High Fidelity Mathematical Modeling of the DA42 L360

Transcription

1 High Fidelity Mathematical Modeling of the DA42 L360 PREPARED BY: Aditya, Ron PAGE i of viii

2 Contents 1. INTRODUCTION General Parameters Wing Parameters Control Surface Parameters Engine Thrust Vector Aerodynamic Center Approximation Aircraft Sign Convention LONGITUDINAL MODELING Introduction Method Model Equations Results Gear Up Flaps Up Gear Up Flaps Approach Gear Down Flaps Up Gear Down Flaps Approach Gear Down Flaps Landing LATERAL MODELING Introduction Method Model Equations Results Gear Up Flaps Up Gear Up Flaps Approach Gear Down Flaps Up Gear Down Flaps Approach Gear Down Flaps Landing STATIC DERIVATIVE DETERMINATION Method Coefficient of Lift Coefficient of Drag Coefficient of Pitching Moment NON-LINEAR AERODYNAMICS Coefficient of Lift Coefficient of Drag Coefficient of Pitching Moment VALIDATION Method Short Period Pitching Oscillations Validation Gear Up Flaps Up Gear Up Flaps Approach Dutch Roll Validation Gear Up Flaps Up Gear Down Flaps Approach... 73

3 6.3.3 Gear Down Flaps Landing Stall AIR DATA SYSTEMS AND FILTERING SOURCES CITED: APPENDICES Appendix A: Data Channel Definitions Appendix B: (PID of Longitudinal Coefficients) Linear Regression Method Output Error Method Appendix C: (Navion 6 DOF Hand Trimming Code): PAGE iii of viii

4 1. INTRODUCTION Just as one can characterize a particular chemical by the constituent ingredients, one can also characterize an aircraft s feel by its stability derivatives. Stability derivatives define how an aircraft responds to a stabilizing or destabilizing perturbation which may include wind shear, control surface input, and angle of attack, just to name a few. Within the last few decades, quantum leaps in the ability to obtain high-accuracy estimations of these stability derivatives have become possible primarily due to increases in computing power, instrumentation sensitivity and collective knowledge of the tools and techniques required. Classical, timeconsuming parameter identification techniques such as measurements of free oscillations and steady-state error can be supplemented by time-effective mathematical modeling using computers. The rising cost of a robust flight test program only further warrants the need to obtain high-fidelity aircraft mathematical modeling. Parameter identification (PID) is the process of determining the stability and control derivatives from timehistory flight test data and comparing the response of a simulated math model to the measured aircraft response. The goal of this text is to provide a validated Simulink mathematical model of the Diamond DA42 L360 aircraft and to present all results in tabulated and graphical forms. Stability derivatives of the following aircraft configurations were obtained in order to establish a well-rounded mathematical model: I. Flaps Retracted, Gear Up II. Flaps Approach, Gear Up III. Flaps Retracted, Gear Down IV. Flaps Approach, Gear Down V. Flaps Landing, Gear Down The outputs of the mathematical model will quantitatively illustrate the impact of landing gear position and flap deployment on the aircraft s stability derivatives. Logic diagrams and lookup tables can then be inputted into the Simulink model to alter aircraft configuration in flight. A standard computer joystick will interface with the model to allow the user to fly the aircraft in a flight simulation program (FlightGear) in real-time. Geometric data of the aircraft was obtained from released manufacturer technical publications and real flight test data was used as inputs to a dynamic Simulink model that simulates aircraft flight characteristics as a result of pilot input, aircraft configuration and aircraft states. Chapter 1 of this report provides basic DA42 L360 data that was used in the Simulink model. A figure is provided to illustrate the calculation of the Mean Aerodynamic Center (MAC) and the engine thrust application vector. 1

