Computers and Mathematics with Applications

Transcription

1 Computers and Mathematics with Applications 5 2) Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: wwwelseviercom/locate/camwa Solution of nonlinear pull-in behavior in electrostatic micro-actuators by using He s homotopy perturbation method Arif Rafiq a,, Muhammad Yousaf Malik b, Tariq Abbasi a a Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan b Department of Mathematics, Quaid-e-Azam University, Islamabad, Pakistan a r t i c l e i n f o a b s t r a c t Article history: Received 6 May 2 Received in revised form January 2 Accepted January 2 Keywords: He s homotopy perturbation method Pull-in MEMS Micro-actuator Electrostatic actuator The work presented here is about the nonlinear pull-in behavior of different electrostatic micro-actuators He s homotopy perturbation method HPM) is applied to solve different types of micro-actuators like Fixed Fixed beam and Cantilever beam actuators Simulated results are presented for further analysis Also the obtained results compare well with the literature 2 Elsevier Ltd All rights reserved Introduction Microelectromechanical system MEMS) is a relatively new application in the area of mathematical modeling and simulation Different types of electrostatic actuators are used in MEMS but beam type electrostatic actuators have also been frequently applied A major problem in using beam type actuators is their nonlinear pull-in behavior Petersen [] was the first person who identified this nonlinear pull-in behavior of an electrostatic micro-actuator Mullen and Hsu [2] presented a coupled non-linear elastic-electrostatic boundary element method BEM) for electro-statically driven actuators The extraction of material properties by performing the measurement of pull-in voltage and capacitance voltage together was discussed by Chan et al [] Using the Raleigh Ritz method, Fujita and Ikoma [4] gave an approximate solution for electrostatic field analysis Mullen et al [5] have used the FEM to predict the load deflection and buckling behavior of microfabricated beams A combination of linear and nonlinear behavior under high voltage deformation was presented by Li and Aluru [6] The design and performance of cantilever beams and fixed curved electrodes were given in [7] The coupling effect between the electrostatic force and elastic deformation was given in [8] Kuang and Chen [] employed the modified Adomian decomposition method to analyze the non-linear pull-in behavior of different types of micro-actuators He s homotopy perturbation method HPM) [ 6] seems to be very convenient and effective to solve nonlinear initial and boundary value problems He [] presented a detailed review about the HPM method A comparison between the Adomian decomposition method and HPM showed that the HPM which does not need small parameters in the equations, therefore is more effective and simple Moreover, HPM can be used for solving non-linear problems of different types ie nonlinear oscillatory problems [7], singular IVP s of Lane Emden type [8], and electrostatic potential problems [] Corresponding author address: arafiq@comsatsedupk A Rafiq) 88-22/$ see front matter 2 Elsevier Ltd All rights reserved doi:6/jcamwa24

2 2724 A Rafiq et al / Computers and Mathematics with Applications 5 2) L p f o s Cantilever Beam x v δ Electrode y Fig A cantilever type beam electrostatic micro-actuator In the present work, HPM is applied to solve a non-linear pull-in behavior problem of different types of micro-actuators The obtained approximate analytical solutions for non-linear equations provides the evidence of the usefulness of He s homotopy perturbation method Simulated results are to provide a comparative study with [] 2 Actuator devices For simplicity the concentration is focused on models Fixed Fixed and Cantilever beam) discussed in [] by Kuang and Chen 2 Cantilever beam Fig shows the cantilever beam type actuator suspended above a fixed electrode with a fixed boundary condition An external voltage V is applied between beam and fixed electrode The small deflection theory of the cantilever beam is discussed in [2] By using this the following non-linear differential equation [ ] d 4 yx) = F ) dx 4 s yx)) 2 is obtained with the distributed load P e x, V) and dimensionless static deflection yx) of a cantilever beam, with x = x, F = o b V 2 L b, b = b and s = s 2) L b 2 E I L b L b In Eq ), E is the effective beam material modulus, Ix) is the moment of inertia for the cross sectional area of the beam, o is the dielectric constant of air, b is the beam width, s is the distance between the cantilever beam and the fixed electrode, t is the beam thickness, L b is the length of the beam and in Eq ) fringing is ignored Boundary conditions for the cantilever beam actuator are yx) =, y x) = at x = y x) =, y x) = at x = ) 22 Fixed Fixed beam Due to the fact that the Fixed Fixed beam actuator is suspended above the ground plane, electrostatic pressure P e x, V) is created to pull the deformable beam towards the ground electrode The Fixed Fixed beam was pulled suddenly into the electrode after the voltage exceeded the critical pull-in voltage see Fig 4) Consider a non-dimensional and nonlinear differential equation of the following type [ d 4 gx) = P dx 4 e x, V) = F gx)) + 65 ] 4) 2 bgx) This expresses a uniform Fixed Fixed beam actuator with the first order fringing correction, where b = b L b, gx) = gx) L b, { gx) is the gap above the ground electrode and F r = F 65 bgx)) } is called the fringing field correction [7] 5)

