Abstract

The dynamic behavior of a microelectromechanical system (MEMS) parallel and electrically coupled double-layers (microbeams) based resonator is investigated. Two numerical methods were used to solve the dynamical problem: the reduced-order modeling (ROM) and the perturbation method. The ROM was derived using the so-called Galerkin expansion with considering the linear undamped mode shapes of straight beam as the basis functions. The perturbation method was generated using the method of multiple scales by direct attack of the equations of motion. Dynamic analyses, assuming the above two numerical methods were performed, and a comparison of the results showed good agreement. Finally, a parametric study was performed using the perturbation on different parameters and the results revealed different interesting features, which hopefully can be useful for some MEMS based applications.

1. Introduction

Microelectromechanical systems (MEMS) were mostly developed during the industrial revolution in the late of 20th century [1]. From that moment, scientists have continuously investigated the potential of these devices to apply them for different applications. Nowadays, their existence became indispensable in many fields [2, 3] and the continuous development hopefully will lead to more importance in future.

When dealing with MEMS based devices, interesting dynamical behaviors can be arising due to the nonlinearity of those devices [4–6]. Among them is their hysteresis (softening and hardening) behavior during the resonance phenomenon. The hardening effect shifts the linear resonance profile to the right and hence produces higher frequencies for the higher amplitude values [7]. A main source of this effect for the clamped-clamped microbeam is the mid-plane stretching. On the other hand, the opposite can be said for the softening effect, which depends on the quadratic nonlinearity effect that may arise from the nonlinear electrical forces [8].

Solving nonlinear dynamical behaviors for MEMS devices is fundamental, since it helps in accurately characterizing and designing them to obtain the desired features quickly and effectively. However, the resolution could be sometime cumbersome and many researchers have struggled to find an effective way to tackle this problem [9–12]. The key techniques in solving such nonlinear problems involve reducing the order of the partial differential governing equation, which sometime is difficult to be solved, into ordinary differential equations, which are much easier to deal with. Different approaches are suggested to solve equation; the reduced-order modeling (ROM) and the perturbation analysis are among those approaches.

In recent years, many researchers extensively used reduced-order modeling (ROM) techniques to obtain the structural behavior of MEMS microstructures [13, 14]. It became nowadays among the most common numerical methods used in the MEMS community. In 2003, Younis et al. [15] presented a new ROM (macromodel) to solve for the nonlinear partial differential equation. They found that their method is more attractive than the finite element method in terms of accuracy and cost. Nayfeh et al. [16] reviewed the work of ROM in MEMS devices. They classified the ROM into two main categories: node and domain methods.

Perturbation method is also another approach used in the investigation. Turner and Andrews [17] used the perturbation method to obtain an approximation for the nonlinear resonance frequency of a microbeam. They modeled the problem by using a spring mass model, and they included a cubic restoring force to represent the mid-plane stretching. Younis and Nayfeh [6] used the method of multiple scales, as a perturbation method, in a model that accounted for the electrical loads (both DC and AC) and for the mid-plane stretching for a clamped-clamped single-microbeam resonator.

In a previous study [18], we examined the static behavior of a double-layers based clamped-clamped microbeams MEMS based actuator configuration. This work considers the nonlinear dynamical behaviors for this MEMS structure by using the ROM and the perturbation method. This is due to the different features this device has as it is compared with the single microbeam. For example, the double-microbeam configuration proved its capability in reducing the actuation and the pull-in voltage, and nowadays there are many available investigations that used this type of configuration [19–23].

We will start our present numerical investigation by discussing first the problem formulation. Then, the reduced-order model for double-layers MEMS based resonator will be derived and followed by the analysis for the perturbation method. Afterward, a comparison will be carried out for each method to validate both approaches. Finally, a parametric study will be done to investigate some dynamical features for the microactuator.

2. Problem Formulation

The proposed model is shown in Figure 1. The suggested MEMS device is mainly made of two parallel electrically actuated microbeams. Both microbeams are clamped-clamped and actuated by both a static DC voltage (of amplitude ) and a harmonic AC bias (of amplitude ).

Figure 1 

3D schematic of a MEMS double-layers MEMS based resonator configuration.

