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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