Wind Tunnel-Based Aerodynamic Model Identification for a Tilt-Wing, Distributed Electric Propulsion Aircraft

I. Introduction lectric vertical takeoff and landing (eVTOL) vehicle concepts are gaining popularity in the aerospace industry as a E means of enabling Urban Air Mobility (UAM), a future transportation method. UAM vehicles require precise hover and efficient cruise capabilities as well as the ability to safely transition between flight regimes. eVTOL aircraft are a hybrid between traditional fixed-wing and rotary-wing aircraft utilizing certain attributes of each class of vehicle.

Fixed-wing aircraft contribute longer endurance, better efficiency, and the ability to operate at high speeds. Rotorcraft have the ability to takeoff and land vertically, hover, and precisely maneuver in confined areas. eVTOL aircraft also uniquely utilize distributed electric propulsion (DEP) technology, which has broadened the traditional aeronautical vehicle design space and resulted in numerous unique vehicle designs [1, 2].

While eVTOL vehicles’ hybrid configurations are convenient from an operational standpoint [ 3 ], there are many research areas required to be addressed prior to introduction of operational eVTOL vehicles for a UAM mission [ 4 ].

eVTOL technical challenges include airworthiness certification, air traffic management, pilot-operator interface, handling qualities, simplified vehicle operations, contingency management, vehicle autonomy, and flight controls strategies.

One important enabling tool for many eVTOL research efforts is an accurate vehicle flight dynamics model, but the current modeling state-of-the-art for eVTOL vehicles heavily relies on low-fidelity conceptual design tools, such as NDARC [ 5 ] and VSPAero [ 6 ], or application of high-fidelity CFD with reduced computational demands to gain calculation speed [ 7 ]. Consequently, development of high-fidelity aerodynamic models for eVTOL aircraft is a crucial need; however, eVTOL-specific aerodynamic model strategies are largely unexplored. Thus, development of high-fidelity aerodynamic models for eVTOL configurations is a new, critical area of research where novel aerodynamic modeling strategies are needed to appropriately represent pertinent aerodynamic phenomena specific to these unique vehicles.

eVTOL aircraft aerodynamic modeling is a challenge due to several vehicle attributes which hamper model development. These features include many control surfaces and propulsors, propulsion-airframe interactions, high- incidence angle propeller aerodynamics, vehicle instability, rapidly changing aerodynamics through transition, and large flight envelopes that need to be characterized by a global aerodynamic model. Most past eVTOL modeling work has used analytical and semi-empirical models to develop vehicle models for research applications, however, these are low-fidelity methods which drastically simplify highly complex aerodynamics. Ref. [ 8 ] highlights many of these efforts for a variety of different hybrid vehicles including tilt-wing, tilt-rotor, tail sitters, and dual-propulsion system configurations. Due to ∗ the wide range of vastly different eVTOL configurations currently under development , aerodynamic modeling strategies will also require some degree of tailoring to an individual vehicle of interest in accordance with the desired modeling fidelity.

The subset of eVTOL aircraft of interest in this work are tilt-wing aircraft with wing mounted propellers. The operational advantages resulting from the DEP, tilt-wing design include delayed onset of stall in transition and control surface effectiveness at low airspeed, both due to the propeller slipstreams blowing over a majority of the wing [ 9 ].

Efficiency benefits also emerge from the use of DEP technology [ 10 , 11 ]. The disadvantages of tilt-wing aircraft include sensitivity to wind at low speed due to the upward wing orientation, requiring powerful actuators to rotate the wing, and possible flow separation from the wing in transition [ 9 ]. Several previous studies have developed dynamic models for single tilt-wing [ 12 – 19 ] and tandem tilt-wing concepts [ 20 – 24 ]. Many of these efforts develop models using analytical methods with a select few utilizing wind tunnel data.

The direct predecessor to the vehicle studied in this work was a subscale, tilt-wing, tilt-tail, DEP, vertical takeoff and landing aircraft called the GL-10 [ 25 ]. Several different scaled variants of the GL-10 vehicle were developed and tested to enable research in aerodynamic modeling [ 26 , 27 ], simulation development [ 25 ], flight controls [ 28 ], and flight testing [ 29 , 30 ]. The complexity associated with the GL-10 aircraft inspired an ongoing effort to develop a process to efficiently develop aerodynamic models for arbitrarily complex aerospace vehicles, referred to as Rapid Aero Modeling or RAM [ 7 , 31 , 32 ]. The RAM process utilizes design of experiments (DOE) theory [ 33 ] and a unique sequential modeling algorithm to develop aerodynamic models meeting a user-defined desired level of fidelity. Several lessons learned from the GL-10 aircraft study and key parts of the RAM process informed the development of the aerodynamic modeling strategies presented in this work. This paper is also complemented by other current eVTOL aircraft research [34–40].

This work builds on previous tilt-wing aircraft modeling studies to develop a high-fidelity aerodynamic model for the Langley Aerodrome No. 8 (LA-8) aircraft using wind tunnel data. Consequently, the following discussion focuses on methods used to develop mathematical models from experimental data, termed aircraft system identification [ 41 – 43 ].

eVTOL vehicles exhibit aerodynamic characteristics similar to both fixed-wing and rotary-wing aircraft, but system ∗ Information available online at https://evtol.news/aircraft [accessed November 2020]

identification approaches used for either type of vehicle do not independently translate to modeling eVTOL vehicles.

This manuscript seeks to provide a thorough development and assessment of new system identification-based approaches for eVTOL aircraft modeling. One major objective is to propose and justify tilt-wing, DEP-specific definitions of modeling explanatory variables and response variables based on vehicle attributes.

The paper is organized as follows: Section II presents salient background information informing model development strategies and introduces the LA-8 aircraft. The wind tunnel data gathering efforts are described in Sec. III. Section IV describes pertinent system identification techniques, followed by postulation of vehicle-specific aerodynamic modeling strategies in Sec. V. Section VI provides sample modeling results accompanied by discussion of results, and overall conclusions are summarized in Sec. VII.

II. Background This section presents an overview of experimental fixed-wing and rotary-wing aircraft modeling techniques, which helps guide the development of the tilt-wing, DEP aircraft modeling approach described in the remainder of the paper.

The section concludes with a description of the LA-8 aircraft.

A. Fixed-Wing Aircraft Modeling Aerodynamic modeling for fixed-wing aircraft is conventionally performed by developing data tables or functional representations of dimensionless aerodynamic force and moment coefficients as a function of aircraft states and controls.

For subsonic aircraft, the dimensionless aerodynamic force and moment coefficients are conventionally expressed as a function of angle of attack 훼 , angle of sideslip 훽 , dimensionless angular rates ˆ 푝 , ˆ 푞 , ˆ 푟 , and control surfaces deflections, such as elevator 훿 , aileron 훿 , and rudder 훿 positions. The dimensionless forces and moments are termed response 푒 푎 푟 variables, or dependent variables; the airflow angles, angular rates, and control surface deflections are termed the explanatory variables, or independent variables. Wind tunnel testing can be used to develop either tabulated or functional representations of the force and moment coefficients through a variety of test techniques [ 44 ], whereas flight testing is typically used to develop functional representations of the force and moment coefficients [ 41 , 43 ]. The dimensionless force and moment coefficients in the aircraft body-axes are defined as 푋 푌 푍 퐿 푀 푁 퐶 = , 퐶 = , 퐶 = , 퐶 = , 퐶 = , 퐶 = (1) 푥 푦 푧 푙 푚 푛 ¯ 푞푆 ¯ 푞푆 ¯ 푞푆 ¯ 푞푆푏 ¯ 푞푆 ¯ 푐 ¯ 푞푆푏 1 2 where 푋 , 푌 , 푍 , 퐿 , 푀 , and 푁 are the measured aerodynamic forces and moments, ¯ 푞 = 휌푉 is the freestream dynamic pressure, 푆 is the wing area, 푏 is the wingspan, and ¯ 푐 is the mean aerodynamic chord. The forces are also commonly expressed in the stability-axes for modeling where lift coefficient 퐶 and drag coefficient 퐶 replace 퐶 and 퐶 .

퐿 퐷 푧 푥 Fixed-wing aircraft system identification techniques are well-developed for standard problems and have been applied successfully to numerous aircraft configurations [45, 46].

Although widely successful for fixed-wing aircraft, the conventional fixed-wing modeling methodology cannot be applied in the same way to modeling eVTOL vehicles. Firstly, nondimensionalization by dynamic pressure ¯ 푞 is not valid for vehicles that are propulsion-dominated and experience significant airframe-propulsion interaction. Propeller aerodynamics, for example, scale with the dynamic pressure experienced by the individual propeller blades, as opposed to freestream dynamic pressure. Due to these aerodynamic scaling differences, testing must be performed at multiple different flight conditions to properly identify the aerodynamics changes across the flight envelope. Another modeling consideration is that eVTOL vehicles will operate at hover which makes modeling in terms of angle of attack 훼 and angle of sideslip 훽 become undefined based on their definitions, − 1 훼 = tan ( 푤 / 푢 ) (2) ( ) 푣 − 1 훽 = sin (3) √ 2 2 2 푢 + 푣 + 푤 where 푢 , 푣 , and 푤 are the body-axis velocity components. Airflow angles are also less physically meaningful in the high wing-angle transition regime at low airspeed. Additionally, dimensionless force and moment coefficients are singular in hover due to division by zero dynamic pressure (see Eq. (1) ), and stability frame response variables are meaningless due to their dependence on 훼 .

B. Rotary-Wing Aircraft Modeling Rotorcraft system identification follows different conventions from fixed-wing modeling, but is also well defined in the literature for helicopters and tilt-rotor variants [ 42 ]. Contrary to fixed-wing aircraft system identification, rotorcraft system identification is generally restricted to developing mathematical models using flight test data because wind tunnel testing is precluded by difficulties in scaling subscale rotary-wing vehicles and facility limitations [ 47 ]. Body-axis force and moment parameters are also generally estimated in their dimensional form due the differences in scaling between rotor and fuselage aerodynamics. Stability-axes and wind-axes become undefined in hover, so modeling is generally only performed in the body-axes for rotorcraft.

Rotorcraft system identification efforts most often develop linear models at a reference flight condition. These point models are only valid at the flight condition where they are identified and assume that the complex coupling of rotorcraft aerodynamics can be represented in linear differential equations. The explanatory variables used for estimation are generally body-axis velocity components 푢 , 푣 , 푤 , angular rates 푝 , 푞 , 푟 , pilot control inputs, and rotor states, such as flapping, lead-lag, inflow, coning, engine dynamics, etc., depending on the design of the vehicle and the desired bandwidth of the developed model. Formulation in terms of body-axis velocity components, as opposed to airflow angles 훼 and 훽 , allows the state variables to be defined in hover and reflects that fuselage angle of attack and angle of sideslip are less physically meaningful for describing rotorcraft aerodynamics. Rotorcraft modeling problems commonly use pilot collective 훿 , longitudinal cyclic 훿 , lateral cyclic 훿 , and pedal deflection 훿 because rotor collective 푐표푙 푙표푛 푙푎푡 푝푒푑 and cyclic blade pitch angles are challenging to measure. Models are commonly expressed in the form of a transfer function or state-space model with added time delay parameters to account for unmodeled higher-order dynamics [ 42 ].

Similar to fixed-wing aircraft, rotorcraft system identification approaches cannot be applied in the same way to modeling eVTOL vehicles. Firstly, significant airframe-propulsion interactions and rapid aerodynamic variation with flight condition for eVTOL vehicles are not accurately captured using linear rotorcraft modeling techniques, requiring extension to nonlinear methods. Secondly, pilot control positions are not acceptable for modeling because there is a far greater number of control surfaces and propulsors than pilot inputs. Furthermore, many rotor states, such as flapping, lead-lag, and coning, are not as relevant to eVTOL vehicles and it would be challenging to measure or estimate these parameters for each propulsor. These states are not necessary to capture dominant eVTOL aerodynamic dependencies due to the smaller diameter of distributed propellers and reduced mechanical complexity compared to articulated rotors. For vehicles with fixed-pitch rotors, modeling can be performed using a propeller rotational speed measurement, whereas vehicles with variable-pitch rotors will require both rotational speed and blade angle measurements.

C. Experimental Aircraft The modeling approaches developed in this paper for tilt-wing, DEP aircraft are described in the context of the Langley Aerodrome No. 8 (LA-8) [ 34 ]. The LA-8, pictured in Fig. 1, is a subscale, tandem tilt-wing, VTOL, DEP configuration intended to be a testbed for eVTOL technology. The LA-8 was developed at NASA Langley Research Center as one of several eVTOL concepts intended to explore their unique flight characteristics and resolve implementation challenges to help bring similar full-scale vehicles into mainstream operation.

(a) LA-8 front view (b) LA-8 rear view Fig. 1 LA-8 mounted in the NASA Langley 12-Foot Low-Speed Tunnel.

The LA-8 is equipped with 20 control effectors, including two rotating wings, four elevons, four single-slotted Fowler flaps, two ruddervators, and eight electric motors/propellers. A diagram of the propulsors and control surface definitions is shown in Fig. 2. The front and rear wing tilt angles are denoted 훿 and 훿 . The control surface deflections are 푤 푤 1 2 denoted 훿 , 훿 , 훿 , 훿 for elevons; 훿 , 훿 , 훿 , 훿 for flaps; and 훿 , 훿 for ruddervators. Wing, elevon, flap, and 푒 푒 푒 푒 푓 푓 푓 푓 푟 푟 1 2 3 4 1 2 3 4 1 2 ruddervator deflections are defined positive trailing edge downward. The propulsor rotational speeds are denoted 푛 , 푛 , 1 2 ..., 푛 . Propellers 1, 3, 6, and 8 rotate clockwise and propellers 2, 4, 5, and 7 rotate counterclockwise, as viewed from the rear. All propellers are 16-inch diameter, 8-inch pitch, fixed-pitch, 3-bladed propellers.

𝒏 𝒏 𝟑 𝟐 𝒏 𝒏 𝟏 𝟒 𝜹 𝒘 𝟏 𝜹 𝒙 𝜹 𝒆 𝜹 𝜹 𝒃 𝒇 𝟏 𝟏 𝒇 𝒆 𝟐 𝟏 𝒏 𝒏 𝟔 𝟕 𝒚 𝒏 𝟓 𝒃 𝒏 𝟖 𝜹 𝒘 𝟐 𝜹 𝒆 𝜹 𝜹 𝜹 𝟑 𝒆 𝒇 𝒇 𝟒 𝟒 𝟑 𝜹 𝜹 𝒓 𝒓 𝟏 𝟐 Fig. 2 LA-8 propulsor and control surface definitions.

