Arbitrary distribution and nonlinear modal interaction in coupled nanomechanical resonators

TL;DR

This paper develops a comprehensive analytical model for vibrational modes in nanomechanical resonators with arbitrary cantilever profiles, and investigates nonlinear modal interactions relevant for experimental device optimization.

Contribution

It introduces a general one-dimensional formulation for analyzing vibrational spectra and nonlinear interactions in coupled nanomechanical resonators with arbitrary cantilever configurations.

Findings

Analytical secular equation for frequency spectrum of resonators.

Validation of the model against specific cantilever length profiles.

Analysis of nonlinear modal coupling using perturbation methods.

Abstract

We propose a general one-dimensional {\em continuous} formulation to analyze the vibrational modes of antenna-like nanomechanical resonators consisting of two symmetric arrays of cantilevers affixed to a central nano-beam. The cantilever arrays can have arbitrary density and length profile along the beam. We obtain the secular equation that allows for the determination of their frequency spectrum and illustrate the results on the particular examples of structures with constant or alternating cantilever length profiles. We show that our analytical results capture the vibration spectrum of such resonators and elucidate key relationships that could prove advantageous for experimental device performance. Furthermore, using a perturbative approach to treat the nonlinear and dissipative dynamics of driven structures, we analyze the anharmonic coupling between two specific widely spaced modes of the coupled-element device, with direct application to experiments.