The spreading residue harmonic balance method for nonlinear vibration of an electrostatically actuated microbeam

Transcription

1 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September The spreading reside harmonic balance method for nonlinear vibration of an electrostatically actated microbeam J. L. Pan W. Y. Zh * College of Mathematics Physics and Information Engineering Zhejiang Normal University Jinha Zhejiang P.R. China To cite this article J.L. Pan W.Y. Zh The spreading reside harmonic balance method for nonlinear vibration of an electrostatically actated microbeam Nonlinear Sci. Lett. 8()()-. Copyright sian cademic Pblisher Ltd. Jornal Homepage: bstract In this paper the spreading reside harmonic balance method (SRHBM) is tilized to solve the nonlinear vibration of an electrostatically actated microbeam. Unlike the tradition harmonic balance method all previos reside errors are introdced to calclate the harmonic approximation in the SRHBM scheme. Nmerical simlations are carried ot in order to demonstrate the effectiveness of this method and the reslts are compared with HM soltions and act soltions showing the present method is effective and reliable. Keywords: nonlinear vibrations spreading reside harmonic balance method electrostatically actated microbeam. Introdction The stdy of electrostatically actated micro-electro-mechanical systems (MEMS) is a branch of mechanics. MEMS are widely applied in areas of electronic medical indstrial atomotive aerospace and other different fields [-]. However microbeams are the important MEMS inclding micro-switches transistors resonant micro-sensors and so on[]. The typical electrostatic micro-switch is consisted of a pair of fixed electrodes and an electrostatically actated microbeam. Firstly becase the potential difference is generated between the two electrodes the electrostatic force is indced to generate static electricity. Secondly de to the gap between the electrode and the microbeam the intermoleclar force is created. ccording to the above two conditions the electrostatically actated microbeam cases vibration. The motion of electrostatically actated microbeam is a nonlinear vibration which can be transformed into nonlinear differential eqations by the modeling process. Nonlinear differential eqations are hard to find ot the act soltion. Therefore in recent years a nmber of scholars have begn to stdy the approximate soltion with different methods. For ample Liao et al. [] tilized an optimal homotopy analysis method to cope with nonlinear differential problem. The variational iteration method was sed to stdy the nonlinear differential eqation by Rezazadeh et al. [8]. F et al. * Corresponding athor. zwy@zjn.cn (Zh)

2 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September [9] acqired approximate soltion of the MEMS by the energy balance method. nd they discovered that the accracy of the reslts obtained by the energy balance method was very high. Sbseqently Qian et al. [] adopted the homotopy analysis method to obtain the approximate soltion for the nonlinear vibration of an electrostatically actated microbeam. It was fond that the approximate soltion was in accord with the act soltion. Recently J et al. [] have researched approximate soltion of the strongly nonlinear systems devised by the global reside harmonic balance method. Meanwhile Xiao et al. [] and Go et al. [] proposed the reside harmonic balance method to discss approximate soltions. The reside harmonic balance method by Leng et al. [] was introdced to deal with the approximate analytical soltions of copled systems. Go et al. [] tended the reside harmonic balance method. They presented the spreading reside harmonic balance method (SRHBM) to solve the approximate soltions of the tapered beams model. In addition some scholars have frther analyzed some special dynamical phenomena for the nonlinear model. Rezazadeh et al. [8] discovered that when the applied DC voltage is eqal or greater than the pll-in vale it is located in the state of instability. Bt when the C voltage is applied the freqency of the micro beam can be stable. Qian et al. [] stdied the an electrostatically actated microbeam and the dynamical phenomena of the microbeam by the homotopy analysis method. The framework of this paper is strctred as follows. The MEMS mathematical model is given in Section. Section depicts that the SRHBM is applied to solve the nonlinear system. The nmerical amples are cited to demonstrate the accracy of the SRHBM and in Section. Finally the research reslts are smmarized in Section.. Mathematical Formlation n electrostatically actated microbeam with niform length l width b b h thickness h electrodes load V and initial gap g is shown in Figre. In non-dimensional form the differential eqation for the electrostatically actated microbeam can be obtained as follows [8] Figre Model for an electrostatically actated microbeam where d d a a a a a a a d () ξ d N V a ξ a ξ a ξ a

