping limits are in the range .3 < ^Sp for the B-1 and B-52H types of vehicles.

The short period frequency is also important in the pilot's percep- tion of a "good flying" vehicle. As the frequency decreases, the pilot must introduce a phase lead into his commanded inputs to maintain precise control. At the upper end of the acceptable short period frequency spectrum the pilot experiences abrupt control requirements to maintain precise vehicle attitudes.

Figure 4.1 shows the appropriate frequency requirements for the B-1 and B--52H. The unaugmented vehicle values are shown appropriately placed within the level 1 acceptable boundaries. This graph is taken from the reference 16 section on longitudinal specifications.

4.2 Variations in Handling Qualities In each control law described in chapter 3 the short period frequency and damping was adjusted within the boundaries specified above.

Lists of the variations induced on each vehicle are outlined in Table 4.1 for the B-62H and Table 4.2 for the B-1. Certain cases are again not included in the lists because they did not apply to this investiga- tion. Tables 4.3 and 4.4 list the data according to decreasing RQI.

The expected trend is substantiated by these cases. Increasing damping and increasing frequency generate better rides.

To further verify these expected trends full state feedback was used to increase the short period frequency while maintaining the same damping. Figure 4.2 shows the apropos change in load factor for the increased frequencies. The ride is definitely improved with increased frequency.

The trend for damping is the same. Table 4.5 shows the data for the B -52H sorted with respect to damping. Table 4.6 shows the same sorting for the B-1. Increased rigid body damping generally produces a better RQI.

4.3 Ride guality for Increased DamRing of Elastic Modes On the B-52H a full state feedback program was implemented to find the effect of increasing the coupled damping on the elastic modes to a .1 minimum value. Table 4.7 shows the changes involved. The elastic frequencies were held constant at the bare airframe coupled value throughout these cases.

Figure 4.3 shows the percent change in ride experienced for increased elastic damping for the two short period cases of Table 4.7.

At this juncture the expected trend is reversed. Increased damping on the elastic, modes produced an eight to ten percent worse load factor at most stations than the same case without increased elastic damping.

Y

a

V i

U

C tlJ C a

L

LL.

a

•r a 4J s- ..0 tyk y 1. "0 100 Figure 4.1 Short Period Frequency Requirements TABLE 4.1; B-52H Handling Quality Variations - B-52H wnSP SAS ASP CASE # TYPE 2.806

1 6 .5157

2 e .4040 2.635 2.970

6 .6170

3.126 4 e .7120 2.521 5 C* .3403 .5140 2.779 6 C* .4820 2.751 7 C* 8 .5060 2.806 C* 9 .8560 3.402 C* .3590 2.754 10 a/ e .7230 3.272 11 a/e .7060 3.497 12 e/e .5280 2.982 0/e .2970 2.677 14 0/6 24 Full State .5157 2.806 .3250 2.806 25 Full State 2.806 .8176 26 Full State .5157 2.806 27 Full State .3250 2.806 28 State Full .8176 2.806 29 Full State .8176 3.000 30 Full State .8175 3.400 31 Full St;:te TABLE 4.2: B-1 Handling Quality Variations B-1 SAS 9SP Wn,P CASE # TYPE 2.790 I e .4708 .6551 2.981 2 a e ,8240 3.168 4 a .2650 2.594 C* .3220 5 2.483 2.504 6 C* .3100 C* .5230 2.685 8 .7160 2.862 C* .6140 3.013 9 C* 3.204 10 C* .8080 2.881 I1 C* .4340 .6120 3.084 12 C* C* .5840 3.160 3.570 C* .8890 I4 2.955 0/6 .4230 .7660 3.288 16 0/e .6950 3.550 17 8/a A 0/0 .8170 3.817 2.790 .4716 19 Full State .4707 2.790 20 Full State .4707 2.790 21 Full State .4707 2. i90 22 Full State .6551 2,790 23 Full State 2.980 .6132 24 Full State .8240 3.168 34 Full State 2.790 .4708 35 Full Stake TABLE 4.3: B-52H RQI Sorting Results RQI B-52H DAMPING NATURAL CASE # FREQUENCY 5 1.36964 .3403 2.521 14+ .2970 2.677 1.30822 28t 1.23552 .3250 2.806 29f .8176 2.806 1.20517 2 1.20014 .4040 2.635 25 2.806 1.18949 .3250 10 2.754 1.14014 .3590 26 .5157 2.806 1.09571 27 1.08706 .5157 2.806 7 1.04233 .4820 2.751 8 2.806 1.02555 .5060 1 1.00000 .5157 2.806 6 2.779 .98127 .5140 30 .95562 .8176 3.0'?0 2.970 3 .90211 .6170 13 .89950 .5280 2.982 3.126 4 .85491 .7120 .8560 3.402 9* .84659 .81461 .7230 3.272 I1 31 .81285 .8176 3.400 3.497 12 + .79921 .7060 as it was an *This case does not follow the general trend versus destabilizing gain effects investigation into stabilizing for C* feedback.

Elastic mde suppression cases which do not follow the trend and will be discussed separately.

section C.5 plots.

+These cases are included in the Appendix C,

TABLE 4.4: B-1 RQI Sorting Results

DAMPING NATURAL

B-1 RQI

FREQUENCY

CASE #

22f 5.66966 .4707 2.790

21 t 5.61414 .4707 2.790

4 + 1.06240 .2650 2.594

2.881

11 1.04007 .4340

3.160

13* 1.03435 .5840

6* 2.504

1.02579 .31001

5* 1.02025 .3220 2.483

15 1.00105 .4230 2.955 12*

1.00095 .6120 3.084

1.00045 .4708 2.790

1.00000 2.790

1 .4708

20 .99986 .4707 2.790

19 .99985 .4716 2.790

2.980 24 .97706 .6132

9 3.013

.97535 .6410

.655I 2.790

23 .97443

2 .96498 .6551 2.981

34 .96300 .8240 3.168

IG .95037 .8080 3.204

.8240 3.168

3 .94063

17 .94061 .6950 3.550

16 .93706 .7760 3.288

2.862

S .93601 .7160

3.817

18+ .92292 .8170

is destabilizing gain effects.

*Cases involve stabilizing

'Elastic mode suppression cases.

load factor plots in Appendix C,

}These cases are shown as

section C.5.

S. =.817

.4DDD

Z,^4b

W"sp 1.2

C3

x

=

.617

ESP

.3GOO

,^ 3, 000

Y =.

U

0)0;.5 .

70V Cr CD .2000 r6 ^ J

5 1 10 1 ^ S `1 20 25 30 35

40 45 (METERS)

.0000 -

0. Sao. 60 0. 900.

1200. 1500. 180a. (INCHES)

BODY STATION

Figure 4.2 B-52H, Effect of Increased Natural Frequency. Mach .55, Altitude 610 m (2000 ft).

RQI Sorted By Damping Value TABLE 4.5: B-52H NATURAL DAMPING RQI B-52H CASE # rREQUENCY 2.677 .2970 14 1.3082 2.806 1.1895 .3250 2.806 .3250 1.2355 2.521 .3403 1.3696 2.754 .3590 10 1.1401 2.635 :4040 2 1.2001 2.751 .4830 7 1.0423 2.806 .5060 1.0256 2.779 .9813 .5140 2.806 .5157 27 1.0877 2.806 1.0000 .5157 2.982 .5280 13 .8995 2.970 .6170 3 .9021 3.497 .7060 12 .7992 3.126 .7120 .8549 3.272 .7230 .8146 3.400 .8176 .8128 3.000 .9556 .8176 2.806 .8176 29 1.2052 1.0957 .8176 2.806 3.402 .8560 9 .8466 -, .