5 1.3 Control Surface Parameters 1.4 Engine Thrust Vector TABLE 4: CONTROL SURFACE PARAMETERS Surface Area, Total (ft 2 ) Deflection ( ) Aileron 7.1 TE up 25º, ± 2º TE down 15º,+ 2º/- 0 º Elevator 7.1 TE up 15.5º, ± 0.5º TE down 13º, ± 1º Elevator Trim Tab Included in elevator + 17º, ± 5º (Nose up at elevator 10 up) - 35º, ± 5º (Nose down at elevator 10 up) Rudder 8.4 Left 27º, ± 1º Right 29º, ± 1º Rudder Trim Tab Included in rudder + 34º, ± 5º (Trim RH at rudder 20 LH) Flaps, Cruise 0, + 2 /- 0 Flaps, Approach º, + 4º/- 2 Flaps, Landing 42º, +3º/- 1º + 18º, ± 5º (Trim LH at rudder 20 LH) TABLE 5: ENGINE THRUST VECTOR TABLE 6: THRUST APPLICATION VECTOR Left Engine = (.,., ) Right Engine = (.999,.044,0) Left Engine = (.,.,. ) Right Engine = (53.92, 64.76, 10.06) 1.5 Aerodynamic Center Approximation The aerodynamic center of the DA42 L360 was geometrically determined using scaled drawings of the aircraft from the manufacturer s technical publications. The lengths of the tip chord and root chord were added to the wing root and tip, respectively, to find the location of the mean aerodynamic chord (MAC). The aerodynamic center (AC) was approximated to be 25% of the MAC, and, assuming a perfectly symmetrical structure, the aerodynamic center (AC) was projected onto BL0. The location of the aerodynamic center was validated by ensuring it falls between the manufacturer s suggested forward and aft CG limits. The AC location was further confirmed during model validation in chapter 6. Figure 1 below provides a high-level overview of the AC and CG locations: 3

6 2. LONGITUDINAL MODELING 2.1 Introduction Longitudinal modeling of the DA42 aircraft was performed by analytically solving for the longitudinal stability derivatives. Several large files of raw DA42 flight test data were obtained with 227 columns of data each representing a data channel that corresponds to a specific parameter. An interface control document (ICD) is provided in Appendix A: Data Channel Definitions for the reader to track the mapping of data channels within the flight test data. It should be noted that each flight test data file represents a specific aircraft configuration, i.e. flaps down gear up, flaps approach gear up, etc. Perhaps one of the most important requirements for obtaining correct stability derivatives is to prevent the state variables from tracking one another. In real life, angle of attack and angle of sideslip track well when the flight path angle is close to zero. According to the Advisory Group for Aerospace Research & Development: The [ ] investigations have shown that an accurate identification of the stability and control derivatives is guaranteed only if the aircraft is excited by input signals which fulfill certain frequency requirements. Parameter Identification [3-10] Tracking can be limited by stimulating the natural frequencies of the aircraft (phugoid and short period for longitudinal motion) via control surface input. In lieu of experimentally determining the exact natural frequency of the particular aircraft and precisely inputting controls to stimulate it, a maneuver can be implemented. The maneuver consists of a stick input of three counts, followed by an opposite input of two counts and two one-count inputs in the same fashion. A step, doublet or chirp input could have been used to perform the maneuvers but the probability of stimulating ϣ decreases. Preliminary examination of the flight test data concluded that longitudinal maneuvers were performed at various angles of attack (AOA) once the aircraft was in a trimmed state. Either through manual pilot input or automated control systems, performing maneuvers provides a good compromise of practicality and precision. AGARD also advocates the maneuver for parameter identification: As the step contains energy only at lower frequencies it cannot excite the higher frequency natural modes. On the other hand, the aircraft rapidly departs from the linear flight regime when excited by the step. For these reasons the step is unfit for parameter identification. The doublet excites a particular band at a higher frequency. By the choice of Δt the peak of the power spectral density can be shifted to the range of the higher frequency natural oscillations. However, the natural frequencies are not known exactly since they are calculated from the a-priori-values of the derivatives. On the other hand they may vary due to a change of the flight conditions. As the doublet is a relatively narrow-band signal it may happen that the natural modes are not excited effectively [ ] these difficulties do not occur when the aircraft is excited by a wide-band signal like the "3211"-input. Parameter Identification [3-3] 6

7 modeling and the residuals are provided below to illustrate the results obtained using the equation error solver with only the most significant regressors: FIGURE 4: CL MATCH MODELING FIGURE 5: CL RESIDUALS Although CL residuals do not reflect pure instrumentation noise towards the end of the data capture, adding additional regressors did not improve the equation error results; large errors and expanded confidence intervals were realized by doing so. Removing as a regressor of decreased the total error of all regressors from 14.2% to 2.7%. Plots of the match modeling and residuals of with removed are provided below: FIGURE 6: CL MATCH MODELING (REDUCED) FIGURE 7: CL RESIDUALS (REDUCED) The non-dimensional coefficient changed from to by removing as a regressor, however, the coefficient of differed by only.002. residuals more closely tracked the state response by removing as a regressor: figures 8 and 9 illustrate the high residuals slightly before locations of high amplitude. Regressors were individually added and removed within each configuration in effort to decrease error and narrow the confidence interval of the coefficients. Once appropriate regressors were chosen and each 8

