文档介绍：6 Parametric oscillator
Mathieu equation
We now study a different kind of forced pendulum.
Specifically, imagine subjecting the pivot of a simple frictionless pendulum
to an alternating vertical motion:
rigid rod
This is called a “parametric pendulum,” because the motion depends on a
time-dependent parameter.
Consider the parametric forcing to be a time-dependent gravitational field:
g(t) = g0 + λ(t)
The linearized equation of motion is then (in the undamped case)
d2β g(t)
+ β= 0.
dt2 l
The time-dependence of g(t) makes the equation hard to solve. We know,
however, that the rest state
β= β˙= 0
is a solution. But is the rest state stable?
We investigate the stability of the rest state for a special case: g(t) periodic
and sinusoidal:
g(t) = g0 + g1 cos(2γt)
Substituting into the equation of motion then gives
d2β
+ γ2 [1 + h cos(2γt)] β= 0 (14)
dt2 0
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where
2
γ= g /l and h = g /g 0.
0 0 1 0
Equation (14) is called the Mathieu equation.
The excitation (forcing) term has amplitude h and period
2αα
T = =
exc 2γγ
On the other hand, the natural, unexcited period of the pendulum is
2α
Tnat =
γ0
We wish to characterize the stability of the rest state. Our previous methods
are unapplicable, however, because of the time-dependent parametric forcing.
We pause, therefore, to consider the theory o