III. Wind Tunnel Testing The aerodynamic models presented in this paper were developed using wind tunnel data collected at the NASA † Langley 12-Foot Low-Speed Tunnel . The LA-8 wind tunnel tests used for this work included an isolated propeller test and a full-scale, powered-airframe LA-8 test using the same vehicle that will be used for flight testing. Testing at the flight vehicle full-scale circumvents the need to scale using similitude relationships, which are challenging for rotorcraft and typically limit vehicle wind tunnel testing [ 47 , 48 ]. An airframe-only test, without propellers, was also performed but not used for model development due to difficulty in superimposing data in the transition flight regime. An overview of the individual tests used for model development is given next.

A. Isolated Propeller Testing Propeller aerodynamics for eVTOL vehicles are complex due to the large range of operational flight conditions, compounded by the presence of many propulsors. An isolated propeller test was conducted to obtain a better understanding of LA-8 propeller behavior in isolation and develop mathematical models for the propeller aerodynamics intended to be used in concert with data from other LA-8 wind tunnel entries to develop the vehicle aerodynamic model.

The test conditions for the isolated propeller testing included dynamic pressure ranging from 0 to 6 psf (corresponding to a freestream velocity of 0 to 71 ft/s), motor speed ranging from approximately 1500 to 6000 RPM, and angle of ◦ ◦ incidence relative to the propeller disk ranging from 0 to 180 . Combinations of dynamic pressure, motor speed, and incidence angle were tested in a one-factor-at-a-time (OFAT) manner for both the clockwise and counterclockwise rotating propellers. The collected data cover nearly the full range of expected flight conditions and are used to augment the powered-airframe LA-8 wind tunnel test described next. The propeller testing methodology and experimental findings are further described in Ref. [ 39 ]. The test data were subsequently used to develop a propulsion system model that characterizes propeller aerodynamics across the wide range of operational flight conditions expected to be encountered by the LA-8 [40].

B. Powered-Airframe Testing Wind tunnel tests to capture complex eVTOL vehicle nonlinear aerodynamics and interactions is a challenging undertaking. The LA-8 aircraft, with eight propulsors, ten control surfaces, and two rotating wings, as well as three static flight condition variables, defined by either 푉 , 훼 , 훽 or 푢 , 푣 , 푤 , results in 23 different experimental factors. Due to the † Information available online at https://researchdirectorate.larc.nasa.gov/12-foot-low-speed-tunnel-12-ft-lst/ [accessed November 2020] large number of factors, traditional static OFAT wind tunnel testing is not practical for developing models describing the complex nonlinear aerodynamics and vehicle interactions of eVTOL aircraft. Experiments designed using DOE theory, however, can efficiently scale to a large number of factors allowing wind tunnel tests to be completed in a reasonable amount of time while supporting identification of interactions between all factors. DOE-based wind tunnel tests were used previously to characterize the GL-10 aircraft [ 26 , 27 ] and is the approach used for the LA-8 powered-airframe testing [38].

DOE-based testing provides a statistically rigorous experimental design methodology supplying rich information content in a relatively compact data set. The model development process also benefits from additional properties of the DOE theory including orthogonality , randomization , replication , blocking , and sequential testing [ 31 , 33 ]. Orthogonal experimental factors aid the model structure identification and parameter estimation process by ensuring low candidate regressor correlation. Randomization of test points reduces the effects of systematic measurement errors and extraneous factors—errors are reflected in the parameter variance rather than corrupting the parameter estimates. Replication of data points provides insight into the measurement facility noise characteristics. Blocking minimizes the effects of unmodeled nuisance factors. Sequential testing allows collecting only the minimum amount of data necessary to meet desired modeling fidelity.

While DOE-based testing has several advantages compared to OFAT testing, particularly for eVTOL vehicles, initial OFAT testing may be needed to help define the ranges of certain factors for DOE testing [ 38 ]. Before performing DOE testing, an OFAT test of the LA-8 vehicle was performed to define the nominal flight envelope and find trim points where longitudinal forces were balanced and pitching moment was zero [ 37 ]. Trimming was primarily accomplished by varying motor speed and wing angle to prevent control surfaces from being near their physical limits. The ranges of factors for the DOE test were subsequently specified in accordance with the nominal flight envelope, providing improved data density for modeling over the full flight envelope.

Static DOE wind tunnel testing for the LA-8 was performed at eight dynamic pressure settings from 0 to 5 psf (freestream airspeed of approximately 0 to 65 ft/s), where the factor ranges at each dynamic pressure reflected the values expected to be seen in flight at that condition. Contrary to subsonic fixed-wing aircraft tests which are typically performed at one dynamic pressure setting, testing was performed at multiple dynamic pressure settings due to the large contributions of both the fixed and rotating vehicle components. Fixed-wing aircraft forces and moments are traditionally nondimensionalized by dynamic pressure ¯ 푞 and aircraft geometric characteristics. Propeller forces and moments are nondimensionalized by air density 휌 , squared rotational speed 푛 , and propeller geometric characteristics while being highly dependent on airspeed in terms of advance ratio 퐽 . These differences in aerodynamic scaling suggest testing at a variety of wind tunnel dynamic pressure settings is required.

The experimental factors specified for DOE testing at each different tunnel dynamic pressure setting were angle of attack 훼 , angle of sideslip 훽 , wing angles 훿 , 훿 , elevon deflection angles 훿 , 훿 , 훿 , 훿 , flap deflection angles 훿 , 푤 푤 푒 푒 푒 푒 푓 1 2 1 2 3 4 1 훿 , 훿 , 훿 , ruddervator deflection angles 훿 , 훿 , and motor pulse width modulation (PWM) command 휂 , 휂 ,..., 휂 , 푓 푓 푓 푟 푟 1 2 8 2 3 4 1 2 resulting in 22 independently varied factors. The factor ranges for angle of attack, angle of sideslip, and wing angle are shown in Fig. 3, with derived parameters of 푧 -axis velocity 푤 = 푉 sin 훼 cos 훽 and 푦 -axis velocity 푣 = 푉 sin 훽 also displayed. The data points show the upper and lower limit for each variable against the tested dynamic pressure setting, and the connecting lines reflect the modeled flight envelope. While wing angles 훿 and 훿 were varied independently 푤 푤 1 2 during testing, the ranges of values were identical for all testing, so the wing angle is displayed with a generic label 훿 .

푤 Fig. 3 LA-8 DOE test state and wing orientation factor ranges against dynamic pressure setting.

After the ranges of factors for each test condition were determined, the powered-airframe wind tunnel experiment was designed using the RAM experimental design process based on DOE theory. For each experiment at different

dynamic pressure settings, a series of five DOE blocks was designed to acquire the data necessary to identify increasingly ® complex aerodynamic models. Block design was accomplished with the aid of Design-Expert , a commercially ‡ available statistical software package. The blocks are as follows: (1) face-centered design (FCD), (2) nested FCD [ 49 ], (3) I-optimal design for quadratic models, (4) I-optimal design for up to cubic models, and (5) I-optimal design used as validation data. The I-optimal blocks are designed to minimize the integrated prediction variance over the range of factors, which reduces prediction error for the identified models [ 33 ]. Figure 4 shows a two-dimensional slice of the 22-factor space for the experiments designed at ¯ 푞 = 3 . 5 psf; Fig. 4a shows Blocks 1-2 and Fig. 4b shows Blocks 3-5. Although 훼 and 훽 are shown in the figure, similar plots would be obtained for all of the other experimental factors varied during the test.

(a) Blocks 1 and 2 (FCD and nested FCD designs) (b) Blocks 3, 4, and 5 (I-optimal and validation designs) Fig. 4 Two-dimensional slice of the designed LA-8 DOE factor space for 휶 and 휷 at ¯ 풒 = 3 . 5 psf.

IV. System Identification Methodology An overview of the methods used for model structure development, collinearity analysis, parameter estimation, and validation are discussed in the following sections; a comprehensive presentation of these techniques can be found in Ref. [ 41 ]. The model structure identification and parameter estimation methods used for this work were adapted from § the System IDentification Programs for AirCraft (SIDPAC) software toolbox. SIDPAC is a collection of programs ® written in MATLAB that can be tailored for a particular modeling effort. This flexibility was beneficial for refinement of model development strategies for eVTOL aircraft.

A. Model Structure Determination Development of an adequate model structure is one of the most challenging aspects of aerodynamic modeling for eVTOL aircraft. eVTOL vehicles share overlapping characteristics with both fixed-wing and rotary-wing aircraft, as well as complex vehicle-specific phenomena such as high incidence angle propeller aerodynamics, DEP, and propulsion-airframe interactions, which must be represented in the model structure. Consequently, suitable definitions of modeling explanatory and response variables are unclear and the expected model structure is not well defined due to limited previous research in this area. Furthermore, the presence of a larger number of candidate regressors compared to conventional aircraft modeling problems leads to numerical conditioning issues, large data processing times, and a requirement for more user insight. Multiple techniques were investigated to develop the model structure, including stepwise regression [ 41 , 50 ] and multivariate orthogonal function modeling [ 41 , 51 ]. Stepwise regression was used to produce the results presented in this paper.

The stepwise regression algorithm used here for model structure development is based on the algorithm described in Ref. [ 50 ]. This algorithm is a combination of forward selection and backwards elimination of candidate regressors where a single regressor is either added to or removed from the model at each iteration. The procedure is started with only a bias parameter included in the model structure. The first step is adding the candidate regressor with the highest ‡ Information available online at https://www.statease.com/software/design-expert/ [accessed November 2020] § Information available online at https://software.nasa.gov/software/LAR-16100-1 [accessed November 2020] correlation to the unmodeled portion of the response variable 푟 into the model. The process is continued by adding 푖 푡 ℎ excluded candidate model terms with the highest 푟 into the model. For the 푖 model term excluded from the model 푖 structure, 푟 is calculated as: 푖 푇 ( 풗 − ¯ 푣 ) ( 풗 − ¯ 푣 ) 푖 푖 푧 푧 푟 = √ √ (4) 푖 푇 푇 ( 풗 − ¯ 푣 ) ( 풗 − ¯ 푣 ) ( 풗 − ¯ 푣 ) ( 풗 − ¯ 푣 ) 푖 푖 푖 푖 푧 푧 푧 푧 푡 ℎ Here, 풗 is the 푖 model term residual vector resulting from being regressed on by terms included in the current model, 푖 with mean denoted ¯ 푣 ; 풗 is the difference between the measured response variable and the response modeled by the 푖 푧 current regressors in the model, with mean denoted ¯ 푣 .

푧 At each stepwise regression iteration, terms included in the model are considered to be removed from the model if their partial 퐹 -statistic, 퐹 , falls below a cutoff threshold 퐹 ( 훼 , 1 , 푁 − 푛 ) prescribed by a partial 퐹 -test for significance 0 푝 푝 푖 푡 ℎ at an 훼 significant level with 푛 included model regressors. For the 푖 model term included in the current iteration of 푝 푝 the model structure, 퐹 is calculated as 푖 ˆ 휃 푖 퐹 = (5) 푖 ˆ 푠 ( 휃 ) 푖 ˆ ˆ where 휃 is the respective parameter estimate and 푠 ( 휃 ) is the respective parameter variance. A modified stepwise 푖 푖 regression procedure was also incorporated to ensure key linear regressors were added to the model before incorporating nonlinear and cross-model terms [41, 50].

The stepwise regression algorithm can be run automatically or manually. Due to the abundance of candidate regressors and large number of model terms needed to describe complex eVTOL aerodynamic phenomena, each model required many iterations to converge to an adequate model structure. For this reason, the stepwise regression algorithm was run automatically until the remaining excluded model terms did not surpass the partial 퐹 -statistic cutoff value when added to the model. However, due to the aerodynamic complexity and large number of candidate regressors, the automated algorithm was found to produce models deemed by analysts to require further adjustments to model terms.

The automated algorithm was effective in predicting dominant terms that should be included in the model, but was more challenged to determine which borderline terms with similar statistical modeling metrics were worthy of inclusion in the model structure (a task that would be more obvious to a subject matter expert based on physical insight). This results in both model terms excluded from the model that should be included and other model terms included in the final model that should be excluded. For example, if three out of four terms describing the interaction of an elevon with its closest propulsor are included in the model, consideration should be given to adding the fourth elevon-propulsor interaction term. Conversely, if the model includes an interaction term for control effectors far apart on the vehicle lacking physical justification, consideration should be given to removing the term from the model. Most often these borderline model terms are either just above or below the statistical cutoff thresholds. For these reasons, it is useful to add user insight to the modeling process while still utilizing the efficiency gained through automation due to the size of the modeling problem.

To address the desire for modeling efficiency, as well as addition of subject matter expert user insight when needed, a partially-automated modified stepwise regression (PAMSWR) process was refined and utilized for model development.

The automated algorithm is first run to develop a baseline model. After termination of the automatic algorithm, a subject matter expert is given the ability to add insight into the modeling process by adding or removing model terms based on physical vehicle insight and statistical metrics introduced above as well as additional metrics such as coefficient of determination 푅 , predicted square error (PSE), and predicted residual error sum of squares (PRESS). The PAMSWR approach is particularly useful for eVTOL vehicle modeling because the automated process helps to expedite the model development process, but the manual model adjustment at the end allows for modeling insight from a subject matter expert.

Part of the model structure determination process is to select a pool of candidate regressors to be evaluated for inclusion in the model structure. Due to the large number of LA-8 test factors (22), the DOE design specified in Sec. III.B only supports up to quadratic model terms with two-factor interactions terms (Quadratic+2FI). An example of Quadratic+2FI complexity candidate model terms for an arbitrary three-factor study involving explanatory variables 2 2 2 of 훼 , 훽 , and 훿 would be 훼, 훽, 훿 , 훼훽 , 훼훿 , 훽훿 , 훼 , 훽 , and 훿 . From this list of candidate regressors, one can infer that the number of candidate model terms would become large for a study with many factors. For a 22-factor study with Quadratic+2FI complexity, there are 275 candidate model terms.

