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Course: ASE 167, Fall 2009
School: University of Texas
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Assignment Computer 3 Numerical Integration of the Equations of Motion for Gliding Flight Gregory R. Whitney David Hughling TA: Eduardo Gildin November 5, 2001 University of Texas at Austin ASE 167M Table of Contents PROGRAM IDENTIFICATION..................................................................................................................1 PURPOSE OF THE...

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Assignment Computer 3 Numerical Integration of the Equations of Motion for Gliding Flight Gregory R. Whitney David Hughling TA: Eduardo Gildin November 5, 2001 University of Texas at Austin ASE 167M Table of Contents PROGRAM IDENTIFICATION..................................................................................................................1 PURPOSE OF THE PROGRAM..................................................................................................................2 MATHEMATICAL TECHNIQUE..............................................................................................................2 PROGRAM USAGE......................................................................................................................................5 PROGRAM LISTING....................................................................................................................................5 DATA RUNS/TEST CASES..........................................................................................................................6 FIGURE 1: VELOCITY VS. DISTANCE FOR GLIDING FLIGHT OF THE LEAR 23 SBJ.........................................................6 FIGURE 2: ALTITUDE VS. DISTANCE FOR GLIDING FLIGHT OF THE LEAR 23 SBJ.........................................................7 FIGURE 3: FLIGHT PATH ANGLE VS. DISTANCE FOR GLIDING FLIGHT OF THE LEAR 23 SBJ...........................................8 FIGURE 4: ALTITUDE VS. TIME FOR GLIDING FLIGHT OF THE LEAR 23 SBJ................................................................9 FIGURE 5: GLIDING DISTANCE VS. TIME FOR GLIDING FLIGHT OF THE LEAR 23 SBJ..................................................10 FIGURE 6: FLIGHT PATH ANGLE VS. TIME FOR GLIDING FLIGHT OF THE LEAR 23 SBJ................................................11 FIGURE 7: VELOCITY VS. TIME FOR GLIDING FLIGHT OF THE LEAR 23 SBJ..............................................................12 APPENDIX A: HARDWARE AND SOFTWARE USED........................................................................13 APPENDIX D: PROGRAM SOURCE CODE..........................................................................................15 GLIDING_FLIGHT.M........................................................................................................................................15 DERIV.M.......................................................................................................................................................18 ATMOS.M......................................................................................................................................................19 Program Identification Program Assignment: Programmer: Date: Requirements: Storage Required: gliding_flight.m Gregory R. Whitney November 5, 2001 Computer running MATLAB 7 KB Special I/O Requirements: None To plot the data, run gliding_flight.m. There is no input required. 