HansBu
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- 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.
$$\frac{d^2\theta}{dt^2}=-\frac{a_d(t)}{L}\sin\theta$$
But how can I show that the pendulum is stable for such case? I am totally clueless.