b) For the case of internal resonance x 1 % 2x 2 , determine a zero order multiple
scales expansion that is valid for small but finite amplitudes.
c) Can there be coupling between the x and the h motions in the linear case? In the
nonlinear?
Problem 4.2 A uniform rod of mass m and length l is hanging from a cart
(Fig. P4.2). The cart has negligible mass, is restricted by a spring with stiffness k,
and loaded by an external force F(t).
a) Show that the governing equations of motion are:
€ u þ ~
x
2
1 u þ
1
2
€ h cos h À
1
2
_
h
2 sin h ¼ FðtÞ;
€ h þ ~
x
2
2 sin h þ
3
2
€ u cos h ¼ 0;
ð4:117Þ
where u = x/l, ~
x
2
1 = k/m and ~
x
2
2 = 3 g/2 l.
b) Expand the equations of motion to order three for small but finite values of
u and h, and determine the linear natural frequencies and mode shapes (x i , u i ).
c) Decouple the linear part of the expanded equations of motion by using a modal
transformation of coordinates x = Uy, where x = {u h}
T and U = [u 1 u 2 ].
d) For the case F(t) = Q cos(Xt), perform a first-order multiple scales analysis to
the stage where secular terms should be eliminated. List all possible external and
internal resonances.
Fig. P4.1
Fig. P4.2
4.9 Problems
261