Kapitza’s Pendulum

One Pendulum Less

A few weeks ago we looked at how to balance an inverted pendulum as part of a 3 part series on balancing inverted pendulums. Part 2 is still being worked on, but here’s part 3, an exploration of how shaking an inverted pendulum up and down really fast can make it stable.

• Balancing an inverted pendulum – Part 1
  
• Balancing an inverted pendulum on a cart – Part 2
  
• Kaptiza’s pendulum – Part 3
  

Note: I’m planning on doing something a bit different for part 2, but if you want a more traditional treatment of nonlinear dynamics, I recommend exploring Nonlinear Dynamics and Chaos by Strogatz

Kapitza’s Pendulum

So our setup is similar to the traditional pendulum with one small change. Instead of one end of the pendulum being fixed at the origin, it vibrates  vertically with the following velocity

where a is the amplitude of its vibration and γ is the frequency of vibration.

Let’s now find the kinetic energy

and potential energy of this pendulum

Which will allow us to form the Lagrangian

We can then simplify the Lagrangian by removing the portions that are solely functions of time. This leaves us with

using the Euler – Lagrange Equation

we get the following equation of motion

And now we can simulate it:

Note: I added some dampening to the simulation to mimic real wold behavior from air resistance. Try Re-Deriving the equations of motion with some friction. Hint: you’ll want to use a Rayleigh Function

So Why is it Stable?

When this phenomena was first discovered in the early 1900’s, it couldn’t be explained. The answer, provided 50 years later by Russian Nobel Laureate Pyotr Kapitza, ended up birthing the field of vibration dynamics. It’s a bit of a long derivation, but we can separate our motion into the “slow motion”, φ, from the swinging of the pendulum, and the “rapid motion”, ξ, from the oscillating of the base.

By expanding this into the second derivative we get

The next portion is more involved, but involves separating  the variables and solving for ξ by assuming φ is constant, and then averaging the remaining terms over time. We can then use that to get an effective potential energy of the system below

A stable point in our system is any point where the effective potential is a minimum. It can be a local or a global minimum. We can analytically find these points by taking two derivatives, but let’s do it visually using a gif where we vary the frequency of the system instead until we find a stable point at θ=π, which corresponds to where the pendulum is inverted.

We can see that a minimum forms at pi right around when the frequency is 50 Hz. As we increase the frequency, our well keeps increasing. 

Here’s the code used to generate the figures in this post.

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