Mechanical filtering in forced-oscillation of two coupled pendulums

TL;DR

This paper analyzes the forced oscillation behavior of two coupled damped pendulums, revealing multiple resonance frequencies, complex dynamics, and damping effects, including a unique frequency where the driven pendulum's amplitude vanishes.

Contribution

It provides an analytical solution for the steady-state amplitudes and phases of coupled pendulums under forcing, highlighting the existence of four resonance frequencies and damping's critical role.

Findings

Four resonance frequencies near mode frequencies are identified.

Damping significantly alters the amplitude response and can suppress the driven pendulum's motion.

A specific driving frequency causes the driven pendulum's amplitude to become zero.

Abstract

Forced oscillation of a system composed of two pendulums coupled by a spring in the presence of damping is investigated. In the steady state and within the small angle approximation we solve the system equations of motion and obtain the amplitudes and phases of in terms of the frequency of the sinusoidal driving force. The resonance frequencies are obtained and the amplitude ratio is discussed in details. Contrary to a single oscillator, in this two-degree of freedom system four resonant frequencies, which are close to mode frequencies, appear. Within the pass-band interval the system is shown to exhibit a rich and complicated behaviour. It is shown that damping crucially affects the system properties. Under certain circumstances, the amplitude of the oscillator which is directly connected to the driving force becomes smaller than the one far from it. Particularly we show the existence of a driving frequency at which the connected oscillator's amplitude goes zero.