Abstract
In the present study, dynamic pullin instability of electrostaticallyactuated microbeams is investigated through proposing the nonlinear frequency amplitude relationship. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). Influences of vibrational amplitude as well as different parameters on dynamic pullin voltage are investigated. It is demonstrated that two terms in series expansions is sufficient to produce an acceptable solution of the mentioned microstructure. The simulations from numerical methods verify the validity of the analytical procedure.
Iteration Perturbation Method; Frequency – amplitude relationship; Dynamic pullin voltage; Microbeam Vibration; Pullin instability
Application of Iteration Perturbation Method in studying dynamic pullin instability of microbeams
Hamid M. SedighiI, ^{*} * Author email: h.msedighi@scu.ac.ir, hmsedighi@gmail.com ; Farhang Daneshmand^{II, III, IV}; Amin Yaghootian^{I}
^{I}Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran
^{II}Department of Mechanical Engineering, McGill. University, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
^{III}Department of Bioresource Engineering, McGill. University, 21111 Lakeshore Road, SainteAnnedeBellevue, Quebec, Canada H9X 3V9
^{IV}School of Mechanical Engineering, Shiraz University, Shiraz, Iran
ABSTRACT
In the present study, dynamic pullin instability of electrostaticallyactuated microbeams is investigated through proposing the nonlinear frequency amplitude relationship. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). Influences of vibrational amplitude as well as different parameters on dynamic pullin voltage are investigated. It is demonstrated that two terms in series expansions is sufficient to produce an acceptable solution of the mentioned microstructure. The simulations from numerical methods verify the validity of the analytical procedure.
Keywords: Iteration Perturbation Method, Frequency amplitude relationship, Dynamic pullin voltage, Microbeam Vibration, Pullin instability.
1 INTRODUCTION
The application of the microscale devices is continuously growing and the microelectromechanical systems (MEMS) have become the interesting area of research in recent years. The application of actuated MEMS includes accelerometers, micropumps, microresonators and manipulators (Grandinetti et al., 2012). In recent years, several researches have been developed on the nonlinear behavior of MEMS/NEMS devices (Sedighi, 2014; Rahaeifard et al., 2013; Tadi Beni et al., 2012; Abdi et al., 2011; Sedighi et al., 2014; Tadi Beni and Abadyan, 2013; Soroush et al., 2010; Sedighi and Shirazi, 2013; Ansari et al., 2012). In the dynamic analysis of microsystems, electrostatic forces cause the relationship between the input excitation and the output response to be nonlinear which is performed by applying voltage between the microbeam and the substrate. The microbeam deflection increases by increasing the actuation voltage. At a specific voltage namely pullin voltage instability happens and the microstructure drops to the substrate.
Rahaeifard et al. (2013) investigated the dynamic behavior of microcantilevers under suddenly applied DC voltage based on the modified couple stress theory using numerical and analytical approaches. Stability of a functionally graded (FG) microbeam, based on modified couple stress theory (MCST), subjected to nonlinear electrostatic pressure and thermal changes have been studied by Zamanzadeh et al. (2013). Sedighi and Shirazi (2013) presented a new asymptotic procedure to predict the nonlinear vibrational behavior of microbeams predeformed by an electric field. The impact of vibrational amplitude on the dynamic pullin instability and fundamental frequency of actuated microbeams has been investigated by Sedighi (2014) via introducing the second order frequency  amplitude relationship. The nonlinear governing equation of microbeam predeformed by an electric force including the fringing field effect, based on the strain gradient elasticity theory was considered and the predicted results of the strain gradient elasticity theory was compared with the outcomes arise from the classical and modified couple stress theory. Sedighi et al., (2014) investigated the dynamic pullin instability of vibrating microbeams undergoing large deflection under electrosatically actuation based on the modified couple stress theory. The modeling and analysis of an opticallyactuated, bistable MEMS switches have been studied by Kumar and Rhoads (2012). They investigated the influence of various system and excitation parameters, including the applied axial load and optical actuation profile, on the system's transient response. Sabater and Rhoads (2012) analyzed the control mechanisms of selfexcited oscillators, founded