Stability of a Driven Pendulum

[HansBu]

Homework Statement

I am having confusion with regards to the proof of my problem. This involves a pendulum with harmonically driven pivot and the task is to show that the pendulum is stable in the inverted position when the amplitude of the driving acceleration is sufficiently high. For reference, consider the problem below.

> Consider a pendulum with harmonically driven pivot. The equation of motion is
$$\frac{d^2\theta}{dt^2}=-\frac{g+a_d(t)}{L}\sin\theta$$
where $$a_d(t)=A_0\sin(2\pi t/T_d)$$ is the time-varying acceleration of the pivot. Show that when the amplitude of the driving acceleration is sufficiently high $$A_0\gg g$$ the pendulum is stable in the inverted position i.e., if $$\theta(t=0)\approx180°$$.

Relevant Equations

$$\frac{d^2\theta}{dt^2}=-\frac{g+a_d(t)}{L}\sin\theta$$
where $$a_d(t)=A_0\sin(2\pi t/T_d)$$ is the time-varying acceleration of the pivot.

I understand that when $$A_0 \gg g$$, the g term in the equation of motion can be dropped. The equation of motion then becomes
$$\frac{d^2\theta}{dt^2}=-\frac{a_d(t)}{L}\sin\theta$$

But how can I show that the pendulum is stable for such case? I am totally clueless.

 

[mitochan]

Is there any condition on $T_d$? Say $T_d=\infty$ the inverted pendulum is unstable.

 

[wrobel]

looks like an ill - understood Kapitza's pendulum

 

[HansBu]

Is there any condition on $T_d$? Say $T_d=\infty$ the inverted pendulum is unstable.

Hi, mitochan! There were no specified conditions for $T_d$ given in the problem.

 

[HansBu]

looks like an ill - understood Kapitza's pendulum

It is quite consistent with the mechanics of Kapitza's pendulum. I presume that something is wrong in the problem, right?

 

[wrobel]

I do not know.

Anyway stability of the Kapitza pendulum is proved by means of KAM theory. All this field is far beyond undergraduate courses.