B. Parameter Estimation Ordinary least-squares regression is used to estimate a vector of 푛 unknown model parameters 휽 for a given model 푝 풚 = 푿휽 [41]. Here 풚 is the length 푁 model response vector and 푿 is a 푁 × 푛 matrix consisting of column vectors of 푝 regressors assumed to be measured without error. The regression equation, including a measurement of the response variable 풛 , corrupted by constant variance, zero-mean, and uncorrelated measurement error 흂 , is given as: 풛 = 푿휽 + 흂 (6) For least-squares parameter estimation, the optimal estimate of the unknown parameters 휽 is determined by minimizing the cost function: 푇 퐽 ( 휽 ) = ( 풛 − 푿휽 ) ( 풛 − 푿휽 ) (7) It follows that the solution to compute an optimal estimate of the unknown parameters is ( ) − 1 푇 푇 ˆ 휽 = 푿 푿 푿 풛 (8) ˆ where 휽 is a vector of 푛 estimated parameters. Assuming uncorrelated measurement errors and an adequate model 푝 ˆ ˆ structure is used to compute a modeled response variable history ˆ 풚 = 푿 휽 , a length 푛 vector of standard errors 풔 ( 휽 ) 푝 ˆ corresponding to the estimated parameters 휽 is given as: √ ( ) 푇 [ ] ( ) ( 풛 − ˆ 풚 ) ( 풛 − ˆ 풚 ) − 1 푇 ˆ 풔 ( 휽 ) = diag 푿 푿 (9) 푁 − 푛 푝 A characteristic of modeling eVTOL aircraft is the presence of many regressors and associated parameter estimates included in the final models. Thus, even when implemented on modern computers, numerical best practices are emphasized for solving the regression problem. This includes scaling regressor measurements to be the same order of ( ) − 1 푇 magnitude for performing calculations and using a robust inversion technique to compute 푿 푿 , such as singular value decomposition.

C. Data Collinearity Assessment Data collinearity is defined as a correlation between regressors high enough to cause corrupted model identifica- tion [ 41 ]. Data collinearity will cause difficulty in both model structure determination and parameter estimation because the effects of certain regressors on the response cannot be distinguished. Model structure identification is corrupted by candidate regressor correlation, particularly for the large number of candidate model terms associated with modeling eVTOL aircraft, because an algorithm is more inclined to include model terms that lack physical reality or exclude model terms describing significant aerodynamic phenomena. Parameter estimation algorithms cannot differentiate between highly correlated model terms, resulting in inaccurate parameter estimates and uncertainties from the poorly conditioned estimation problem. For these reasons, it is important to assess the presence of data collinearity in the candidate regressor selection and the final aerodynamic model equations.

Correlation between two regressors can be assessed using the correlation coefficient. The correlation coefficient 푟 푖 푗 between two regressor measurement histories, 풙 and 풙 , is defined as 푖 푗 푇 ( 풙 − ¯ 푥 ) ( 풙 − ¯ 푥 ) 푖 푖 푗 푗 푟 = √ (10) 푖 푗 √ 푇 푇 ( 풙 − ¯ 푥 ) ( 풙 − ¯ 푥 ) ( 풙 − ¯ 푥 ) ( 풙 − ¯ 푥 ) 푖 푖 푖 푖 푗 푗 푗 푗 where ¯ 푥 and ¯ 푥 are the respective mean values. A correlation coefficient value of zero means the signals are uncorrelated, 푖 푗 or orthogonal, and an absolute correlation coefficient of one indicates that the signals are completely correlated. A correlation coefficient between regressors with magnitude greater than 0.9 indicates that data collinearity problems may be encountered [41, 43].

Collinearity assessment is useful for confirming a choice of modeling candidate regressors from a given data set are sufficiently decorrelated for model identification. Figure 5 shows the correlation coefficients between all Quadratic+2FI candidate regressors for the designed 22-factor test matrix used for LA-8 model development. This figure shows that Quadratic+2FI regressors derived from the experimental test factors are sufficiently decorrelated and provides a basis for comparison to other candidate regressor choices explored in future sections.

Fig. 5 Correlation of Quadratic+2FI experimental factor candidate regressors (in coded units, see Sec. IV.E).

D. Model Validation Model validation is an examination of model adequacy using data withheld from the model development process.

Regression methods minimize the summation of squared modeling residuals between modeled and measured response, so inspection of modeling fit metrics and modeling residuals alone does not provide information about the model predictive capability. Assessment of model performance using validation data not used for modeling provides a more reliable estimate of model prediction accuracy. Validation assessment can be performed by comparing the measured response for validation data to the response predicted by the model for the same explanatory variable inputs. Further assessment is performed by analyzing the prediction residuals between the measured and predicted response, 풆 = 풛 − ˆ 풚 . Here, it is useful to compare modeling and prediction residuals because a significant increase in the spread of prediction residuals compared to modeling residuals is a way of diagnosing an improper model. Plots of residuals over a measurement history should appear as white noise with a magnitude below a level deemed acceptable for a particular modeling effort.

Further residual distribution statistics can be computed, including the standard deviation and root-mean-square error (RMSE).

Residuals and their statistical properties can be given further interpretability by normalization. The error normalization metric used in this work is the range of response variable measurements used to develop the model, range ( 풛 ) = 풛 − 풛 . Range normalization provides a fair comparison between prediction error metrics for different 푚푎푥 푚푖푛 response variables used for aircraft modeling because longitudinal responses are generally biased above or below zero and lateral-directional responses are generally centered about zero. The normalized residual vector is defined as: 풛 − ˆ 풚 ∗ 풆 = (11) range ( 풛 ) Similarly, the normalized root-mean-square modeling error (NRMSE) is defined as: √ 푇 1 ( 풛 − ˆ 풚 ) ( 풛 − ˆ 풚 ) NRMSE = (12) range ( 풛 ) 푁 A prediction error metric defined using critical binomial analysis of validation residuals is particularly useful as a quantitative measure of the model adequacy. For this analysis, each validation data point is considered to either pass or fail relative to a prediction error threshold. Failed trials can indicate model inadequacy or measurement error. The binomial test provides a threshold to determine when the number of failures is statistically significant. For this metric, the prediction error level associated with the number of successful trials being equal to the critical binomial number B can be used for comparison to a pre-defined level of acceptable modeling error to determine model adequacy. This ∗ prediction error threshold is denoted here as 푒 . For the experiments designed for this work, there were 75 validation 푐푣 points acquired to test the prediction capability of each model. At the 95% confidence level, the binomial test with a ¶ 1% inference error has a critical binomial number of B = 66 . This can be interpreted as allowing no more than nine validation residuals to exceed the prediction error threshold for an adequate model. The process of computing the 95% ∗ prediction error metric interval used for this work is to calculate the normalized prediction residual vector 풆 using ∗ Eq. (11) and then sort the absolute value of 풆 in ascending order: ∗ ∗ 풆 = sort (| 풆 |) (13) 푎 ¶ ® A method of computing the critical binomial number B is using the binomial inverse cumulative distribution function in MATLAB , where the syntax in this case would be binoinv ( 0 . 01 , 75 , 0 . 95 ) .

∗ ∗ Using the ascending absolute normalized prediction residual vector 풆 , the prediction error metric 푒 is provided as 푎 푐푣 푡 ℎ ∗ the B value of 풆 vector: 푎 ∗ ∗ 푒 = 풆 (B) (14) 푐푣 푎 Further explanation of critical binomial analysis of residuals and justification for using this metric to assess prediction error is given in Ref. [ 52 ]. This metric has been used in several previous aircraft DOE wind tunnel testing studies and is a key component of the aforementioned RAM process [7, 31, 32, 53].

E. Design of Experiments Modeling Considerations As described in Sec. III.B, there are many benefits to using DOE compared to OFAT approaches, particularly for testing of complex eVTOL vehicles. While the DOE approach to experimental design and data collection is a well-suited modeling approach, there are certain important considerations associated with using the data for estimation of response surface models.

A key benefit of the DOE approach is that the experimental factors are optimized to minimize prediction error and have low correlation, aiding the modeling process. When orthogonality is incorporated into the experimental design, as is the case for this work, a coded variable representation of the modeling explanatory variables maintains orthogonality among candidate model terms. Coded variables are transformed from their original units, also called natural or engineering units, using a bias and scale factor so that all variables lie in the interval [− 1 , + 1 ] using the equation 푥 − ( 푥 + 푥 )/ 2 푚푖푛 푚푎푥 ˜ 푥 = (15) ( 푥 − 푥 )/ 2 푚푎푥 푚푖푛 where 푥 is the original regression variable and ˜ 푥 is the coded variable. Due to the orthogonality in coded variable space, it is important to perform modeling in terms of coded variables, as opposed to variables defined in engineering units. Variables in their original units, or multiplied by a scale factor, are not guaranteed to be orthogonal, even for orthogonal designs [ 33 ]. Use of variables in their engineering units can, therefore, result in significant correlation between candidate model terms. While correlation between linear model terms remains the same between using natural variables and coded variables, factor interactions terms may become highly correlated, adversely affecting model identification (see Sec. IV.C). Figure 6 shows the correlation coefficient metric for test factor Quadratic+2FI candidate regressors defined in engineering units for the 22-factor DOE test. Coded variable regressors exhibit low correlation favorable for modeling (Fig. 5), whereas regressors in engineering units show significant correlation (Fig. 6), which would be expected to impair model identification.

Fig. 6 Correlation of Quadratic+2FI experimental factor candidate regressors in engineering units.

The orthogonality property in the experimental design also limits using certain alternative definitions of modeling explanatory variables than were used as test factors in an experiment. This is important because practical testing limitations may require a different choice of factors for testing than used for modeling regressors. Certain transformations are permissible, as long as they are nearly proportional to the factors for which the experiment was designed. However, many transformations are detrimental to model identification due to interaction terms becoming severely correlated.

This concept will be revisited later in Sec. V.A and Sec. V.B.

While models defined in coded units are suitable for model predictions and simulation, models in engineering units are generally desired because of their physical meaning relating to stability and control derivatives. Consequently, an additional consideration for modeling from DOE data is transforming models in coded units to models in engineering units. The transformation is not trivial when nonlinear and cross model terms are included in the model because coded units for explanatory variables are defined using both a bias and scale factor. For this transformation to be consistent, the model must be hierarchical meaning that if higher-order terms appear in the model, the complementary lower-order

terms must also be included. A potential downside to enforcing a hierarchical model is the requirement for additional model terms, increasing the total number of model terms, which can be accompanied by decreased accuracy in model parameters and increased prediction error [ 33 ]. A practical strategy that was found to work well for developing models in coded and engineering units was to first determine the model structure and parameter estimates using coded variables, enforcing model hierarchy; then, as a last step, least-squares parameter estimation was again performed using the same identified model structure with regressors in their engineering units (using the numerical conditioning strategies stated previously in Sec. IV.B). This approach was successful because parameter estimation did not appear to be significantly affected by whether estimation for a hierarchical model was performed with coded or natural variables. Conversely, it was determined essential to use coded units for model structure identification to retain candidate regressor orthogonality.

This approach resulted in models in coded and engineering units with identical prediction capability.

V. Aerodynamic Modeling Approaches Aerodynamic modeling for this effort focuses on developing a polynomial representation of the aerodynamic forces and moments as a function of vehicle state and control variables. Two approaches were hypothesized and tested to investigate modeling for eVTOL aircraft. The first approach discussed is a conventional procedure where factors under test, or close variants, are added to a universal candidate regressor pool and the model is identified from the powered-airframe DOE wind tunnel data. The second approach utilizes identified isolated propeller models to inform full-airframe model identification. The modeling approaches developed herein apply relevant parts from both fixed-wing and rotary-wing modeling methodologies as well as incorporate strategies specific to tilt-wing, DEP aircraft. For this ∗ study, the goal was to develop models minimizing prediction error, where a value of 5% or less for 푒 was considered 푐푣 to be adequate.

Aerodynamic modeling for tilt-wing, DEP aircraft requires a different approach, compared to conventional fixed- wing and rotary-wing aircraft modeling approaches outlined in Sec. II.A-II.B. eVTOL vehicles can be considered a fixed-wing/rotary-wing hybrid suggesting that a combination of the two modeling methodologies will facilitate suitable model development. Accordingly, the modeling approaches defined here largely seek to merge appropriate fixed-wing and rotary-wing modeling attributes with certain new strategies to develop a modeling methodology for LA-8 and other similar vehicles. Adopted from rotorcraft modeling, the response variables are defined as the dimensional body-axis aerodynamic forces and moments 푋 , 푌 , 푍 , 퐿 , 푀 , and 푁 , as opposed to nondimensional aerodynamic force and moment coefficients 퐶 , 퐶 , 퐶 , 퐶 , 퐶 , and 퐶 . Furthermore, the explanatory variables for aerodynamic states are defined in 푥 푦 푧 푙 푚 푛 terms of the body-axis velocity components 푣 and 푤 , as opposed to angle of attack 훼 and angle of sideslip 훽 . These choices facilitate a modeling strategy valid from cruise to hover. Adopted from fixed-wing modeling, a generally nonlinear polynomial expansion modeling approach is used to capture the nonlinear aerodynamic effects including airframe-propulsion interactions.

Certain attributes of tilt-wing, DEP aircraft require specific modeling techniques not gleaned from fixed-wing or rotary-wing system identification. Both fixed-wing and rotary-wing modeling approaches also do not translate well to a vehicle with many propulsion elements which individually interact with lifting surfaces and control surfaces. The rotating wings add additional challenges not seen in tilt-rotor designs because the propellers, wings, and wing-fixed control surfaces all change orientation with respect to the modeling frame of reference in the body-axes. Each different combination of wing angle orientation can be interpreted as a vehicle configuration change. One way of handling this complexity is to develop a different aerodynamic model at each combination of wing angle settings, in addition to flight condition defined by ¯ 푞 for wind tunnel testing. This method would be ideal when a transition wing angle schedule has been defined. However, the identified aerodynamic model may be used to inform the transition wing angle schedule, as is the case for the present vehicle. The presence of two independently rotating wings adds further complication. In this case, developing a new aerodynamic model at numerous possible combinations of wing angle throughout transition becomes impractical due to the large increase in the number of test points required, and therefore, it is necessary to include wing angle explanatory variables in modeling rather than a configuration parameter.

A. Approach I: Modeling Using Only Powered-Aircraft Test Data and a Standard Regressor Definition The first approach for modeling the LA-8 aircraft, referred to as “Approach I,” uses the powered-airframe DOE wind tunnel test described in Sec. III.B for model identification. Models developed from DOE testing conventionally evaluate the factors under test as candidates for explanatory variables. However, due to unique characteristics of eVTOL aircraft, test facility integration limitations, and convenience, analysis was instead performed by redefining certain explanatory variables for modeling. For example, testing was performed by varying 훼 and 훽 because of ease of envelope definition and simplified integration into a wind tunnel test apparatus. While testing was performed with experimental factors of 훼 and 훽 , modeling was performed using body-axis velocity components 푣 and 푤 , following rotorcraft system identification convention. Since body-axis velocity components are closely related to airflow angles, this variable change does not affect the ability to identify a model from the DOE data since the regressors retain most of their designed orthogonality (see Sec. IV.E). Similarly, testing was performed by varying motor PWM commands, but modeling was performed using measured propeller rotational speed. Propeller speed is more relevant to describe propeller aerodynamics and the ‖ relationship between PWM command and propeller speed can change significantly due to nuisance factors , such as motor temperature. While the PWM command to propeller speed relationship does not follow a linear trend, particularly at incidence [ 40 ], low regressor correlation was still sufficiently maintained when transferring the variables to the coded variable space, making this approach justifiable. Figure 7 shows the correlation coefficient metric for Quadratic+2FI candidate regressors computed for the explanatory variables specified in this section. The correlation for equivalent levels of model complexity is only slightly higher than the correlation between commanded experimental factors shown in Fig. 5. This suggests validity in the proposed modeling approach.