8 2.4.4 Gear Down Flaps Approach The PID solution of the gear down flaps approach configuration appears similar to the gear down flaps up configuration, however, the residuals plotted in FIGURE 39 show correlation to the CM matching. Derivative estimation was found to improve by using the output error method. FIGURE 38: CM MATCHING, EQUATION ERROR FIGURE 39: CM RESIDUALS, EQUATION ERROR FIGURE 40: CM MATCHING, OUTPUT ERROR FIGURE 41: CM RESIDUALS, OUTPUT ERROR 23

9 3. LATERAL MODELING 3.1 Introduction As was done in chapter 2 to determine the DA42 longitudinal stability derivatives, the lateral derivatives were analytically determined using the same methods. Because each flight test data file corresponding to a specific airplane configuration contained a longitudinal maneuver followed by a lateral maneuver, the lateral data was cut using Matlab scripts. This method proved useful for group coordination to ensure the same data was being analyzed each time the SIDPAC GUI was run. The succeeding sections of chapter 3 will highlight the method, the reasoning justifying the method, and the ultimate results of performing lateral parameter identification on the DA42 aircraft. 3.2 Method The lateral stability derivatives were found using the same methods as were used for the longitudinal derivatives. Raw flight test data files were loaded into Matlab and the SIDPAC toolbox was used to trim individual lateral maneuvers performed at varying angles of attack. The derivatives were initially solved using the equation error method and regressors were selected by comparing and contrasting the results obtained from each iteration. In addition to equation error, the output error method was also used to obtain the lateral stability derivatives. Although similar results were obtained, the math model ultimately received and validated the output error PID results in accordance with the project requirements. 3.3 Model Equations Whether the equation error or output error method is used within SIDPAC, only the individual stability derivatives are provided; the stability coefficients must be manually calculated using standard equations. The following equations were solved within the Simulink model to calculate the respective lateral stability coefficients: = = = Results The output error method provided exceptional matching of lateral stability derivatives as compared to equation error. Matching and residuals plots are included below for each configuration to demonstrate the accuracy of the solver. For the gear down flaps landing configuration, the average percent error of the lateral derivatives using output error was roughly 9.5% whereas the average percent error of the longitudinal derivatives is 7.28%. The increased average error of the lateral derivatives can be attributed to the greater number of lateral 31

10 3.4.3 Gear Down Flaps Up Figures 90 and 91 represent matching and residuals using the equation error solver. The residuals represent the derivative response, thus, the matching is insufficient. Figures 92 and 93 represent matching and residuals of the same maneuver using the output error technique. It is evident that substantially better results are obtained using the output error technique. FIGURE 89: CL MATCHING, EQUATION ERROR FIGURE 90: CL RESIDUALS, EQUATION ERROR FIGURE 91: LATERAL MATCHING, OUTPUT ERROR FIGURE 92: LATERAL RESIDUALS, OUTPUT ERROR Table 31 provides the raw PID derivative values at various AOAs using the SIDPAC toolbox. Although the values are naturally in the body reference frame, they were not converted to the wind reference frame as the maneuvers were performed roughly at 0 degrees of sideslip and small angle approximation is acceptable. 42

11 4. STATIC DERIVATIVE DETERMINATION 4.1 Method The coefficients for lift, drag, and pitching moment have been determined for each flap and gear configuration for the DA42 aircraft. Most static coefficients have been provided directly from the flight test data set; however, the lift, drag, and pitching moment variables presented here have been calculated based on other supplementary information. Only the linear portions of the data are presented here. All data corresponding to post stall can be found in Chapter 5. The following equations define the coefficients of lift and drag: = C cos( ) + sin ( ) = sin( ) cos ( ) Where CZ and CX are defined as: = = + 2 Where T1 and T2 correspond to the average thrust levels of engines 1 and 2, respectively. CX and CZ can also be extracted directly from the flight test data, however equating these static coefficients from flight conditions was shown to produce more reasonable results for lift and drag. Pitching moment curves were determined from CM α and CM0 values implemented in the DA42 model for each alpha. The following equation details pitching moment: = + Most flight test data is available for limited angles of attack. CL, CD, and CM were fit using these limited angles and then extrapolated to the maximum angle of attack. In some cases, small modifications were made near αmax to incorporate the nonlinear portion of each coefficient. The following figures show the lift, drag, and pitching moment curves for each evaluated configuration as a function of angle of attack. 57