1 Purpose of the Program The purpose of this program was to plot the performance of a Lear 23 (SBJ) during gliding flight. Since maximizing ground distance is often the goal in glide situations, the glide equations of motion were numerically integrated along the maximum range path using the 4th Order Runge-Kutta method. This iterative method is required to find the maximum endurance (final time) and the corresponding maximum range defining the performance of the Lear 23 during gliding flight. Mathematical Technique To calculate the maximum range and endurance of the Lear 23 the equations of motion for non-steady flight in the vertical plane are modified for gliding flight. Enforcing the glide condition, T = 0, the equations of motion for gliding flight become x = V cos h = V sin V = -g/W [D + W sin ] = g/VW [L - W cos ] (1) (2) (3) (4) At optimum flight conditions for gliding flight, the velocity of the SBJ is low enough so that we can use the a parabolic drag polar CD = CDo + KCL2 (5) For maximum range, the lift-to-drag ratio (E = CL/CD) is maximum. Using equation (5) 2 to substitute for CD, E becomes a function of CL: E = CL/CDo + KCL2 Differentiating E(CL) with respect to CL, and setting the derivative equal to zero CDo + KCL2 CL(2KCL)/(CDo + KCL)2 = 0 By simplifying equation (7), it can be seen that to maximize E CDo KCL2 = 0 From here, the lift and drag coefficients for maximum range (CL*, CD*) are CL * = C Do K (9) (10) (8) (7) (6) CD* = 2K(CL*)2 Maximum range lift and drag are defined as D = CD* (h)V2S L = CL* (h)V2S (11) (12) Substituting equations (11-12) into equations (1-4) the equations of motion become x = V cos h = V sin - g 1 SV 2 C * + W sin V = D W 2 (13) (14 [ [ ] ] (15) = g 1 * SV 2 C L - W cos WV 2 (16) 3 Equations (13-16) are the non-steady gliding flight equations of motion along the maximum range path. These equations are numerically integrated for a descent from 10,000 feet to sea level using the 4th Order Runge-Kutta method. Runge-Kutta is mathematically represented by xn+1 = xn + 1/6 (k1 + 2k2 + 2k3 + k4) where k1 = f(xn,tn)t k2 = f(xn+ k1, tn+ t)t k3 = f(xn+ k2, tn+ t)t k4 = f(xn+k3,tn+t)t (17) To solve for the maximum range, the initial state of the aircraft is represented in the following way: X(0) = [ x(0) h(0) V(0) (0)]T where x(0) = 0 ft h(0) = 10,000 ft v(0) = 2W * (h(0))C L S w * (18) (0) = - C D * CL 4 Program Usage Input format: There is no input for this program Output format: This program outputs 7 plots describing the gliding flight performance of the Lear 23 (SBJ). The six plots are 1. Altitude vs. Time 2. Altitude vs. Distance 3. Distance vs. Time 4. Flight Path Angle vs. Time 5. Flight Path Angle vs. Distance 6. Velocity vs. Time 7. Velocity vs. Distance Error Provisions: None User Instructions: 1. Start MATLAB 2. Change directory to Z:\My Documents\167M\lab6\ 3. At the prompt, type "gliding_flight" 4. To save the figures, simply click "Save As" and choose a filename and directory Program Listing See the source files, starting on page 15, for the actual program codes. 5 Data Runs/Test Cases The following plots show the gliding flight performance for the Lear 23 (SBJ) at a weight of 11,000 pounds and a starting altitude of 10,000 ft. Figure 1: velocity vs. distance for gliding flight of the Lear 23 SBJ 6 Figure 2: altitude vs. distance for gliding flight of the Lear 23 SBJ 7 Figure 3: flight path angle vs. distance for gliding flight of the Lear 23 SBJ 8 Figure 4: altitude vs. time for gliding flight of the Lear 23 SBJ 9 Figure 5: gliding distance vs. time for gliding flight of the Lear 23 SBJ 10 Figure 6: flight path angle vs. time for gliding flight of the Lear 23 SBJ 11 Figure 7: velocity vs. time for gliding flight of the Lear 23 SBJ 12 Appendix A: Hardware and Software Used This lab was done using a home PC and running Windows 2000 and MATLAB 6.1. Appendix B: Analytical Results The analytical solutions to gliding flight are derived by making the following assumptions: << 1 V = = 0 By applying the above assumptions to equations (1-4), we get the following equations: x =V h = V 0 = D + W 0 = L -W (B-1) (B-2) (B-3) (B-4) therefore, if we solve for h(x), V(x), and (x) we get the analytical solutions: ( x) = - C D * * CL (B-5) (B-6) (B-7) h( x) = ( x f - x 0 ) + h0 V ( x) = 2W * (h( x))C L S 13 Appendix Discussion C: of Simulation Figures 1-3 show the results of the numerical integration compared to the analytical results. For figure 1, the velocity versus distance plot, the numerical analysis has an initial transient response whereas the analytical result is a line of constant slope. The difference between the numerical and analytical results is small