Abstract
This paper investigates the dynamic pullin instability of vibrating microbeams undergoing large deflection under electrosatically actuation. The governing equation of motion is derived based on the modified couple stress theory. Homotopy Perturbation Method is employed to produce the high accuracy approximate solution as well as the secondorder frequency amplitude relationship. The nonlinear governing equation of micro beam vibrations predeformed by an electric field includes both even and odd nonlinearities. The influences of basic nondimensional parameters on the pullin instability as well as the natural frequency are studied. It is demonstrated that two terms in series expansions are sufficient to produce high accuracy solution of the microstructure. The accuracy of proposed asymptotic approach is validated via numerical results. The phase portrait of the system exhibits periodic and homoclinic orbits.
Dynamic pullin instability; Modified couple stress theory; Homotopy Perturbation Method; Frequency  amplitude relationship; Homoclinic orbit
Dynamic pullin instability of geometrically nonlinear actuated microbeams based on the modified couple stress theory
Hamid M.Sedighi^{*} * Author email: hmsedighi@gmail.com ; Maziar ChanGizian; Aminreza NoghrehaBadi
Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran
ABSTRACT
This paper investigates the dynamic pullin instability of vibrating microbeams undergoing large deflection under electrosatically actuation. The governing equation of motion is derived based on the modified couple stress theory. Homotopy Perturbation Method is employed to produce the high accuracy approximate solution as well as the secondorder frequency amplitude relationship. The nonlinear governing equation of micro beam vibrations predeformed by an electric field includes both even and odd nonlinearities. The influences of basic nondimensional parameters on the pullin instability as well as the natural frequency are studied. It is demonstrated that two terms in series expansions are sufficient to produce high accuracy solution of the microstructure. The accuracy of proposed asymptotic approach is validated via numerical results. The phase portrait of the system exhibits periodic and homoclinic orbits.
Keywords: Dynamic pullin instability, Modified couple stress theory, Homotopy Perturbation Method, Frequency  amplitude relationship, Homoclinic orbit.
1 INTRODUCTION
With development of modern technology, micro and nanoelectro mechanical systems have shown enormous popularity on engineering and industry. The application of microelectromechanical systems (MEMS) devices especially the electrically actuated MEMS devices which require low actuation voltage levels are continuously growing. Actuated MEMS devices are extensively used in inkjet printers, switches, gyroscopes, chemosensors and so on. Recently several numerical and experimental studies have been conducted on the pullin instability and dynamic behavior of MEMS devices (Ansari et al., 2013; Zhang and Fu, 2012; Rajabi and Ramezani, 2013; Nayfeh et al., 2005; He et al., 2009; Sedighi and Shirazi, 2013; Jia et al., 2011; Younis and Nayfeh, 2003; Mobki et al., 2013; Rahaeifard et al., 2013; Caruntu et al., 2013; Batra et al., 2008; Moghimi Zand and Ahmadian, 2009; Nabian et al., 2013). However, the amplitude dependence of nonlinear frequency and pullin instability has not been developed, till present.
A distributed sizedependent model based on the modified strain gradient elasticity theory (MSGT) was developed by Ansari et al. (2013) in order to investigate the pullin instability of circular microplates subjected to the uniform hydrostatic and nonuniform electrostatic actuations. They employed step by step linearization of the equation of motion and utilized the generalized differential quadrature (GDQ) method to solve the problem numerically. A new model for a viscoelastic beam based on the simplified couple stress theory was developed by Zhang and Fu(2012). They investigated the effect of the beam size on the instantaneous pullin voltage, durable pullin voltage and pullin delay time of the system. Rajabi and Ramezani (2013)studied the dynamic behaviour of the micro scale nonlinear beam model based on strain gradient elasticity. They demonstrated that by increasing the beam thickness, the strain gradient effect on increasing the natural frequency decreases and geometric nonlinearity plays the main role on its trend. Nayfeh et al. (2005) reviewed the development of reducedorder models (node and domain methods) for MEMS devices and discussed the advantages and disadvantages of each implementation.
