Investigation of a Bandpass Filter Based on Nonlinear Modal Coupling via 2:1 Internal Resonance of Electrostatically Actuated Clamped-Guided Microbeams

Abstract

Introduction and Purpose

Nonlinear modal coupling via internal resonances and parametric excitation in MEMS resonators has been widely used to improve the functionality and performance of a wide range of potential applications. In this paper, the bandpass filter based on a 2:1 internal resonance between the second and third modes of a clamped-guided microbeam with electrostatic activation is proposed for the first time. Due to internal resonance, the amplitude–frequency response of the second mode showed a rectangle-like bandpass behaviour when the third mode is externally excited around the resonance by sweeping external excitation frequency.

Methods

In this study, numerical time integration and the method of multiple scales are used to confirm the bandpass characteristics in the amplitude–frequency response curve of the second mode of the beam.

Results

The computational results showed that the mid-plane stretching nonlinearity, excitation amplitude, and damping have a significant impact on the bandpass behaviour. We also presented the centre frequency and 3 dB bandwidth of the filter as a function of the mid-plane stretching nonlinearity parameter and a combination of excitation amplitude and damping. Moreover, we found the bandpass frequency bandwidth 261.64−1130 Hz for various axial excitation amplitude and damping combinations using a particular beam dimension and material properties from the literature. Specifically, the bandpass filter exhibited a 3 dB bandwidth of 954.97 Hz with a centre frequency of 143.09 kHz for a set of beam dimensions and material properties.

Conclusions

The internal resonance-based bandpass filter demonstrated here illustrates a novel method for building filters using a single MEMS resonator. Here, we consider electrostatically actuated clamped–guided beams, but the concept can be applied to any MEMS resonator that exhibits modal coupling based on internal resonance. In addition, it is more durable than contemporary electronic filters and is temperature-robust. The work presented here might potentially serve as a guide for investigating more MEMS resonators to create bandpass filters based on internal resonances.

Appendix

Derivation of Governing Equation

We consider the dynamic response of a clamped-guided microbeam to electrostatic force. The guided end of the beam is modelled as a linear axial spring. We assume that the cross sections remain plane during transverse bending. By utilising moment balance and the relationship between shear force and moment, we find the coupled two partial differential equations corresponding transverse $w\left(x,t\right)$ and axial $u\left(x,t\right)$ deflections [43]

$$\rho A\ddot{u} - EAu^{\prime\prime} = EA\left( {\frac{{\left( {w^{\prime}} \right)^{2} }}{2}} \right)^{\prime}$$

(14)

$$\rho A\dddot w + EIw^{{\prime\prime\prime}} = EA\left( {u^{\prime}w^{\prime} + \frac{1}{2}\left( {w^{\prime}} \right)^{3} } \right)^{'} + F.$$

(15)

The axial natural frequency is much higher than the transversal natural frequency. Hence, the inertia term $\ddot{u}$ in Eq. 14 can be ignored and the axial deformation becomes

$$\begin{array}{*{20}r} \hfill {u^{\prime\prime} = - \left( {\frac{{\left( {w^{\prime}} \right)^{2} }}{2}} \right)^{\prime} .} \\ \end{array}$$

(16)

To obtain axial deflection, we integrate Eq. 16 twice with respect to $x$

$$\begin{array}{c}u=-\frac{1}{2}{\int }_{0}^{x}{\left({w}^{\mathrm{^{\prime}}}\right)}^{2}dx+x{C}_{1}\left(t\right)+{C}_{2}\left(t\right),\end{array}$$

(17)

where ${C}_{1}\left(t\right)$ and ${C}_{2}\left(t\right)$ are constants of integration. To obtain these constants, we use clamped and guided (restrained by a linear spring) boundary conditions for the axial motion. Hence, boundary conditions can be expressed as

$$\begin{array}{cc}& u\left(0,t\right)=0 \\ & EA\left({u}^{\mathrm{^{\prime}}}+\frac{1}{2}{\left({w}^{\mathrm{^{\prime}}}\right)}^{2}\right)+{K}_{a}u=0\, \mathrm{at}\, x=L,\end{array}$$

(18)

where $L$ is undeformed beam length and ${K}_{a}$ stiffness of axial spring.

Substituting Eq. 18 into Eq. 17 gives

$$\begin{gathered} C_{1} \left( t \right) = \frac{{K_{a} }}{{2\left( {EA + K_{a} L} \right)}}\mathop \int \limits_{0}^{L} \left( {w^{\prime}} \right)^{2} dx\,{\text{and}} \hfill \\ C_{2} \left( t \right) = 0. \hfill \\ \end{gathered}$$

(19)

Substituting Eq. 19 into Eq. 17 and then the outcome into Eq. 15 yield

$${\rho A\ddot{w} + EIw^{{\prime\prime\prime}} = \frac{{K_{a} EA}}{{2\left( {EA + K_{a} L} \right)}}\left( {\mathop \int \limits_{0}^{L} w^{{\prime2}} dx} \right)w^{\prime\prime} + F.}$$

(20)

Equation 20 is the nonlinear governing equation of motion that is typically used for beam vibration with mid-plane stretching. Here, $F$ is electrostatic force per unit length in a parallel-plate capacitor [43]

$$\begin{array}{c}F=\frac{\epsilon b{\left({V}_{dc}+{V}_{ac}\mathrm{cos}\left(\Omega t\right)\right)}^{2}}{2{\left(g-w\right)}^{2}},\end{array}$$

(21)

where $g$ is the electrode gap between the beam and the substrate, ${V}_{dc}$ is the DC voltage, and ${V}_{ac}$ and $\Omega$ the AC harmonic voltage amplitude and frequency, respectively.