I - ^... q . -.- TABLE 4.6- B-1 RQI Sorted By Damping Value B-1 RQI DAMPING NATURAL CASE # FREQUENCY 4 1.0624 .2650 2.594 1.0258 6 ,3100 2.504 1.0202 5 .3220 2.483 15 1.0011 .4230 2.955 11 1.0401 .4340 2.881 5.6141 .4707 2.790 .9999 .4707 2.790 5.6697 .4707 2.790 35 1.0004 .4708 2.790 1.0000 .4708 2.790 19 .9998 .4715 2.790 1.0344 .5840 3.160 1.0009 .6120 3.084 .9771 .6132 2.980 9 .9754 .6410 3.013 2 .9649 .6551 2.981 .9744 .6551 2.790 17 .9406 .6950 3.550 8 .9360 .7160 2.862 16 .9370 .7760 3.288 10 .9504 .8080 3,204 .9229 .8170 3.817 .9630 .8240 3.168 3 .9406 .8240 3.168 45.00 rill ' ED 30.00 W,,

U

^,

S P

LL-

Cas e #28

.34e 2.81 r--)

Q

Case #25 81.

Z.

.325 J 19.00 rill w CD

.00

2.S j Case #1 ,51&

z

U

H

,9i7 2 . 81 Case #29

LU

U

W-15.00

Case #25

.817 z.8i

.^^

3.0 0 Case #30

(METERS) -30.00 0. 300.

600. 900.

1200. 1500. 1800.

{ TNCHES 1

BODY STATION

Figure 4.3 B-52H Full State Elastic Damping Changes. Mach .55, Altitude 610 m (2000 ft).

TABLE 4.7: B-52H Damping Changes for Elastic Modes B-52H SHORT DAMPING CASE # PERIOD SORT RQI FREQUENCY PERIOD MODE 1 MODE 2 MODE 3 MODE 25 2.806 .3250 .16 .08 .10 .06 1.189 26 2.806 .8176 .16 .08 .01 .06 1.095 28 2.806 .3250 .16 .10 .10 .10 1.230 29 2.806 .10 .10 .8176 .16 .10 1.200 This RQI trend reversal can be explained by the fact that the RMS elevator deflections were higher for the increased elastic mode damping cases. A check of the contributions to the load factor values showed that the elastic modes were damped more and actually contributed less to the overall ride. However, the rigid body RMS levels increased as a penalty and produced the degraded ride. A similar set of cases was run on the B-1 with identical results. This phenomenon deserves more research attention. There may be fatigue and/or RQ tradeoffs that would enhance stvuctural life by useful cancellation of ride contribu- tions due to elastic interactions.

Cases #21 and #22 in the B-1 parameterizations represent arbitrar- ily chosen increases in the coupled elastic damping values. The effect noted on the B-52H is much more pronounced on the B-1. Table 4.8 shows the B-1 damping change parameters and lists the RQI associated with each case.

TABLE 4.8: 3-1 Damping Changes for Elastic Modes B-1 SHORT MPING DA CASE # PERIOD SHORT RQI FREQUENCY MODE 3 MODE 4 PERIOD MODE 1 MODE 2 2.790 .4708 .009 .199 1.00 1 .047 .022 .090 .199 5.61 21 2.790 .4708 .100 .022 2.790 5.66 22 .4708 .200 .022 .100 .199 Again the increased rigid body accelerations far outweighed any gain in ride supplied by increased damping of the elastic modes.

Furthermore the coupling with the elastic modes from the increased rigid body ride actually destroyed the favorable damping affect in the elastic contributions to the load factors. The rigid body load factor increased by a factor of ten when Case #1 was compared to Case #21.

RQ relationships to the short period handling qualities are very important. The investigation has shown that increased damping and/or frequency have produced generally favorable effects on the ride.

Attempts to damp elastic modes using the primary elevator control resulted in degraded RQ because of the increased rigid body accelera- tions introduced.

-i } Chapter 5 RIDE QUALITY SENSITIVITY TO RELAXED STATIC STABILITY 5.1 Aircraft Sta _ tic Stability The definition of longitudinal static stability can be related to a pitch stiffness quality as in Etkin's book, reference 5. If a vehicle is disturbed from an equilibrium angle of attack in a positive (nose up) direction, the total moment generated, M na, is negative and thus tends a to return the vehicle to equilibrium. This case represents positive static stability. Figure 5.1 ,gives a pictorial representation of the forces and moments involved. The pitching moment is really a function cc (: of the stability derivative Cm a , where Ma = p m . All stability 2IDz derivatives used in this study are defined in consonance with reference 6. A summary of the stability derivatives used in this chapter is given in the symbols listing a •^ the front of this thesis.

Vehicles have traditionally been designed with positive static stability and cargo loading restrictions which maintain a minimum degree of longitudinal stability. Vehicles then need varying amounts of lift generated by the horizontal stabilizer to maintain cruise condition level flight. This results in a drag penalty that all vehicles in cruise conditions have had to accept to date. If, however, we were to of modify the position the center of gravity and the wing-body aero- dynamic center such that Cm. -+ 0, then less lift would be required to offset the moment due to the wing-body combination. Only a small lift, if any, would be required from the horizontal tail. This reduces the tail drag and results in better cruise efficiency. Increased cruise efficiency begets fuel economy. As fuel prices rise and availability declines, RSS will therefore be required as a design point on future vehicles.

Mq- I tail lift odvnamic center mic center meters 5.2 Relaxed Static Stability RSS refers to the relaxation of pitch stability requirements. The resulting fuel economy is one factor that new generation vehicles will have to incorporate in order that they remain competitive or even practical.

RSS -:an be implemented in several ways. First the tail surface area can be reduced. This der7reases the moment restoring farce and has the artificial effect of moving the neutral point toward the cg. Its value is purely theoretical. Second the center of gravity can be artificially moved toward the tail by fuel or cargo management. This method could be used on present vehicles. lastly the engines and other major structural members of preliminary designs may be repositioned for better performance. Preliminary design RSS offers the most efficient implementation in my opinion. Any of these implementations may result in a neutral or statically unstable vehicle. The first two methods represent the techniques that will be used in this study.

RSS is basically a rigid body phenomnon, so only the short period equations for the rigid vehicle will be used to derive the "relaxed conditions". More elaborate models are possible using a variation of the technique developed by Swaim and Fullman in reference 12.

5.3 Equations of Motion in Stability Derivative Format The time domain, short period, longitudinal equations of motion for rigid vehicles are given in 5.I: A W

w -- Z w w - U z d a e + Zwwg

e

(5.I )

M kr - Mww - Mq 6 + = M s + Mw g g w v^ + S + Mwwg e e where w(t) Perturbation plunge velocity, positive in positive z axis direction wg (t) Gust velocity, positive in negative z axis direction O(t) Perturbation pitch angle, measured from the x--y plane, positive nose up 6 e Elevator deflection in radians, negative in B-1 and B-52 data for nose up changes If we lei; w = Ua and solve for accelerations on the left hand side, then 5.2 results: de U 0 a Z U 0 Z U U^ n 6 Se w + }^ a (5.2) --N1U 1 e - %U M q a Ma MW V+Mq j ag e --9a olving foron the left hand side and noting that q g = a g we get 5.3: 0 rs e zw l zs /U z 'ag - + e +M w

a U (%z

w ) (U%+M q ) (%Za +M6 ) U / %zw+Mw) %U+MQ e e 119.

L _ (5.3) Equation 5.3 is the familiar control Form 5.4: (5.4) x = Ax+Qu+A g x The numerical values for all of these coefficient matrices are known in the B-1 and B-52H equations. From 5.3 it is easy to deduce that CL.,, C O , CM are the stability C E a, Cma , Cmq , CL q , CE &e , and Cm se derivative coefficients to be considered under RSS. Matching the appropriate coefficient combinations to the numerical equations of motion, and assuming that Z remains constant, we can generate the basic ,., slues for all of these coefficients. In preliminary design, the aero- dynamics group would provide estimation for these derivatives after the planfonn and type of airfoil were established. Generation of the co- efficients as described above was required in this work because the data did not include these numbers.

