Abstract

This paper implements the higher order Hamiltonian method to analyze an electrostatically actuated nonlinear micro beam-based micro electro mechanical oscillator. First, second and third approximate solutions are obtained, and the frequency responses of the system are compared with energy balance method solution and previously solved Variational Approach (VA) and exact solution. After driving the equation of motion based on the Euler-Bernoulli beam theory, Galerkin method has been used to simplify the nonlinear equation of motion. Higher order Hamiltonian approach has been used to solve the problem and introduce a design strategy. Phase plane diagram of electrostatically actuated micro beam has plotted to show the stability of presented nonlinear system and natural frequencies are calculated to use for resonator design. According to the numerical results, the second approximate is more acceptable and results show that one could obtain a predesign strategy by prediction of effects of mechanical properties and electrical coefficients on the stability and free vibration of common electrostatically actuated micro beam.

Keywords:
Higher order Hamiltonian; MEMS; Electrostatically actuated micro beam; Free Vibrations; Euler-Bernoulli theory; Galerkin method

1 INTRODUCTION

Micro Electro Mechanical Systems (MEMS) are used in various engineering fields such as aerospace, optical and biomedical engineering and are enormously used in applications such as micro-switches, transistors, accelerometers, pressure sensors, micro-mirrors, micro-pumps, micro-grippers and bio-MEMS. (Davis, Green et al. 1998Davis, J. J., M. L. H. Green, H. Allen O. Hill, Y. C. Leung, P. J. Sadler, J. Sloan, A. V. Xavier and S. Chi Tsang (1998). "The immobilisation of proteins in carbon nanotubes." Inorganica Chimica Acta 272(1-2): 261-266., Wang and Musameh 2003Wang, J. and M. Musameh (2003). "Carbon nanotube/teflon composite electrochemical sensors and biosensors." Analytical chemistry 75(9): 2075-2079., Atashbar, Bejcek et al. 2004Atashbar, M. Z., B. Bejcek, S. Singamaneni and S. Santucci (2004). Carbon nanotube based biosensors. Sensors, 2004. Proceedings of IEEE, IEEE., Lin, Taylor et al. 2004Lin, Y., S. Taylor, H. Li, K. A. S. Fernando, L. Qu, W. Wang, L. Gu, B. Zhou and Y.-P. Sun (2004). "Advances toward bioapplications of carbon nanotubes." Journal of Materials Chemistry 14(4): 527-541.,Gu, Elkin et al. 2005Gu, L., T. Elkin, X. Jiang, H. Li, Y. Lin, L. Qu, T.-R. J. Tzeng, R. Joseph and Y.