### The curious behaviour of the Wilberforce Spring.

The Wilberforce Spring (often called the Wilberforce pendulum) is a simple mechanical device that illustrates the conversion of energy between two forms. It comprises a weight attached to a spring that is free to stretch up and down and to twist about its axis.

Wilberforce spring [image from Wikipedia Commons].}

In equilibrium, the spring hangs down with the pull of gravity balanced by the elastic restoring force. When the weight is pulled down and released, it immediately oscillates up and down.

However, due to a mechanical coupling between the stretching and torsion, there is a link between stretching and twisting motions, and the energy is gradually converted from vertical oscillations to axial motion about the vertical. This motion is, in turn, converted back to vertical oscillations, and the cycle continues indefinitely, in the absence of damping.

The conversion is dependent upon a resonance condition being satisfied: the frequencies of the stretching and twisting modes must be very close in value. This is usually achieved by having small adjustable weights mounted on the pendulum.

There are several videos of a Wilberforce springs in action on YouTube. For example, see here.

Equations of Motion

The simplest way to get the equations is from the Lagrangian ${L = T - V}$. The kinetic energy is due to the up-and-down motion of the mass and its rotation

$\displaystyle T = \frac{1}{2}\left( m \dot z^2 + I \dot\theta^2 \right)$

where ${z}$ is the vertical distance from equilibrium and ${\theta}$ the twisting angle. The potential energy comes from the elastic restoring force in the vertical and the torsional restoring force

$\displaystyle V = \frac{1}{2}\left( k z^2 + \kappa \theta^2 \right)$

Most importantly, there is a coupling term. This arises because, when the spring is stretched it also tends to uncoil along the direction of the wire. This induces a twisting motion. Likewise, if the spring is twisted, coiling up, it seeks to shorten itself, inducing an up-down motion. The coupling potential energy may be modelled by a term

$\displaystyle C =- \epsilon z\theta$

Now the complete Lagrangian is

$\displaystyle L = \frac{1}{2}\left[ ( m \dot z^2 - k z^2 ) + ( I \dot\theta^2 - \kappa \theta^2 ) \right] + \epsilon z\theta$

The equations of motion may be written now:

$\displaystyle \begin{array}{rcl} m \ddot z + k z &=& \epsilon\theta \\ I \ddot\theta + \kappa\theta &=& \epsilon z \end{array}$

The ${z}$ equation, for the up-and-down motion, has solutions oscillating with frequency ${\omega_z = \sqrt{k/m}}$ and an additional term proportional to ${\theta}$. The ${\theta}$-equation, for the twisting motion, has a solution oscillating with frequency ${\omega_\theta = \sqrt{\kappa/I}}$ and an additional term proportional to ${z}$.

If the system is in resonance we have ${\omega_z \approx \omega_\theta}$. Then each variable has two components with frequencies that are almost equal.

The figure shows a solution of the equations: the blue curve is ${z}$ and the red one is ${\theta}$.

A solution of the Wilberforce spring equations. Blue curve: ${z}$ Red curve: ${\theta}$.

To understand how such a solution behaves, we look at the phenomenon of beats.

Beats

If two musical tones of almost equal frequency are sounded together, a low-frequency fluttering sound is heard. This is easily explained in terms of combined sine-waves. Suppose the two tones are at frequencies ${\omega-\epsilon}$ and ${\omega+\epsilon}$. Let us add the two sine waves together:

$\displaystyle \sin[(\omega-\epsilon)t] + \sin[(\omega+\epsilon)t] = 2 \sin[\omega t] \cos[\epsilon t ]$

The first factor is a sinewave of frequency ${\omega}$, the average frequency of the two waves. The second factor (the cosine) is a low-frequency envelope. We plot the two components and their sum in the figure below.

Two sine waves of frequency 9.5 Hz and 10.5 hZ and their sum.

We see how the beating effect resembles the behaviour of each variable (${z}$ or ${\theta}$) for the Wilberforce spring.