5.4 Static Stability Effects on Stability_ Carivatives Relaxed How do these terms Mary as tail volume coefficient or cg are changed to induce RSS? 7o keep the analytics tractable in the investiga- tion, it is assumed .that the vehicle will maintain the same lift coefficient, weight, and cruise conditions. These assumptions are reasonable considering the accuracy of the original data. They are also reasonable if changes in the tail surface area or cg position are used that do not significantly alter the overall vehicle performance.

l USAF DATCOM (reference 7) descriptions of the stability derivatives were used to generate the computerized equations of motion for the RSS cases. Table 5.1 lists the stability term equations for the 8-52H.

Table 5.2 has the B-1 stability coefficient changes. Recalling equation 5.3, it is seen that the baseline or "unrelaxed" values of Z w and Mq are explicitly specified by the actual vehicle equations of motion. The remaining derivative values can be found by simultaneous solution of the matched coefficient equations. These values can be subjected to artifi- cial RSS implementation, and by reversing the above procedure, a new set of equations of motion can be derived.

TABLE 5.1: B-52H RSS Stability Derivatives -3 M w = 8.365 x 10 - 4.655 x 107[St][Fcg] -3 Mw = -4.150 x 10 2.768 x 10-10[St][Fcg]2 7 [S`] [ F 12 = 4.217 - 6.915 x - M q 10 F cg = 66.55 - [x,,, .25][22.961 Z a = --3.756 x 10 - '[S t] + [ M^ = -16.058 - 1.228 x 10 - '[S t ] x cg + .856][12.129 - 4.239 x 10-''(St)] e Zw = -1.352 Note: S t is tail planform surface area x is the cg position from the wing leading edge normalized by cg the mean aerodynamic chord Insufficient data was available in the Rockwell International docu- ments (reference 15) to completely specify the derivatives for the B-1 from the equations of motion as given. As an example, the tail lift coefficient derivative with respect to alpha was assumed to be the nominal 2w value because it was not specified its the Rockwell document.

r This assumption was not critical in the study since it did not affect trend information except as a linear scaling factor. However, any actual design implementation of this technique would require the specifi- cation of this derivative.

TABLE 5.2: B-1 RSS Stability Derivative Coefficients [C L + C D ] 3.905 a Cm = 3.588 - 2.164 x 10-4[St][FcgI M Cm. = 5.288 - 1.611 x 10-5[StIEFca12 a Cm = 23.045 - 5.479 x 10-5[StI[Fcg19 q -5[5t] = 1.895 x 10 C se = -2.528 [5.462 x 10 -` ^ _ (x -.25)(15.33)][S t ] + 6.148 x 10 -2 cg CM e (xcg-.25)15.33 Fcg = 46.45 - (x cg - .25)(15.33) Note: S t and x cg are defined as in Table 5.1 5.5 Relaxed Static Stability Rigid Body dn1y Effects Using the same algorithm for RQ, but rigid body 2 degree of freedom information only, a number of load factor plots were run for the B-52N.

Figure 5.2 shows the B-52N results for decreasing static stability.

Note the RQ effect that the pitch acceleration term is contributing less as the cg maven closer to the neutral point. Finally, at neutral stabil- ity, the only acceleration effect is the vertical acceleration generated by the cg movement.

G G M/SEC FT/SEC .14 -- .044 .09 ,028 • r r r • ^ rr U - ^ r

d t f

d

-- ^ .ter, • .^^ .. .^ • .^ . ^^ • ' Q

r r ^

r

N

. . C g @ . . • chard, 600 ft tail 40% /A^ A/ .04 f 012 cg @ chard 40% chord 30% cg @ .......

chard 25% - cq @ 45 (METERS) 35 5 15 25 A I 1400 1800 (INCHES) b0 0 0 1 0 0 1 0 -- FUSELAGE STATION lo Elastic Dynamics. Y, Figure 5.2 B--52H RSS, Rigid Body On Ln w

5.6 Relaxed Static Stability With Restored Handling utilities

difference between

For the first time in this study the significant

the elastic properties of the B-1 and B-52H will affect the outcome.

Figure 5.3 shows the B--52H load factor curves for the rigid body plus 4 mode case and for the rigid body only. Even at the lower end of the curve, the elastic contribution to the ride is a maximum of 15%.

Figure 5.4 shows the same unaugmented situation for the B-1, however the rigid body contribution is now the minor one. To be more exact, at body ride is only 5% of the total station 72 the rigid body contribution to B-1 ride. At other points on the fuselage the rigid body contribution is higher. Yet, the net effect is very little rigid body contribution to the overall ride on the B-1 for this flight condition.

The RSS vehicle equations of motion we re augmented by full state feedback systems to restore the original unaugmented airframe handling characteristics for both vehicles. The resulting load factor curies were computed and compared to the bare airframe load factors.

Table 5.3 shows the RQI for each case and its associated data. In general the restored handling characteristics generated worse ride con- ditions because of the elevator activation required and its effect on the rigid body dynamics. Case #33 deserves special attention. The cg was 3.4 feet further aft in this case. The restored handling ride was actually better than the bare airframe ride as shown by the RQI.

Figure 5.5 shows the Case #33 rigid and rigid plus four modes load factor curves in comparison to the bare airframe vehicle. mote in the figure the four modes and the rigid body have favorably interacted because the fully flexible load factor curve is lower in several areas.

This favorable interaction has been prophesied and discussed in the literature but no practical technique is yet available for utilizing it.

The favorable ride effects neat- the tail are due to decreased moment arm effects in the moment equation. For a fairly rigid fuselage, RSS with restored handling characteristics generally degraded the B-52H ride.

One phenomena on the less elastic B-52H went unnoticed but proved to be important on the B-1. As the static stability is reduced on the B-1 the rigid body dynamic effect produced by the elastic modes is

G G

FT/SEC MISEC

.5000

.4000

CD

x

ar

.3000

1f^^ U-

M

t7 ^y ^y ^y .2000 J J CE

z .1000

+s (METERS) .0000 U. JUU.

bUU.

900, 1200.

1500.

1800, (INCHES)

BODY STATION

Figure 5.3 B-52H Unaugmented Vehicle. Mach .55, Ln Ln Altitude 610 m (2000 ft).

C ti G G Ln FT/SEC M1SEC .2000 .060 .1500 .045 0 Rigid body plus four modes Rigid body plus three modes ^— a v body plus two modes Rigid + c Rigid body plus one mode o Rigid body only .1000 C .030 o

J

U] .0500 .0155 5 0

20 25 30 35 40 45 (METERS)

0.0000 0. 300. 600. 900. 1200.

1500. 1800. (INCHES)

BODY STATION Figure 5.4 B-1 Unaugmented Vehicle Load Factors.

ME:. , - .85, I Altitude 30 m (100 ft).

i r • f

r

V G FT/SEC MISEC

.1000 T

3Q

.0800--.24

U

U

CE E.[_.

ED .01300 .18

U

_J J cC

U

N

^- .0400--.12

w

Unaugmented Vehicle ua d body plus four modes Rigi .0200 ,Q Case #33 Piaid body only

5 10 15 25 30 35 40 45 ( METERS J

.0000

0. 300. 600. 900. 1200. 1500. 1800. (INCHES)

BODY STATION Figure 5.5 5-52H RSS, Restored Handling, Case #33. Mach .55, Altitude 610 m (2000 fit).

u, V F TABLE 5.3: RSS B-52H With Restored Handling Qualities Ride RQI B-52H cg S EFFECT SAS RESTORED VALUE t t CASE # (Fuselage Station) `n A SP Wn CSP SP SP 32 856.0 .5158 .9991 900 2.805 .5156 2.805 33 897.3 900 2.585 .5158 .8893 .4940 2.805 34 952.4 2.264 .4690 2.805 .5158 1.3222 35 1.5183 952.4 600 .3208 3.796 .7264 Unstable 36 3.005 897.3 600 .5403 Stable .5840 1.3918 37+ 856.0 2.805 .5158 1.3553 600 1.153 .764 apparently reduced. Figure 5.6 shows rigid body only plots from the un- augmented B-1 and from the RSS B-1 with handling restored. The rigid body line segment denoted by triangles for the RSS vehicle had consist- ently less curvature throughout the RSS B-1 data. Insufficient curva- ture and fidelity was generated on the B-52H rigid body load factor to generalize about this finding. In any case the apparent reduced coupl- ing with the elastic modes generated a slightly better RSS B-1 ride (with restored handling characteristics) as shown by Table 5.4 with the RQI. Since the flight conditions were quite different no further generalizations about RSS should be made at this point. Research is needed into the coupling affects before generalizations about the elastic RSS can be attempted.