12 4.3 Coefficient of Drag To extrapolate the remaining drag data from the limited flight test data, a parabola was fit to the known data and the remaining linear regime was appended to create the full drag bucket. Figure 139 shows the drag curves for each configuration vs. angle of attack. There are some nonlinearities that exist in the drag model that occur around CDmin. These have been captured here. FIGURE 138: LINEAR REGIME DRAG COEFFICIENT VS. ANGLE OF ATTACK FOR ALL CONFIGURATIONS Drag data for the pre-stall portion of the flight regime have been tabulated and can be found in Table 38. The data here only represents the drag curve up to the maximum angle of attack. This table is an abbreviation of the final drag curves and may not contain all nonlinearities shown in the figure above. Alpha TABLE 36: ABBREVIATED LOOKUP TABLE FOR LINEAR CD FOR ALL CONFIGURATIONS Gear Up, Flaps Up Gear Up, Flaps Approach Gear Down, Flaps Up Gear Down, Flaps Approach Gear Down, Flaps Landing

13 5. NON-LINEAR AERODYNAMICS Non-linear aerodynamic behavior has been determined for the Diamond DA-42 aircraft from flight test data for the flaps up, gear up configuration. The provided flight test data for the stall regime for this configuration was presented up to an angle of attack of 18 degrees. The data for each coefficient was fit to the end of the each linear range for other configurations based on various factors including engineering judgment. Coefficient and lift, drag, and pitching moment have been determined based on this nonlinear flight test data and the linear ranges determined by static analysis. 5.1 Coefficient of Lift The raw flight test data for the coefficient of lift was fitted to a third order polynomial using engineering judgment and educated knowledge of general aviation stall curves; this includes accounting for low Reynold s numbers effects. In a low Reynold s number aircraft such as a DA42, the stall break in lift will be gradual and the loss of lift post stall will slowly increase. Hysteresis effects were also considered when choosing the manner in which to fit the raw lift data. Figure 141 presents the raw stall flight test data overlaid with the fit stall data curve. 1.5 CL vs Raw Stall Data Reduced Stall Data (deg) FIGURE 140: RAW CL STALL DATA AND FIT CL CURVE The determined stall data was then plotted against the original PID data for the gear up, flaps up configuration to check for validity. The stall data joined cleanly with the original PID data indicating the fit was suitable for further investigation and validation. Figure 142 shows the correlation between the chosen stall data curve and the original flaps up gear up PID data 61

14 6. VALIDATION Validation is a necessary step to quantitatively verify the simulation behaves like the physical aircraft it represents. Because aircraft simulators are often used in a training environment, their accuracy is fundamentally paramount. In addition, parameter identification can be used to facilitate the flight test process and reduce the associated risk. The FAA provides general guidelines within Title 14, Part 60 to ensure accuracy of these simulators is maintained. This chapter is devoted to demonstrating the DA42 L360 simulator meets all requirements set forth by the instructor by visually confirming the aircraft responds properly to a known input. Further validation will be demonstrated via manually flying the aircraft in FlightGear with a joystick and observing the aircraft response. Table 44 below provides a high level overview of the Part 60 requirements for simulator validation with respect to natural frequency response: Part 60 Requirement 2.c.9.a. 2.c.9.b. 2.c d.7. Natural Frequency Phugoid Dynamics Phugoid Dynamics Short Period Dynamics Dutch Roll, (Yaw Damper OFF) Numerical Requirement ±10% period, ±10% of time to 1/2 or double amplitude or ±.02 of damping ratio ±10% period, Representative damping ±1.5 pitch angle or ±2 /sec pitch rate, ±0.10g acceleration. ±0.5 sec or ±10% of period, ±10% of time to 1/2 or double amplitude or ±.02 of damping ratio. ±20% or ±1 sec of time difference between peaks of bank and sideslip Flight Profile Cruise Cruise Cruise Cruise, and Approach or Landing Validation Requirement The test must include whichever is less of the following: Three full cycles (six overshoots after the input is completed), or the number of cycles sufficient to determine time to 1/2 or double amplitude. CCA: Test in Nonnormal control state The test must include whichever is less of the following: Three full cycles (six overshoots after the input is completed), or the number of cycles sufficient to determine representative damping. CCA: Test in Nonnormal control state CCA: Test in Non-normal control state. Record results for at least 6 complete cycles with stability augmentation OFF CCA: Test in Non-normal control state. TABLE 42: PART 60 VALIDATION PARAMETERS FOR PHUGOID, SHORT PERIOD & DUTCH ROLL 6.1 Method Unfiltered validation data of maneuvers stimulating the dutch roll and short period frequencies were inputted into the validation wrapper of the Simulink model. A second, identical set of this data was used as the base model input to which the aircraft was trimmed using the JJ trim and JJ lin trimming functions. A +/- 5% upper and lower limit was applied to the lateral and longitudinal states to visually display the allowable tolerance 68