3 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September a N α dξ dξ a N α dξ dξ a α ξ ξ d d () where is maximm amplitde. nd the overdot indicates the derivative with relative to t while and denotes the derivative with relative to. The fnction in eqation () can be given ξ ξ ξ. by. pplication of the SRHBM Based on the working programs of the spreading reside harmonic for eqation () we mst qote a new independent variable t and eqation () trns into a a a a a a a ( ) ( ). () where and indicate the derivative with relative to and is the anglar freqency. De to the system withot damping effects the periodic soltion is considered. In eqation () the soltion can be showed by a set of base fnctions cos k k. () nd the steady state soltions are represented by p p p p () where p is a parameter and i i are nknown. Nt we mainly analyze the harmonic approximations with varios orders... The zeroth-order harmonic approximation To satisfy the initial conditions in eqation () the initial gess of for the zeroth-order harmonic approximation is chose to cos t. () Sbstitte eqation () into eqation () then the residal term is yielded R a a a a a a a a a a a a cos 8 8 a a a a a a a cos

4 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September a a a cos a cos. ccording to Galerkin procedre the right hand side of eqation () can not contain seclar term cos. In order to get ot of this term we let its coefficient be zero. Then we obtain a a a a a a a. (8) 8 8 By eqation (8) we can draw the nknown freqency a 8a a a. (9) a 8a a t the moment the zeroth-order approximation periodic soltion can be gained as follows cos. t () ().. The first-order harmonic approximations Sbstitting eqation () into eqation () and presenting the coefficients of parameter p yields φ a a a a a a a a a a a. a It is noted that eqation () is linear with respect to and take b cos cos () where b is nknown. R (). ccording to eqation () we Sbstitting eqation () into eqation () and improving the accracy the residal term is introdced. So we shold consider Eqation () yields R φ. R () 9 R ab a b a b a b a b a b a b a 8 a a cos a a a

5 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September a b a b a b a a a b 9 a a b 9a b 8 a a cos a a b a b a b a a b a b cos a a b a a b a b a b cos cos9. cos and cos voiding the eqation () containing seclar terms and spposing their coefficients be zeros we obtain two linear eqations containing two nknowns and b. We can get () b ( a a 8a) M N where 8 a 8 a a 9 a a a a M N () M a a a a a N 8 aa aa aa 8 aa 8 aa 8 aa aa aa aa aa 9 aa 9 a 9 a a a a a a a 8a Therefore the first-order harmonic approximation freqency and periodic soltion are obtained as follows b cos cos b. () t. ()..The second-order harmonic approximations We can se the same method to acqire the second-order harmonic approximations. Sbstitting eqation () into eqation () and eqating the coefficients of the p yields φ a a a a a a a a a a a a a 8

6 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September a a a a a a a a a a a. Becase of the soltion the form of eqation () we choose b cos cos b cos cos (8) (9) where b and b are nknown to be determined. Sbstitting eqation (9) into eqation (8) and considering the following eqation we obtain the R. cos cos cos R φ R () Let the coefficients be zeros then we obtain three linear eqations. ccording to these eqations we can solve three nknown b b and. Ths the second-order harmonic approximation freqency and periodic soltion are obtained as follows k t. () Then the kth-order harmonic approximation freqencies and periodic soltions are presented k k k and k () k k k k k k k k k k a cos cosi k i i k cos k. (). Reslts and Discssion In this part in order to demonstrate the effectiveness of the SRHBM we compare the act soltion the energy balance method EBM the homotopy analysis method HM and the second-order spreading reside harmonic balance approach SRHB. The act soltion is compted throgh the nmerical techniqe. Table takes on several corresponding reslts for the varios amplitdes of vibration and the different parameters N α V. In Table the errors are calclated between the three methods and the act soltion. 9

7 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September Table Comparison of the act and approximate freqencies corresponding to varios parameters Case N α V EBM HM SRHB Table Comparison of the accracy of approximate freqencies with act soltions Case N α V EBM % HM % SRHB % For Case in Table we notice that the error is.8% for the energy balance method. Meanwhile the error is.% for the homotopy analysis method. However the error of the second-order SRHBM is %. Similarly from the Case Case and Case we observe that the errors of the SRHBM are mch less than the relative errors of the homotopy analysis method and the energy balance method. Therefore the approximate soltions obtained by the SRHBM are more accrate. o SRHB soltion act soltion Figre Comparison of the approximate and act soltions for Case : phase crve and time history response Figre - represent the phase portrait and time history responses for for different cases. mong them the circles indicate the second-order SRHBM approximate soltions. The solid lines denote the act soltions. From the phase crve of Figre - we distinctly discover that the SRHBM approximate soltions are in good agreement with the act soltions. We know the system has a periodic soltion. From the time history responses of Figre - we observe that the system is a periodic system. Particlarly we draw the time history crve of t in.. By narrowing the range it is frther proof that the presented approximate soltions are in line with the act soltions. In Figre - we discssed the reslts are obtained by V for. Nt we let the vale of V