TABLE 5.4: RSS B-1 With Restored Handling Qualities B-1 cg St RSS EF= FECT SAS RESTORED RQI CASE # (Fuselage Station) mn ASP `On ASP SP SP 27 1088.8 .9965 497.4 2.510 2.793 .474 .464 28 1125.6 .9714 2.793 .474 497.4 2.070 .464 29 1061.2 .9772 400 1.514 2.793 .474 .663 30 2.793 .474 1061.2 450 .516 .9966 2.263 31 1088.8 2.793 .474 .839 .9696 1.040 32+ 1125.6 2.803 .475 .9664 425 .908 .857 factor plots are shown in C, section C.5.

+ Load Appendix G G FT/SEC MISEC

.5000

1.5

.4000

x

.3000

Q

LL-

° .2000

UD

,1oaa

13 Unaugmented vehicle

A RSS vehicle

5 10 15 20

25 30 35

40 45 (METERS)

. 0000

t --F- 300. 600. 0.

900. 1200, 1500. 1800.

(INCHES)

BODY STATION Rigid Body Load Factor Comparison.

Figure 5.6 B-1 U1 to In my opinion, the one generalization allowable concerning the B--1 is that RSS did not degrade the ride at this flight condition. At any other flight condition where the rigid body effects might contribute more to the load factor curve, the findings should be similar to the B-52H RSS. In addition inclusion of the SMCS (structural mode control system) on the B-1 will change the percentage contribution to overall ride by the rigid body.

5.7 Relaxed Static Stability Sum - mary RSS has a definite effect on RQ. This study has shown that the effect is not favorable when the original handling characteristics are restored on a more rigid aircraft. On the B-1 the restored handling resulted in a slightly better ride. This result may be modified when RSS is utilized in preliminary design or when appropriate elastic changes can be incorporated in RSS implementation on the B-1. A specific study should be made concerning the possible tradeoffs of increased RSS with restoration to different ',andling qualities. It would seem that some optimum RSS level exists as a design point for specific handling qualities.

The overwhelming percentage of t )tal ride on the B-52H was due to the rigid body effects. Therefore, a larger effect was induced by RSS.

By contrast, the B-1 had a relatively small RQ contribution (compared to the total) due to rigid body dynamics and was not as sensitive to the RSS effect. To completely verify the total B-1 effects the Swaim technique for estimating the coupling effects of the RSS implementation would be required as a minimum. This would insure that the decoupling effect mentioned above was not due to omission of changes to elastic mode aerodynamic terms due to RSS.

Chapter 6 RESULTS, CONCLUSIONS, AND RECOMMENDATIONS 6.1 Ride Quality Index Results and Recommendations The utility of the RQI has been demonstrated in studying RQ under the influence of control law changes and variations of handling charac- teristics. The marketing appeal for a ride metric should be high since another decision index could now be placed at the consumer manager's disposal with minimum manufacturers' cost. Most companies do this kind of marketing now but there is no common denominator from which to judge the comparative ride quality that is being purchased.

Further study under the guidance of an aerospace industry marketing expert would clarify practical usage questions. For the USAF a study to find an avionics multiplier for specification values on the marketing index would give better guidance than now exists. A maximum operational avionics load factor value would provide an additional design point for the RQI.

6.2 B--52H and B-1 Ride Quality Conclusions and Recommendations In summarizing the RQ investigations on the B--52H and B-1, I would like to re-emphasize the fact that no attempt was made to optimize the vehicle RQ. The technique demonstrated herein certainly would lend itself to a quadratic optimal performance or penalty function approach toward finding an optimal RQ feedback gain.

RQ is essentially independent of the control law type for equiva- lent closed loop dynamics. This finding was reasonable based on the current experience of control experts and was consistent on both of the test vehicles. It- -z import lies in the inference that the different numerator dynamics for two different control laws generate the same load factor output.

RQ is very sensitive to Kandling qualities criteria. Higher frequency and damping provide better RQ assuming no control, resonance, or flutter problems occur on a specific vehicle. This finding was consistent for both vehicles as well.

effect on RQ in that it RSS by itself has a slightly favorable The over- rigid body RMS load factor curve.

decreases the slope of the all effect is a decrease in the energy imparted to the aircraft aft of g ig. This finding the cg. An increase occurs in the energy ahead of the was the same for both of the vehicles.

RSS with restored handling qualities degraded the RQ on the B-52H.

This result was basically due to the higher RMS deflections on the elevator that were required to restore the handling characteristics desirEd. This effect was not evident with the B-1 flight condition evaluated in this study.

RSS slightly improved the B-1 overall RQ. Since RSS was treated as a rigid body phenomena no strong conclusion can be drawn from this find- ing. A preliminary interpretation would suggest that a favorable decoupling effect occurred in the highly elastic B-1 case, resulting in a better ride. An investigation of this interpretation is certainly justified. More importantly this investigation would be economical and straightforward with the elastic derivative synthesis process developed by Swaim and Fullman.

These conclusions raise other important questions which must be answered before a complete RQ criterion can be generated for large flexible vehicles of the future. A few of the important questions follow.

In the next ten to fifteen years retro-fit will be utilized by the airlines and USAF to update their airplanes. In order that RSS could be implemented as a fuel saving measure, several questions must be ficed analytically as well as experimentally. First, is it possible to imple- ment RSS by cargo or fuel management without appreciably changing the free--free elastic mode shape curves? This question is especially important on a highly flexible vehicle where the RQ is very sensitive to the mode shapes and the time history of the generalized coordinates.

Second, if the shapes do change, what is the maximum cg movement that { could be implemented without degrading the ride?

From the preliminary design point of view, will RSS demonstrate the sane load factor trends for new concept vehicles? This question in- volves the entire CCV concept and could affect major design decisions su:h as engine placement, configuration, and so on.

Another design question revolves around the favorable interaction of elastic modes for RCS. Is it possible to invoke maximum cancellation of significant elastic modes with the rigid body or other modes by utilizing a single control? Could this be done by artificially increas-

ing the activity of a favorable mode?

Since the interaction phenomena exists, perhaps RSS with an active ride quality control system would generate more favorable cancellations.

This might lead to significantly reduced load factor levels.

6.3 Summar While this study ranged over a wide variety of handling character-- istics and common control laws, it examined only two vehicles at two different flight conditions. Substantiating data is needed from other sources, vehicles, and flight conditions before conclusions and recommendations could be made for a general design logic. The ride quality metric should prove to be a valuable index for future parametric comparisons of this type.

REFERENCES a 1. Bryson, A.E. and Ho, Yu-Chi, Applied- 0 Optimal Control, John Wiley & Sons, 1975.

2. Gelb, Arthur, et al, Applied- 0 t p i1na1_ Estimation, MIT Press, 1974.

3. Makers, D.T., "A Blended Feedback Variable for Control Augmentation Systems", AFFDL-TR-67-87, Wright-Patterson AFB, Ohio, 1967.

Perkins, W.R. and Cruz, J.B., Jr., Engineering of Dynamic Systems, tems, 4.

Wiley & Sons, 1969.

Etkin, B., Dynamics of Atmospheric Flight, John Wiley & Sons, Inc., 5.

1972.

mics and 6. McRuer, D., Ashkenas, I., and Graham, D., Aircraft Dynamics Automatic Control, Princeton University Press-7-1-973.

Finck, R.D., Principal Investigator, Douglas Aircraft Division, 7.

