Analysis of response to thermal noise in electrostatic MEMS bifurcation sensors

Abstract

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.

Appendices

Appendix

A: Stochastic seculars terms in the third order

White noise excites a wide spectrum of frequencies, while the secular terms only result from specific frequency components. We can write additive noise $\xi_{1} (T_{0} ){\kern 1pt}$ as the summation of secular and nonsecular components as follows:

$$ \xi_{1} (T_{0} ){\kern 1pt} { = }A_{{\xi_{1} }} \exp (i\omega_{0} T_{0} ) + \overline{A}_{{\xi_{1} }} \exp ( - i\omega_{0} T_{0} ){ + }NST $$

where NST represents terms that do not produce secular terms.

To eliminate the secular terms, the coefficients $A_{{\xi_{1} }}$ and $\overline{A}_{{\xi_{1} }}$ should be set equal to zero. Since the conjugation part is represented by “cc”, only $A_{{\xi_{1} }} { = }\xi_{1} (T_{0} )\exp ( - i\omega_{0} T_{0} )$ appears in Eq. (16) of the manuscript.

For multiplicative noise $u_{0} \eta_{e} (T_{0} )$, since $u_{0} = Aexp(i\omega_{0} T_{0} ) + \overline{A}exp( - i\omega_{0} T_{0} )$, per Eq. (14), there are two cases in which $u_{0} \eta_{e} (T_{0} )$ generates secular terms:

• 1. When $\eta_{e} (T_{0} )$ is in the vicinity of $\omega_{0} { = }0$ (DC component). In this case, only $u_{0}$ generates secular term. To eliminate it, the coefficient $A\eta_{e} (T_{0} )$ should be set to zero.

• 2. When $\eta_{e} (T_{0} )$ is in the vicinity of $2\omega_{0}$ (second harmonic). In this case, $\eta_{e} (T_{0} )$ can be expanded as $\eta_{e} (T_{0} ){ = }A_{{\eta_{e} }} \exp (i2\omega_{0} T_{0} ) + \overline{A}_{{\eta_{e} }} \exp ( - i2\omega_{0} T_{0} ){ + }NST$. Here, $u_{0} \eta_{e} (T_{0} )$ will generate secular terms by

  $$ u_{0} \eta_{e} (T_{0} ){ = }\overline{A}A_{{\eta_{e} }} exp(i\omega_{0} T_{0} ){ + }A\overline{A}_{{\eta_{e} }} exp( - i\omega_{0} T_{0} ) + NST $$

The total coefficient in the resulting secular term of $\exp (i\omega_{0} T_{0} )$ is $\overline{A}\eta_{e} (T_{0} )e^{{ - 2i\omega_{0} T_{0} }}$.The secular term about $\exp ( - i\omega_{0} T_{0} )$ is represented by $cc$.

To sum up, eliminate the seculars terms in the third order yields

$$ \begin{gathered} 2{\text{i}}\omega_{0} (A^{\prime} + \frac{\mu }{2}A) + (3\alpha_{3} - \frac{{10\alpha_{2}^{2} }}{{3\omega_{0}^{2} }})A^{2} \overline{A} - \frac{1}{2}\lambda \exp (i\sigma T_{2} ) - \xi_{1} (T_{0} )e^{{ - i\omega_{0} T_{0} }} \hfill \\ - A\delta \eta_{e} (T_{0} ) - \overline{A}\delta \eta_{e} (T_{0} )e^{{ - 2i\omega_{0} T_{0} }} = 0 \hfill \\ \end{gathered} $$

B: Long-time integration (LTI)

The equations of motion were integrated (utilizing fourth-order Runge–Kutta scheme) for 1000 excitation periods $T = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \Omega }} \right. \kern-\nulldelimiterspace} \Omega }$, and the maximum displacement over the last $200T$ was evaluated as the amplitude.

C: The MCS procedures

In Fig. 5, each simulation consisted of integrating the approximate equation of motion, Eq. (7), for $20000T$ and collecting the amplitude of the flow over the last $4000T$ and repeating this procedure 100 times. The PDF was then obtained by partitioning the target region of the amplitude space $a \in [0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\mu {\text{m}}]$ into 200 sample intervals and counting the number of times the flow visits each sampling interval.

In Fig. 6a, for each initial condition, the equation of motion, Eq. (7), was integrated for $20000T$ and the amplitude of the flow over the last $4000T$ was collected. The PDF was obtained by counting the number of times the flow for all six sets of initial conditions visited 200 sampling intervals uniformly distributed over the amplitude space $a \in [0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\mu {\text{m}}]$.