8 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September become larger then the corresponding phase crves are presented in the Figre where V is and respectively. From the image we can nderstand that when V takes a larger vale its approximate soltion and act vale are also very similar. o SRHB soltion act soltion Figre Comparison of the approximate and act soltions for Case : phase crve and time history response o SRHB soltion act soltion Figre Comparison of the approximate and act soltions for Case : phase crve and time history response o SRHB soltion act soltion Figre Comparison of the approximate and act soltions for Case : phase crve and time history response

9 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September o SRHB soltion act soltion Figre Comparison of the approximate and act soltions of phase crve for N V ; N V.. Conclsions The analytical techniqe is called the spreading reside harmonic balance method (SRHBM) in this paper. To prove the accracy of this approach the SRHBM is tilized to deal with the nonlinear vibrations of an electrostatically actated microbeam. dopting the SRHBM it is fond that the approximate soltions are in cellent agreement with the act soltions. The SRHBM is different to the pertrbation method and this approach has no se for small parameters. Compared with the EBM and HM the approximate soltions of the SRHBM converge to the act soltions rapidly. In general it is believed that the SRHBM is more practicality and validity. Ultimately we think that the SRHBM is effective and can be applied to other strongly nonlinear vibration systems. cknowledgements: The athors grateflly acknowledge the spport of the National Natral Science Fondations of China (NNSFC) throgh grant No.88. References [] S. E. Lyshevski MEMS and NEMS: Systems Devices and Strctres. IEEE Electrical Inslation Magazine ()(): -. [] R. Osiander M. nn Garrison Darrin J.L. Champion MEMS and Microstrctres in erospace pplications Boca Raton: Taylor & Francis. [] R. C. Batra M. Porfiri D. Spinello Electromechanical model of electrically actated narrow microbeams Jornal of Microelectromechanical Systems ()(): -89. [] M. Moghimi Zand M. T. hmadian Vibrational analysis of electrostatically actated microstrctres considering nonlinear effects Commnications in Nonlinear Science and Nmerical Simlation ()(9): -8. [] M. Moghimi Zand M. T. hmadian pplication of homotopy analysis method in stdying dynamic pll-in instability of microsystems Mechanics Research Commnications ()(9): [] X. L. Jia J. Yang S. Kitipornchai Free vibration of geometrically nonlinear micro-switches nder electrostatic and Casimir forces Smart Materials & Strctres 9()():-. [] S. J. Liao n optimal homotopy-analysis approach for strongly nonlinear differential eqations Commnications in Nonlinear Science and Nmerical Simlation (8)(): -. [8] G. Rezazadeh H. Madinei R. Shabani Stdy of parametric oscillation of an electrostatically actated microbeam sing variational iteration method pplied Mathematical Modelling

10 J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September ()(): -. [9] Y. M. F J. Zhang L.J. Wan pplication of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS) Crrent pplied Physics ()(): 8-8. [] Y. H. Qian D. X. Ren S. K. Lai S. M. Chen nalytical approximations to nonlinear vibration of an electrostatically actated microbeam Commnications in Nonlinear Science and Nmerical Simlation ()(): 9-9. [] P. J. J X. Xe Global reside harmonic balance method for large-amplitde oscillators of a nonlinear system pplied Mathematical Modelling 9()(): 9-. [] P. J. J Global reside harmonic balance method for a nonlinear oscillator with discontinity pplied Mathematical Modelling 9 ()(): 8-. [] M. Xiao W. X. Zheng J. D. Cao pproximate pression of a fractional order Van der Pol oscillator by the reside harmonic balance method Mathematics and Compters in Simlation 89()(): -. [] Z. J. Go X. Y. Ma Reside harmonic balance soltion procedre to nonlinear delay differential systems pplied Mathematics and Comptation ()(): -. []. Y. T. Leng Z. J. Go H. X. Yang Reside harmonic balance analysis for the damped Dffing resonator driven by a van der Pol oscillator International Jornal of Mechanical Sciences ()(): 9-. [] Z. J. Go W. Zhang The spreading reside harmonic balance stdy on the vibration freqencies of tapered beams pplied Mathematical Modelling (-)(): 9-. [] Y. H. Qian W. K. Li L. Shen S. P. Chen symptotic nalytical Soltions of anelectrostatically ctated Microbeam Base on Homotopy nalysis Method International Jornal of Mathematics and Comptational Science()(): 9-. Biographical Notes W. Y. Zh is a professor at Zhejiang Normal University. His crrent research interest mainly covers in nmber theory geometry and dynamic system. He has pblished more than articles in ISI-listed jornals. Competing Interests ll athors declare that there are no competing interests regarding the pblication of this paper. thors Contribtions ll athors contribted eqally to this work.