The governing equations of motion for upper and lower microbeams of the MEMS multilayers based resonator are given, respectively, as follows:and the associated clamped-clamped boundary conditions areThe functions and are, respectively, the in-plane deflections of the upper and lower microbeam, respectively. For convenience, the equation of the system will be normalized while considering the following nondimensional parameters:where is a time scale parameter and is chosen to be .

Therefore, substituting (4) into (1)–(3) will givewhere the nondimensional parameters defined in (5)-(6) are given as follows:

3. Reduced-Order Modeling (ROM)

Reduced-order modeling (ROM) for the above described nonlinear problem can be obtained by discretizing (5) and (6) using the Galerkin expansion. Therefore, the deflection for the lower and upper microbeams will have the following forms, respectively:where are the linear undamped mode shapes of a clamped-clamped microbeam, which are orthogonal. After that, (5) will be multiplied by and (6) will be multiplied by since this will help in reducing the computational cost. Next, the outcome will be multiplied by and then integrated from to to give the following ROM equations:

4. Perturbation Analysis

Numerical solutions of the ROM dynamical equations provide good results for low nonlinearity values. However, when tuning the system’s geometrical parameters as well as the forcing amplitudes, one can increase the source of the nonlinearity, leading to nonlinear frequency responses which cannot be fully captured by ROM. As a result, perturbation theory was investigated to find the resonance of the double-microbeams especially for any high nonlinearity factors. The method of multiple scales was used by direct attack of the equations of motion [6, 24]. So, the variable for the time scales and its derivatives were defined as follows: Next, the damping coefficient and the forcing amplitude were scaled so that their nonlinearity effect will be balanced in the modulation equations [6, 24] as follows:where is a bookkeeping parameter.

We seek a solution for the lower and upper electrodes in the following form, respectively:where and are the static components of the microbeams’ deflection and and are their dynamic components for the lower and upper microbeams, respectively.

Substituting (14) and (15) into (5) and (6) givesThe electrical forces and are expanded around () and (), respectively, using a Taylor-series expansion. Further, the expression of will be approximated as . Using the aforementioned expressions and canceling the static equations and neglecting all small terms while equating the coefficients of like powers of , the following equations are obtained:  For the upper microbeam, that is, (16),(i)order :(ii)order :(iii)order : where is a linear differential operator defined for any two functions ( and ) by and where For the upper microbeam, that is, (17),(i)order : (ii)order :(iii)order : where is a linear differential operator defined for any two functions ( and ) byAssuming no internal resonances situations, the solutions of (18) and (23) are assumed to consist of only the directly excited modes, , since the indirectly excited modes will die out in the presence of damping. As a result, the solutions of the dynamic components and of the lower and upper microbeams, respectively, are where is a complex-valued function that is determined by imposing the solvability condition at the third order, the overbar denotes the complex conjugate, and and are the natural frequency and corresponding eigenfunction of the directly excited modes, respectively. Here, the complex-valued function is considered to be the same for both microbeams, while the mode shapes will compensate for the resulting difference. This will help in obtaining the eigenvalue problem. Substituting (27) and (28) into (18) and (23) we obtain the following nonlinear coupled eigenvalue problem (EVP):Next, the solutions and will be substituted in the second-order equations (), which will givewhereAssume the particular solution of and aswhere the functions are the solutions of the boundary value problems:and where is the Kronecker delta operator and the two linear differential operators and are defined asIntroduce the detuning parameter to describe the nearness of the excitation frequency to the fundamental natural frequency of the excited mode asThe solutions of the first and second order are then substituted in (20) and (25), leading towhere NST stands for the nonsecular terms and the functions are defined as Hence, we need to eliminate the secular terms in (36) by seeking a particular solution free of secular terms in the form:Then, (38) are substituted into (36). Next, the coefficients of in the resultant equations will be equated. The outcome of two equations will be then multiplied by two adjoints functions and , respectively, and then integrated by parts to transfer the derivatives from to the adjoints, leading toBy adding (39) and rearranging the terms, we can obtain the two adjoints equations governing both functions and as follows:which are the same as the first-order equations (29) so we call them self-adjoint equations. The solvability condition can be then obtained asUsing Euler’s formula for and , then separating the real and imaginary parts of the solvability condition will give the following two coupled equations:whereBy squaring both sides of (42) and (43) and then adding the results, while remembering that (), we getSince the quality factor is related to the damping coefficient () byit can be replaced in the solvability equation (45) to examine the effect of the quality factor on the system dynamics.