3.3 Main Example: Pendulum with an Oscillating
Support
As a first object for demonstrating nonlinear analysis we consider a pendulum that
is hinged at a harmonically oscillating support. Despite its simplicity, this system
exhibits a rich dynamic behavior. It will allow many of the concepts, phenomena
and tools of nonlinear analysis to be introduced without cluttering the explanations
in inessential algebra.
We start by exploring the free nonlinear oscillations of the pendulum. First
qualitatively, allowing the concepts of phase planes, singular points, singular-point
stability and local behavior to be introduced. Next quantitatively, introducing
various methods of perturbation analysis. Then we turn into determining the forced
response, exploring further the potentials of perturbation analysis. The concepts of
limit cycles and limit cycle stability appear, as do several other terms describing
nonlinear phenomena, e.g., nonlinear frequency response, frequency locking, amplitude jumps, subharmonics and superharmonics.
The pendulum with an oscillating support is known to exhibit chaotic behavior
for certain ranges of physical parameters; However, since chaos is generally a
global phenomenon it is not revealed by the local methods described in this chapter.
Only ordered, non-chaotic motions will be considered, whereas the study of chaotic
motions will have to wait until Chap. 6.
3.3.1 Equation of Motion
Fig. 3.5 shows the pendulum, characterized by mass m, length l, and instantaneous
angle of rotation h(t). The pendulum is subjected to a gravity field g, and to a
viscous damping moment Àcl
2 _
h. The position u(t) of the hinged support oscillates
harmonically at a prescribed amplitude ql and frequency X.
The equation of motion is conveniently set up using Lagrange’s equation for
single-DOF non-conservative systems (cf. Sect. 1.5.1):
d
dt
@L
@ _
h
À
@L
@h
¼ Q; L ¼ T À V;
ð3:8Þ
The kinetic energy T, potential energy V, and generalized non-conservative force
Q become, respectively, with x = lcosh – u and y = lsinh:
T ¼
1
2
m_ x
2
þ
1
2
m_ y
2
¼
1
2
m l
2 _
h
2
þ _
u
2
þ 2l _
h _
u sin h
;
V ¼ Àmgx ¼ Àmg l cos h À u
ð
Þ ;
Q ¼ Àcl
2 _
h:
ð3:9Þ
102
3 Nonlinear Vibrations: Classical Local Theory
Support
As a first object for demonstrating nonlinear analysis we consider a pendulum that
is hinged at a harmonically oscillating support. Despite its simplicity, this system
exhibits a rich dynamic behavior. It will allow many of the concepts, phenomena
and tools of nonlinear analysis to be introduced without cluttering the explanations
in inessential algebra.
We start by exploring the free nonlinear oscillations of the pendulum. First
qualitatively, allowing the concepts of phase planes, singular points, singular-point
stability and local behavior to be introduced. Next quantitatively, introducing
various methods of perturbation analysis. Then we turn into determining the forced
response, exploring further the potentials of perturbation analysis. The concepts of
limit cycles and limit cycle stability appear, as do several other terms describing
nonlinear phenomena, e.g., nonlinear frequency response, frequency locking, amplitude jumps, subharmonics and superharmonics.
The pendulum with an oscillating support is known to exhibit chaotic behavior
for certain ranges of physical parameters; However, since chaos is generally a
global phenomenon it is not revealed by the local methods described in this chapter.
Only ordered, non-chaotic motions will be considered, whereas the study of chaotic
motions will have to wait until Chap. 6.
3.3.1 Equation of Motion
Fig. 3.5 shows the pendulum, characterized by mass m, length l, and instantaneous
angle of rotation h(t). The pendulum is subjected to a gravity field g, and to a
viscous damping moment Àcl
2 _
h. The position u(t) of the hinged support oscillates
harmonically at a prescribed amplitude ql and frequency X.
The equation of motion is conveniently set up using Lagrange’s equation for
single-DOF non-conservative systems (cf. Sect. 1.5.1):
d
dt
@L
@ _
h
À
@L
@h
¼ Q; L ¼ T À V;
ð3:8Þ
The kinetic energy T, potential energy V, and generalized non-conservative force
Q become, respectively, with x = lcosh – u and y = lsinh:
T ¼
1
2
m_ x
2
þ
1
2
m_ y
2
¼
1
2
m l
2 _
h
2
þ _
u
2
þ 2l _
h _
u sin h
;
V ¼ Àmgx ¼ Àmg l cos h À u
ð
Þ ;
Q ¼ Àcl
2 _
h:
ð3:9Þ
102
3 Nonlinear Vibrations: Classical Local Theory