-P. Sun (2005). "Single-walled carbon nanotubes displaying multivalent ligands for capturing pathogens." Chemical Communications (7): 874-876., Osiander, Darrin et al. 2005Osiander, R., M. A. G. Darrin and J. L. Champion (2005). MEMS and microstructures in aerospace applications, CRC press., Balasubramanian and Burghard 2006Balasubramanian, K. and M. Burghard (2006). "Biosensors based on carbon nanotubes." Analytical and bioanalytical chemistry 385(3): 452-468., Allen, Kichambare et al. 2007Allen, B. L., P. D. Kichambare and A. Star (2007). "Carbon nanotube field‐effect‐transistor‐based biosensors." Advanced Materials 19(11): 1439-1451., Yogeswaran and Chen 2008Yogeswaran, U. and S.-M. Chen (2008). "A Review on the Electrochemical Sensors and Biosensors Composed of Nanowires as Sensing Material." Sensors 8(1): 290., Yogeswaran, Thiagarajan et al. 2008Yogeswaran, U., S. Thiagarajan and S.-M. Chen (2008). "Recent Updates of DNA Incorporated in Carbon Nanotubes and Nanoparticles for Electrochemical Sensors and Biosensors." Sensors 8(11): 7191., Jain and Goodson 2011Jain, A. and K. E. Goodson (2011). "Thermal microdevices for biological and biomedical applications." Journal of Thermal Biology 36(4): 209-218.). MEMS are merged devices that join electrical and mechanical components. The study of dynamic and static behavior of atomic force microscope (AFM) cantilevers and vibration control of AFM cantilevers are one of the challenges that coupled electrical and mechanical components (Korayem, Sadeghzadeh et al. 2011Korayem, M., S. Sadeghzadeh and A. Homayooni (2011). "Semi-analytical motion analysis of nano-steering devices, segmented piezotube scanners." International Journal of Mechanical Sciences 53(7): 536-548., Korayem, Sadeghzadeh et al. 2012Korayem, M., S. Sadeghzadeh and A. Homayooni (2012). "Coupled dynamics of piezo-tube and microcantilever in scanning probe devices and sensitive samples imaging." Micro & Nano Letters, IET 7(9): 986-990., Korayem, Homayooni et al. 2013Korayem, M., A. Homayooni and S. Sadeghzadeh (2013). "Semi-analytic actuating and sensing in regular and irregular MEMs, single and assembled micro cantilevers." Applied Mathematical Modelling 37(7): 4717-4732., Korayem, Karimi et al. 2014Korayem, M., A. Karimi and S. Sadeghzadeh (2014). "GDQEM Analysis for Free Vibration of V-shaped Atomic Force Microscope Cantilevers." International Journal of Nanoscience and Nanotechnology 10(4): 205-214.). Ghalambaz et al. (Ghalambaz, Ghalambaz et al. 2015Ghalambaz, M., M. Ghalambaz and M. Edalatifar (2015). "Nonlinear oscillation of nanoelectro-mechanical resonators using energy balance method: considering the size effect and the van der Waals force." Applied Nanoscience: 1-9.) studied the effects of the van der Waals attractions, Casimir force, the small size, the fringing field, the mid-plane stretching, and the axial load on the oscillation frequency of resonators.

