Pendulum Chain Isolation

What is modelled

N equal masses hang in series. Each link has the restoring stiffness of a pendulum, k = m g / L, and viscous damping. The ground moves sinusoidally; the payload is the last mass. The frequency response is found from a complex tridiagonal system, and the time response from a fourth-order Runge-Kutta integration of the same chain.

Equations

• Stage resonance f0 = (1/2π) √(g/L).
• Single stage T = √((1 + (2ζr)²)/((1 - r²)² + (2ζr)²)), r = f/f₀.
• Chain: (-ω² + K_i) xi - K xi-1 - K xi+1 = 0 with K = k + i c ω.
• Far above resonance the amplitude falls as f-2N, i.e. 12 dB per octave per stage.

Checks run in the page

• One stage against the closed form.
• Time-domain steady amplitude against the transfer function.
• The 2^(2N) roll-off ratio.
• The pendulum period against a nonlinear integration.

Honest simplifications

Horizontal motion only, small angles for the chain, equal masses, viscous damping and no cross-coupling, tilt or vertical stage. Real platforms add vertical springs and active control; this is a teaching model, not an instrument design.

• Press Single pendulum then Triple stage: the attenuation at 2 Hz grows by orders of magnitude.
• Drive below resonance (Microseism 0.2 Hz): isolation fails and the payload moves with the ground.
• Lower the damping and watch the resonance peak in the left graph.