Mode localization and sensitivity in weakly coupled resonators

TL;DR

This paper investigates how mode localization and sensitivity in weakly coupled resonators can be enhanced using energy analysis, especially in systems with higher degrees of freedom and asymmetry, without relying on perturbation methods.

Contribution

It introduces a non-perturbative energy analysis approach to quantify sensitivity and linear range enhancements in complex coupled resonator systems with asymmetry.

Findings

Sensitivity scales as /ppa^{n-1} in higher degrees of freedom systems.

Asymmetry improves both sensitivity and linear range.

Energy analysis provides accurate expressions beyond perturbation approximations.

Abstract

Localization of normal modes is used in recent microelectromechanical systems (MEMS) technologies with orders of magnitude improvements in sensitivity. A pair of eigenvalues veer, or avoid crossing each other, as a single parameter of a vibrating system is varied. While it is well-known that the sensitivity ($s$) of modal amplitude ratio varies with strength of coupling ($\kappa$) as $s \propto\kappa^{-1} $ in the case of two identical coupled oscillators, recently, we showed that asymmetry $\alpha$ will also influence sensitivity according to $s \propto (\alpha \kappa)^{-1}$. Here, we show that further enhancements in sensitivity is possible in higher degrees of freedom ($n$) systems using energy analysis. In the case of $n-2$ uniformly coupled oscillators embedded between two oscillators, we show that $s \propto \alpha^{-1}\kappa^{1-n}$, if the blocked resonance spectra of the embedded oscillators and the end oscillators are well-separated. We also show that asymmetric coupled oscillators also enhance linear range in addition to sensitivity when compared to their symmetric counterparts. We do not use a perturbation approach in our energy analysis; hence the sensitivity and linear range expressions derived have a wider range of accuracy.