Problem 4.3 A uniform rod of length l 2 and mass m is hanging from a massless
chord of length l 1 , as shown in Fig. P4.3.
a) Show that the governing equations are
l 1 € h 1 þ
1
2
l 2 € h 2 cosðh 2 À h 1 Þ À
1
2
l 2 _
h
2
2 sinðh 2 À h 1 Þ þ g sin h 1 ¼ 0;
1
3
l 2 € h 2 þ
1
2
l 1 € h 1 cosðh 2 À h 1 Þ þ
1
2
l 1 _
h
2
1 sinðh 2 À h 1 Þ þ
1
2
g sin h 2 ¼ 0:
ð4:118Þ
b) Determine the linear natural frequencies of the system.
c) Assuming small but finite amplitudes, determine all possible conditions of
internal and external resonance.
d) Choose l 2 /l 1 to produce a three-to-one internal resonance, and determine for this
case a uniformly valid first-order multiple scales expansion.
Problem 4.4 A rigid beam with mass m and moment of inertia I is supported in a
gravity field g by linear springs having stiffnesses k 1 and k 2 (Fig. P4.4). The center
of gravity G of the system can move only vertically.
a) Show that the equations of motion are:
m€ x þ ðk 1 þ k 2 Þx þ ðk 1 l 1 À k 2 l 2 Þ sin h ¼ Àmg;
I € h þ ðk 1 l 1 À k 2 l 2 Þx cos h þ
1
2
ðk 1 l
2
1 þ k 2 l
2
2 Þ sin 2h ¼ 0:
ð4:119Þ
b) Determine the linear natural frequencies of the system.
c) Assume small but finite amplitudes and determine all possible conditions of
internal and external resonance.
d) Determine a uniformly valid first-order multiple scales expansion that includes
the case of modal coupling.
Fig. P4.3
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4 Nonlinear Multiple-DOF Systems: Local Analysis