USAF St ability and Control DATCOM, Air Force Flight Dynamics Labora^`ory,Wright-Patterson AFB, Ohio, 1960, revised 1975.

8. Heath, R.E., II, "State Variable Model of Wind Gusts", AFFOL/FGC--TM-72-12, Wright--Patterson AFB, Ohio, 1972.

Rustenburg, J.W., "Ride Quality Design Criteria for Aircraft With 9.

Active Mode Control Systems", ASD-TR-72-64, Wright-Patterson AFB, Ohio, 1972.

'10. Rustenburg, J.W., "Development of Tracking Error Frequency Response Functions and Aircraft Ride Quality Design Criteria for Vertical and Lateral Vibration", .-ASD-TR-70-18, Wright-Patterson AFB, Ohio,

1971.-

Jacobsen, I.D. and Rudrapatna, A.N., "Models of Subjective Response 11.

To In-Flight Motion Data", Technical Report 403209, Short--Haul Air Transportation Program, University of Virginia, 1973.

Swaim, R.L. and Fullman, D.G., "A Unique Formulation of Elastic 12.

Airplane Longitudinal Equations of Motion", to be published, Purdue University, West Lafayette, IN, 1976.

13. 9-52 CCV Control System Synthesis", by Boeing Company, Wichita Division, AFFDL-TR-7 4-92, Volume II, Wright-Patterson AFB, Ohio, 1975.

14. "Critical Analysis of 5-52 Stability Augmentation and Flight Control Systems For Improved Structural Life", Parts I-V, D3-6434-1 thru -5, Boeing Company, Wichita, KA, 1965.

15. Wykes, J.H., 9-1 Flexible Vehicle Equations of Motion For Ride Quality, Terrain Following, and Handling Qualities Studies", Internal Document TFD-71-430--1, North American Rockwell, Los Angeles, CA, 1971, revised 1973.

16. Chalk, C.R. et al., "Background Information and User Guide for MIL-F-8785B^ASG), 'Military Specification - Flying Qualities of Piloted Airplanes'," AFFDL-TR-69-72, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, Ohio, 1969.

if

Appendix A LONGITUDINAL EQUATIONS OF MOTION A.1 General Description and Assumptions.

The vector differential equations A.1 and A.2 provide the origin of the mathematical model for elastic vehicles.

d = F (A.1) d t dH = M (A.2) dt where p Linear momentum vector H Angular momentum vector F Resultant sum of all externally applied forces M Sum of all applied torques.

With the assumptions that: r 1) the earth is an inertial reference in space 2) the airframe is initially a rigid structure mass and mass distributions of the vehicle are constant 3) 4) the vertical or Xz plane is a plane of symmetry 5) perturbations from the cruise conditions allots small angle assumptions 6) quasi-steady flo g is sufficient to describe the aerodynamic perturbations 7) the vehicle is at cruise-level conditions 8) the flight path is over a flat surface earth The Laplace transformed longitudinal equations of motion are given in A.3: -Xqs+gcos Y O u s-Xu -Xis-Xw - Z u s--Zws-Z w -Us-Z g s+gSin Y O w - _M u --Mws^-M^^ sz --sm e (A.3) X Xis+Xw Xu ae e Zqs Z^s+Zw U Zae Z u9 Mqs M S Mu Mws+Mw-- U w9 e The stability axis sign convention is depicted in Figure A.7. Note that the vertical gust velocity is defined positive in the negative Z axis direction thereby inducing a positive angle of attack.

^9 y'NY x i e o _ — " — Horizon cg 4 z,Nz Figure A.l Vehicle Stability Axis Sign Convention.

• For our purposes the perturbation of forward velocity has an extremely small effect on vertical load factors. Hence, the first equa- tion is dropped and the short period, rigid body equations of motion remain. To these equations we add the equations of motion corresponding to the generalized coordinates of the four lowest frequency elastic modes of each vehicle. The instantaneous generalized coordinate value of a particular mode multiplied by the mode shape yields the instanta- neous displacement of the mode at that point. Utilizing this orthogonal M CO --Z^

-Us -Z^ -Z w(s)

(s-ZW ) -ZC2 -Mws-Mw (s2-Mqs) -Mi --M -M -M e(s) 2 3 k

0 -Fib - - F

- F iW ib (S2+2^jwjs'}-W2) -Fib ^, (S) 2 3 F -F2& 3 E2(S) - 2W 0 -F 2 ^ (52}2^2W2S+W2) (S2 - F 3w 0 -F3^ -F3 93(s) -F3^ }243W3S}W3) i 2 //

- F 4 W 0 -4 -F4, -F4

1 52+Z ^ W ! } ) E(S) L} S+W 2 3 z z e Md Mg Mw e Flde l F1 W9(s) e ts) + 0 (A.4) F es F2w 0 1 49(s) Fa d F3w 0 e F4W 0 F46 e where F i represents the generaliz A,' force change of the ith mode due to a change in the state j.

F L

i

mode model, which is most common in aeroelasticity, and equation A.3, we can write the elastic vehicle equations as shown in equation A.4. Re- writing the system as a 10th order set of first order linear dif-Feren- tial equations, equation A.4 becomes the familiar control equation A.5.

The approximation w = ua is used to transform the vertical velocity to the perturbation angle of attack. Note that the states in A.5 are arranged in a slightly different manner compared to the Laplace domain system.

g * (A.5) x* = A*x* + B*u + M*x where x* ' = L1 ^ 2 a 9 Z 2 93 9 4 13 k4j *' L g = a g x qg] L A.2 Turbule nce Model The Dryden power spectral density representation of turbulence is modeled as a set of three first order linear differential equations.

This model is due to Heath and is derived in reference 8.

The temporal frequency representations of the angle of attack and pitch gust power spectra are: 3 (LUw,) L L 1 + - a ^,) - —w --T- T a g w g U [1 + ( L m )2]' ID ag 4b 1,1 q ( Tru) RMS vertical gust intensity (.30 m/sec (1 ft/sec) here a g throughout this investigation) Gust scale which depends on the altitude L U Average velocity of the vehicle h Wing span Table A.1 shows the appropriate values for these factors at the / flight conditions used in this study.

TABLE A.1: Gust Specifications For The Study Vehicles Factor B-1 Value B-52H Value .30 m/sec (1 ft/sec) .30 m/sec(1 ft/sec) 6w 9 Lw 30 m (100 ft) 533 m (1750 ft) U 289 m/sec (949.45 ft/sec) 186 m/sec (608.8 ft/sec) 56 m (185 ft) b 42 m (136.68 ft) Modeling the gusts above as a white noise input to a linear system, the spectra can be represented as a system of linear, first order dif- ferential equations shown in equation A.6: a91 a91 f (A.6) ag = [ A9 ^ a G*n G g q where n is scalar unit white noise in the time domain Using A.6 as the forcing system we can now rewrite A.5 with A.6 appended as follows: x* A* M* x* B* + u + n (A.7) 0 G* xg 0 xg A Simplifying this simultaneous set into single matrix elements, the general equations of motion for both vehicles are written in the form A.8: (A.8) x = Ax+Bu+Gn where x' = Cz E 3 ^' a k, (A.9) e Z z Zs ^' ag,' ag7 qgj LI , yf A.3 B-52H Equations of Motion The B-52H equations were derived from the 18 degree of freedom time domain equations in reference 14. The four lowest frequency elastic modes were used in this study. The eigenvalues are shown in Table A.2.

The 4A*M*], B*, and G* matrices derived from the documents supplied by Boeing Company are given by Tables A.3, A.4, and A.5 respectively.

Table A.6 specifies the Ag gust coefficient matrix which is appended to the vehicle state equations in equation A.7. The mode shapes, associ- ated locations, and distance to the cg (in feet) are given in Table A.7.