The improved macromodel of the fixedfixed microbeambased of MEMS capacitive switch was developed by He et al. (2009) to predict the electromechanical behaviors of electrically actuated MEMS capacitive switch. Their model accounted for moderately large deflections, dynamic loads, axial stress induced by the midplane stretching and the residual stress. Sedighi and Shirazi (2013) presented a new asymptotic procedure to predict the nonlinear vibrational behaviour of classical microbeams predeformed by an electric field using Parameter expansion method. Jia et al. (2011) investigated the pullin instability of microswitches under the combined electrostatic and intermo lecular forces. They accounted for the effect of axial residual stress, the force nonlinearity and geometric nonlinearity in their research and solved the governing equation using the differential quadrature method.
An investigation into the response of a resonant microbeam to an electric actuation was presented by Younis and Nayfeh (2003). They employed the nonlinear model to account for the midplane stretching, a DC electrostatic force, and an AC harmonic force and discussed the effect of the design parameters on the dynamic response of the microstructure. Mobki et al. (2013) studied the mechanical and bifurcation behavior of a capacitive microbeam suspended between two conductive stationary plates. They used a modified nonlinear massspring model in order to study the global stability of the fixed points and showed the homoclinic and heteroclinic orbits by plotting phase plane diagrams. Rahaeifard et al. (2013) investigated the dynamic behavior of microcantilevers under suddenly applied DC voltage based on the modified couple stress theory. They utilized multiple scales method for analytical analysis and their numerical approach was based on a hybrid finite element/finite difference method. Caruntu et al. (2013) employed the reduced order model (ROM) method to investigate the nonlinearparametric dynamics of electrostatically actuated microelectromechanical systems (MEMS) cantilever resonators under soft alternating current (AC) voltage of frequency near half natural frequency. They found that the fringe effect affects significantly the behavior of the MEMS resonator especially for the narrower microcantilevers. Batra et al. (2008) studied the vibrations of narrow microbeams predeformed by an electric field with the electromechanical model that estimated the electrostatic fringing field due to both the finite width and the finite thickness of the microbeam. Moghimi Zand et al (2009) studied the dynamic pullin instability of microbeams subjected to step voltages using homotopy analysis method under the consideration of electrostatic force and midplane stretching. Nabian et al. (2013) examined the stability of a functionally graded clampedclamped microplate subjected to hydrostatic and electrostatic pressures and illustrated that the microsystem undergoes a saddle node and homoclinic bifurcations.
The current work intends to compute the secondorder frequencyamplitude relation in a micro clampedclamped beam due to electrostatic voltage. Recently, considerable progresses had been made in asymptotic approximate solutionsof nonlinear differential equations (Sedighi et al., 2013; Sedighi and Shirazi, 2012; Sedighi et al., 2012). There have been several approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations such as Parametrized Perturbation Method (PPM) (Barari et al., 2011), Energy Balance Method (Ghadimi et al., 2012), Variational Iteration Method and Hamiltonian Approach (HA) (Sedighi et al., 2012), Laplace Transform Method (Rafieipour et al., 2012), MaxMin Approach (He, 2008), Homotopy Analysis Method (HAM) (Sedighi et al., 2012), Parameter Expansion Method (Sedighi et al, 2012), Iteration Perturbation Method (IPM) (He, 2001) and Homotopy Perturbation Method (HPM) (He, 1999). It is well known that while the perturbation methods provide the most versatile tools for the nonlinear analysis of engineering problems, they have also some limitations. In order to overcome these drawbacks, combining the standard homotopy and the perturbation method, known as the Homotopy Perturbation Method (HPM), improves the drawbacks of both approaches. He (1999) developed the homotopy perturbation method for solving a variety of problems including the linear and the nonlinear as well as the initial and the boundary value problems by merging two aforementioned techniques. Benefiting from easily computable components and rapid convergence, it has been applied to a wide class of functional equations.
The aim of the present article is to investigate the dynamic pullin instability of geometrically nonlinear actuated microbeams by introducing the secondorder frequency amplitude relation. The nonlinear equation of motion is derived based on the modified couple stress theory using Hamilton's principle. The effect of vibrational amplitude and system parameters on the pullin instability and natural frequency is studied via HPM. In this direction, analytical expressions for vibrational response of actuated microbeams are presented. The proposed analytical method demonstrates that two terms in series expansions is sufficient to obtain a highly accurate solution of microbeam vibration.