nonetheless. Figure 2, the height versus time plot, also shows that the numerical result differs from the analytical. The reason for this can be seen from equation (B-6). In equation (B-6), the analytical height is shown to depend linearly on gamma; gamma is shown to be constant by equation (B-5). As seen in figure 3, the numerical result for the flight path angle resembles a damped oscillator, and is not constant. Since gamma is not constant, the change in height is slightly different from the theoretical prediction. Figure 3, the gamma versus distance plot, shows an interesting difference between the numerical and analytical results. For the analytical gamma it was assumed that the magnitude of gamma would be very small, so a constant gamma was used to simplify the solutions. The results of the iteration process have shown that even though gamma changes are small, less than .006 rad, the values do oscillate for a while before settling to a steady state value of approximately -0.072 rad; a damped oscillator! It is also interesting to note that the final numerical gamma does not match the analytical value of 0.075 rad. Figure 3 basically shows that the analytical assumption of constant flight path angle is a good one, but the oscillatory motion of the aircraft is not discovered until the results are numerically integrated. 14 Appendix D: Program Source Code Gliding_Flight.m function gliding_flight %These variables are needed in deriv also, so make them global global W S Cd_max Cl_max K %Define some constants S = 232; %[ft^2] Cd0 = 0.02; K = 0.07; W = 11000; %[lbf] Cl_max = sqrt(Cd0/K); Cd_max = 2*K*Cl_max^2; %Define the initial conditions t = 0; x = 0; h = 10000; %[ft] [temp,press,rho,sos]=atmos(h); v = sqrt((2*W)/(rho*S*Cl_max)); %[ft/s] gamma = -Cd_max/Cl_max; %This is a constant dt = 2; %[sec] i = 1; while h > 0.0001 %Define the State Vector at t=0 %X = [x(t);h(t);v(t);gamma(t)] X = [x;h;v;gamma]; %Call Ode45 to integrate the EOM's [t, X_new] = ode45('deriv',[t,t+dt], X); %Create a vector for each variable that will be plotted plotted_x(i) = X_new(size(t,1),1); plotted_h(i) = X_new(size(t,1),2); plotted_v(i) = X_new(size(t,1),3); plotted_gamma(i) = X_new(size(t,1),4); h_analyt(i) = (-Cd_max/Cl_max)*(plotted_x(i))+10000; [temp,press,rho_analyt,sos]=atmos(h_analyt(i)); v_analyt(i) = sqrt((2*W)/(rho_analyt*S*Cl_max)); gamma_analyt(i) = -Cd_max/Cl_max; 15 plotted_time(i) = t(size(t,1)); %Redefine the state variables for a new X x = X_new(size(t,1),1); h = X_new(size(t,1),2); v = X_new(size(t,1),3); gamma = X_new(size(t,1),4); t = t(size(t,1)); if h > .0001 Time(i) = t; %Save the time so that we know how to step back once h < 0 i = i + 1; elseif h < -.0001 %If the altitude is negative, then the dt is changed %and the state variables need to be stepped back dt = 0.5*dt; t = Time(i-1); x = plotted_x(i-1); h = plotted_h(i-1); v = plotted_v(i-1); gamma = plotted_gamma(i-1); else break; end end plot(plotted_x(:),plotted_h(:)); hold on plot(plotted_x(:),h_analyt(:),'r'); hold off title('Altitude vs Distance for Gliding Flight'); xlabel('Distance [ft]'); ylabel('Altitude [ft]'); figure plot(plotted_x(:),plotted_v(:)); hold on plot(plotted_x(:),v_analyt(:),'r'); hold off title('Velocity vs Distance for Gliding Flight'); xlabel('Distance [ft]'); ylabel('Velocity [ft/s]'); figure plot(plotted_x(:),plotted_gamma(:)); title('Flight Path Angle vs Distance for Gliding Flight'); hold on plot(plotted_x(:),gamma_analyt(:),'r'); 16 hold off xlabel('Distance [ft]'); ylabel('Flight Path Angle [rad]'); figure plot(plotted_time(:),plotted_x(:)); title('Gliding Distance vs Time for Gliding Flight'); xlabel('Time [s]'); ylabel('Distance [ft]'); figure plot(plotted_time(:),plotted_h(:)); title('Altitude vs Time for Gliding Flight'); xlabel('Time [s]'); ylabel('Altitude [ft]'); figure plot(plotted_time(:),plotted_v(:)); title('Velocity vs Time for Gliding Flight'); xlabel('Time [s]'); ylabel('Velocity [ft/s]'); figure plot(plotted_time(:),plotted_gamma(:)); title('Flight Path Angle vs Time for Gliding ...

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