Physics Explanation

Equation of Motion

Assume a massless support and rod, with no friction. We define the following variables:

• \( t \): the time.
• \( \omega \): the frequency that the support oscillates at.
• \( A \): the amplitude that the support oscillates with.
• \( y = A\cos (\omega t) \): the distance of the support from its initial position.
• \( \ell \): the length of the pendulum.
• \( \theta(t) \): the angle the pendulum makes with the vertical.

Let's proceed using Newton's Laws, although Lagrangian Mechanics works equally well here. Consider the frame of reference of somebody on the support of the pendulum. The vertical acceleration here is \( \ddot{y} = - \omega^2 \cos (\omega t)\), so this person feels a downward acceleration of \( g - \omega^2 \cos (\omega t) \). The tangential acceleration is that times \( \sin \theta \), so: \[ \ell \ddot{\theta} = (g - \omega^2 \cos (\omega t))\sin \theta .\]

We can then approximately solve this differential equation.

Exercise: Why does it stay up?

Hint: the frequency of the support plays a big role. For example, you'll note that in in this case with 10 Hz, the pendulum falls down. But if we increase the frequency to 20 Hz, the stick comfortably stays up. This agrees with our intuition that the stick would fall over if \( \omega = 0\).

Second Hint: it may help to consider the average value of \( \ddot\theta \). Note that in addition to its large oscillations, \( \theta \) undergoes small oscillations with frequency \( \omega \).

Further Reading

See "On the dynamic stabilization of an inverted pendulum" by Butikov (2001) for a detailed discussion.

Technology

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