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MEAM 211 Converting 2nd order ODEs to 2 1st order ODEs Example MEAM 211 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector MEAM 211 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector Write state equations MEAM 211 Converting 2nd order ODEs to 2 1st order ODEs Example Define the state vector Write state equations With initial conditions MEAM 211 Solving Equations of Motions for Particles Example Particle in 3-D subject to thrust, gravity and drag State vector is 6x1 6 State Equations MEAM 211 MATLAB Example A ball is thrown upward against gravitational attraction and air resistance with an initial velocity of 30 meters/second. The air resistance opposes the velocity and is proportional to the square of the velocity. The acceleration is: a = - g – cv2 sign(v) where g = 9.81 meter/sec2 and c = 0.001 1/meter. Solve for the position and velocity of the particle as a function of time through a six second time interval. main.m timeInterval=[0, 6]; x0=[0; 30]; % interval for integration % initial position = 0, velocity = 30 intFn ('vertical', timeInterval, x0); function xdot=vertical(t, x); c=0.001; g=9.81; % c=0.001 1/m % g=9.81 m/sec/sec xdot1 = x(2); % derivative of position = velocity xdot2 = -g - c*x(2)*abs(x(2));% acceleration vertical.m xdot=[xdot1; xdot2]; MEAM 211 Example Figure SA2.2.1 (p. 91) Schematic of ejection seat test device. The seat is moving at 600 knots when it is ejected with a specified acceleration a-bt2 for 800 msec. A chute is released to slow the chair down until the chair reaches a speed of 100 knots. Then a parachute is deployed. [Assume the parachute drag is 10 times the drag of the first (drogue) chute.] MEAM 211 Ejection Seat T= 0: pilot pulls handle, T=0.15 secs: seat clears cockpit; T=0.5-0.8 secs: seat/man separator fires; T = 200-400 seconds; main parachute deploys MEAM 211 if t < 0.8 % this is phase I acceleration=a-b*t*t; xdot(3) = -acceleration*sin(pi/9); % sin(20 deg) = sin % This function computes the rate of change (pi/9 rads) % of position and velocity for the ejection seat xdot(4) = acceleration*cos(pi/9); % cos(20 deg) = cos (pi/9 rads) % else % phase II or III % x(1) = position (y); if speed > v_crit % this is phase II % x(2) = velocity (ydot); xdot(3)=-c*speed*xdot(1); % xdot(4)=-c*speed*xdot(2)-g; else % we are in phase III k1=1.1508122; % conversion from knots to miles per hour if x(2) > 0 % the ejection seat is in the air k2=1.4666249; % conversion from miles per hour to ft/sec. xdot(3)=-10*c*speed*xdot(1); k=k1*k2; % conversion from knots to feet/second xdot(4)=-10*c*speed*xdot(2)-g; v_crit=k*100; % speed at which Phase III starts else % the ejection seat has hit the ground xdot(3)=0; c=0.003; % c=0.003 1/ft xdot(4)=0; g=32.2; % g=32.2 ft/sec/sec xdot(1)=0; a=16*g; % a=16g xdot(2)=0; b=9*g; % b= 9g end end end xdot(1) = x(3); % derivative of x position = x velocity xdot(2) = x(4); % derivative of y position = y velocity xdot=xdot'; % need to transpose the vector to get speed=sqrt(x(3)^2+x(4)^2); % xdot to be a column vector function xdot=ejectionSeat(t, x);