ME580
Nonlinear Systems
Home work #1
Due: Sept. 7, 2010
1.
Consider the second-order linear system presented in class:
mx  cx  kx  F0 cos(t );
m, c, k  0.
Non-dimensionalize the above system to

x  2n x  n2 x  ( F0 / m)cos(t ),
and find the complete solution for arbitrary initial conditions of x(0)  x0 , x (0)  x0 , assuming that
0   1.
Now, specialize the solution in (a) to the undamped case of   0 , and set x0 to zero. This solution
consists of two frequencies and we now want to explore the nature of this solution.
i) Let x0  0.5, F0 / m  1.0, n  4.0 and   0.5 . Plot the solution
x(t) as a function of time; Plot the
solution in the phase plane ( ( x, x ) plane); Now pick the initial condition x 0 such that no contribution
comes from the free response, and again plot the solution
plane. Comment on all these solution plots.
ii) Repeat the work in part (i) above with n  0.5 and
x(t) as a function of time as well as in phase
  4.0 .
2.
In this problem, we make a small excursion into the nonlinear world for the second-order system.
Don't worry that we have not yet covered it in the class.
Consider an approximate version of the equation governing the motion of a harmonically excited
pendulum:

x  n2 ( x  x3 / 6)  F0 cos(t ) ,
and explore the possibility of finding an approximate periodic (really only harmonic) solution at the
excitation frequency by assuming the solution to be
x(t)  Acos(t)  Bsin(t) ,
where A and B are constants to be determined. Substitute this solution form in the differential equation,
expand the functions of (t ) in a Fourier series, and then compare coefficients of cos(t) and sin(t)
on the two sides of the resulting expression. Note that this gives two simultaneous nonlinear equations in
the two unknowns A and B. Their solutions provide the so-called first approximations to the possible
periodic solutions of the appropriate type!!.
3.
Consider the system

x  n2 ( x  x2 / 6)  F0 cos(t ) ,
and explore the possibility of finding an approximate periodic (really only harmonic) solution at the
excitation frequency by assuming the solution to be
x(t )  A0  A cos(t )  B sin(t ) ,
where A0, A and B are constants to be determined. Substitute this solution form in the differential
equation, expand the functions of (t ) in a Fourier series, and then compare the appropriate coefficients
on the two sides of the resulting expression. Note that this gives three (!!) simultaneous nonlinear
equations in the three unknowns A0, A and B. Their solutions provide an approximation to the possible
periodic solutions of the appropriate type!!.
4.
Consider the piece-wise linear system
mx  k1 x  0 for x  0,
mx  k2 x  0 for x  0.
Find the solution for an arbitrary initial condition x(0)  A  0, x(0)  0, and write the solution for
both x > 0 and x < 0. Show that this solution is periodic. Find the period of this solution. Does it depend
on the ‘amplitude’ of the initial condition?