Intrinsic intricacy of nonlinear vibration problem of MEMS forces numerical solutions instead of exact analytical responses. Shooting method (Abdel-Rahman, Younis et al. 2002Abdel-Rahman, E. M., M. I. Younis and A. H. Nayfeh (2002). "Characterization of the mechanical behavior of an electrically actuated microbeam." Journal of Micromechanics and Microengineering 12(6): 759.), δ- perturbation method (He 2003He, J.-H. (2003). "Homotopy perturbation method: a new nonlinear analytical technique." Applied Mathematics and computation 135(1): 73-79.), differential quadrature method (Kuang and Chen 2004Kuang, J.-H. and C.-J. Chen (2004). "Dynamic characteristics of shaped micro-actuators solved using the differential quadrature method." Journal of Micromechanics and Microengineering 14(4): 647.), Lindstedt-Poincaré method (He 2002He, J.-H. (2002). "Modified Lindstedt-Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant." International Journal of Non-Linear Mechanics 37(2): 309-314.), integral equation method (Pouya 1997Pouya, M. S. V. (1997). NUMERICAL ANALYSIS OF SECOND ORDER WAVE DIFFRACTION ON TWO-DIMENSIONAL BODIES. The Seventh Asian Congress of Fluid Mechanics, Allied Publishers.), homotopy analysis method (HAM) (Beléndez, Beléndez et al. 2008Beléndez, A., T. Beléndez, A. Márquez and C. Neipp (2008). "Application of He's homotopy perturbation method to conservative truly nonlinear oscillators." Chaos, Solitons & Fractals 37(3): 770-780.), variational approach (VA) (He 2007He, J.-H. (2007). "Variational approach for nonlinear oscillators." Chaos, Solitons & Fractals 34(5): 1430-1439.), Max-Min approach (He 2008He, J.-H. (2008). "Max-min approach to nonlinear oscillators." International Journal of Nonlinear Sciences and Numerical Simulation 9(2): 207-210., Zeng 2009Zeng, D.-Q. (2009). "Nonlinear oscillator with discontinuity by the max-min approach." Chaos, Solitons & Fractals 42(5): 2885-2889., Zeng and Lee 2009Zeng, D. and Y. Lee (2009). "Analysis of strongly nonlinear oscillator using the max-min approach." International Journal of Nonlinear Sciences and Numerical Simulation 10(10): 1361-1368.) and Energy Balance Method (He 2002He, J.-H. (2002). "Preliminary report on the energy balance for nonlinear oscillations." Mechanics Research Communications 29(2): 107-111.) are some of the numerical and approximate analytical approaches could be addressed. Ganji, Azimi et al. (Ganji, Azimi et al. 2012Ganji, D. D., M. Azimi and M. Mostofi (2012). "Energy balance method and amplitude frequency formulation based simulation of strongly non-linear oscillators." Indian journal of Pure and Applied Physics 50(9): 670-675.) applied the Energy Balance Method (EBM) and Amplitude Frequency Formulation (AFF) to govern the approximate analytical solution for motion of two mechanical oscillators. They showed that in comparison with the fourth order Runge-Kutta method, their solution ismore comfortable and useful for solving strong non-linear oscillators. Ganji and Azimi. (Ganji and Azimi 2012Ganji, D. D. and M. Azimi (2012). "Application of max min approach and amplitude frequency formulation to nonlinear oscillation systems." UPB Scientific Bulletin 74(3): 131-140.) used the Max-Min Approach (MMA) and Amplitude Frequency Formulation (AFF) to derive the approximate analytical solution for motion of nonlinear free vibration of conservative, single degree of freedom systems, and they concluded that both methods have the same results. The results showed these methods are very convenient for solving nonlinear equations and also can be utilized for a wide range of time and boundary conditions for nonlinear oscillators. Yildirim, Saadatnia et al (Yildirim, Saadatnia et al. 2011Yildirim, A., Z. Saadatnia, H. Askari, Y. Khan and M. KalamiYazdi (2011). "Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach." Applied Mathematics Letters 24(12): 2042-2051.) applied the Hamiltonian approach to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quantic nonlinear oscillator. Obtained results were completely inagreement with the approximate frequencies and the exact solution. H. Askari et al. (Askari 2013Askari, H. (2013). "Application of higher order Hamiltonian approach to nonlinear vibrating systems." Journal of Theoretical and Applied Mechanics.) utilized the higher order Hamiltonian approach to elicit approximate solutions for the model of buckling of a column and mass-spring system. Y. Khan and M. Akbarzade. (Khan and Akbarzade 2012Khan, Y. and M. Akbarzade (2012). "Dynamic analysis of nonlinear oscillator equation arising in double-sided driven clamped microbeam-based electromechanical resonator." Zeitschrift für Naturforschung A 67(8-9): 435-440.) used variational approach, Hamiltonian approach, and amplitude-frequency formulation to analysis of nonlinear oscillator equation arising in double-sided clamped microbeam-based electromechanical resonator. Qian, Ren et al. (Qian, Ren et al. 2012Qian, Y., D. Ren, S. Lai and S. Chen (2012). "Analytical approximations to nonlinear vibration of an electrostatically actuated microbeam." Communications in Nonlinear Science and Numerical Simulation 17(4): 1947-1955.) utilized the homotopy analysis method (HAM) to derive analytical approximate solutions for nonlinear vibration of an electrostatically actuated microbeam and for verifying the accuracy of this approach, they compared their method with other analytical and exact solutions.