Fig. 7 Correlation of Approach I Quadratic+2FI candidate regressors in coded units.

To summarize, the modeling explanatory variables for this approach were defined to be the body-axis velocity components 푣, 푤 in ft/s, propeller speed 푛 , 푛 , ..., 푛 in revolutions per second, wing angle 훿 , 훿 in radians, elevon 1 2 8 푤 푤 1 2 deflection 훿 , 훿 , 훿 , 훿 in radians, flap deflection 훿 , 훿 , 훿 , 훿 in radians, and ruddervator deflection 훿 , 훿 in 푒 푒 푒 푒 푓 푓 푓 푓 푟 푟 1 2 3 4 1 2 3 4 1 2 radians. The response variables are defined as the dimensional aerodynamic forces 푋, 푌 , 푍 in lbf and moments 퐿, 푀, 푁 in ft-lbf acting on the aircraft in body-fixed axes. An independent model was developed at each dynamic pressure condition tested.

One advantage of modeling Approach I is its generality, where the only limitation in describing the vehicle static aerodynamic characteristics is the adequacy of modeling with the particular selection of candidate regressors.

Additionally, this approach avoids assumptions of model superposition validity (i.e. combining aerodynamic models for different aircraft components, such as propellers and wings) since all model parameters are estimates from a comprehensive data set where all states and controls are varied. The final models are also in a compact form allowing easy use for a variety of applications. A second modeling approach, described next, strives to add additional fidelity and simulation advantages utilizing supplementary propeller wind tunnel testing to develop the aerodynamic model.

B. Approach II: Modeling Using a Combination of Isolated Propulsion and Powered-Airframe Testing An alternative modeling approach to the preceding section’s methodology, referred to as “Approach II,” is to combine isolated propeller models with full-airframe models. Isolated propeller aerodynamics across an eVTOL aircraft flight envelope are highly complex and, at incidence, produce significant off-axis forces and moments in addition to axial thrust and torque. For example, in Ref. [ 40 ], 31 model terms are identified to characterize the aerodynamic forces and ◦ ◦ moments produced by a single isolated propeller for airflow incidence angles ranging between 0 to 60 . Consequently, the complexity of propeller aerodynamics for many different propulsors increases the difficulty of identifying all necessary model terms from a powered-airframe test alone, suggesting possible merit for pursuing a hybrid propeller and full-airframe modeling approach.

The LA-8 propeller aerodynamic model was developed using the isolated propeller test described in Sec. III.A. A 푝 푝 푝 푝 푝 푝 functional representation of thrust components 푇 , 푇 , 푇 and torque components 푄 , 푄 , 푄 in a propeller-fixed axis 푥 푦 푧 푥 푦 푧 system were identified using the dimensionless thrust and torque coefficients as the response variables. The propeller ‖ For this reason, it is essential to acquire a direct measurement of propeller rotational speed for modeling, as opposed to relying on a calibration curve between motor PWM command and propeller speed.

thrust and torque coefficients are defined as 푝 푝 푝 푝 푝 푝 푇 푄 푇 푇 푄 푄 푦 푦 푥 푧 푥 푧 퐶 = , 퐶 = , 퐶 = , 퐶 = , 퐶 = , 퐶 = (16) 푇 푇 푇 푄 푄 푄 푥 푦 푧 푥 푦 푧 2 4 2 4 2 4 2 5 2 5 2 5 휌푛 퐷 휌푛 퐷 휌푛 퐷 휌푛 퐷 휌푛 퐷 휌푛 퐷 where 휌 is the air density, 푛 is the propeller speed in revolutions per second, and 퐷 is the propeller diameter.

The propeller modeling explanatory variables were the normal components of advance ratio 퐽 , the tangential 푥 components of advance ratio 퐽 , and the propeller blade Reynolds number 푅푒 , defined respectively as: 푧 푉 cos 푖 푝 퐽 = (17) 푥 푛퐷 푉 sin 푖 푝 퐽 = (18) 푧 푛퐷 휌푣 푐 푝 푅푒 = (19) 휇 In these equations, 푉 is the freestream airspeed, 푖 is the propeller incidence angle relative to the oncoming airflow (see 푝 Fig. 8), 휇 is the air viscosity, 푣 is the linear speed of the propeller blade, and 푐 is the propeller chord length, with other 푝 parameters defined previously. Following the definitions in Refs. [ 54 , 55 ], the propeller blade speed and propeller chord are calculated at 75% of the propeller blade length. The LA-8 propeller aerodynamic model development is described in further detail in Ref. [40].

𝑥 𝑝 𝑖 𝑉 𝑝 𝑧 𝑝 Fig. 8 Propeller incidence angle definition and coordinate system.

Using the aerodynamic model developed for the isolated propellers, the aerodynamic conditions at the eight vehicle propeller disk centers were used to estimate forces and moments produced by each of the propellers in the 푡 ℎ powered-airframe DOE test data. Each calculated 푘 propeller forces and moments were transferred from the propeller center in the propeller-fixed frame to the aircraft modeling reference location in the body-fixed frame, as shown in 푡 ℎ Ref. [ 40 ]. The 푘 propeller forces and moments transferred to the modeling reference location in the body-fixed 푏 푏 푏 푏 푏 푏 frame are denoted 푇 , 푇 , 푇 , 푄 , 푄 , and 푄 . The estimated propeller forces and moments for each of the eight 푥 푦 푧 푥 푦 푧 푘 푘 푘 푘 푘 푘 propellers were then subtracted from the measured forces and moments to compute an estimate for the non-propulsive ˆ ˆ ˆ ˆ ˆ ˆ forces and moments experienced by the aircraft, denoted 푋 , 푌 , 푍 , 퐿 , 푀 , and 푁 , as follows: 8 8 8 8 8 8 ’ ’ ’ ’ ’ ’ 푏 푏 푏 푏 푏 푏 ˆ ˆ ˆ ˆ ˆ ˆ 푋 = 푋 − 푇 , 푌 = 푌 − 푇 , 푍 = 푍 − 푇 , 퐿 = 퐿 − 푄 , 푀 = 푋 − 푄 , 푁 = 푋 − 푄 (20) 푥 푦 푧 푥 푦 푧 푘 푘 푘 푘 푘 푘 푘 = 1 푘 = 1 푘 = 1 푘 = 1 푘 = 1 푘 = 1 These forces and moments with propulsion contributions removed, with respective units of lbf and ft-lbf, are defined as the response variables for modeling Approach II.

Similar to Approach I, explanatory variables used for modeling included body-axis velocity components 푣, 푤 in ft/s, wing angle 훿 , 훿 in radians, elevon deflection 훿 , 훿 , 훿 , 훿 in radians, flap deflection 훿 , 훿 , 훿 , 훿 in radians, 푤 푤 푒 푒 푒 푒 푓 푓 푓 푓 1 2 1 2 3 4 1 2 3 4 and ruddervator deflection 훿 , 훿 in radians. To model propulsion-airframe interactions and correct for the fact that the 푟 푟 1 2 presence of the vehicle will have some effect on the forces and moments produced by the propellers, it is important to include a propulsion explanatory variable, even though the main propulsion effects are described by the propeller models.

One choice is to use propeller speed 푛 , 푛 , ..., 푛 in revolutions per second, as was the method used in Approach I. An 1 2 8 푝 alternative approach is to use the estimated axial thrust for each propeller 푇 as an explanatory variable (referred to 푥 푘 henceforth as 푇 for simplicity). The primary propulsion-airframe interactions for blown wing aircraft are theoretically 푘 proportional to the slipstream dynamic pressure . Slipstream dynamic pressure, denoted ¯ 푞 , is the theoretical dynamic 푠푠 푡 ℎ pressure located behind a propeller derived from momentum theory [56, 57]. For the 푘 propeller, 1 푇 푘 ¯ 푞 = 휌푉 + (21) 푠푠 푘 2 퐴 The slipstream dynamic pressure consists of the sum of freestream dynamic pressure ¯ 푞 = 휌푉 and propeller disk 휋 loading 푇 / 퐴 , where 퐴 = 퐷 is the propeller disk area. A visualization of the propeller slipstreams present on the 푘 LA-8 vehicle is given in Fig. 9. The slipstream dynamic pressure is an important quantity because it relates to the control authority of control surfaces and wings interacting with the propeller slipstream. Since a new model is developed at each dynamic pressure for this study and propeller area is a constant, axial thrust for each propeller is the only quantity governing these interactions, and thus, 푇 , 푇 , ..., 푇 in lbf are selected as the propulsor explanatory variables for this 1 2 8 approach. Following the logic presented in Sec. V.A, including these quantities as explanatory variables is permissible because the correlation between modeling terms remains low, as shown in Fig. 10.

ഥ 𝒒 ഥ 𝒒 𝒔𝒔 𝒔𝒔 ഥ 𝒒 ഥ 𝒒 𝟐 𝟑 𝒔𝒔 𝒔𝒔 𝟏 𝟒 𝒙 𝒃 𝒚 ഥ 𝒒 ഥ 𝒒 𝒃 𝒔𝒔 𝒔𝒔 𝟔 𝟕 ഥ 𝒒 ഥ 𝒒 𝒔𝒔 𝒔𝒔 𝟓 𝟖 Fig. 9 Visualization of the propulsor slipstreams on LA-8.

Fig. 10 Correlation of Approach II Quadratic+2FI candidate regressors in coded units.

Modeling Approach II offers multiple additional advantages compared to Approach I. Firstly, Approach II allows for correction of propeller flow conditions when the vehicle model is implemented into a dynamic simulation with nonzero angular velocity [ 40 ]. This attribute is important for superimposing dynamic aerodynamic effects with the static models developed herein. Approach II also offers some meaningful extrapolation capability because the primary propulsion forces and moments are modeled in their dimensionless form. Additionally, utilization of individual propeller models facilitates modeling additional vehicle complexity unable to be described by the candidate regressor pool defined for modeling from the powered-airframe test alone. Disadvantages of Approach II include the additional resources needed to perform supplementary propeller testing and the assumption that superposition of propeller models with full-airframe models is acceptable.

The explanatory variables and response variables defined for each specific approach are summarized in Table 1. The following section presents modeling results for these two LA-8 modeling approaches.

Table 1 Summary of explanatory variables and response variables for the developed modeling approaches Approach I Explanatory Variables 푣, 푤, 푛 , 푛 , 푛 , 푛 , 푛 , 푛 , 푛 , 푛 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 1 2 3 4 5 6 7 8 푤 푤 푒 푒 푒 푒 푓 푓 푓 푓 푟 푟 1 2 1 2 3 4 1 2 3 4 1 2 Response Variables 푋, 푌 , 푍, 퐿, 푀, 푁 Approach II Explanatory Variables 푣, 푤, 푇 , 푇 , 푇 , 푇 , 푇 , 푇 , 푇 , 푇 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 , 훿 1 2 3 4 5 6 7 8 푤 푤 푒 푒 푒 푒 푓 푓 푓 푓 푟 푟 1 2 1 2 3 4 1 2 3 4 1 2 ˆ ˆ ˆ ˆ ˆ ˆ Response Variables 푋, 푌 , 푍, 퐿, 푀, 푁 VI. Results This section presents sample results for the aerodynamic models identified for the LA-8 aircraft. The two aerodynamic modeling approaches described in the previous section were used to develop models at the eight different dynamic pressures tested. The results presented here only consider the models identified at ¯ 푞 = 3 . 5 psf. A future comprehensive report is expected to be published by the authors documenting all models identified for the LA-8 aircraft.

A. Model Identification Results As described in Sec. IV, the modeling process was facilitated using the PAMSWR procedure and least-squares regression. The final model terms, parameter estimates, and standard errors for the identified models at ¯ 푞 = 3 . 5 psf for each dimensional body-axis force and moment using modeling Approach I are given in Table 4 and Table 5, at the conclusion of this paper. Similar results for the modeling Approach II full-airframe models for each dimensional body-axis force and moment with estimated propulsion effects removed are given in Table 6 and Table 7. Sample clockwise propeller models used for Approach II are given in Ref. [40].

A few interesting characteristics about identified model terms are highlighted. One clear feature is that the number of parameters in the models is far greater than many conventional aircraft modeling problems. Another observation is that significant propulsor-wing angle interaction is reflected in every model equation. Control surface-wing and control surface-propulsion interactions are also clearly represented. Control surface-wing and control surface-propulsion interactions are reflected in 푥 -axis force, rolling moment, and yawing moment, but it is apparent that control surface interactions have limited contributions for pitching moment. Control surfaces appear to only have significant interactions with the wings they are affixed to or with the corresponding propulsor(s) displayed graphically in Fig. 9. Comparing the two modeling approaches, the interaction terms are similar, but Approach II has fewer propulsion terms because most of the isolated propeller aerodynamics are captured by the separate propulsion model.

Another characteristic to note is that the models contain significant lateral-directional asymmetries that are not apparent from the LA-8 vehicle configuration. This is a result of manufacturing differences between the clockwise and counterclockwise propellers, which resulted in a significant difference in thrust production between the propeller variants [ 39 ]. Since the propulsion-only and propulsion-airframe interaction effects are significant, this propulsion asymmetry is manifested in many model terms. Consequently, lateral-directional forces and moments have nonzero values for symmetric control inputs, and trim solutions require asymmetric control surface deflections and/or propulsor speed.