3 A Rafiq et al / Computers and Mathematics with Applications 5 2) Also for simplicity residual stress is zero in the Fixed Fixed beam The boundary conditions for the Fixed Fixed beam are gx) = g o L b = g, g x) = at x = gx) = g o L b = g, g x) = at x = where g o = is the initial gap 6) He s homotopy perturbation method He s homotopy perturbation method was established as very effective, simple and convenient when solving nonlinear initial and boundary value problems Consider the following general nonlinear differential equation Fq) hθ) =, θ Ω Subject to the boundary conditions G q, q ) =, θ Γ η where F is a general differential operator, G is a boundary operator, hθ) is a known analytic function, Γ is the boundary of the domain Ω Split the operator F as usual into L, the linear part and N, the non-linear part to modify Eq 7) as Lq) + Nq) hθ) = ) we can construct the following homotopy by using homotopy technique, vθ, p) : Ω [, ] R which satisfies Hv, p) = p)[lv) Lq )] + p[fv) hθ)] =, p [, ] ) we can also write it as Hv, p) = Lv) Lq ) + plq ) + p[nv) hθ)] = ) where p [, ] is an embedding parameter and q is the initial approximation to 7) which satisfies the boundary conditions Hv, ) = Lv) Lq ) = Hv, ) = Fv) hθ) = The changing process of p from zero to unity is just that of vθ, p) changing from q θ) to qθ) This is called deformation, and also, Lv) Lq ) and Fv) hθ) are called homotopic in topology If the embedding parameter p p ) is considered as a small parameter, applying the classical perturbation technique, we can naturally assume that the solution of equations Eqs 2a) and 2b) can give a power series in p, ie 7) 8) 2a) 2b) v = v + pv + p 2 v 2 + p v + and setting p = results in the approximate solution of Eq 7) as q = lim v = v + v + v 2 + p ) 4) The convergence of series 4) has been proved by He [2] The major advantage of He s homotopy perturbation method is that the perturbation equation can be freely constructed in many ways therefore it is problem dependent) as homotopy in topology Also the initial approximation can freely be selected 4 Numerical analysis 4 Cantilever beam HPM is used to solve Eq ) and the deflection for a cantilever beam is represented as follows Construct the following homotopy for Eq ) subject to the boundary conditions )

4 2726 A Rafiq et al / Computers and Mathematics with Applications 5 2) yx) pf x 4 s + 2 yx) 2 s + y 2 x) s 4 Assume the solution of 5) in the form + 4 y ) x) = 5) s 5 yx) = y + py + p 2 y 2 + p y + 6) Substituting 6) in 5) and collecting terms of the same power of p gives 4 y x) = p : x 4 y x) =, y 7) x) = at x = y x) =, y x) = at x = 4 y x) F p : x 4 s + 2 y x) + y 2 x) + 4 y x) ) = 2 s s 4 s 5 y x) =, y 8) x) = at x = y x) =, y x) = at x = 4 y 2 x) F 2 y x) + 6 y x)y x) + 2 y 2 x)y ) x) = p 2 : x 4 s s 4 s 5 y 2 x) =, y ) 2 x) = at x = y 2 x) =, y 2 x) = at x = 4 y x) F 2 y 2x) y x)y 2 x) + y x)y 2 + x)) y x) + y2 x)y 2 x) ) ) = p : x 4 s s 5 s 4 y x) =, y x) = at x = 2) y x) =, y x) = at x = After solving the above system y x) = 2) y x) = F s 2 [ 24 x4 6 x + 4 x2 ] y 2 x) = F [ 2 s 5 26 x8 252 x x6 6 x + ] 6 x2 y x) = F [ 7 s x2 7 x x 5 x + 4 x8 4 5 x ) 45 x6 y 4 x) and y 5 x) are given in Appendix and the approximate solution will be yx) = 22) 2) 44 x ] 2268 x2 24) y n x) = y x) + y x) + y 2 x) + y x) + 25) n= 42 Application of cantilever beam A rectangle poly silicon beam as shown in Fig is used for the cantilever micro-actuator The beam length is L b = 5 µm The thickness and the width of beam are t = 6 µm and b = 46 µm respectively The maximum height of the actuator is s = 22 µm Also the non-dimensional parameter in this case is F = , s = 22 5, b = 46 5 where the Young s modulus for the beam material ie poly silicon) is, E = 6 GPa, V = V