Fu et al. (Fu, Zhang et al. 2011Fu, Y., J. Zhang and L. Wan (2011). "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)." Current applied physics 11(3): 482-485.) applied the Energy Balance Method (EBM) to study a nonlinear oscillation problem in the micro beam model. They governed equation of free vibration of a micro beam, based on the Euler- Bernoulli hypothesis and also compared the results with fourth-order Runge-Kutta method.

H. Rafieipour et al. (Rafieipour, Lotfavar et al. 2013Rafieipour, H., A. Lotfavar and A. Masroori (2013). "ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR VIBRATION OF MICROELECTROMECHANICAL SYSTEM USING HE'S FREQUENCY AMPLITUDE FORMULATION." IJST 37(M2): 83-90.) used the He's frequency amplitude method and presented an analytical closed form solution. Obtained results were in a good agreement with numerical methods.

Bayat et al. (Bayat, Bayat et al. 2014Bayat, M., M. Bayat and I. Pakar (2014). "Nonlinear vibration of an electrostatically actuated microbeam." Latin American Journal of Solids and Structures 11(3): 534-544.) investigated He's Variational Approach (VA) to solve nonlinear vibration of an electro statically actuated clamped- clamped micro beam that was equivalent to the first order of higher Hamiltonian method (Yildirim, Askari et al. 2012Yildirim, A., H. Askari, M. K. Yazdi and Y. Khan (2012). "A relationship between three analytical approaches to nonlinear problems." Applied Mathematics Letters 25(11): 1729-1733.). They demonstrated that VA can be a good candidate for precise periodic solution of nonlinear systems. Final results of mentioned works are listed in table 1.

This research investigated high order approximate solutions by using higher order Hamiltonian method (He 2010He, J.-H. (2010). "Hamiltonian approach to nonlinear oscillators." Physics Letters A 374(23): 2312-2314.) for solving a non-linear dynamic problem, in order to have a highly accurate numerical approximation. Contrarily to some recent researches such as (Bayat, Bayat et al. 2014Bayat, M., M. Bayat and I. Pakar (2014). "Nonlinear vibration of an electrostatically actuated microbeam." Latin American Journal of Solids and Structures 11(3): 534-544.), we showed that the second order is extremely close to the EBM solution and exact solution. The methodology of the higher order Hamiltonian for solving an ordinary differential equation with strong power nonlinearity is presented. Numerical comparisons and results were carried out to confirm the rightness and accuracy of the applied method.

To use higher order Hamiltonian approach, a clamped-clamped micro beam is modeled that placed between two completely fixed electrodes. Deriving the dimensionless equation of motion and separation with assumed mode method, first and higher order approximation of Hamiltonian of system have proposed and then, natural frequency calculated for each case. Finally, we show the effects of various parameters on the frequency of electrostatically actuated micro beam, concluding Hamiltonian approach is completely efficient and agreeable.

2 MATHEMATICAL MODEL

Figure 1 and 2 depict the clamped-clamped micro beam with lengthl width h, constant thickness 'b', initial gap g₀ and electrostatic applied voltage V. The micro beam is doubly clamped and placed between two completely fixed electrodes. Applied voltage is due to the electric field that could be divided into two parts; a DC polarization and an AC electric field.

Applying an AC electric field or a periodic mechanical load results in dynamic deflection and vibration of the micro beam (Younis and Nayfeh 2003Younis, M. and A. Nayfeh (2003). "A study of the nonlinear response of a resonant microbeam to an electric actuation." Nonlinear Dynamics 31(1): 91-117.). For more design options and facilities, computational studies are essential beside experiments. On the other hand, there are not exact (analytical) closed form solutions for all boundary conditions of mechanical systems. As a good alternative, by applying the Galerkin Method (GM) and utilizing the classical beam theory, the free vibration problem of MEMS could be solved.