Modeling performance statistics, including 푅 , PRESS, PSE, normalized root-mean-square error for modeling data ∗ NRMSE , and standard deviation of normalized modeling residuals 휎 are given in Table 2 for Approach I and Table 3 푚 푚 for Approach II. It should be noted that 푅 , PRESS, and PSE for Approach II are calculated using the modeling fit ˆ ˆ ˆ ˆ ˆ ˆ to its unique response variables, 푋 , 푌 , 푍 , 퐿 , 푀 , and 푁 , defined in Sec. V.B. Accordingly, direct comparison between Approach I and Approach II cannot be made based on these model fit metrics because a substantial part of the response variations are already described by the isolated propeller models, which is not reflected in these metrics. The NRMSE 푚 ∗ and 휎 metrics in Table 3 are calculated using the modeling fit to the total forces and moments to facilitate a fairer 푚 comparison to the results of Approach I. To aid interpretation of results, Table 2 and Table 3 also list the number of model terms 푛 ; the range of total force and moments in the estimation data, range ( 푧 ) ; the maximum absolute 푝 푚 total force or moment value in the estimation data, max (| 푧 |) ; and the standard deviation of the normalized measured 푚 ∗ response for replicate data points 휈 , which gives an estimate of the measurement error in testing. Modeling metrics 푚 indicate a good fit to the data for both approaches, with comparable metrics reflecting similar values. However, modeling metrics are not always consistent with actual model prediction capability. Prediction error metrics are considered a superior measure of modeling success for prediction (see Sec. IV.D) and are discussed in the next section.

Table 2 Approach I modeling and prediction metrics at ¯ 풒 = 3 . 5 psf Metric 푋 푌 푍 퐿 푀 푁 푅 99.3 97.1 98.1 93.3 97.5 98.5 PSE 4.42 0.519 28.3 17.5 54.6 8.76 PRESS 441 191 7940 9640 18600 1770 푛 52 37 37 43 35 48 푝 range ( 푧 ) 59.1 14.3 116 109 186 78.0 푚 max (| 푧 |) 43.9 7.26 108 56.0 96.7 45.7 푚 ∗ 휈 [%] 0.331 1.55 0.520 0.917 0.526 0.204 푚 NRMSE [%] 1.12 3.14 2.54 2.87 2.41 1.71 푚 NRMSE [%] 1.35 3.52 2.92 3.84 2.79 2.57 푣 ∗ 휎 [%] 1.12 3.14 2.55 2.88 2.41 1.71 푚 ∗ 휎 [%] 1.33 3.50 2.72 3.21 2.74 2.58 푣 ∗ 푒 [%] 1.89 5.39 4.16 5.62 4.16 3.75 푐푣 Table 3 Approach II modeling and prediction metrics at ¯ 풒 = 3 . 5 psf Metric 푋 푌 푍 퐿 푀 푁 푅 98.7 96.0 96.7 86.8 95.2 94.5 PSE 2.71 0.426 25.1 14.7 43.9 3.18 PRESS 484 202 10000 10400 21900 1590 푛 47 31 35 39 33 45 푝 range ( 푧 ) 59.1 14.3 116 109 186 78.0 푚 max (| 푧 |) 43.9 7.26 108 56.0 96.7 45.7 푚 ∗ 휈 [%] 0.331 1.55 0.520 0.917 0.526 0.204 푚 NRMSE [%] 1.21 3.27 2.88 3.01 2.63 1.62 푚 NRMSE [%] 1.54 3.79 3.01 4.13 2.83 2.28 푣 ∗ 휎 [%] 1.21 3.27 2.88 3.01 2.63 1.62 푚 ∗ 휎 [%] 1.51 3.77 2.93 3.62 2.76 2.29 푣 ∗ 푒 [%] 2.47 6.00 4.58 7.03 4.02 2.53 푐푣 B. Model Validation A test of model prediction capability using data not considered for model estimation is the best way to quantify modeling success for this study because the objective is to minimize prediction error. Table 2 and Table 3 list prediction metrics computed by comparing the modeled response to measured response data not used for modeling. The prediction error metrics listed are the normalized root-mean-square error for prediction data, NRMSE ; standard deviation of 푣 ∗ ∗ normalized validation residuals, 휎 ; and binomial analysis of residuals prediction error metric, 푒 . The NRMSE and 푣 푣 푐푣 ∗ 휎 are useful because they facilitate direct comparison to the equivalent metrics for modeling data. Observing that the 푣 prediction version of the metrics hold similar values or only slightly increase compared to the equivalent modeling data metrics suggests that a model with good predictive capability has been identified. The binomial analysis of residuals ∗ prediction error metric 푒 quantifies the level of error in the models by defining a 95% prediction error metric interval.

푐푣 ∗ Seeing that all 푒 values are roughly 5% or less indicates good models have been identified, given the experimental 푐푣

facility used for wind tunnel data collection [31].

Figure 11 shows a comparison of select model validation metrics for modeling Approach I and Approach II. The

∗

metrics displayed are the binomial analysis of residuals prediction error metric 푒 , normalized root mean square error

푐푣

for validation data NRMSE , and the number of terms in the full-airframe model, 푛 . For Approach II, the calculated

푣 푝

model predictions are for the total dimensional forces and moments which requires use of both the identified propeller models and full-airframe models. Overall, the prediction error levels and trends are similar between the two different modeling approaches, but Approach II generally has fewer terms in the full-airframe model because significant propulsor complexity is captured in the isolated propeller model.

Fig. 11 Comparison of model validation metrics for both modeling approaches at ¯ 풒 = 3 . 5 psf.

∗

For further model response analysis, Fig. 12 shows a history of normalized modeling residuals 푒 and normalized

푚

∗ ∗

validation residuals 푒 , as well as the binomial analysis of residuals prediction error metric bounds, ± 푒 . The residuals

푣 푐푣

appear to be mostly white noise indicating that the dominant deterministic aerodynamic effects are reflected in the identified models. The modeling and prediction residuals also appear to have similar magnitude, supporting the claim that a good predictive model has been identified.

VII. Conclusions

eVTOL vehicles present new challenges for aircraft modeling and are currently an important area of research. eVTOL vehicle characteristics overlap with fixed-wing and rotary-wing aircraft, but also include complex vehicle-specific phenomena. Consequently, eVTOL aircraft modeling strategies require significant aerodynamic insight from flight dynamics and statistical engineering subject matter experts. The mathematical models developed to describe eVTOL aerodynamics must readily describe many control effectors and complex interactions while also being amendable to drastically changing aerodynamics at numerous different flight conditions across a wide flight envelope. The result is a large modeling problem that must be identifiable within cost and time constraints. This is complicated by numerical issues, large processing time, and substantial user insight needed to produce adequate models.

When using the models developed in this work, certain attributes should be considered. Firstly, the presented models only contain information about the static aerodynamics and contain a quasi-steady assumption, where aerodynamics at the current point in time are only dependent on the current states and controls. Identification of dynamic aerodynamic coefficients dependent on vehicle angular rates and the history of the explanatory variables will be needed to improve model predictive capability in dynamic maneuvering. Additionally, the models only capture two-factor interaction effects; a higher number of factor interactions worthy of inclusion in the model may be present for a vehicle of this complexity. However, experimentation to make these terms identifiable is a challenging task. Also, due to the dimensionality in the models, extrapolation to different flight conditions is not recommended. Finally, lateral-directional asymmetries in the models reflecting physical differences between the vehicle propellers will require attention for trimming and controls.

Multiple aerodynamic modeling approaches have been proposed and evaluated for modeling a tandem tilt-wing, distributed electric propulsion aircraft. One approach utilized data from a powered-airframe, 22-factor DOE wind tunnel

(a) Results for modeling Approach I (b) Results for modeling Approach II

Fig. 12 Normalized modeling and prediction residuals at ¯ 풒 = 3 . 5 psf.

test at multiple dynamic pressure settings to develop a model of the dimensional forces and moments exerted on the aircraft. A second approach sought to produce a higher fidelity model utilizing wind tunnel-derived isolated propulsion models in concert with a full-airframe model identified using the same powered-airframe DOE wind tunnel test. Both approaches proposed a unique set of modeling explanatory variables and response variables, tailored specifically to tilt-wing, DEP aircraft. Many control effectors and complex vehicle interactions result in a large number of potential model terms and challenges in postulating an adequate model structure. A practical and efficient model structure identification strategy was proposed and shown to be effective. Final models were shown to have good predictive capability and small normalized model fit error. This work provides progress in eVTOL aircraft modeling research using experimental techniques, but future eVTOL modeling studies are anticipated to further refine aerodynamic model development methodologies.

Acknowledgments

This research was funded by the NASA Aeronautics Research Mission Directorate (ARMD) Transformational Tools and Technologies (TTT) project. Testing support was provided by Ronald Busan, David Hatke, Earl Harris, Sue Grafton, and Wes O’Neal. Photography support was provided by Lee Pollard. LA-8 vehicle support and insight was provided by David North, Gregory Howland, Steven Geuther, and Robert McSwain. Discussions about the GL-10 aircraft with Paul Rothhaar, Barton Bacon, and Kasey Ackerman assisted postulation of model development approaches. Additional conversations about this work with Eugene Morelli, Jacob Cook, and many other colleagues at NASA Langley Research Center are acknowledged and greatly appreciated.

Parameter Estimate Tables

Table 4 Approach I Aerodynamic Models for 푿 , 풁 , and 푴 at ¯ 풒 = 3 . 5 psf

ˆ ˆ ˆ ˆ ˆ ˆ

Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 )

− 1 − 3 0 − 2 − 1 − 2

푋 + 3.29 × 10 ± 8.62 × 10 푍 − 3.61 × 10 ± 4.53 × 10 푀 − 2.67 × 10 ± 6.94 × 10

푤 푤 푤 − 2 − 2 − 2 − 3 − 1 − 2

푋 − 2.24 × 10 ± 2.02 × 10 푍 − 3.07 × 10 ± 7.86 × 10 푀 + 1.05 × 10 ± 1.20 × 10

푛 1 푛 1 푛 1 − 1 − 2 − 2 − 3 − 1 − 2

푋 + 1.06 × 10 ± 2.10 × 10 푍 − 4.31 × 10 ± 8.31 × 10 푀 + 1.76 × 10 ± 1.27 × 10

푛 푛 푛 2 2 2 − 3 − 2 − 2 − 3 − 1 − 2

푋 + 3.17 × 10 ± 1.83 × 10 푍 − 5.74 × 10 ± 7.35 × 10 푀 + 1.95 × 10 ± 1.12 × 10

푛 푛 푛 3 3 3 − 2 − 2 − 2 − 3 − 1 − 2

푋 + 5.57 × 10 ± 2.05 × 10 푍 − 3.01 × 10 ± 7.89 × 10 푀 + 1.20 × 10 ± 1.21 × 10

푛 푛 푛 4 4 4 − 2 − 2 − 2 − 3 − 1 − 2

푋 + 6.63 × 10 ± 2.90 × 10 푍 − 3.40 × 10 ± 9.70 × 10 푀 − 1.26 × 10 ± 1.49 × 10

푛 푛 푛 5 5 5 − 2 − 2 − 2 − 3 − 2 − 2

푋 + 6.54 × 10 ± 2.31 × 10 푍 − 3.02 × 10 ± 8.46 × 10 푀 − 6.51 × 10 ± 1.30 × 10

푛 푛 푛 6 6 6 − 2 − 2 − 2 − 3 − 2 − 2

푋 + 6.16 × 10 ± 2.54 × 10 푍 − 2.55 × 10 ± 8.89 × 10 푀 − 8.24 × 10 ± 1.36 × 10

푛 푛 푛 7 7 7 − 1 − 2 − 2 − 3 − 1 − 2

푋 + 1.06 × 10 ± 2.50 × 10 푍 − 3.59 × 10 ± 8.91 × 10 푀 − 1.15 × 10 ± 1.36 × 10

푛 푛 푛 8 8 8 1 0 − 1 0 1 0

푋 + 2.69 × 10 ± 1.33 × 10 푍 − 6.59 × 10 ± 5.41 × 10 푀 + 4.47 × 10 ± 8.01 × 10

훿푤 훿푤 훿푤 1 1 1 1 0 1 0 1 0

푋 + 2.89 × 10 ± 1.47 × 10 푍 − 8.11 × 10 ± 6.03 × 10 푀 − 9.11 × 10 ± 8.97 × 10

훿푤 훿푤 훿푤 2 2 2 − 3 − 1 0 0 0 − 1

푋 − 3.05 × 10 ± 1.12 × 10 푍 + 1.68 × 10 ± 1.04 × 10 푀 + 2.45 × 10 ± 4.83 × 10

훿푒 훿푒 훿푒 1 1 1 − 1 − 1 0 0 0 − 1

푋 + 1.01 × 10 ± 1.12 × 10 푍 + 1.90 × 10 ± 1.06 × 10 푀 + 2.67 × 10 ± 4.84 × 10

훿푒 훿푒 훿푒 2 2 2 − 1 − 1 0 − 1 0 − 1

푋 − 1.12 × 10 ± 1.10 × 10 푍 − 1.59 × 10 ± 3.16 × 10 푀 − 2.71 × 10 ± 4.84 × 10

훿푒 훿푒 훿푒 3 3 3 − 1 − 1 0 − 1 0 − 1

푋 − 3.67 × 10 ± 1.10 × 10 푍 − 3.79 × 10 ± 3.17 × 10 푀 − 6.71 × 10 ± 4.85 × 10

훿푒 훿푒 훿푒 4 4 4 0 − 1 0 − 1 1 0

푋 + 1.63 × 10 ± 6.68 × 10 푍 − 6.85 × 10 ± 7.98 × 10 푀 + 1.28 × 10 ± 1.22 × 10

훿 푓 훿 푓 훿 푓 1 1 1 − 1 − 1 0 − 1 0 0

푋 + 5.90 × 10 ± 6.30 × 10 푍 − 6.84 × 10 ± 7.91 × 10 푀 + 9.54 × 10 ± 1.21 × 10

훿 푓 훿 푓 훿 푓 2 2 2 0 − 1 1 − 1 1 0

푋 + 1.78 × 10 ± 6.70 × 10 푍 − 1.32 × 10 ± 7.93 × 10 푀 − 2.14 × 10 ± 1.22 × 10

훿 푓 훿 푓 훿 푓 3 3 3 0 − 1 1 − 1 1 0

푋 + 3.04 × 10 ± 7.28 × 10 푍 − 1.22 × 10 ± 7.87 × 10 푀 − 1.74 × 10 ± 1.21 × 10

훿 푓 훿 푓 훿 푓 4 4 4 − 2 − 3 0 − 1 0 − 1

푋 2 + 2.32 × 10 ± 2.91 × 10 푍 − 1.42 × 10 ± 2.63 × 10 푀 − 2.25 × 10 ± 4.04 × 10

훿푟 훿푟 푤 1 1 − 4 − 4 0 − 1 0 − 1

푋 2 + 9.07 × 10 ± 1.24 × 10 푍 − 1.91 × 10 ± 2.65 × 10 푀 − 2.66 × 10 ± 4.06 × 10

훿푟 훿푟 푛 2 2 − 5 − 4 − 2 − 2 0 − 1

푋 2 + 5.26 × 10 ± 1.29 × 10 푍 2 + 5.28 × 10 ± 1.12 × 10 푀 − 4.55 × 10 ± 1.92 × 10

