Parametric Excitation of Discrete Nonlinear Systems With Many Degrees of Freedom

TL;DR

This paper develops a theoretical framework for understanding the nonlinear dynamics of large arrays of coupled MEMS and NEMS oscillators under parametric excitation, revealing unique wave-number dependent bifurcations and hysteresis effects.

Contribution

It derives a novel amplitude equation from first principles that includes nonlinear gradient terms, explaining complex bifurcation behavior in coupled nonlinear oscillators.

Findings

Amplitude equation captures slow dynamics near parametric oscillation threshold.

Unique wave-number dependent bifurcation similar to Faraday wave instability.

Predicted strong hysteresis in response to drive amplitude.

Abstract

The response of a large array of coupled nonlinear oscillators to parametric excitation is studied, motivated by the growing interest in the nonlinear dynamics of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). Using a multiscale analysis, we derive an amplitude equation that captures the slow dynamics of the coupled oscillators just above the onset of parametric oscillations. The amplitude equation that we derive here from first principles contains uncommon nonlinear gradient terms which yield a unique wave-number dependent bifurcation similar in character to the behavior known to exist in fluids undergoing the Faraday wave instability. We suggest a number of experiments with nanomechanical or micromechanical resonators to test the predictions of our theory, in particular the strong hysteretic dependence on the drive amplitude.