ME580
Nonlinear Systems
Home work #1
Due: Sept. 7, 2010
1.
Consider the secondorder linear system presented in class:
0
cos(
);
mx cx kx
F
t
m
,
c
,
k
0.
Nondimensionalize the above system to
2
0
2
(
/
)cos(
),
nn
x
x
x
F m
t
and find the complete solution for arbitrary initial conditions of
00
(0)
,
(0)
x
x x
x
,
assuming that
0
1.
Now, specialize the solution in (a) to the undamped case of
0, and set
0
x
to zero.
This solution
consists of two frequencies and we now want to explore the nature of this solution.
i) Let
0.5,
/
1.0,
4.0
n
x
F
m
and
0.5
.
Plot the solution
x
(
t
) as a function of time; Plot the
solution in the phase plane (
( , )
xx
plane); Now pick the initial condition
x
0
such that no contribution
comes from the free response, and again plot the solution
x
(
t
) as a function of time as well as in phase
plane.
Comment on all these solution plots.
ii) Repeat the work in part (i) above with
0.5
n
and
4.0
.
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 Fall '10
 NA
 Nonlinear system, nonlinear systems

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