푤 푤 훿푤 푛 1 − 4 − 4 0 − 1 − 1 − 2

푋 2 + 7.35 × 10 ± 1.12 × 10 푍 + 2.80 × 10 ± 1.25 × 10 푀 + 3.91 × 10 ± 3.90 × 10

푤 훿푤 푛 훿푤 푛 1 1 1 − 4 − 4 − 1 − 2 − 1 − 2

푋 2 + 3.75 × 10 ± 1.21 × 10 푍 − 2.36 × 10 ± 2.55 × 10 푀 + 4.42 × 10 ± 4.05 × 10

푛 훿푤 푛 훿푤 푛 1 1 2 1 − 4 − 4 − 1 − 2 − 1 − 2

푋 2 + 3.09 × 10 ± 1.80 × 10 푍 − 2.97 × 10 ± 2.65 × 10 푀 + 4.60 × 10 ± 3.67 × 10

푛 훿푤 푛 훿푤 푛 2 1 3 1 − 4 − 4 − 1 − 2 − 1 − 2

푋 + 3.82 × 10 ± 1.49 × 10 푍 − 3.30 × 10 ± 2.41 × 10 푀 + 3.49 × 10 ± 3.88 × 10

푛 푛 3 훿푤 1 푛 4 훿푤 1 − 4 − 4 − 1 − 2 2 1

푋 2 + 3.60 × 10 ± 1.57 × 10 푍 − 2.35 × 10 ± 2.53 × 10 푀 2 − 1.87 × 10 ± 1.09 × 10

푛 훿푤 푛 4 1 훿푤 7 1 − 5 − 4 1 0 0 − 1

푋 2 + 8.69 × 10 ± 1.58 × 10 푍 2 + 9.62 × 10 ± 7.89 × 10 푀 + 4.86 × 10 ± 1.94 × 10

푤 훿푤 푛 훿푤 2 8 1 − 1 − 2 0 − 1 − 1 − 2

푋 + 1.59 × 10 ± 2.95 × 10 푍 + 3.90 × 10 ± 1.27 × 10 푀 − 1.42 × 10 ± 4.90 × 10

푤 훿푤 푤 훿푤 푛 훿푤 1 2 5 2 − 2 − 3 − 1 − 2 − 1 − 2

푋 − 4.27 × 10 ± 6.37 × 10 푍 − 1.05 × 10 ± 3.20 × 10 푀 − 5.64 × 10 ± 4.33 × 10

푛 훿푤 푛 훿푤 푛 훿푤 1 1 5 2 6 2 − 1 − 3 − 1 − 2 − 1 − 2

푋 − 1.12 × 10 ± 6.93 × 10 푍 − 4.29 × 10 ± 2.83 × 10 푀 − 5.10 × 10 ± 4.47 × 10

푛 훿푤 푛 훿푤 푛 훿푤 2 1 6 2 7 2 − 1 − 3 − 1 − 2 − 1 − 2

푋 − 1.12 × 10 ± 6.07 × 10 푍 − 3.94 × 10 ± 2.92 × 10 푀 − 1.77 × 10 ± 4.54 × 10

푛 훿푤 푛 훿푤 푛 훿푤 3 1 7 2 8 2 − 2 − 3 − 1 − 2 1 0

푋 − 6.23 × 10 ± 6.41 × 10 푍 − 1.23 × 10 ± 2.97 × 10 푀 − 3.95 × 10 ± 5.01 × 10

푛 훿푤 푛 훿푤 훿푤 훿푤 4 1 8 2 1 2 1 0 2 0 2 1

푋 2 − 3.37 × 10 ± 2.06 × 10 푍 2 + 1.60 × 10 ± 7.73 × 10 푀 2 + 1.95 × 10 ± 1.10 × 10

훿푤 훿푤 훿푤 1 2 2 − 2 − 3 − 2 − 2 1 0

푋 − 6.14 × 10 ± 8.21 × 10 푍 − 4.15 × 10 ± 1.26 × 10 푀 − 1.47 × 10 ± 2.98 × 10

푛 훿푤 푛 훿푒 표 5 2 1 1 − 1 − 3 − 2 − 2

푋 − 1.35 × 10 ± 7.02 × 10 푍 − 5.31 × 10 ± 1.25 × 10

푛 훿푤 푛 훿푒 6 2 4 2 − 1 − 3 0 0

푋 − 1.47 × 10 ± 7.34 × 10 푍 + 2.63 × 10 ± 1.98 × 10

푛 7 훿푤 표 − 2 − 3

푋 − 6.40 × 10 ± 7.63 × 10

푛 훿푤 8 2 1 0

푋 2 − 4.35 × 10 ± 2.04 × 10

훿푤 0 − 1

푋 − 2.05 × 10 ± 3.82 × 10

훿푤 훿푒 1 1 0 − 1

푋 − 2.62 × 10 ± 3.81 × 10

훿푤 1 훿푒 2 0 − 1

푋 − 1.59 × 10 ± 3.87 × 10

훿푤 훿푒 2 3 0 − 1

푋 − 2.72 × 10 ± 3.83 × 10

훿푤 훿푒 2 4 − 2 − 3

푋 − 3.42 × 10 ± 7.70 × 10

푛 훿 푓 2 1 0 − 1

푋 − 7.26 × 10 ± 9.73 × 10

훿푤 1 훿 푓 1 − 2 − 3

푋 − 2.27 × 10 ± 6.94 × 10

푛 훿 푓 3 2 0 − 1

푋 − 6.42 × 10 ± 9.67 × 10

훿푤 훿 푓 1 2 − 2 − 3

푋 − 4.90 × 10 ± 8.07 × 10

푛 훿 푓 6 3 0 − 1

푋 − 8.36 × 10 ± 9.64 × 10

훿푤 훿 푓 2 3 − 2 − 3

푋 − 5.42 × 10 ± 8.45 × 10

푛 훿 푓 7 4 1 − 1

푋 − 1.14 × 10 ± 9.54 × 10

훿푤 훿 푓 2 4 1 0

푋 − 4.03 × 10 ± 1.62 × 10

표

Table 5 Approach I Aerodynamic Models for 풀 , 푳 , and 푵 at ¯ 풒 = 3 . 5 psf

ˆ ˆ ˆ ˆ ˆ ˆ

Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 )