upon electromagneticallyactuated microbeams. Their study specifically focused on the characterization of nonlinear behaviors arising in isolated mutuallycoupled oscillators. Ramezani (2012) used Hamilton's principle to derive the nonlinear governing equations of motion and boundary conditions of micro scale Timoshenko beam model based on a general form of strain gradient elasticity theory. He demonstrated that both strain gradient and geometric nonlinearity effects increase the beam natural frequency. Abdi et al. (2011) employed the modified couple stress theory to model the size effect on the pullin instability of electrostatic nanocantilevers in the presence of dispersion (Casimir/van der Waals) forces. Ansari et al. (2012) investigated the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) based on the strain gradient elasticity theory and von Karman geometric nonlinearity. Rhoads et al. (2013) presented the highly nonlinear dynamic behavior of a new class of parametrically excited, electromagnetically actuated microcantilevers. They studied the frequency response behavior and bifurcation analysis of the micro scale system using perturbation method. Chouvion et al. (2012) reviewed several approaches for calculating semiinfinite support loss in microelectromechanical system resonators undergoing inplane vibration. Rajabi and Ramezani (2012) presented a micro scale nonlinear beam model based on strain gradient elasticity. They indicated that in a microbeam having a thickness to length parameter ratio close to unity, the strain gradient effect on increasing the natural frequency is predominant and by increasing the beam thickness, this effect decreases and geometric nonlinearity plays the main role on increasing the natural frequency. The effects of the open crack on the static and dynamic pullin voltages of an electrostatically actuated fixedfixed and cantilever microbeam has investigated by Motallebi et al. (2012). Thermoelastic damping (TED) in a microbeam resonator with a pair of piezoelectric layers bonded on its upper and lower surfaces have been studied by Vahdat et al. (2012). They demonstrated that thickness of the piezoelectric layers and application of DC voltage can affect the TED ratio and the critical thickness value of the resonator. Daneshmand and Amabili (2012) described a mechanics model of an anisotropic microtubule to predict the coupled frequencies of microtubulecytoplasm system including the effect of the surrounding cytoplasm. They developed the displacement representation of the firstorder shear deformation shell theory for orthotropic materials for modeling the microtubule.
Recently, new asymptotic methods have been increasingly developed in order to solve nonlinear differential equations (ShiJun Liao, 2004; Sedighi and Shirazi, 2012; Sedighi et al., 2012a, 2012b). There have been several approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations such as Energy Balance Method (Ghadimi et al., 2012), Variational Iteration Method (He, 2007; Sedighi et al., 2012), and Hamiltonian Approach (HA) (He, 2010; Sedighi and Shirazi, 2013), Amplitude  frequency formulation (He, 2008), MaxMin approach (He, 2008), Homotopy Analysis Method (HAM) (Liao, 2003 and 2004), Parameter Expansion Method (Wang and He, 2008; He, 2002), Homotopy Perturbation Method (HPM) (Yazdi, 2013), Iteration Perturbation Method (He, 2001; Sedighi et al., 2013). The Iteration Perturbation Method (IPM), which is developed by JiHuan He (2001), provides an effective and efficient tool for solving an extensive range of nonlinear equations.
The present article attempts to indicate the impact of vibrational amplitude on dynamic pullin instability of actuated microbeams by introducing the nonlinear frequency  amplitude relationship. In this direction, analytical expressions for vibrational responses of cantilever and clampedclamped micro actuated beams are presented. The results presented in this paper exhibit that the analytical method is very effective and convenient for nonlinear vibration for which the highly nonlinear governing equations exist. The proposed analytical method demonstrates that two terms in series expansions is sufficient to obtain a highly accurate solution of microbeam vibration. Finally, the influences of amplitude and significant parameters on the pullin instability behavior are studied.
2 EQUATION OF MOTION
Consider an actuated microbeam suspended above a rigid substrate and under electrostatically actuated voltage as shown in Fig. 1. The cantilever and clampedclamped micro beams have length , thickness , width , density , moment of inertia and a modulus of elasticity . The air initial gap is and an attractive electrostatic force which originates from voltage causes the microbeam to deform.