5 A Rafiq et al / Computers and Mathematics with Applications 5 2) e+7 2e+7 e+7 8e+6 6e+6 4e+6 2e Legend HPM x Adomian Fig 2 Adomian vs HPM for cantilever beam up to terms) So, Eqs 2) 24) gives y x) = y x) = x x x 2 [ ] y 2 x) = 5 x x x x + 677x x x x x y x) = x x x x x 2 and hence 25) implies x x x 4 yx) = x x x x x 26) x x 2 + The non-dimensional tip deformation for this cantilever beam actuator will be y) = for V and it can be improved by increasing the number of approximating terms The graphs Figs 2 and ) show the comparison between [] and the present solution It is observed that after taking more approximating terms, HPM gives better results than the Adomian method Fig 2 shows that the graph of HPM and Adomian method overlaps for some terms, but if we take more approximating terms, HPM proves much better than the Adomian method as shown in Fig 4 Fixed Fixed beam The Fixed Fixed beam deflection equation after employing the HPM can be expressed as [ d 4 gx) = pf dx4 g x)) + 65 ] 2 bgx) Assume the solution in terms of 27) gx) = g x) + p g x) + p 2 g 2 x) + p g x) + 28) Substituting 28) in 27) and comparing with respect to p 4 g x) = p : x 4 g x) = g, g 2) x) = at x = g x) = g o, g x) = at x =

6 2728 A Rafiq et al / Computers and Mathematics with Applications 5 2) Legend x HPM Adomian Fig HPM vs Adomian for more term approximation Deformable Fixed-Fixed Beam L b t V gx) P o x,v) g Fixed Ground Electrode Dielectric Structure Fig 4 A Fixed Fixed beam type electrostatic micro-actuator 4 g x) 65 + F p x 4 bg x) + ) g 2 : x) = g x) = g, g x) = at x = g x) = g, g x) = at x = 4 g 2 x) F 2 g x) p 2 x 4 g : x) + 65 g ) x) = bg 2 g 2 x) = g, g 2 x) = at x = g 2 x) = g, g 2 x) = at x = 4 g x) + F g 2x) p x 4 g 4 : x) 2 g 2x) g x) 65 g 2 2x) g + 65 x) ) bg 2 x) bg x) = g x) = g, g x) = at x = g x) = g, g x) = at x = ) ) 2) The solution of the above system will be, g x) = 25 g x) = F [ b 5b + ) x 2 ) x ) ] x 4 2b ) 4)

7 A Rafiq et al / Computers and Mathematics with Applications 5 2) { 2b 2 + ) ) } b 456b F 2 F 576 b x b + 25 { b 2 + ) ) } 2875b F 2 44b 2 F 456 b b 625 g 2 x) = b { + b) F} x4 25 b 48 b F 2 + b) x { b F 2 + b) )} b x { b F 2 + b) b )} x g x) and g 4 x) are given in Appendix and the general approximate solution will be gx) = g n x) = g x) + g x) + g 2 x) + g x) + 6) n= x 5) 44 Application of Fixed Fixed beam In this example beam length L b = 25 µm, beam width is b = 5 µm the thickness is t = µm and initial gap g = µm The non-dimensional parameter in this case are F = 26 6, g =, b = 5 = and applied voltage is V = V 5 Applying the above data for the Fixed Fixed beam, the solution Eqs ) 5) becomes g x) = 25 [ ] g x) = x x x 4 [ ] g 2 x) = x x x x x x x x g x) = x x x x x x 24545x 2 and hence from 6) the deflection is approximated under V as x x + 8 x 4 gx) = x x x x x x 24545x 2 + 7) The non-dimensional tip deformation in this case will be g) = 6 for V and it can be improved by increasing the number of approximating terms The graphs Figs 5 and 6) show that the graphs of HPM and the Adomian method have the same pattern The graph Fig 7) gives a good comparison between [] and the present solution 5 Conclusions In the present work we have reanalyzed the simulated results of pull-in behavior by Kuang and Chen [] for Fixed Fixed and Cantilever beam type micro-actuators It has been observed that HPM is more handy, trouble-free and well-organized to solve the nonlinear behavior of micro-actuators The nonlinear deflection equation is easily modified by the homotopy technique and the solution involved no complexity of functions

8 27 A Rafiq et al / Computers and Mathematics with Applications 5 2) x -4e+ -8e+ -2e+2 Legend HPM Fig 5 Approximation by HPM x -4e+ -8e+ -2e+2-6e+2 Legend Adomian Fig 6 Approximation by the Adomian method Numerical approximate results have been obtained for both Fixed Fixed and Cantilever beam models In the end, simulated numerical results are to provide a comparative study with [] More complicated models can be worked out by using the homotopy perturbation method Appendix y 4 x) := F x x x x x x 2x x8 22 x ) / 5 x6 557 y 5 x) := F x x s x x F x s + F 4 x s x x 6 8

