MAE142_Solution3

# MAE142_Solution3 - MAE 142 Homework Set#3(Winter 2011...

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MAE 142 Homework Set #3 (Winter 2011) Problem 1 (10 pts) Develop a state-space model for the van der Pol oscillator. ¨ y − 1 y 2  ˙ y y = 0 = constant Define your state vector such that: x 1 = y t x 2 = ˙ y t Problem 2 (20 pts) Compare the roll rate (p) and bank angle (phi) time response of the F-16 airplane to the F-4 airplane. F-4: repeat the class lecture example to create the roll rate and bank angle time history plots for a 10 deg aileron doublet. F-16: use the provided F-16 model to simulate the airplane response with the same 10 deg aileron doublet, while beginning from the following level-flight condition: altitude = 35,000 ft, airspeed = 850 ft/s, angle-of-attack = 1.88 deg, elevator = -1.78 deg, and thrust = 1600 lb. Problem 3 (10 pts) Find a linear model for the AUV. The control variables (u vector) include buoyancy, thrust, and all four tail fin deflections. Assume a level-flight equilibrium condition with speed = 1.5 m/s, depth = 50 m, buoyancy = 184.5 N, angle-of- attack = -1.06 deg, thrust = 8.2 N, fin_left = 4.3 deg, fin_right = 0.8 deg.

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MAE 142 Homework #3 SOLUTIONS 1. Let x 1 = y and x 2 = ˙ y . This implies that x 2 = ˙ x 1 . The van der Pol equation can be rewritten now in terms of x 1 and x 2 : ¨ y - μ ( 1 - y 2 ) ˙ y + y = 0 ˙ x 2 - μ ( 1 - x 2 1 ) x 2 + x 1 = 0 Combining all of the above information: ˙ x 1 = x 2 ˙ x 2 = μ ( 1 - x 2 1 ) x 2 - x 1 2. 0 2 4 6 8 10 12 14 16 18 20 -500 -400 -300 -200 -100 0 100 200 time [sec] roll rate (p) [deg/sec] F4 F16 1

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if (time(i) > 6.0) da_cmd = -10*pi/180; end if (time(i) > 9.0) da_cmd = 0.0; end x(13,1) = 1600; % Engine thrust [lb] x(14,1) = -1.78*pi/180; % Elevator deflection [rad] x(15,1) = da_cmd; % // set aileron command x = RK4_for_F16func(x,dt); % // integrate end % figure(1) plot(time,x_saveF4(1,:),time,x_saveF16(4,:)*180/pi),grid on; xlabel(’time [sec]’); ylabel(’roll rate (p) [deg/sec]’); legend(’F4’,’F16’); % figure(gcf+1)
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