Stationary Large Amplitude Dynamics of the Finite Chain of Harmonically Coupled Pendulums

TL;DR

This paper analytically describes large-amplitude stationary oscillations in finite chains of coupled pendulums, revealing complex zone structures and resonances that differ from the infinite or long-wavelength approximations, with numerical validation.

Contribution

It provides a novel analytical framework for nonlinear normal modes at arbitrary amplitudes in finite pendulum chains, surpassing previous infinite or small-amplitude models.

Findings

Long-wavelength approximation is inadequate at large amplitudes.

Complex zone structures with multiple resonances are identified.

Numerical simulations agree well with analytical predictions.

Abstract

We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically-coupled pendulums without any restrictions to their amplitudes (excluding a vicinity of $\pi$). Although this model has numerous applications in different fields of physics it was studied earlier in the infinite limit only. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We underline that the long-wavelength approximation, which is described by well- known sine-Gordon equation leads to inadequate zone structure for the amplitude order of $\pi/2$ even if the chain is long enough. The extremely complex zone structure at the large amplitudes corresponds to lot of resonances between nonlinear normal modes even with strongly different wave numbers. Due to complexity of the dispersion relations the more short wavelength modes can possess the smaller frequencies. The numerical simulation of the dynamics of the finite-length chain is in a good agreement with obtained analytical predictions.