Assume that the beam considered here, be the Euler  Bernoulli beam. By incorporating von Karman nonlinearity, large deflections, large rotations and small strains of the narrow microbeam are taken into account. The governing equation of motion of micro actuated beam is expressed as follows (Moghimi Zand and Ahmadian, 2009):
Where represents the axial force. By introducing the following nondimensional variables
the nondimensional nonlinear governing equation of motion can be written as follows:
Assuming , where is the first eigenmode of the microbeam vibration and can be expressed as:
where and are the roots of characteristic equations for the first two eigenmodes of clampedclamped and cantilever micro beams. It should be pointed out that for cantilever microbeam vibrations the stretching term as well as the axial load is equal to zero. Applying the BubnovGalerkin procedure yields the following nondimensional nonlinear governing equation of motion:
where the parameters to for clampedclamped as well as cantilever microbeam have been described in the ^{Appendix A} Appendix A .
3 APPROXIMATION BY THE ITERATION PERTURBATION METHOD
The Iteration Perturbation Method proposed by He (2001) is constructed based on perturbation technique coupling with iteration method. This method is valid not only for weakly nonlinear problems but also for strongly nonlinear differential equations. Consider the nonlinear equation (5), in order to obtain an iteration perturbation solution of the governing equation, an artificial parameter should be introduced as:
Equation (6) can be approximated by:
where is the initial approximate solution. The initial solution can be assumed in the form, where is the unknown angular frequency and should be determined. Substituting into equation (6) yields
Equation (8) can be rewritten in the following form
assuming that
Substituting equations (10) and (11) into (9), and equating the coefficients of the same power of , the following differential equation for can be obtained
If the term exists in the right hand side of equation (12), the secular term will appear in the final solution. Therefore, the coefficient of this term in (12) should be equal to zero, so we have
similarly, the following differential equation for can be obtained
No secular term in the second equation for yields:
thereby, solving equations (11) and (15) with for fundamental frequency gives the following frequency amplitude relationship for actuated microbeam vibrations as:
Solving Eq. (12) for gives the following second order approximation for as:
4 RESULTS AND DISCUSSION
To indicate the validity of proposed solution by IPM, the analytical solutions at the side of corresponding numerical results have been plotted. As can be seen in Fig. 2, the second order approximation of using analytical methods for both clampedclamped and cantilever microbeam show good agreement with numerical results from fourthorder RungeKutta method. The numerical values of system parameters used for asymptotic analysis are described in ^{Appendix B} Appendix B .
Analytical simulation of the dimensionless dynamic pullin voltage of clampedclamped and cantilever microbeam versus nondimensional amplitude are depicted in Figs. 3 to 5. The effect of dimensionless axial force parameter on the dynamic pullin voltage as a function of initial amplitude is illustrated in Fig. 3. It is obvious that the dynamic pullin voltage increases by increasing the axial force parameter. It appears from Fig. 3 that the dynamic pullin voltage decreases when the normalized amplitude increases. Fig. 3 also shows that the effect of normalized amplitude on the dynamic pullin voltage is more considerable than its effect on the axial force parameter. Fig. 4 shows the characteristic curves of dynamic pullin voltage of double clamped microbeam for various values of dimensionless parameter . Fig. 4 indicates that the dynamic pullin voltage increases as the parameter increases. In addition, the impact of dimensionless parameter is more considerable when the normalized amplitude increases from zero to unity.
The influence of nondimensional parameter on the dynamic pullin voltage of cantilever and double clamped actuated microbeams are depicted in Figs. 5 and 6. The comparison between the values of pullin voltages reveals that the dynamic pullin voltage for cantilever microbeam is less than its value for double clamped one. From Figs. 5 and 6, it is concluded that the increase in the parameter causes decrease in the dynamic pullin voltage. Furthermore, it is clear that the effect of dimensionless parameter is more considerable when the normalized amplitude decreases.
5 CONCLUSIONS
A modern powerful analytical approach called Iteration perturbation method was employed to establish the frequency  amplitude relationship of vibrating actuated microbeams. The influence of vibrational amplitude on pullin instability and dynamic pullin voltage was investigated in this research. The accuracy of the obtained analytical solutions is verified by numerical results. It is indicated that two terms in series expansions is sufficient to produce an acceptable accurate solutions. Finally, the significant effects of nondimensional parameters on the dynamic pullin voltage were investigated.
Appendix A
For clampedclapmed micro beam the defined parameters are:
For cantilever micro beam the defined parameters are:
Appendix B
System parameters used for numerical analysis
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Appendix A
Appendix B
Publication Dates

Publication in this collection
13 Mar 2014 
Date of issue
Dec 2014