The nonlinear partial differential equation of the transverse motion regarding the effect of mid-plane deformation could be expressed as (Rao 2007Rao, S. S. (2007). Vibration of continuous systems, John Wiley & Sons.):

Where w(x, t) is the transverse deflection,E is the Young's modulus, v is the Poisson's ratio and is the effective modulus of the micro beam. The quantity of E changes with different thicknesses of the micro beam as follows (Rafieipour, Lotfavar et al. 2013Rafieipour, H., A. Lotfavar and A. Masroori (2013). "ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR VIBRATION OF MICROELECTROMECHANICAL SYSTEM USING HE'S FREQUENCY AMPLITUDE FORMULATION." IJST 37(M2): 83-90.):

symbolizes the tensile or compressive axial load and is related to the discrepancy of both thermal expansion coefficient and crystal lattice period between substrate and the micro beam. q(x, t) is normalized motivating force that derived from electrostatic excitation as (Pelesko and Bernstein 2002Pelesko, J. A. and D. H. Bernstein (2002). Modeling Mems and Nems, CRC press.):

Where ε_v = 8.85 pF/m is the dielectric constant of the interface. Boundary conditions are as follows;

The following dimensionless parameters are used to normalize equation (1);

Then, dimensionless boundary conditions could be written as:

Based on presented formulas, dimensionless equation of motion could be implemented for MEMS resonators by the following equation;

By using the assumed modes method, dimensionless deflection solution of Eq. (9) could be introduced as;

Where ϕ_i(ξ) is the i^th Eigen function of micro beam that fulfills the appropriate boundary conditions,u_i(τ) is the i^th time dependent deflection coordinate and n is the supposed degrees of freedom of the micro beam.

To solve Eq. (9), we consider a single degree of freedom model (n = 1) and deflection functionW(ξ, τ) is assumed to be as:

The trial function is

This function satisfies the boundary conditions.

Then by substitution of presented functions to the dimensionless equation of motion and integrating from 0 to 1, dimensionless equation of motion changes to (Fu, Zhang et al. 2011Fu, Y., J. Zhang and L. Wan (2011). "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)." Current applied physics 11(3): 482-485.):

Where

3 SOLUTION PROCEDURE

For the following general oscillator

Where u and t are generalized dimensionless displacement and dimensionless time andA is amplitude of oscillator. Based on the variational principle, by implementing the semi-inverse method (He 1997He, J.-H. (1997). "Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics." International Journal of Turbo and Jet Engines 14(1): 23-28., He 2004He, J.-H. (2004). "Variational principles for some nonlinear partial differential equations with variable coefficients." Chaos, Solitons & Fractals 19(4): 847-851.) and He's method (He 2007He, J.-H. (2007). "Variational approach for nonlinear oscillators." Chaos, Solitons & Fractals 34(5): 1430-1439., Bayat and Pakar 2011Bayat, M. and I. Pakar (2011). "Application of He's Energy Balance Method for nonlinear vibration of thin circular sector cylinder." Int. J. Phys. Sci 6(23): 5564-5570.), variation parameter could be written as;

Where T = 2π/ω is period of the oscillator and =f(u). Thus, Hamiltonian of presented problem could be expressed as;

Then defining a new function as;

By choosing any arbitrary point like ωt = π/4, and setting, an approximate frequency-amplitude relationship could be obtained. Such approach is much simpler and has been widely used (Jamshidi and Ganji 2010Jamshidi, N. and D. Ganji (2010). "Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire." Current Applied Physics 10(2): 484-486.). The accuracy of such location method, however, strongly depends upon the chosen location point. To overcome the shortcomings of the energy balance method, a new approach based on Hamiltonian has been suggested (He 2010He, J.-H. (2010). "Hamiltonian approach to nonlinear oscillators." Physics Letters A 374(23): 2312-2314.). Differentiating the Hamiltonian leads to natural frequency of the system;

For more convenience, a new function (u) defined as;

And then for natural frequencies of the system, one could use the following relation;

From Eq. (21) we can obtain approximate frequency-amplitude relationship of a nonlinear oscillator (Shou 2009Shou, D.-H. (2009). "Variational approach for nonlinear oscillators with discontinuities." Computers & Mathematics with Applications 58(11): 2416-2419., He 2010He, J.-H. (2010). "Hamiltonian approach to nonlinear oscillators." Physics Letters A 374(23): 2312-2314.). For current special problem, we have following Hamiltonian equation:

