Mode Localization and Sensitivity in Weakly Coupled Resonators

Abstract

Localization of normal modes in weakly coupled resonators is used in the recent MEMS sensors to achieve orders of magnitude improvement in sensitivity. It is known that the sensitivity (S) of the modal amplitude ratio varies with the strength of the normalized coupling (κ) as S ∝ κ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> in the case of two identical coupled oscillators, and as S ∝ κ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> in the case of three coupled oscillators with the middle oscillator of a much higher natural frequency. We have recently shown that an asymmetry measure (α) will also affect the sensitivity according to S ∝ α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . Here, we prove, using an energybased analysis, that in the general case of n ·· 2 uniformly coupled oscillators chained between two terminal oscillators, S ∝ α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> κ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-n</sup> , if the blocked resonance spectra of the embedded oscillators and the end oscillators are well-separated. We further show that coupled asymmetric oscillators will also enhance the operating linear range, besides the increased sensitivity when compared with their symmetric counterparts, in a three coupled oscillators system. Our sensitivity and linear range expressions, not based on a (small) perturbation approach, have a wider range of accuracy. These analytical results are validated through numerical computation in the illustrative cases of three and four cell resonator chains.