TABLE A.2: B-52H Bare Airframe Coupled Eigenvalues Dampin g Mode Roots _ Frequency 2.8066 .5157 Short Period -1.447 * 2.404i f 1 .919 5.679i 5.7531 .1598 2 .959 } 11.513i 11.5530 .0830 .0112 3 .140 ± 12.470i 12.4710 14.8270 .0553 11 .820 ± 14.804i TABLE A.4: B--52H Control Matrix B* (10x1) 0 0 0 -5.52E - 02 -3.99E + 00 3.36E + 01 -1.28E + 00 9.12E + 00 -2.92E + 01 TABLE A.5: B-52H White Noise Matrix, G* (3xl) 4.33710E - 03 1.00000 1.67804E - 03 52H Gust Matrix, Ag (3x3) TABLE A.6 B- -3.4789E - 01 0 0 -3.4789E - 01 0 -2.4648E - 04 -2.5846E + 00 -6.3775E - 04 -8.9916E - 01 V N TABLE A.3: B -52H [A* :M*] Matrix (10x13) L J 8 TI O1 a n4 r12 n3 ag J ag qg nz n4 ;3 1 +00 1+n0OD n n 0 0 n n n n n n n n n 0 0 n n n n n 0 n n i,nOn^+nn n n n n n a n n 0 n n i^nnnn+on o 0 n n n n n n [ ► n (IOOD+ 0 0 n 1 * -6.041D-04 -1.352U+On -R.073D_o2 1.406n-01 9949RD-nl -a,ali,n_na _? ,A4?n -01 4x076D-o3 n -1.15 2 2U+On 0

9 lKnn-03

692950-03 t ;6U+On -1.718D+00 R.305D-01 -6.Ob6n-01 -7w5 -4,ARnn+00 +n0 +0n - 1.71nD q ,A47n 6@321D - 02 n " 7 .5560 - 0? - +nt -2.217D+00 4,293n -1,01 4U +02 -1.n78D-nl +0? -1.8430;O1 - ^,Z14Ah .. 1, 77 ,7n + n1 n -1.014u+O9 0 _7,177n-ol l.09BD+oo -4,74in+nn - 2.442D

- 3.629U - 01

+01 ?5() +n0

1.i p A n -9.719D-ol 4.2470 + 00 -2.4

-A.P7 1 4n..n1 +0?

- n -3.629u+01 0 o,nr,nn-0? -4.337D_04 ^^'= - 1,^Qnn +nn -89578D-02 3,074r)+00 7.820u+On lo447D+00 ^,n7^.n+0n -1.548D+n?

^-^ - q.n7ih..n1 7. q ?Ou 0 n +On - -'A 71Qn-01 49690n - nl 1. lln- n 1 -7.347D-02 3.837U+D1 - 4.RlRD+no -7,iinn+nl

7. 12:3D+On - 2 ,222n +02 -

b n -3vA37U+01 -1 .604D;On «-a,7f,Rn_nt i ,-n41n+On L^ Ca .

C Table A,7: 6-52H Mode Shapes

Fuselage Distance

Made 3 Mode 4 Mode 2 Mode 1 From cg Station - .2520n0 ,340000

200.nnnonn 0 .37ROOn

.322non -.1950n0 .24n000 300.nnnnnn -.0200n0 .2770n O n 0 ,1^^000 'iA, 000000 r4nnonn -.0390nO o -.143

400-

-.100000 , nP 1000 -.0600n0 .237non 500-r4nnonn -.063000 ^n^5poo -0080000 •208nOO 600.nnnnnn

7oo.nnnnnn

-.lu00n0 .1830on -.032000 1^,nnnnh0 800. nnnnnn -.1240no •16?nbn -.013000 -.055000 4.15^^S^h7 -sOU3ono -,Q7 ?000 900.nnnonn -.147nnO -147n0n 00550mo -, -17.005000 nnnono -X175000 .127000 -.079000 1000, -,n7nooa 1100.nnnonn -,2U3on0 .lip000 -,0160n0 - . n 4gO -.04nOnn OO ..?A,F,^,hch P 1200.nnn0nn -.2340nO .104000

-*2b5noo -. 07

-^r,nnnnn0 13on,nnnnnn .100000 2000 -,00h000 .095000 -,1150no ,045000 1400.nnonnn -.300n00 -5 ^, hR^,^ h7 -.3380n0 .090non -,1650no .11h000 150060nnnnn ..:3150n0 .OR7nn0 2:30000 ,2nS000 -6P. nnnnno 1600- nnnono ...415nn0 .ORhnnn -.3onano , 3 nS 000 .7m,1141111 17ononnnonn ,42n000 -.455000 .OR9n0n -,3br.Ono

-7Q. F,AAAf ", 1800.nnnnnn

V W A.4 B-1 Equations of Motion The low level penetration flight condition for the 3-1 was supplied by the B-1 System Program Office at Wright-Patterson AFB from Rockwell International documents. The values used in these equations of motion were based on preliminary aerodynamic analyses, but are closely repre- sentative of the vehicle now flying. The time domain format for the longitudinal equations shown below was decoupled from the complete vehicle set given in reference 15 The eigenvalues for the longitudinal set are shown in Table A.B.

TABLE A.8: 3-1 Bare Airframe Coupled Eigenvalues __ Mode Roots Fr^ency Damp i n ± Shot Period -1.3136 2.4617i 2.750 .4708 1 - .6335 ± 13.2574i 13.2'72 .0477 i 21.3522i 2 .0221 - .4715 21.357 3 - .2038 ± 22.0188i 22.020 .0093 4 -4.4861 ± 22.0144i 22.467 .1997 The values for the unaugmented vehicle equations are given by the [A*M*], B*, and G* matrices in Tables A.9, A.10, and A.11 respectively.

No structural mode control system dynamics were included in this study.

Table A.12 gives the B-1 flight condition values for the gust coeffi- cients A,Table A.13 shows the mode shape information for the B-1.

TABLE A.10: B-1 Control Matrix, B* (10x1) 0 0 0 -2.8840E - 01 -1.5025E + 01 -2'.2303E + 01 --2.1523E + 02 1.0785E - 01 6.1356E + 02 TABLE A.li: B-1 White Noise Matrix, G* (3xl) 3.0669E - 02 1.0000E + 00 5.6210E - 03

M*] Matrix (1 N13)

[A*: TABLE A.9: B--1 X14 a nl '12 L Tr 1 cXg ^3 n 2 agl gg n 1.000 0+00 0 0 n O 0 r n u 0 0 n 0 U n 0 n [^ .0000+00 0 0 0 n 0 0 0 G

0 1.0aOFI+c0 0 0 0

0 0 0 0 0 0 0 0 1.n00D+00 0 4nU'-TT3'=^:r's^T14^T^ -03 -4.346D-03 -1.2030+TM S r 03 9.5250 - 02 4.9180 -8.9-400 - 0 -1.204n+00 5.4720-n2 98n -Oh 59DD-03 1.526n-04 - 4. g +00 +nC -7.4380-03 r2n-01 -9.171D-OP --7.059D -2.0180 -1.842D-01 -1.248D-01 2. n = 0 -7.049D+nO -1.91rD+no 9720-02 6.963n-03 -2.nO4D-03 2.088D+01 -7.3630+02 -1.3330+n? -1.139+7+00 -1.7740+02 -1.568D+01 6.I g O n +01 0 -7.3710+02 -1.679D+02 -2.7700-01 9.159D-01 --1..9770-01 1,300o+03 1.63313+rrl 2.351-0-III 3D+00 1.1249+01 1.4{37[3+03. -5,061n+02 6.r g 0 1.2960+n3 2990?D+01 -8.596D+00 1.131n-ol 1.1a0D-+7l - - 4.Rmin +02 3.457.+00 7.577D+02 5.595D+ni -1.219[7-01 6.990D+00 3.45Bn+Dl

r ^

0 7.5A4D+02 6.180D+OI ,iA5r -Di 7840+00 -8.465n-oi -2.

