hw4sol.pdf - George W Woodruff School of Mechanical...

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George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology ME6401 HW SET #4 Solutions ___________________________________________________________________________ 1) Text Problem 9.1. Y(s)/U(s)=C(sI-A) -1 B+D=1/(s+2)=>The system is BIBO stable The eigenvalues of A are -2, 1, -1. Therefore the system is unstable. 2) The equation of motion of the pendulum shown assuming small angles is given by 0 )) ( 1 ( 2 2  t a n n where the vertical acceleration a is assumed to be a sinusoidal input of frequency : a(t)=0.5 sin t where 0  <10. . a) Express the model in the state space form x A x ) ( t by a suitable choice of state variables. function dx=mathiu(t,x,flag,wn,w,q) % State Space representation of the 2nd order Matu Equation % Called by ode45.m A=[0 1;-wn^2*(1-0.5*sin(w*t)) -0.2*wn]; dx=A*x; b) Write the a MATLAB function to compute the fundamental matrix ) 0 , ( ) ( t t X for this system. Plot the elements of X ( t ) vs. t , 150 0 t assuming n =1 rad/sec, =0.1, and =2 rad/sec. Determine the values of for which the system is asymptotically stable. %Script file to call ode45 to get state transition matrix %of the Mathieu equation in Homework #4 problem 2 for different %values of parameter q %initial conditions to be used X0=eye(2); %Time-Span Tspan=[0:0.1:150]; % Set w parameter to vary from 0 to 10 W=[0:0.1:10]; for i=1:101 m a L
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