15 6.2 Short Period Pitching Oscillations Validation Gear Up Flaps Up FIGURE 151: ELEVATOR INPUT TO MODEL FROM FLIGHT TEST DATA FIGURE 152: ALPHA RESPONSE WITH BOUNDS FIGURE 153: PITCH RATE RESPONSE WITH BOUNDS FIGURE 154: THETA RESPONSE WITH BOUNDS 70

16 6.3.3 Gear Down Flaps Landing FIGURE 167: RUDDER INPUT TO MODEL FROM FLIGHT TEST DATA FIGURE 168: SIDESLIP RESPONSE WITH BOUNDS FIGURE 169: ROLL RATE RESPONSE WITH BOUNDS FIGURE 170: YAW RATE RESPONSE WITH BOUNDS 74

17 7. AIR DATA SYSTEMS AND FILTERING The filtering of data is used to simplify or correct the results produced from sensors. Often times, the data has inaccuracies, or noise, that can throw off the measurement of a variable. In aircraft, spacecraft, and missile design, a very common filtering method is the Kalman filter. It recursively uses streams of noisy/inaccurate measurement data to produce an optimal estimate of the core system state. The Kalman filter is especially useful for aerospace applications as it can run in real time using only the present input measurements and the previously calculated state and its uncertainty matrix; no additional past information is required. Using these fundamental ideas, the Kalman filter was applied to a maneuver from the DA42L s approach (gear down, flaps approach) flight test data in order to find the moment coefficient (CM) stability derivatives in real time. The CM longitudinal model equation used is shown below: = The CM value, calculated using the moment equation for a rigid aircraft, is used as the measurement input. Even with a high noise variance on 0.1 (mean at 0) in the measured values the Kalman Filter produces an excellent estimate. FIGURE 171: CM DERIVATIVES ESTIMATES WITH +/-20% BOUND OF THE ORIGINAL DERIVATIVE Another important part of a Kalman filter is the variance or uncertainty in the predicted result. Because an initial covariance is applied to the input measurements, the state variable results are initially very inaccurate. As the 76

18 Appendix B: (PID of Longitudinal Coefficients) Linear Regression Method Accepted Form: Linear Regression Result: = ( ) + = 5.064( ).742 = ( ) + ( ) =.5058( ) ( ) = ( ) + ( ) =.0866( ) ( ) = ( ) + ( ) + =.0105( ).00167( ) = ( ) + =.0011( ).1726 : PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX P( 1 ) E E [ , ] 1 P( 2 ) E E [ , ] 10 : PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX P( 1 ) E E [ , ] 1 P( 2 ) 5.951E E [ 0.570, ] 10 P( 3 ) E E [ , ] 100 P( 4 ) E E [ , ] 1000 : PARAMETER ESTIMATE STD ERROR % ERROR 95 % CONFIDENCE INTERVAL INDEX P( 1 ) 8.655E E [ 0.084, ] 1 P( 2 ) 6.159E E [ 0.002, ] 10 P( 3 ) E E [ , ] 100 P( 4 ) 7.169E E [ 0.698, ]

19 Appendix C: (Navion 6 DOF Hand Trimming Code): clear all clc %Initial attitude initbank = 0; initpitch=.1425; inithead=0; %Trim Velocity components initu = 130; initv = 0; initw = initpitch*initu; %Initial rates initp=0; initq=0; initr=0; %Initial Position initnorth=0; initeast=0; initalt=10000; % NAVION stability derivatives % CL cla=4.44; cladot=0; clq=3.8; clde=0.355; clo=0.41; % CD cda2=0.33; cdo=0.05; % CM cma=-0.683; cmde=-0.923; cmadot=-4.36; cmq=-9.96; %CY cyb = ; cydr = 0.157; %CL - Roll clb = ; clp = ; clr = 0.107; clda = ; 90