− 1 − 3 − 1 − 2 − 1 − 2

푌 − 5.70 × 10 ± 8.45 × 10 퐿 − 5.48 × 10 ± 3.10 × 10 푁 − 2.03 × 10 ± 2.55 × 10

푣 푣 푣 − 3 − 3 − 2 − 2 − 1 − 3

푌 − 1.01 × 10 ± 1.21 × 10 퐿 + 2.21 × 10 ± 1.03 × 10 푁 + 2.84 × 10 ± 3.60 × 10

푛 푛 푛 1 1 1 − 2 − 3 − 3 − 2 − 1 − 3

푌 − 1.66 × 10 ± 1.28 × 10 퐿 + 3.12 × 10 ± 1.09 × 10 푁 + 1.13 × 10 ± 4.63 × 10

푛 2 푛 2 푛 2 − 2 − 3 − 2 − 3 − 1 − 3

푌 + 1.61 × 10 ± 1.13 × 10 퐿 − 1.62 × 10 ± 9.71 × 10 푁 − 1.25 × 10 ± 4.09 × 10

푛 푛 푛 3 3 3 − 3 − 3 − 2 − 2 − 1 − 3

푌 − 4.78 × 10 ± 1.21 × 10 퐿 − 3.15 × 10 ± 1.04 × 10 푁 − 2.77 × 10 ± 3.62 × 10

푛 푛 푛 4 4 4 − 3 − 3 − 2 − 2 − 1 − 3

푌 + 9.48 × 10 ± 1.49 × 10 퐿 + 6.94 × 10 ± 1.05 × 10 푁 + 3.22 × 10 ± 4.46 × 10

푛 푛 푛 5 5 5 − 3 − 3 − 3 − 2 − 1 − 3

푌 − 7.94 × 10 ± 1.30 × 10 퐿 + 5.71 × 10 ± 1.11 × 10 푁 + 1.41 × 10 ± 4.73 × 10

푛 6 푛 6 푛 6 − 3 − 3 − 4 − 2 − 1 − 3

푌 + 6.89 × 10 ± 1.37 × 10 퐿 − 2.23 × 10 ± 1.18 × 10 푁 − 1.31 × 10 ± 5.03 × 10

푛 푛 푛 7 7 7 − 3 − 3 − 2 − 3 − 1 − 3

푌 − 9.50 × 10 ± 1.38 × 10 퐿 − 9.17 × 10 ± 9.60 × 10 푁 − 3.32 × 10 ± 4.09 × 10

푛 푛 푛 8 8 8 − 2 − 2 − 1 0 0 − 1

푌 − 4.37 × 10 ± 4.84 × 10 퐿 − 9.74 × 10 ± 1.13 × 10 푁 + 1.73 × 10 ± 5.22 × 10

훿푒 훿푒 훿푒 1 1 1 − 2 − 2 0 0 0 − 1

푌 − 9.44 × 10 ± 4.86 × 10 퐿 + 1.64 × 10 ± 1.15 × 10 푁 − 2.04 × 10 ± 5.37 × 10

훿푒 2 훿푒 2 훿푒 2 − 1 − 2 0 0 0 − 1

푌 − 1.58 × 10 ± 4.85 × 10 퐿 − 1.88 × 10 ± 1.45 × 10 푁 + 1.58 × 10 ± 6.25 × 10

훿푒 훿푒 훿푒 3 3 3 − 1 − 2 0 0 0 − 1

푌 + 3.03 × 10 ± 4.87 × 10 퐿 − 4.19 × 10 ± 1.36 × 10 푁 − 2.09 × 10 ± 5.80 × 10

훿푒 훿푒 훿푒 4 4 4 0 − 1 0 − 1 0 0

푌 − 1.18 × 10 ± 1.23 × 10 퐿 + 7.12 × 10 ± 8.68 × 10 푁 + 4.26 × 10 ± 1.33 × 10

훿 푓 훿 푓 훿 푓 1 1 1 0 − 1 0 − 1 0 0

푌 + 1.06 × 10 ± 1.22 × 10 퐿 − 7.89 × 10 ± 8.57 × 10 푁 − 1.23 × 10 ± 1.25 × 10

훿 푓 2 훿 푓 2 훿 푓 2 0 − 1 0 0 − 1 0

푌 + 1.62 × 10 ± 1.22 × 10 퐿 + 6.88 × 10 ± 2.92 × 10 푁 + 7.84 × 10 ± 1.33 × 10

훿 푓 훿 푓 훿 푓 3 3 3 0 − 1 0 0 0 0

푌 − 1.71 × 10 ± 1.21 × 10 퐿 − 2.85 × 10 ± 3.22 × 10 푁 − 3.91 × 10 ± 1.45 × 10

훿 푓 훿 푓 훿 푓 4 4 4 − 1 − 1 0 0 0 − 1

푌 − 9.42 × 10 ± 1.35 × 10 퐿 − 3.93 × 10 ± 4.27 × 10 푁 + 2.87 × 10 ± 1.22 × 10

훿푟 훿푤 훿푟 1 1 1 − 1 − 1 1 0 0 − 1

푌 + 8.52 × 10 ± 1.27 × 10 퐿 − 1.69 × 10 ± 7.32 × 10 푁 − 2.81 × 10 ± 1.22 × 10

훿푟 훿푤 훿푟 2 2 2 − 1 − 1 − 2 − 2 0 0

푌 − 7.41 × 10 ± 6.09 × 10 퐿 + 3.95 × 10 ± 2.63 × 10 푁 − 4.35 × 10 ± 1.88 × 10

훿푤 푤 훿푤 1 1 0 − 1 − 2 − 2 0 0

푌 + 3.60 × 10 ± 8.35 × 10 퐿 + 6.86 × 10 ± 1.37 × 10 푁 + 3.42 × 10 ± 2.18 × 10

훿푤 푛 훿푒 훿푤 2 1 1 2 − 2 − 3 − 2 − 2 − 2 − 3

푌 + 1.65 × 10 ± 3.74 × 10 퐿 − 6.92 × 10 ± 1.36 × 10 푁 − 3.74 × 10 ± 5.86 × 10

푤 푛 훿푒 푛 훿푒 4 2 1 1 − 3 − 3 − 2 − 2 − 2 − 3

푌 − 9.69 × 10 ± 1.67 × 10 퐿 + 9.12 × 10 ± 1.74 × 10 푁 + 4.09 × 10 ± 5.81 × 10

푛 훿푟 푛 훿푒 푛 훿푒 2 1 5 3 4 2 − 3 − 3 − 2 − 2 − 2 − 3

푌 + 9.91 × 10 ± 1.54 × 10 퐿 − 8.85 × 10 ± 1.60 × 10 푁 − 1.66 × 10 ± 7.50 × 10

푛 훿푟 푛 훿푒 푛 훿푒 3 2 8 4 5 3 − 1 − 2 − 1 − 2 − 2 − 3

푌 − 1.01 × 10 ± 2.29 × 10 퐿 + 1.32 × 10 ± 3.75 × 10 푁 + 2.89 × 10 ± 6.80 × 10

푣 훿푤 푛 훿 푓 푛 훿푒 1 6 3 8 4 − 2 − 3 − 1 − 2 − 2 − 2

푌 − 1.94 × 10 ± 3.94 × 10 퐿 − 1.46 × 10 ± 3.97 × 10 푁 − 8.06 × 10 ± 1.53 × 10

푛 훿푤 푛 훿 푓 푛 훿 푓 1 1 7 4 2 1 − 3 − 3 − 1 − 2 − 2 − 2

푌 − 9.28 × 10 ± 4.07 × 10 퐿 + 4.80 × 10 ± 2.75 × 10 푁 + 4.73 × 10 ± 1.38 × 10

푛 훿푤 푛 훿푤 푛 훿 푓 2 1 1 1 3 2 − 2 − 3 − 1 − 2 − 2 − 2

푌 + 1.64 × 10 ± 3.68 × 10 퐿 + 3.88 × 10 ± 2.87 × 10 푁 − 6.99 × 10 ± 1.60 × 10

푛 3 훿푤 1 푛 2 훿푤 1 푛 6 훿 푓 3 − 2 − 3 − 1 − 2 − 1 − 2

푌 + 2.20 × 10 ± 3.90 × 10 퐿 − 3.64 × 10 ± 2.62 × 10 푁 + 1.03 × 10 ± 1.69 × 10

푛 훿푤 푛 훿푤 푛 훿 푓 4 1 3 1 7 4 − 1 − 2 − 1 − 2 − 1 − 2

푌 − 1.74 × 10 ± 2.31 × 10 퐿 − 4.44 × 10 ± 2.73 × 10 푁 + 5.54 × 10 ± 6.88 × 10

푣 훿푤 푛 훿푤 푣 훿푤 2 4 1 1 − 3 − 3 − 1 − 2 − 1 − 2

푌 + 2.54 × 10 ± 4.91 × 10 퐿 + 1.06 × 10 ± 2.77 × 10 푁 − 2.18 × 10 ± 1.17 × 10

푛 훿푤 푛 훿푤 푛 훿푤 5 2 1 2 1 1 − 2 − 3 − 1 − 2 − 1 − 2

푌 + 2.45 × 10 ± 4.35 × 10 퐿 + 1.35 × 10 ± 2.84 × 10 푁 − 1.70 × 10 ± 1.22 × 10

푛 6 훿푤 2 푛 2 훿푤 2 푛 2 훿푤 1 − 2 − 3 − 1 − 2 − 1 − 2

푌 − 2.92 × 10 ± 4.49 × 10 퐿 − 1.45 × 10 ± 2.59 × 10 푁 + 2.12 × 10 ± 1.12 × 10

푛 훿푤 푛 훿푤 푛 훿푤 7 2 3 2 3 1 − 2 − 3 − 2 − 2 − 1 − 2

푌 − 1.24 × 10 ± 4.56 × 10 퐿 − 9.99 × 10 ± 2.71 × 10 푁 + 2.58 × 10 ± 1.17 × 10

푛 훿푤 푛 훿푤 푛 훿푤 8 2 4 2 4 1 0 − 1 − 1 − 2 0 − 1

푌 2 − 5.25 × 10 ± 8.17 × 10 퐿 + 4.29 × 10 ± 3.47 × 10 푁 − 5.02 × 10 ± 7.61 × 10

푛 훿푤 훿푒 훿푤 훿푤 5 2 1 1 − 3 − 4 − 1 − 2 0 − 1

푌 + 3.77 × 10 ± 8.83 × 10 퐿 + 5.05 × 10 ± 3.06 × 10 푁 + 3.92 × 10 ± 7.67 × 10

푣 푤 푛 훿푤 훿푒 훿푤 6 2 2 1 − 1 − 1 − 1 − 2 1 0

푌 + 4.55 × 10 ± 2.98 × 10 퐿 − 5.02 × 10 ± 3.17 × 10 푁 − 1.10 × 10 ± 1.94 × 10

표 푛 훿푤 훿 푓 훿푤 7 2 1 1 − 1 − 2 0 0

퐿 − 3.72 × 10 ± 3.20 × 10 푁 + 6.15 × 10 ± 1.93 × 10

푛 훿푤 훿 푓 훿푤 8 2 2 1 0 0 0 − 2

퐿 − 9.22 × 10 ± 1.79 × 10 푁 + 1.08 × 10 ± 6.91 × 10

훿푒 훿푤 푣 훿푤 3 2 2 1 0 − 1 − 2

퐿 + 1.22 × 10 ± 1.81 × 10 푁 − 1.84 × 10 ± 1.48 × 10

훿푒 4 훿푤 2 푛 5 훿푤 2 1 0 − 1 − 2

퐿 2 + 4.16 × 10 ± 5.73 × 10 푁 − 1.93 × 10 ± 1.30 × 10

푛 훿푤 훿푤 6 2 − 2 − 3 − 1 − 2

퐿 − 3.88 × 10 ± 6.24 × 10 푁 + 1.75 × 10 ± 1.37 × 10

푣 푤 푛 훿푤 7 2 − 1 0 − 1 − 2

퐿 − 4.62 × 10 ± 2.43 × 10 푁 + 1.60 × 10 ± 1.37 × 10

표 푛 훿푤 8 2 0 − 1

푁 − 3.81 × 10 ± 7.73 × 10

훿푒 훿푤 3 2 0 − 1

푁 + 6.35 × 10 ± 7.71 × 10

훿푒 훿푤 4 2 1 0

푁 − 1.29 × 10 ± 1.92 × 10

훿 푓 훿푤 3 2 1 0

푁 + 1.04 × 10 ± 1.90 × 10

훿 푓 훿푤 4 2 − 1 − 1

푁 + 4.13 × 10 ± 9.43 × 10

표

ˆ ˆ ˆ

Table 6 Approach II Aerodynamic Models for 푿 , 풁 , and 푴 at ¯ 풒 = 3 . 5 psf

ˆ ˆ ˆ ˆ ˆ ˆ

Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 )

− 1 − 3 0 − 2 − 1 − 2

ˆ ˆ ˆ

푋 + 2.97 × 10 ± 8.82 × 10 푍 − 3.51 × 10 ± 5.11 × 10 푀 − 3.99 × 10 ± 7.56 × 10

푤 푤 푤 0 − 1 1 0 2 0

ˆ ˆ ˆ

푋 + 8.07 × 10 ± 8.71 × 10 푍 − 7.49 × 10 ± 4.16 × 10 푀 + 1.44 × 10 ± 5.77 × 10

훿푤 훿푤 훿푤 1 1 1 0 − 1 2 0 2 0

ˆ ˆ ˆ

푋 + 5.51 × 10 ± 8.86 × 10 푍 − 1.53 × 10 ± 4.01 × 10 푀 − 1.63 × 10 ± 5.71 × 10

훿푤 훿푤 훿푤 2 2 2 − 1 − 1 − 1 − 1 0 − 1

ˆ ˆ ˆ

푋 + 3.57 × 10 ± 1.39 × 10 푍 − 6.73 × 10 ± 4.97 × 10 푀 + 2.49 × 10 ± 5.26 × 10

훿푒 1 훿푒 1 훿푒 1 − 1 − 1 0 − 1 0 − 1

ˆ ˆ ˆ

푋 + 4.02 × 10 ± 1.39 × 10 푍 − 1.08 × 10 ± 4.84 × 10 푀 + 2.67 × 10 ± 5.27 × 10

훿푒 훿푒 훿푒 2 2 2 − 1 − 1 0 − 1 0 − 1

ˆ ˆ ˆ

푋 + 1.17 × 10 ± 1.35 × 10 푍 − 1.67 × 10 ± 3.56 × 10 푀 − 2.51 × 10 ± 5.27 × 10

훿푒 훿푒 훿푒 3 3 3 − 2 − 1 0 − 1 0 − 1

ˆ ˆ ˆ

푋 − 6.65 × 10 ± 1.40 × 10 푍 − 3.62 × 10 ± 3.57 × 10 푀 − 6.72 × 10 ± 5.29 × 10

훿푒 훿푒 훿푒 4 4 4 − 1 − 1 0 − 1 1 0

ˆ ˆ ˆ

푋 − 5.75 × 10 ± 3.35 × 10 푍 − 6.84 × 10 ± 8.95 × 10 푀 + 1.24 × 10 ± 1.33 × 10

훿 푓 1 훿 푓 1 훿 푓 1 − 1 − 1 0 − 1 0 0

ˆ ˆ ˆ

푋 − 7.79 × 10 ± 3.51 × 10 푍 − 6.83 × 10 ± 8.93 × 10 푀 + 9.65 × 10 ± 1.32 × 10

훿 푓 훿 푓 훿 푓 2 2 2 − 1 − 1 1 − 1 1 0

ˆ ˆ ˆ

푋 − 7.59 × 10 ± 3.37 × 10 푍 − 1.35 × 10 ± 8.90 × 10 푀 − 2.14 × 10 ± 1.32 × 10

훿 푓 훿 푓 훿 푓 3 3 3 − 1 − 1 1 − 1 1 0

ˆ ˆ ˆ

푋 − 3.16 × 10 ± 3.37 × 10 푍 − 1.24 × 10 ± 8.87 × 10 푀 − 1.77 × 10 ± 1.31 × 10

훿 푓 훿 푓 훿 푓 4 4 4 − 1 − 2 0 − 1 0 − 1

ˆ ˆ ˆ

푋 − 1.42 × 10 ± 1.34 × 10 푍 − 1.39 × 10 ± 2.97 × 10 푀 − 2.35 × 10 ± 4.39 × 10

푇 훿푟 훿푟 1 1 1 − 1 − 2 0 − 1 0 − 1

ˆ ˆ ˆ

푋 − 1.25 × 10 ± 1.98 × 10 푍 − 1.97 × 10 ± 2.98 × 10 푀 − 2.68 × 10 ± 4.42 × 10

푇 훿푟 훿푟 2 2 2 − 1 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푋 − 1.70 × 10 ± 1.51 × 10 푍 − 2.54 × 10 ± 6.14 × 10 푀 + 1.36 × 10 ± 9.10 × 10

푇 푇 푇 3 1 1 − 1 − 2 − 1 − 2 − 1 − 1

ˆ ˆ ˆ

푋 − 1.01 × 10 ± 1.48 × 10 푍 − 3.61 × 10 ± 7.43 × 10 푀 + 6.87 × 10 ± 1.10 × 10

푇 푇 푇 4 2 2 − 1 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푋 − 1.17 × 10 ± 1.88 × 10 푍 − 4.26 × 10 ± 5.68 × 10 푀 + 6.35 × 10 ± 8.40 × 10

푇 푇 푇 5 3 3 − 2 − 2 − 1 − 2 − 1 − 1

ˆ ˆ ˆ

푋 − 5.65 × 10 ± 1.94 × 10 푍 − 2.32 × 10 ± 6.84 × 10 푀 + 3.33 × 10 ± 1.01 × 10

푇 푇 푇 6 4 4 − 2 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푋 − 7.78 × 10 ± 2.14 × 10 푍 − 2.23 × 10 ± 5.67 × 10 푀 − 3.75 × 10 ± 8.40 × 10

푇 푇 푇 7 5 5 − 1 − 2 − 1 − 2 − 1 − 1

ˆ ˆ ˆ

푋 − 1.24 × 10 ± 1.61 × 10 푍 − 2.96 × 10 ± 7.33 × 10 푀 − 3.42 × 10 ± 1.09 × 10

푇 푇 푇 8 6 6 1 0 − 1 − 2 − 1 − 1

ˆ ˆ ˆ

푋 2 − 4.17 × 10 ± 1.77 × 10 푍 − 2.35 × 10 ± 7.96 × 10 푀 − 4.77 × 10 ± 1.18 × 10

푇 푇 훿푤 7 7 − 1 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푋 − 1.61 × 10 ± 3.11 × 10 푍 − 2.41 × 10 ± 4.82 × 10 푀 − 2.56 × 10 ± 7.13 × 10

푤 훿푤 푇 푇 2 8 8 1 0 − 2 − 2 0 − 1

ˆ ˆ ˆ

푋 2 − 5.16 × 10 ± 1.78 × 10 푍 2 + 9.37 × 10 ± 1.26 × 10 푀 − 5.74 × 10 ± 2.08 × 10

푤 훿푤 훿푤 푤 1 0 − 1 0 − 1 2 1

ˆ ˆ ˆ

푋 − 2.10 × 10 ± 4.04 × 10 푍 + 3.38 × 10 ± 1.40 × 10 푀 2 − 2.43 × 10 ± 1.21 × 10

훿푤 훿푒 푤 훿푤 1 1 1 훿푤 0 − 1 2 0 0 − 1

ˆ ˆ ˆ

푋 − 2.64 × 10 ± 4.02 × 10 푍 + 1.40 × 10 ± 8.95 × 10 푀 + 5.84 × 10 ± 2.11 × 10

훿푤 1 훿푒 2 훿푤 푤 훿푤 2 0 − 1 0 − 1 1 0

ˆ ˆ ˆ

푋 − 1.61 × 10 ± 4.10 × 10 푍 + 4.42 × 10 ± 1.42 × 10 푀 − 4.35 × 10 ± 5.46 × 10

훿푤 훿푒 푤 훿푤 훿푤 훿푤 2 3 2 1 2 0 − 1 2 0 2 1

ˆ ˆ ˆ

푋 − 3.10 × 10 ± 4.06 × 10 푍 2 + 2.06 × 10 ± 8.76 × 10 푀 2 + 2.04 × 10 ± 1.22 × 10