-4.849D+02 1.6860+nO 399900-n2 -3.292n-04 0^ 1.623E-o2 -9.7000-02 7.112n- 0 1.6970+n0 3.817D-n2 2.4270-03 -2.764n-04 -4.n p 0n-01 I

V

t t TABLE A.12: B-1 Gust Matrix, A 9 (30) -9.495E + 00 0 u --2.256E - 02 -9.495E + 00 0 -1.231E - 01 -5.456E + 00

i6

Table A.13: 5-1 Mode Shapes Distance Fuselage Mode 4 Made 3 Mode 2 Made 1 From c9 Station 7?.000000 x2.433333 1.000000 1.000000 .000nOu 1 .00r1000 100.000000 80.100000 .RF,Oi 00 .900000 . 8-0 5 0-a 0 .$-80-a" 200.000000 71.766667 .Aa5000 .750000 1 500000 .50V000 500.000000 6504 33333 . ;^94D00 .400H-r 400. 0100000 55.100000 .1A7000 - * 075000 .200000 - 110no0a 50x.000000 46.766667 .0140&0 -.2G20-e-Ge 0 600.000000 38.433333 -.058000 -.240r00 -4260000 0 70-0.000000 30.10000-0 -.10000 -.300000 -.2400-0-0 -62affoaa-- M00.060000 21 ,7666(+7 -.1?,5000 -.205000 -.100000 -.22FiD00 900.00000-3 13.433333 - . 120003 -.050000 1000.000000 5.100000 -.105000 - .125000 0 - .16nO O O 1100.OU0000 3.233333 -.070000 0100000 1200.000000 11.566667 -.061000 .200000 -.09n000 1300.000000 -19.900000 .100000 .3601300 -. 041 00-B -. G-39-3-0f) -28.233333 14u0.OUOOOLJ .P05000 .460000 -.121000 .04no00 it)00.000000 -36.566667 . 3 ? 2 0.0-3 X3{35-04$ .7800{30 . -, 19"0-0 1600.OUD000 -44.90n000 .UCOSO0 .020000 1.000000 .22ci000 V V I

Appendix B CASE DOCUMENTATIONS B.1 Discussion outlines of the control law and The tables B.1 and B.3 are complete on the B-52H and B-1 handling quality parameterizations which were run matrices are identified vertical load factor cases. The full state gain by case # in Table B.2 and B.4.

TABLE 8.1: B-52H Case Control List u = -- K x Case # Type SAS ^n Gain 1 Gain 2 ISP SP 2.806 1 a .5157 0.

2 2.635 .2 e .404 3 e .617 2.970 +.2 4 .712 3.126 +.4 -,0005 0.

5 C* .3403 2.521 -.0005 -.3 C* .514 2.779 7 C* .482 2.751 -.9001 .0 +.0303 -.2 8 C* .506 2.806 +.5 9 C't .856 3.402 .0004 2.754 .25 -.2 10 9/9 .359 0/6 .723 3.272 .25 .5 .706 3.497 .75 .6 12 0/9 .25 0/o .528 2.982 .l .25 -.3 14 0/e .297 2.677 15 Full State .5157 2.806 t6 Full State .325 2.ZO6 17 Full State .8176 2.806 18 Full State .5157 2.806 TABLE B.1, cont.

19 Full State .325 2.806 2C State .8176 Full 2.806 21 Full State a Comparison 22 Full State .5157 3.000 23 Full State .5157 3.400 2.806 24 Full State .5157 25 Full State .325 2.806 26 Full State .8176 2.806 27* Full State .5157 2.806 2.806 28* Full State .325 29* 2.806 Full State .8176 30 Full State .8176 3.000 Full .8175 3.400 31 State 32 RSS Test Case .5158 2.805 33 RSS Restored HQ .5158 2.805 2.805 34 RSS Restored HQ .5158 35 RSS Restored HQ .7264 3.796 36 RSS Restored HQ .5840 3.005 2.805 37 RSS Restored HQ .5158 38 RSS No Restoration .5158 2.805 2.640 39 RSS Rigid Only .5470, RSS Rigid Only .5920 2.410 2.160 41 RSS Rigid Only .6550 42 RSS Rigid Only .7480 1.860 2.350 43 RSS Rigid Only .5770• .6290 2.01C 44 RSS Rigid Only .7380 1.700 45 RSS Rigid Only RSS Rigid Only 1.000 2.207 *Increased damping on elastic modes TABLE B.2: B-52H Full State Control Gains KID Case # KI K2 K4 K5 K6 K8 K9 K3 K7 0 0 0 24 0 0 0 0 0 0 0 .28 -.0006 -.007 .002 -.001 25 -.004 .05 -.01 .02 --.34 .01 26 .006 -.03 .54 -.44 .001 --.002 .001 -.09 .03 -.08 .329 -.024 27 -.005 .59 -1.40 - 01 .19 -.02 .001 .25 .009 -.08 .313 -.03 28 -.Ol .73 -1.73 -.85 -.14 -.02 29 .004 .38 -.85 -1.00 .71 -.45 .003 --.08 .35 -.003 30 .008 --.11 .03 -.03 .34 --.52 .001 .015 .002 -.68 .001 .022 -.003 .002 31 .01 -.13 .04 -.04 - .14 RSS RESTORED HQ 32 0 0 0 0 0 0 0 0 .004 -.004 -.005 -.39 -.15 .0001 .005 -.0007 .0004 33 .004 -.06 .58 -.17 .18 3.84 3.69 -.002 -.14 .02 -.01 6.1 .01 35 -.14 2.10 .302 .43 14.05 -.03 -.36 -.03 -3.95 -1.15 --.007 .04 -.007 .0004 36 .02 .15 .11 -.002 .02 .003 -.003 -1.57 -.321 --.0005 .01 -.001 .0004 37 .008 TABLE B.3: B-1 Case Control List u--K Case # Type SAS w Gain 1 Gain 2 4SP SP l e .4708 2.790 0.

2 6 .6551 2.981 +,1

3 6 .8240 3.168 +,2 4 e .2650 2.594 -.1 5 C* .3220 2.483 -.0005 +.4 +.3 6 C* .3100 2.504 -.0004 .5230 2.685 -.0004 +.4 7 C* 8 C* .7160 2.862 -.0004 +.5 9 3.013 +.0001 0.

C* .6410 10 C* .8080 3.204 +.0001 +.1 11 C* .4340 2.881 +.0003 -.3 12 C* .6120 3.084 +.0003 -.2 13 C* .5840 -.4 3.160 +.0005 14 C* .8890 3.570 +.0005 -.2 15 e/6 .4230 2.955 +.1 0.

16 a/6 .7760 3.288 +.1 +.2 6/6 17 .6950 3.550 +.3 +.2 +.3 18 e/6 .8170 3.817 +:", 19 2.790 Full State .4716 20 Full State .4707 2.790 Increased damping 21 Full State .4707 2.790 cp, 3 22 Full State .4707 2.790 Increased damping c1,c3 23 Full State .6551 2.790 2.980 24 Full State .6132 1.0000 0.229 25 Full State Full State .2850 2.594 27 RSS Restored HQ .4740 2.793 28 RSS Restored HQ .4740 2.193 2.793 29 RSS Restored HQ .4740 2.793 30 RSS Restored HQ .4740 cont.