2 MATHEMATICAL MODELING
Consider a doubleclamped microbeam suspended above a rigid plate and under electrostatically actuation voltage as shown in Fig. 1. The actuated microbeam has length L, cross section area A_{0}, height h, width b, density r, moment of inertia I and modulus of elasticity E. The air initial gap is d_{gap} and an attractive electrostatic force which originates from voltage V causes the microbeam to deflect. Assume that the microbeam considered here, be the EulerBernoulli beam. The strain of a material point located at a distance z from the middle plane for the microbeam caused by the large rotation and large displacement of the crosssection is represented as (Sedighi et al., 2012):
Taking into account the linear relation between the stress and strain, we have σ = Eε, The strain energy of the microbeam can be calculated from:
Furthermore, the strain energy of an elastic continuum medium using the modified couple stress theory (MCST) can be mentioned as (Rahaeifard et al., 2013):
where µ represents the shear modulus and l denotes the material length scale parameter, using the descriptions:
The total strain energy of the microstructure incorporating the MCST microbeam model can be expressed as:
The virtual work W performed by the axial and electrical forces incorporating the von Karman type nonlinear strain can be written as(Batra et al., 2008):
where N is the axial force and F_{es} is the electrostatic force per unit length of the micro beam which is described by(Moghimi Zand et al, 2009):
where b = 0.65 for doubleclamped microbeam. The kinetic energy is obtained as follows
Applying the Hamilton's principle leads to:
After some mathematical computations, the nonlinear governing equation of motion for actuated MCST microbeam model is expressed as follows:
By introducing the dimensionless parameters as:
the nondimensional nonlinear equation of the microbeam based on the MCST model can be written as follows:
Using Taylor expansion for electrostatic force in equation (12) results in
Assuming W(ξ, τ) = q (τ)ϕ(ξ), where ϕ(ξ) is the first eigen mode of the clampedclamped beam and can be expressed as:
where λ_{ee} = 4.73 is the root of characteristic equation for first eigen mode. Applying the BubnovGalerkin decomposition method, the nondimensional nonlinear governing equation of motion can be written as follows:
where the parameters β_{0}, ..., β_{4} have been described in the Appendix.
3 BASIC IDEA OF HOMOTOPY PERTURBATION METHOD
Consider the following nonlinear differential equation (He, 1999):
subjected to the following boundary condition:
where A is a general differential operator, B a boundary operator, f (r) is a known analytical function, Γ is the boundary of the solution domain (Ω), and ∂uf/∂t denotes differentiation along the outwards normal to Γ. Generally, the operator A may be divided into two parts: a linear part L and a nonlinear one N. Therefore, Eq. (16) may be rewritten as follows:
In cases where the nonlinear Eq. (16) includes no small parameter, one may construct the following homotopy equation
In Eq. (19), p ∈ [o,i] is an embedding parameter and u_{0} is the first approximation that satisfies the boundary condition. One may assume that solution of Eq. (19) may be written as a power series in p, as the following:
The homotopy parameter p is also used to expand the square of the unknown angular frequency as follows:
where ω_{0} is the coefficient of u(r) in Eq. (16) and should be substituted for the right hand side of Eq. (19). Besides, ω_{1} (i = 1, 2, 3, ...) are unknown parameters. The best approximations for the solution and the angular frequency ω are
Now we apply the homotopy perturbation method on Eq. (15). We construct a homotopy in the following form:
According to the HPM, we assume that the solution of Eq. (24) can be expressed in a series of p;
the coefficient of q is expanded into a series in p in a similar way (He, 1999):
After substituting Eq. (26) and Eq. (25) into Eq. (24), and rearranging based on powers of p terms, we have:
Since the solution of Eq. (27a) is q_{0} = Acos(ωτ), the solution of Eq. (27b) should not contain the socalled secular term cos(ωτ). Substitution of this result into the righthand side of equation (27b) yields:
No secular terms in q_{1}(τ) require eliminating contributions proportional to cos(ωt) on the righthand side of equation (28), we have:
Solving equation (28) for q_{1}(τ) gives the following second order approximation for q(τ) as:
equation (26) for two terms approximation of series respect to p and for p = 1 yields:
substitution of this result into the righthand side of equation (27c) for q_{2}(τ) and eliminating the secular terms proportional to cos(ωτ) results in:
solving equation (32) for the fundamental frequency gives the following secondorder frequency amplitude relationship for actuated microbeam vibrations as:
4 RESULTS AND DISCUSSION
In order to verify the effectiveness of the present modeling and the approximate approach, the obtained results in this work are compared with the results of Rahaeifard et al. (2013) for microcantilever beam vibrations. As can be seen in Fig. 2, the second order approximation for q(t) based on the modified couple stress theory exhibits an excellent agreement with the results obtained by Rahaeifard et al. (2013)using a hybrid finite difference method. It should be noted that in order to achieve the best secondorder approximation with zero initial conditions, q_{0}(τ) = 1 cos(ωτ) is substituted for the first trial solution.