훿푤 훿푒 2 4 훿푤 훿푤 2 2 0 0 − 1 − 1 0 − 1

ˆ ˆ ˆ

푋 − 5.86 × 10 ± 1.03 × 10 푍 − 5.03 × 10 ± 2.02 × 10 푀 + 1.33 × 10 ± 2.99 × 10

훿푤 1 훿 푓 1 훿푤 1 푇 1 훿푤 1 푇 1 0 0 − 1 − 1 0 − 1

ˆ ˆ ˆ

푋 − 5.26 × 10 ± 1.02 × 10 푍 − 2.49 × 10 ± 1.00 × 10 푀 + 2.34 × 10 ± 3.59 × 10

훿푤 훿 푓 훿푒 푇 훿푤 푇 1 2 1 1 1 2 0 0 0 − 1 0 − 1

ˆ ˆ ˆ

푋 − 9.32 × 10 ± 1.02 × 10 푍 − 1.29 × 10 ± 2.43 × 10 푀 + 1.98 × 10 ± 2.78 × 10

훿푤 훿 푓 훿푤 푇 훿푤 푇 2 3 1 2 1 3 1 0 0 − 1 0 − 1

ˆ ˆ ˆ

푋 − 1.03 × 10 ± 1.01 × 10 푍 − 1.24 × 10 ± 1.89 × 10 푀 + 1.36 × 10 ± 3.33 × 10

훿푤 훿 푓 훿푤 푇 훿푤 푇 2 4 1 3 1 4 − 2 − 2 − 1 − 1 0 − 1

ˆ ˆ ˆ

푋 − 8.70 × 10 ± 4.37 × 10 푍 − 8.21 × 10 ± 2.25 × 10 푀 − 2.62 × 10 ± 3.66 × 10

훿푤 1 푇 1 훿푤 1 푇 4 훿푤 2 푇 − 1 − 2 − 1 − 1 0 − 1

ˆ ˆ ˆ

푋 − 1.12 × 10 ± 2.19 × 10 푍 − 4.36 × 10 ± 1.12 × 10 푀 − 2.41 × 10 ± 3.94 × 10

훿푒 푇 훿푒 푇 훿푤 푇 1 1 2 4 2 7 − 1 − 2 0 − 1 0 0

ˆ ˆ ˆ

푋 − 5.07 × 10 ± 5.31 × 10 푍 − 2.03 × 10 ± 2.48 × 10 푀 − 2.75 × 10 ± 1.00 × 10

훿푤 푇 훿푤 푇 표 1 2 2 6 − 1 − 2 0 − 1

ˆ ˆ

푋 − 2.71 × 10 ± 6.68 × 10 푍 − 2.05 × 10 ± 2.67 × 10

훿 푓 푇 훿푤 푇 1 2 2 7 − 1 − 2 1 − 1

ˆ ˆ

푋 − 5.44 × 10 ± 4.14 × 10 푍 − 1.44 × 10 ± 7.49 × 10

훿푤 푇 표 1 3 − 1 − 2

ˆ

푋 − 1.77 × 10 ± 5.17 × 10

훿 푓 푇 2 3 − 1 − 2

ˆ

푋 − 2.01 × 10 ± 4.88 × 10

훿푤 푇 1 4 − 1 − 2

ˆ

푋 − 1.08 × 10 ± 2.45 × 10

훿푒 푇 2 4 − 1 − 2

ˆ

푋 − 1.30 × 10 ± 6.37 × 10

훿푤 푇 2 5 − 2 − 2

ˆ

푋 − 8.65 × 10 ± 3.21 × 10

훿푒 푇 3 5 − 1 − 2

ˆ

푋 − 8.24 × 10 ± 5.38 × 10

훿푤 푇 2 6 − 1 − 2

ˆ

푋 − 3.33 × 10 ± 6.61 × 10

훿 푓 푇 3 6 − 1 − 2

ˆ

푋 − 8.49 × 10 ± 5.89 × 10

훿푤 푇 2 7 − 1 − 2

ˆ

푋 − 4.33 × 10 ± 7.28 × 10

훿 푓 4 푇 7 − 1 − 2

ˆ

푋 − 1.38 × 10 ± 5.47 × 10

훿푤 푇 2 8 − 2 − 2

ˆ

푋 − 7.64 × 10 ± 2.71 × 10

훿푒 푇 4 8 0 − 1

ˆ

푋 − 2.33 × 10 ± 1.76 × 10

표

ˆ ˆ ˆ

Table 7 Approach II Aerodynamic Models for 풀 , 푳 , and 푵 at ¯ 풒 = 3 . 5 psf

ˆ ˆ ˆ ˆ ˆ ˆ

Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 ) Term 휃 ± 푠 ( 휃 )

− 1 − 3 − 1 − 2 − 2 − 2

ˆ ˆ ˆ

푌 − 4.88 × 10 ± 6.88 × 10 퐿 − 5.15 × 10 ± 3.23 × 10 푁 − 8.84 × 10 ± 2.42 × 10

푣 푣 푣 − 1 − 2 0 − 1 − 1 − 1

ˆ ˆ ˆ

푌 − 1.81 × 10 ± 5.01 × 10 퐿 + 2.74 × 10 ± 4.99 × 10 푁 − 1.45 × 10 ± 2.48 × 10

훿푒 훿푒 훿푒 3 1 1 − 1 − 2 0 − 1 − 1 − 1

ˆ ˆ ˆ

푌 + 3.08 × 10 ± 5.04 × 10 퐿 − 2.56 × 10 ± 4.85 × 10 푁 + 3.92 × 10 ± 2.48 × 10

훿푒 훿푒 훿푒 4 2 2 0 − 1 0 − 1 − 1 − 1

ˆ ˆ ˆ

푌 − 1.26 × 10 ± 1.26 × 10 퐿 + 3.37 × 10 ± 6.15 × 10 푁 + 4.95 × 10 ± 2.40 × 10

훿 푓 훿푒 훿푒 1 3 3 0 − 1 0 − 1 − 1 − 1

ˆ ˆ ˆ

푌 + 1.11 × 10 ± 1.26 × 10 퐿 − 9.35 × 10 ± 6.46 × 10 푁 − 1.81 × 10 ± 2.50 × 10

훿 푓 훿푒 훿푒 2 4 4 0 − 1 0 − 1 0 − 1

ˆ ˆ ˆ

푌 + 1.59 × 10 ± 1.26 × 10 퐿 + 7.25 × 10 ± 8.99 × 10 푁 − 2.52 × 10 ± 4.52 × 10

훿 푓 훿 푓 훿 푓 3 1 1 0 − 1 0 − 1 0 − 1

ˆ ˆ ˆ

푌 − 1.68 × 10 ± 1.25 × 10 퐿 − 7.81 × 10 ± 8.94 × 10 푁 + 2.91 × 10 ± 4.72 × 10

훿 푓 훿 푓 훿 푓 4 2 2 0 − 2 1 0 0 − 1

ˆ ˆ ˆ

푌 − 1.51 × 10 ± 5.42 × 10 퐿 + 1.44 × 10 ± 1.21 × 10 푁 − 3.39 × 10 ± 6.00 × 10

훿푟 훿 푓 훿 푓 1 3 3 0 − 2 1 0 0 − 1

ˆ ˆ ˆ

푌 + 1.41 × 10 ± 5.72 × 10 퐿 − 1.17 × 10 ± 1.22 × 10 푁 + 2.67 × 10 ± 6.00 × 10

훿푟 2 훿 푓 4 훿 푓 4 − 2 − 1 0 0 0 − 1

ˆ ˆ ˆ

푌 − 4.85 × 10 ± 2.03 × 10 퐿 − 1.57 × 10 ± 1.43 × 10 푁 + 2.83 × 10 ± 1.15 × 10

훿푤 훿푤 훿푟 1 1 1 0 − 1 1 0 0 − 1

ˆ ˆ ˆ

푌 + 2.34 × 10 ± 4.29 × 10 퐿 − 1.14 × 10 ± 3.14 × 10 푁 − 2.84 × 10 ± 1.16 × 10

훿푤 훿푤 훿푟 2 2 2 − 2 − 3 − 2 − 2 − 1 − 1

ˆ ˆ ˆ

푌 + 1.86 × 10 ± 3.85 × 10 퐿 + 2.98 × 10 ± 2.73 × 10 푁 − 7.34 × 10 ± 5.55 × 10

푤 푤 훿푤 − 3 − 3 − 1 − 2 − 1 − 1

ˆ ˆ ˆ

푌 − 9.85 × 10 ± 8.66 × 10 퐿 + 4.12 × 10 ± 6.13 × 10 푁 + 8.40 × 10 ± 6.37 × 10

푇 1 푇 1 훿푤 2 − 1 − 2 − 2 − 2 − 2 − 2

ˆ ˆ ˆ

푌 − 1.32 × 10 ± 1.05 × 10 퐿 − 7.95 × 10 ± 9.11 × 10 푁 − 6.10 × 10 ± 1.06 × 10

푇 푇 푤 2 2 − 1 − 3 − 3 − 2 − 1 − 2

ˆ ˆ ˆ

푌 + 1.08 × 10 ± 8.00 × 10 퐿 − 6.73 × 10 ± 6.97 × 10 푁 − 4.23 × 10 ± 2.36 × 10

푇 푇 푇 3 3 1 − 2 − 3 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푌 − 3.65 × 10 ± 9.65 × 10 퐿 − 5.15 × 10 ± 6.82 × 10 푁 − 1.07 × 10 ± 3.52 × 10

푇 푇 푇 4 4 2 − 2 − 3 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푌 + 7.71 × 10 ± 8.00 × 10 퐿 + 5.94 × 10 ± 5.68 × 10 푁 + 1.44 × 10 ± 2.67 × 10

푇 푇 푇 3 5 5 − 2 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푌 − 6.20 × 10 ± 1.03 × 10 퐿 + 2.07 × 10 ± 8.94 × 10 푁 + 2.48 × 10 ± 2.64 × 10

푇 푇 푇 6 6 4 − 2 − 2 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푌 + 5.46 × 10 ± 1.12 × 10 퐿 − 1.25 × 10 ± 9.79 × 10 푁 − 6.02 × 10 ± 2.16 × 10

푇 푇 푇 7 7 5 − 2 − 3 − 1 − 2 − 1 − 2

ˆ ˆ ˆ

푌 − 9.40 × 10 ± 6.80 × 10 퐿 − 5.43 × 10 ± 4.82 × 10 푁 − 1.16 × 10 ± 3.45 × 10

푇 푇 푇 8 8 6 − 1 − 2 1 0 − 1 − 2

ˆ ˆ ˆ

푌 − 1.78 × 10 ± 2.39 × 10 퐿 − 1.00 × 10 ± 1.87 × 10 푁 + 1.71 × 10 ± 3.78 × 10

푣 훿푤 2 훿푒 3 훿푤 2 푇 7 0 − 1 1 0 − 1 − 2

ˆ ˆ ˆ

푌 2 − 4.67 × 10 ± 9.14 × 10 퐿 + 1.33 × 10 ± 1.87 × 10 푁 + 5.96 × 10 ± 1.85 × 10

훿푒 훿푤 푇 훿푤 4 2 8 − 1 − 2 1 0 − 1 − 2

ˆ ˆ ˆ

푌 − 1.21 × 10 ± 2.86 × 10 퐿 2 + 4.02 × 10 ± 6.46 × 10 푁 + 5.36 × 10 ± 6.53 × 10

훿푤 푇 푣 훿푤 1 1 훿푤 1 − 2 − 2 − 2 − 3 0 − 1

ˆ ˆ ˆ

푌 − 7.16 × 10 ± 1.42 × 10 퐿 − 3.63 × 10 ± 6.48 × 10 푁 − 4.87 × 10 ± 7.21 × 10

훿푟 푇 푣 푤 훿푒 훿푤 1 2 1 1 − 1 − 2 − 1 − 1 0 − 1

ˆ ˆ ˆ

푌 − 1.34 × 10 ± 3.42 × 10 퐿 + 5.02 × 10 ± 1.01 × 10 푁 + 3.87 × 10 ± 7.23 × 10

훿푤 푇 훿푒 푇 훿푒 훿푤 1 2 1 1 2 1 − 2 − 2 0 − 1 0 − 2

ˆ ˆ ˆ

푌 + 6.25 × 10 ± 1.12 × 10 퐿 + 1.21 × 10 ± 2.01 × 10 푁 + 1.13 × 10 ± 6.55 × 10

훿푟 푇 훿푤 푇 푣 훿푤 2 3 1 1 2 − 1 − 2 0 − 1 0 − 1

ˆ ˆ ˆ

푌 + 1.57 × 10 ± 2.66 × 10 퐿 + 2.38 × 10 ± 2.44 × 10 푁 − 3.15 × 10 ± 7.29 × 10

훿푤 푇 훿푤 푇 훿푒 훿푤 1 3 1 2 3 2 − 1 − 2 0 − 1 0 − 1

ˆ ˆ ˆ

푌 + 1.47 × 10 ± 3.17 × 10 퐿 + 1.11 × 10 ± 2.44 × 10 푁 + 5.55 × 10 ± 7.26 × 10

훿푤 푇 훿푤 푇 훿푒 훿푤 1 4 2 2 4 2 − 1 − 2 0 − 1 1 0

ˆ ˆ ˆ

푌 + 2.00 × 10 ± 3.49 × 10 퐿 − 1.56 × 10 ± 1.90 × 10 푁 − 1.19 × 10 ± 1.82 × 10

훿푤 2 푇 6 훿푤 1 푇 3 훿 푓 3 훿푤 2 − 1 − 2 0 − 1 0 0

ˆ ˆ ˆ

푌 − 2.62 × 10 ± 3.76 × 10 퐿 − 1.10 × 10 ± 1.90 × 10 푁 + 9.13 × 10 ± 1.80 × 10

훿푤 푇 훿푤 푇 훿 푓 훿푤 2 7 2 3 4 2 − 2 − 2 − 1 − 1 − 1 − 2

ˆ ˆ ˆ

푌 + 9.97 × 10 ± 9.16 × 10 퐿 − 4.68 × 10 ± 1.12 × 10 푁 − 2.89 × 10 ± 3.90 × 10

표 훿푒 푇 훿푒 푇 2 4 1 1 0 − 1 − 1 − 2

ˆ ˆ

퐿 − 1.30 × 10 ± 2.24 × 10 푁 − 7.71 × 10 ± 7.79 × 10

훿푤 푇 훿푤 푇 1 4 1 1 − 1 − 1 − 1 − 1

ˆ ˆ

퐿 + 8.39 × 10 ± 1.46 × 10 푁 − 6.41 × 10 ± 1.18 × 10

훿푒 3 푇 5 훿 푓 1 푇 2 − 1 − 1 0 − 2

ˆ ˆ

퐿 + 9.71 × 10 ± 3.04 × 10 푁 − 1.09 × 10 ± 9.47 × 10

훿 푓 푇 훿푤 푇 3 6 1 2 0 − 1 − 1 − 2

ˆ ˆ

퐿 + 2.86 × 10 ± 2.48 × 10 푁 + 3.01 × 10 ± 9.16 × 10

훿푤 푇 훿 푓 푇 2 6 2 3 0 − 1 0 − 2

ˆ ˆ

퐿 − 1.04 × 10 ± 3.34 × 10 푁 + 1.15 × 10 ± 7.38 × 10

훿 푓 푇 훿푤 푇 4 7 1 3 0 − 1 − 1 − 2

ˆ ˆ

퐿 − 3.10 × 10 ± 2.68 × 10 푁 + 3.25 × 10 ± 4.35 × 10

훿푤 2 푇 7 훿푒 2 푇 4 − 1 − 1 0 − 2

ˆ ˆ

퐿 − 6.11 × 10 ± 1.25 × 10 푁 + 1.22 × 10 ± 8.72 × 10

훿푒 푇 훿푤 푇 4 8 1 4 0 − 1 − 1 − 2

ˆ ˆ

퐿 − 3.27 × 10 ± 7.07 × 10 푁 − 1.06 × 10 ± 5.71 × 10

표 훿푒 푇 3 5 − 1 − 1

ˆ

푁 − 4.98 × 10 ± 1.18 × 10

훿 푓 푇 3 6 0 − 2

ˆ

푁 − 1.35 × 10 ± 9.60 × 10

훿푤 2 푇 − 1 − 1

ˆ

푁 + 6.55 × 10 ± 1.29 × 10

훿 푓 푇 4 7 0 − 1

ˆ

푁 + 1.21 × 10 ± 1.04 × 10

훿푤 푇 2 7 − 1 − 2

ˆ

푁 + 1.72 × 10 ± 4.84 × 10

훿푒 푇 4 8 − 1 − 1

ˆ

푁 + 4.36 × 10 ± 2.65 × 10

표

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