The two-degree-of-freedom parametric oscillator: A mechanical experimental implementation

¹ Sorbonne Université - 4 place Jussieu, 75005 Paris, France
² Université de Paris, MSC, UMR 7057 CNRS - 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France

Received: 4 March 2020
Accepted: 12 October 2020

Abstract

We report the observation and quantitative measurements of the parametric resonance of a two-degree-of-freedom harmonic oscillator made of two coupled pendula. When varying the length of one of the pendula, we parametrically excite the whole set. The frequency of this excitation can be tuned and its depth ajusted, enabling us to draw a frequency-depth Cartesian diagram in which the oscillation behaviour, namely finiteness or exponential divergence, can be plotted. As in the well-known case of the one-degree-of-freedom oscillator, the stability and instability domains are demarcated by a tongue centred at twice the oscillator's free frequency. But in the two-degree-of-freedom oscillator case, this tongue has a three-tip fine structure, two tips corresponding to the free oscillator's eigenfrequencies and an extra tip corresponding to the average of both eigenfrequencies. This situation is unravelled theoretically using a classical adaptation of a formalism introduced in quantum optics by Glauber.