Barton's Pendulums

The set-up

A heavy brass pendulum of length L_d and seven light paper pendulums of lengths L_i hang from one rod. When the heavy pendulum swings, its pull moves the rod's support points sideways by X_s = c L_d theta_0, where c is a small coupling constant (the fraction of the driver's motion that reaches the rod).

Equation of each light pendulum

In the frame of the moving support a light pendulum feels an inertial force: phi'' + 2 gamma_i phi' + (g / L_i) sin(phi) = -(c L_d / L_i) theta_0'' cos(phi), with gamma_i = omega_i / (2Q) and omega_i = sqrt(g / L_i).

Resonance and phase

For small angles and a sinusoidal driver theta_0 = Theta sin(Omega t) the steady state is phi = Phi sin(Omega t - delta) with

Phi = (c L_d / L_i) Omega^2 Theta / sqrt((omega_i^2 - Omega^2)^2 + (2 gamma_i Omega)^2), tan(delta) = 2 gamma_i Omega / (omega_i^2 - Omega^2).

Below resonance (Omega < omega_i) the pendulum follows the driver in phase (delta near 0); exactly at resonance it lags by 90 degrees; above resonance it swings opposite to the driver (delta near 180 degrees). The width of the peak is about omega_i / Q, so a high Q makes the response selective.

Free mode

In free mode the driver is itself a damped pendulum, theta_0'' + 2 gamma_0 theta_0' + (g/L_d) sin(theta_0) = 0, started at the chosen amplitude. This is the classic Barton demonstration.

Simplifications

The light pendulums do not push back on the driver (valid when their total mass is far below the driver's, which is why the driver is heavy), and the rod is treated as rigid apart from the small transmitted motion. The coupling c is a model parameter, not measured from a real rod. Air drag is linear damping with the quality factor Q.

1. Free mode: after a few swings the pendulum with the same length as the driver (number 4 at the default) grows the biggest swing; the others stay small and drift in and out of step.
2. Motor mode at 0.6435 Hz: pendulum 4 lags the driver by 90 degrees; look at the phase graph. Pendulums 1-3 (shorter, faster) follow the driver, pendulums 5-7 swing against it.
3. Raise Q to 60: the resonance peak narrows and only one pendulum responds. Lower it to 4: several swing together.
4. Moon gravity: every natural frequency drops by a factor 2.46, the driver slows down, and the match still holds because both change together.
5. Use the view from above: the bobs trace a snapshot of the lag from one pendulum to the next.