3.1 First Order Hamiltonian Approach

With satisfying the initial conditions, utilizing u = A cos ωt as the trial function into equation (22), we obtain;

That leads to;

Then, the frequency-amplitude relationship can be obtained from;

Therefore, after some approximations and simplifications, equation (25) could be solved and the natural frequency could be obtained as;

That is approximately equal to;

The variational approach (Bayat, Bayat et al. 2014Bayat, M., M. Bayat and I. Pakar (2014). "Nonlinear vibration of an electrostatically actuated microbeam." Latin American Journal of Solids and Structures 11(3): 534-544.) and analytical approximate solution (Rafieipour, Lotfavar et al. 2013Rafieipour, H., A. Lotfavar and A. Masroori (2013). "ANALYTICAL APPROXIMATE SOLUTION FOR NONLINEAR VIBRATION OF MICROELECTROMECHANICAL SYSTEM USING HE'S FREQUENCY AMPLITUDE FORMULATION." IJST 37(M2): 83-90.) resulted the same response for this problem.

3.2 Second Order Hamiltonian Approach

To improve the accuracy of this approach, a higher order periodic solution was assumed as time response function as;

Where the initial condition is

Substituting Eq. (29) into Eq. (22), we obtain:

And then the frequency-amplitude relationship can be obtained from following equation;

To obtain natural frequency, substituting equation (29) in (31) as b = A - a, a second order algebraic equation set will be ready to solve to get the natural frequency and values of 'a', 'b' for various values of A andV, some of the results are listed in table 2.

3.3 Third Order Hamiltonian Approach

One could use a third order time response for micro beam as;

Where the initial condition is

Same as the second order Hamiltonian approach, with some mathematical simplification, values of a, b and c could be obtained for various values of A and V such as what listed in table 3.

4 VALIDATIONS, RESULTS AND DISCUSSIONS

4.1 Computational Efficiency

Presented nonlinear algebraic equations are solved by using Wolfram Mathematica software on Intel(R) Core (TM) i5-3230M CPU @ 2.6 GHz processor, includes 6 GB installed memory on a 64-bit operating system. Required time for calculation of natural frequencies was 5 to 10 seconds, 30 to 40 seconds and 3.5 to 4 minutes for first, second and third order Hamiltonian approach respectively. In terms of accuracy and computational efficiency, second order solution was the best.

4.2 Validation

In comparison with previous works, where higher order approximations had not been used, more accurate dynamic response and natural frequencies are observed. Energy Balance Method (EBM) is the best criterion for comparisons. Figure 3 depicts comparison of the dynamic response of a micro beam under an electric excitation (V=24 Volt), with parameters N = 10, a = 24 and A = 0.4 obtained with the first, 2nd and 3rd order Hamiltonian approaches. Figure 4 repeated the comparisons with changing the A value to 0.5.

Table 4 compares frequency commensurate for different parameters of system, obtained from Hamiltonian method and energy balance method (EBM (Fu, Zhang et al. 2011Fu, Y., J. Zhang and L. Wan (2011). "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)." Current applied physics 11(3): 482-485.)). The exact values for some cases are also reported. Accuracy increases with increasing the order of approximations. When the order increases more accurate results are achieved. Increasing applied voltage or initial amplitude leads to more errors. Thus, in the case of larger initial amplitude and applied voltage, higher order approximations would be more useful.

Table 5 lists the effect of increasing the initial amplitude onto the natural frequencies obtained by various approximations.

Dynamic response of micro beam is depicted in figure 4. By reduction of parameters a, b and c for each order, amplitude decreases and it decreases more by increasing the order of approximate solution. Furthermore, second and third order approximations have almost the same range of amplitudes.