TABLE 8.3, 2-M 31 RSS Restored HQ .4740 .4750 2.803 32 RSS Restored HQ Full State .4708 2.790 No q 9 effect Full State .8240 3.168 35 Full State .4708 2.790 Control Gains TABLE B.4: B-1 Full State K7 K8 Case ®rr K1 KL K3 K4 K5 K6 K9 K10 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28.97 .02 -.003 659. -1.04 -.17 -.002 21 -.01 -.16 -.19 27.43 --.002 -1.92 -.21 -.003 .03 -.06 -.24 894.6 22 -.02 .23 -.00003 .0001 --.00007 .001 -1.73 .06 --.08 23 -.0008 .003 .0001 -.00007 .22 -1.72 -.02 -.08 -.00002 24 -.0007 .003 .001 -.000036 .000096 .66 -.16 .0001 .0002 .008 .001 -.01 25 --.002 33.66 -.002 .0186 -.004 566.9 -.50 -.076 26 -.007 --.26 --.19 RSS HO RESTORED .000004 .00003 -.00001 -.00008 .0001 .0001 -.11 -.03 27 -.0001 .002 -.00005 -.00002 .00001 .001 -.28 -.06 .00008 .003 28 --.0002 .004 -.0001 -.00002 .00001 -.001 -.43 -.05 .00007 .004 .0001 29 -.00002 .000006 .00004 -.00005 --.000008 -.21 -.02 -•.00002 .002 .00007 --.0002 .0001 -.00003 .00001 -.003 --.54 -.72 -.0001 -.00008 .005 .0002 .00002 --.09 .0001 -.0002 --.00004 .0004 .0005 -.61 -.0002 .007

Appendix C NUMERICAL ANALYSES C.l Computational Algorithms Figure C.1 shows the logic and names of the subroutines which accomplished all of the computing for these studies. Various versions, of these routines are available in the Purdue University School of Aeronautics and Astronautics Guidance and Control Laboratory. The generalized program RIDEQ can be used to do all general load factor was tasks and is referenced here in terms of subroutine names. RIDEQ developed as an all purpose program by Mr. Dan Raymer. The subroutines in RIDEQ were due to Mr. Andrew Hinsdale. A user guide is available in the Guidance and Control Laboratory for RIDEQ.

C.2 Subroutine TRANSIT This algorithm for computation of the linear covariance equation solution deserves special mention. The technique from reference 2 is programmed in variable dimensions and requires only 20-30 seconds on she Purdue CDC-6500. It uses a step size of 0.08 -- .1 very efficiently.

Iterative checking on the solution accuracy allows early completion if the spectral radii of the matrices are appropriately small.

The physical requirement for a stable final vehicle configuration manifests itself in the requirement that the spectral radius of the A matrix in C.1 be less than unity.

A Efxx') + E{xx') A' + GG' = 0 (C.1) To review the computational procedure, let E{xx`I = P and P = AP + PA S + GG' (C.2) f' fART READ ALL UTPUT INPUT DATA ALL DATA AS READ COMPUTE THE OUTPUT T,T ,T AT, SIt3ILARITY ?MATRIX AND T B,TT INVERSE OUTPUT COMPUTE THE CLOSED LOOP E ,D ,K,k POLY & GAINS COMPUTE THE OUTPUT A" CLOSED LOOP AUGMENTED A MATR IX iJT ^UBRnUTI:'iE nUTP NO IGENVA LIIES Rr,Eirr 5 5S STABLE?

i YES COMPUTE THE SUBROUTINE lCOVARIANGE I TRANS IT IEQUATiON I r -- SOLUTION

r

iCOtMPUTE THE 5iJ[3ROUTINEr LOAD FACTORS E FSLF SAND PLOT Nz

r

STOP/END Fi g ure C.1 RIDEQ Load Factors Algorithm.

Define -A 0 M = .....:...

(C.3) GG A The state transition matrix for C.1 can be represented by: e MT = I + MT M 2 T 2 + ...

( C.4)

+ 2_

Then let: MT = M1 . ..:..° e .

(C.5) M 2 M3 Let T = t 2 -- t l = At and t 2 is the next point in time where P is evaluated P(tz) = M2 M3 + M 3 P(t Z )M 3 (C.6) After a certain number of iterations we minimize off diagonal round-off error by: (C.7) P(ti) - EP(ti) + P - ( t i ) aC27 Substitution into C.2 to test P -).- 0 and prescription of a stopping k precision are the remaining steps for this algorithm.

C.3 Phase Variable Canonical Matrix Calculations The solution of C.1 is sensitive to matrix ill-conditioning in the sense that the matrix A should be reasonably well distributed by element magnitudes. We found that the form C.8 would lead to two or three orders of magnitude error in the final covariance solution matrix.

A double precision routine was written to test TRANSIT under these same conditions. The test case showed little improvement and would have required almost five times more CPU time to complete the computations for the covariance matrix. As a result all TRANSIT computations were done in the physical state variable form. That is, the A matrix was not used in TRANSIT in the phase variable canonical form.

C.4 Transformation Matrix T Another form of the same matrix ill-conditioning appeared in the solution check on the similarity transform, T. After forming the T matrix we formed and printed the ,r .T - 'AT] combination which should have been the phase variable canonical form mentioned above.

All rows and columns of the check case were very accurate except the first three or four columns. These columns showed great sensitivity to the accuracy with which the T matrix was originally formed. This was especially true with respect to the coefficients of the open loop characteristic equation.

It was found that a sufficiently accurate check could be attained if the transformation matrix routine was supplied with 16 digit (single precision) data but was computed in double precision modes. Of course the accuracy of the study vehicle data was not known to this extent.

Hence, the extension of the subroutine hand-off accuracy was a theoreti- cal exercise in computation and had no bearing on the outcome or results of the studies.

C.5 Load Factor Plots for Significant Cases This section contains the plots for significant cases in the required comparisons of the main body of this work. The case number is specified in each of the figure captions.

G

G

FTI S EC W S EC 4 nn?l

^s (METERS)

30. (INCHES)

F _, G G

FT/SEC, MlSEC

Lo CD .1000 .0800 U Cl: .0600 O Cr D ___J .0}00 CE Z .0200 (METERS) .0000 300. 900. 0.

600. 1200. 1500. 1800 . ( I NCHES )

BODY STATION

Figure C.3 B--52H load Factors, Case #6. Mach .55, A1t;tude 610 m (2000 ft).

I

F ' :r

G G

FTISEC ANSEC .1000 .0800 UD

U

Q .0600 LL CC -J .0400 ED

z

.0200 +5 (METERS) .0000 0. 300. 600. 900. 1200. 1500.

1800. (INCHES)

BODY STATION Figure C.4 B-52H Load Factors, Case #12. Mach .55, Altitude 510 m (2000 ft).

G G FTISEC MISEC

N

.1000 .0800 co F- U Q .0600 Q rx rigid .0400 .0200 + (METERS) .0000 o.

1800. (INCHES) 300. 600. 900. 1200. 1500.

BODY STATION Figure C.5 H--52H Load Factors, Case #14. Mach .55, Altitude 610 m (2000 fl;).

G G

FTISEC MISEC

.ioo0 .0800

U

Cr-.moo J

U

H

.01400 CK

W

U] .0200

is (METERS)

.0000

0. saa . 1200. 1Sao. 1800. (INCHES)

ano . BODY STR

STATION dN

Figure C.6 B-52H Load Factors, Case #37. Bach .55, w Altitude 610 m (2000 ft).

^

G

G

LO FTISFC MISFC

,2soa

1.75 .2000 CQ mode Z L0 mode 2 U ,1500 mode 3 Q mode 4 I CE to J .1000 cn a^ .0500 —^ (METERS)

0.0000

-11 0. 300. 600. 900. 1200. 1500. lsoo .

(INCHES)

BODY STATION

Figure C.7 B--1 Load Factors, Case #4. Mach .85, Altitude 30 m (100 ft).

.f UD Of a3 b :4 .1550

U

LL E) J .1000 fF]

A

.0500

(METERS)

O.0000 0.

300. 600.

900. 1200. 1500.

lsoo . (INCHES)

BODY STATION Figure C.8 B--1 Load Factors, Case #18. Mach .85, ILD U1 Altitude 30 m (100 ft).

G G

FTISFC M/SFC

11.5

,4oaa

1.2

^

o Rigid

X Plus made 1 cn + Plus mode 2 A Plus mode 3 ED .3000 +3 Plus mode 4

g

Q

d

J

Cn .2000

[r '

.1000 ^ r

o 45 (METERS) .0000 -

1800 , (INCHES)

1200. 1500.

0. 300. 500. 000.

BODY STATION

Figure C.9 8--1 Load Factors, Case #32. Mach .85, Altitude 30 m (100 ft).