Axisymmetric membrane nano-resonators: A comparison of nonlinear reduced-order models

TL;DR

This paper compares nonlinear reduced-order models for axisymmetric membrane nano-resonators, highlighting the accuracy of Galerkin and Harmonic Balance methods in predicting their nonlinear frequency response and jump-down behavior.

Contribution

It develops an axisymmetric PDE model, derives asymptotic solutions, and compares numerical and reduced-order models to improve understanding of nonlinear dynamics in nano-resonators.

Findings

Galerkin and Harmonic Balance methods accurately predict jump-down behavior.

Numerical solutions show non-constant maximum compliance with load, contradicting asymptotic predictions.

Fitting jump-down points for parameter inference may be sensitive to noise.

Abstract

The shift in the backbone of the frequency--response curve and the `jump-down' observed at a critical frequency observed in nano-resonators are caused by their nonlinear mechanical response. The shift and jump-down point are therefore often used to infer the mechanical properties that underlie the nonlinear response, particularly the resonator's stretching modulus. To facilitate this, the resonators's dynamics are often modelled using a Galerkin-type numerical approach or lumped ordinary differential equations like the Duffing equation, that incorporate an appropriate nonlinearity. To understand the source of the problem's nonlinearities, we first develop an axisymmetric but spatially-varying model of a membrane resonator subject to a uniform oscillatory load with linear damping. We then derive asymptotic solutions for the resulting partial differential equations (PDEs) using the Method of Multiple Scales (MS), which allows a systematic reduction to a Duffing-like equation with analytically determined coefficients. We also solve the PDEs numerically via the method of lines. By comparing the numerical solutions with the asymptotic results, we demonstrate that the numerical approach reveals a non-constant maximum compliance with increasing load, which contradicts the predictions of the MS analysis. In contrast, we show that combining a Galerkin decomposition with the Harmonic Balance Method accurately captures the non-constant maximum compliance and reliably predicts jump-down behaviour. We analyze the resulting frequency-response predictions derived from these methods. We also argue that fitting based on the jump-down point may be sensitive to noise and discuss strategies for fitting frequency-response curves from experimental data to theory that are robust to this.