The effect of normalized parameters on the natural frequency and pullin instability of actuated clampedclamped microbeams have been illustrated in Figs. 3 to ^{10} . Fig.3 shows the characteristic curves of natural frequency for a microbeam as a function of under some assigned values of normalized amplitude A. It is demonstrated that this nondimensional parameter has a significant effect on predicting the pullin phenomenon. It is obvious from Fig. 3 that by increasing the initial amplitude, the pullin instability occurs at the lower values of actuation parameter . In addition, it is found that, the fundamental frequency of actuated microbeam decreases by increasing the applied voltage until the natural frequency vanishes and the microbeam drops to the rigid plate.
Fig. 4 examines how the nonlinear fundamental frequency of actuated microbeam is affected by the axial force parameter f_{i}. As can be observed, the fundamental frequency increases by decreasing initial condition A . Also, it appears from this Fig. that when the normalized amplitude A decreases, pullin instability occurs at the lower values of parameter f_{i}. Furthermore, pullin phenomenon vanishes by increasing the nondimensional parameter f. The influence of nonlinearity parameter κ on the fundamental frequency is investigated in Fig. 5. According to the illustrated results, it is obvious that the fundamental frequency decreases as the parameter κ increases. Moreover, the pullin voltage shifts downward by increasing this parameter.
Fig. 6 represents the impact of length scale parameter h/l on the pullin instability of microsystems. It is clear from the Fig. that,when the beam thickness is in order of the material length scale, the normalized amplitude has no significant effect on the pullin behavior of the structure. The fundamental frequency decreases by increasing the length scale parameter. In addition, as the initial condition increases, pullin phenomenon occurs at lower values of nondimensional parameter h/l. As mentioned earlier, the normalized amplitude and actuation voltage play substantial roles on the pullin behavior of the system. To this end, the effects of these parameters on the dynamic behavior of microbeams are studied by plotting the time history and phase portrait diagrams. Figs. 7 and 8 investigate the nonlinear behavior of the system as a function of initial condition A .It is concluded that the time period of oscillation increases by increasing the normalized amplitude. In the vicinity of pullin point (here A = 0.65 ), a small increase in the amplitude, changes the dynamic behavior of microsystem. In this situation, when the initial amplitude increases, the system loses its stability and drops to the substrate beyond the pullin point. According to Figs. 9 and ^{10} , at less values of actuation voltage, the system exhibits periodic motion around the stable center point in the phase plane. Before pullin point, when the actuation parameter increases, time period of vibration increases. As the actuation parameter approaches to the pullin voltage (here = 19.35 ), the motion trajectories in the phase plane approach to the unstable saddle node. There exists homo clinicorbit which starts from the unstable branch and goes back to the saddle node at the stable one. By increasing the applied voltage and above the pullin voltage, the microbeam becomes dynamically unstable and collapse onto the rigid plate.
4 CONCLUDING REMARKS
In this research, the Homotopy Perturbation Method was employed to solve governing equation of geometrically nonlinear actuated microbeams based on the modified couple stress theory. An excellent analytical solution using asymptotic approach was obtained. The integrity of the obtained analytical solutions is verified in comparison with the results in the literature. The presented results showed thatas the normalized amplitude increases, the pullin phenomenon occurs at lower values of actuation voltage. In addition, the pullin stability disappears by increasing the axial force parameter. The phase plane portrait of the system illustrated the periodic orbits around the stable center point and homoclinic orbit near the unstable saddle node.
Received in 05 Mar 2013
In revised form 06 Aug 2013
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Appendix


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Publication Dates

Publication in this collection
03 Feb 2014 
Date of issue
Oct 2014
History

Accepted
06 Aug 2013 
Received
05 Mar 2013