4.3 Phase Diagram of Micro Beam

Simplifying the convolution of a nonlinear system to a linear wise model could provide a useful view on the stability and controllability of system. For example, assume that a nonlinear MEMS micro beam has a linear wise model as below that includes all nonlinearities on the second part of its dynamics;

Nonlinear term (Ψ_Nonlinear(x, t, u)) could contain a high level of nonlinearity because of including space variables, time and also inputs as operands. It could exert some amazing variations on the dynamics of system. For more clarity, phase diagram of electrostatically actuated micro beam has plotted to demonstrate the effects of nonlinear part of MEMS micro beam. Figures 5 and 6 show compares phase diagram of obtained values by the higher Hamiltonian approach with the energy balance method (EBM). These figures depict the effects of A and Vparameters on the phase plan of the system in second order Hamiltonian method.

Figure 7 shows the effect of parameter A on the phase plan of system for N = 10, a = 24, V = 5 simulated by using the second order of Hamiltonian method. As it can be seen on the figure, by increase in the order of Hamiltonian approach, amplitude parameters (a, b, c) decreased, thus overall amplitude decreases too. When micro beam resonates near the zero point as the basal condition, a notable reduction in the velocity of resonator is observable. This phenomenon disappears immediately after passing from basal condition. This means that dynamics of this nonlinear system also depends on the position of the point that is being measured on the MEMS micro beam. Same analysis describes figure 8, where shows the effect of V parameter on the phase plan of the system for N = 10, a = 24, A = 0.6 simulated by using the second order of Hamiltonian method.

4.4 Free Vibration

Figure 9 shows the effect ofN parameter on natural frequency. It can be observed that the frequency is proportional with N. However, it decreases when initial amplitude (A) increases. The second order Hamiltonian has nearly same response in comparison to the EBM solution, even for higher values of N and amplitudes.

Figure 10 depicts the effect of parameter α on natural frequency of electrostatically actuated micro beam with parametersN = 10, V = 20 and various values for A. It can be observed that the frequency increases with increasing α. Obtained results by the second order Hamiltonian are close to the EBM solution, especially for low amplitudes and α values.

Figure 11 shows the effect of applied voltage on the natural frequency of electrostatically actuated micro beam. It can be seen that the frequency is decreased with increase in the voltage. Second order Hamiltonian is extremely close to EBM solution but they have considerable discrepancies in high amplitudes and voltages.

Nonlinear behavior of system leads to abrupt falling on high applied voltages. Natural frequency decreases dramatically on high voltages. On the other hand, natural frequency increases also with increasing the amplitude. This is due to the effect of more hardening the equivalent linear system of electrostatically actuated micro beam when initial amplitude increases. It is also observed that for higher amplitudes, discrepancy is less than lower values.

Figure 12 shows the effect of modulus of elasticity of electrostatically actuated micro beam on the natural frequency with various initial amplitudes. As demonstrated clearly, natural frequency increases with increasing the modulus of elasticity.

Figure 13 shows the effect of thickness of electrostatically actuated micro beam on the natural frequency with various initial amplitudes. As demonstrated clearly, natural frequency decreases with increasing the thickness of micro beam.

Several simulations and plots could be introduced to consider the fundamental design rquirements before any manufacturing process. Based on the presented examples,the proposed nonlinear model based on Hamiltonian approach is completely efficient and acceptable to find the effects of parameters on the natural frequency and phase plane diagram of electrostatically actuated micro beam.

5 CONCLUSION

Applying the Hamiltonian approach to the nonlinear problem of electrostatically actuated micro beam, this paper studied the effect of various parameters on the dynamic response and phase diagram (stability) of the system. Due to the nonlinear manner of MEMS resonators on sensor design paradigms, this would be used in practical work for more efficient and low cost experiments. Utilized approximate solution converged to the exact solution and obviously demonstrated a good level of accuracy. It was presented that increasing the order of Hamiltonian approach, more agreeable results could be achieved. Briefly, presented approach resulted in below findings;

Finally, Based on presented examples to find the effects of parameters on the natural frequency and phase plane diagram of electrostatically actuated micro beam, it seems that this nonlinear model that was based on Hamiltonian approach is completely efficient and acceptable.

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