9 A Rafiq et al / Computers and Mathematics with Applications 5 2) x e+8 -e+ -5e+ -2e+ -25e+ -e+ Legend Adomian HPM Fig 7 Adomian method vs HPM for the Fixed Fixed beam 2488 x x x 8 582x ) / x 6 x x x s 4 x x 45 and F 5 x s 4 g x) := F F 5 x 2 s 4 75Fb x x 2 F 2 b x 2 F 2 b 24x 8 fb F 2 b x 2 + 6x 7 Fb 2 2x 6 Fb F 2 b x F 2 b x F 2 b x x F 2 b x F 2 b x F 2 b x F x 2 F x 8 F x F F 2 b x x F 2 b 5Fb x x 7 F x F b x 4 x 6 F F 2 b x x F 2 b x F 2 b + Fb x x 6 F 2 b 2 x 7 F 2 b x 8 F 2 b x 6 F 2 b x 7 F 2 b x 8 F 2 b 78x 6 Fb ) / F 2 b x + 684x 7 Fb 52x 8 Fb F225F 2 b F 2 b Fb F 2 b F 2 656Fb 2 6Fb + 24b )x /b F F 2 b F 2 b Fb F 2 b F Fb 2 8Fb + 44b )x 2 /b + 25 g 4 x) := F F 2 b F 2 b

10 272 A Rafiq et al / Computers and Mathematics with Applications 5 2) F 2 b F 2 b F 2 b 2 )x F 2 b F F 2 b F 2 b F 2 b 2 )x F 2 b F F 2 b F 2 b F 2 b 2 )x F 2 b F 2 b F F 2 b F 2 b 4 )x F F 2 b F 2 b F 2 b F 2 b Fb Fb Fb 77Fb 4 )x Fb F 2 b F 2 b F 2 b F F 2 b + 276Fb 4 + 6Fb Fb)x F Fb Fb F 2 b F 2 b 8882Fb F 2 b Fb F 2 b)x F Fb Fb F 2 b 2 525F 2 b Fb F 2 b Fb 7875F 2 b)x F Fb b Fb b F 2 b F 2 b 85Fb F 2 b b Fb F 2 b)x F 2 b F 2 b F 2 b Fb F Fb F 2 b Fb Fb 2 786b b 85274b 2 )x F Fb b F 2 b b 88Fb F 2 b 48875Fb b F 2 b 2 ) / s F 2 b 782Fb 4 )x 6 b F F Fb b Fb b F 2 b F 2 b 66Fb F 2 b 4

11 A Rafiq et al / Computers and Mathematics with Applications 5 2) b 74225Fb F 2 b)x /b F F b b + 54b F 2 b 2286Fb Fb F 2 b F 2 b 2774Fb F 2 b Fb )x 2 /b References [] KE Petersen, Electron Devises ED-25 ) 78) 24 [2] R Mullen, SK Hsu, Boundary elements XU, in: CA Brebbia, JJ Rencis Eds), in: Stress Analysis, vol 2, Elsevier Science, 2, p 57 [] EK Chan, K Garikipati, RW Dutton, J Microelectromech Syst 8 ) 27 [4] H Fujita, T Ikoma, Sensors Actuators A 2-A 2) ) 25 [5] R Mullen, M Mehregany, M Omar, W Ko, Proc IEEE Micro Electro Mech Syst ) 7 [6] G Li, NR Aluru, Sensors Actuators A 2) 278 [7] PM Osterberg, SD Senturia, Microelectromech Syst 6 7) 7 [8] F Shi, P Ramesh, S Mukherjee, Comput Struct 56 7) 76 [] JH Kuang, CJ Chen, Math Comput Model 4 25) 47 [] JH He, Int J Mod Phys B 2 26) 4 [] JH He, Dissertation de-verlag im Internet GmbH, 26 [2] JH He, Comput Methods Appl Mech Eng 78 4) ) 257 [] JH He, Int J Non-Linear Mech 5 2) 7 [4] JH He, Appl Math Comput 5 2) 7 [5] JH He, Phys Lett A 5 26) 87 [6] JH, Int J Mod Phys B 2 26) 256 [7] JH He, DF Moore, H Taylor, M Boutchich, P Boyle, G Mcshane, M Hopcroft, JK Luo, Appl Math Comput 5 24) 287 [8] A Yıldırım, T Öziş, Solutions of singular IVPs of Lane Emden type by homotopy perturbation method, Phys Lett A 6 2) 27) 7 76 [] Li-Na Zhang, Ji-Huan He, Math Problems Eng 26 26), Article ID 8878 [2] R Legtenberg, J Gilbert, SD Senturia, M Elwenspoek, J Microelectromech Syst 6 7) 257