A Chirping Elliptic Rocker

Sitting at the breakfast table, I noticed that a small cereal bowl placed within another larger one was rocking, and that the period became shorter as the amplitude died down. What was going on?

A small bowl with its handles resting on the rim of a larger bowl. The handles are approximately elliptical in cross-section.

The handles of the smaller bowl appeared to be elliptical in cross-section, so I considered how a rigid body shaped like an elliptical cylinder rolling on a flat surface might behave.

The figure below shows the cross-section of an elliptical cylinder. We call the line from the centre C to the point of contact P the contact vector. Its length varies as a function of ${\theta}$ .

Cross-section of a cylinder, elliptical in cross-section.

Two angles are relevant:

${\theta}$ , the slope of the major axis relative to the flat surface.
${\phi}$ , the central angle between the major axis and the contact vector.

The length of the contact vector, ${r}$ , varies as the body rocks back and forth.

Equilibrium

If the cylinder is resting with the contact point at the extremity of the minor axis B, it is in a stable equilibrium. This is evident from the fact that any disturbance will raise the centre point C and therefore increase the potential energy. If the body is disturbed slightly, it will oscillate back and forth about equilibrium. The frequency may be computed in terms of the curvature at B. This, in turn depends on the semi-axes ${a}$ and ${b}$ .

If the cylinder is balanced on its `pointy end’ at A, it is in an unstable equilibrium. Any slight disturbance will cause it to move far away from this state. The potential energy is at a maximum when the body is balanced at A.

Phase portrait of a simple pendulum.

If the energy of the body is less than the resting potential energy at unstable equilibrium, it cannot reach this state and will oscillate or librate. If it has sufficient energy to pass over the unstable equilibrium point, it will continue to roll, albeit in a staggering fashion, in one direction, executing a rotational motion. This is reminiscent of a simple pendulum, where the motions fall into librating and rotating classes. The phase-plane diagram for the elliptic cylinder is not identical, but is topologically equivalent, to the diagram for the pendulum.

Rolling Constraint

Suppose that at stable equilibrium the point B on the ellipse coincides with point O on the underlying plane. Then, because of the rolling constraint, the arc-length from B to P is equal to ${s}$ , the length of the line segment from O to P. This is expressed as an elliptic integral:

$\displaystyle s = a E(x/a,e) = a \int_0^{x/a} \sqrt{\frac{1-e^2\xi^2}{1-\xi^2}} \, \mathrm{d}\xi$

where ${e = \sqrt{(a^2-b^2)/a^2}}$ is the eccentricity.

We can derive the equations of motion by writing down the Lagrangian

$\displaystyle L = T - V = \frac{1}{2}m(\dot\xi^2+\dot\eta^2) + \frac{1}{2} I\dot\theta^2 - mgh$

where ${(\xi,\eta)}$ are cartesian coordinates in the (stationary) space frame and ${h}$ is the height of the centre of mass of the body. Damping may be included by adding a Rayleigh dissipation function to the Lagrangian. Due to the rolling constraint, all the variables can be expressed in terms of a single quantity, the angle ${\theta}$ . The trigonometry is tortuous but elementary.

Chirping Oscillations

It is now straightforward to write down the Euler-Lagrange equations. They may be solved numerically using, for example, Mathematica. The graph below shows the arc-length between B and P as a function of time. It is a damped chirp: a chirp is a signal with increasing frequency. For example, the function ${x = \sin k t^2}$ can be viewed as ${x = \sin \omega(t)t}$ where the frequency ${\omega(t) = k t}$ increases linearly with time.

Variation of arc-length s between point B and the contact point P as the oscillations decrease in amplitude.

The arc-length graph is essentially a damped version of this, i.e., a damped chirp. This behaviour is consistent with the observed increase in frequency of the rocking bowl as the amplitude decreases.

ThatsMaths