Analysis of response to thermal noise in electrostatic MEMS bifurcation sensors

This paper presents an alternative approach to stochastic analysis of nonlinear dynamic systems. It exploits this approach to analyze the response of electrostatic MEMS bifurcation sensors to a combination of deterministic excitation, mechanical-thermal noise, and electrical-thermal noise. The analytical approach combines the methods of multiple scales and stochastic averaging of the amplitude, to derive the stochastic Itô differential equations describing the modulations of the sensor amplitude and phase difference in the presence of thermal noise and the Fokker–Planck–Kolmogorov (FPK) equation governing the stationary probability density function (PDF) of the stochastic response. Good agreement is found between the predictions of the derived modulation equations and the original equation of motion. The scope of the FPK equation applicability to the noise excitation levels is examined. The impact of the additive noise, arising from mechanical-thermal and electrical-thermal noise, on the sensor response is found to dominate that of the multiplicative noise, arising from the electrical-thermal noise. PDFs of the response are used to investigate the stochastic switching between the co-existing orbits of the bifurcation sensor under the interaction between the excitation frequency and noise intensity. We found that the stochastic switching is activated when the margins of stability of both orbits become comparable to the size of noise-driven motions. Variations in the mean and variance of the amplitude within the hysteretic region can be exploited as sensitive indicators of the stochastic switching. Finally, our results suggest the possibility of implementing a novel highly sensitivity ‘noise-aware’ bifurcation sensor that exploits the quantitative change in the mean amplitude (or RMS) of the sensor states within the frequency range of stochastic switching to detect mass change or gas concentration.