Static Analysis of Electrostatically Actuated Micro Cantilever Beam

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1 Available online at Proceia Engineering 51 ( 13 ) Chemical, Civil an Mechanical Engineering Tracks of 3r Nirma University International Conference on Engineering (NUiCONE 1) Static Analysis of Electrostatically Actuate Micro Cantilever Beam Anchit J. Kaneria a,d. S. Sharma b*, R. R. Trivei b a Mechanical Engineering Department, Marwai College of Engineeringi, Rajkot 476, Inia b Mechanical Engineering Department, Institute of Technology, Nirma University, Ahmeab,38481, Inia Abstract Electrostatically actuate micro evices experience a funamental limit on their stable travel range ue to a phenomenon calle as the pull-in Instability. Accurate etermination of pull-in parameters (pull-in isplacement an pull-in voltage) is vital in the esign of electrostatic micro actuators. A systematic metho of analysis of prismatic type of electrostatic beam is iscusse in this paper. Using Galerkin metho static analysis is carrie out. Behaviour of interaction of nonlinear electrostatic force with linear restoring force of the micro cantilever beam is stuie. Static analysis using COMSOL multiphysics finite element package is one to valiate the results. The results are also compare with existing literature. Keywors: Electrostatic Actuator, Pull-in instability, MEMS Nomenclature u(x) Downwar eflection of the micro beam towars the fixe electroe. A Electrostatic area of overlap I Moment of inertia of the beam Initial gap between two electroe g E x V Young s moulus of the micro-beam material Length co-orinate varies from to L Applie actuation voltage 1. Introuction The Micro-Electro-Mechanical Systems (MEMS) is the integration of mechanical elements, actuators, sensors an electronics on a common silicon substrate through the utilization of micro-fabrication technology. The Micro-Electro- Mechanical systems (MEMS) have experience a lot of progress recently. Their light weight, low cost, small imensions, low-energy consumption an urability attracte them wiely. The successful an commercialize MEMS actuators inclue igital micro mirror evice, automotive crash sensors, ink jet printer nozzles, catheter tip pressure sensors, etc. [1]. Newer applications of MEMS technology aboun in the areas that inclue, but are not limite to, the telecommunication, biomeical, semiconuctor, an aerospace inustries. The present ay national an international initiatives in the area of MEMS reveal the ever growing scientific an inustrial interest in this fiel. There are two basic components of MEMS, sensors an actuators. A sensor gathers the information from the environment by utilizing mechanical, thermal, biological, chemical, optical, an magnetic properties. The actuator contains the mechanical members, which are acte upon by various mechanisms like electromagnetic, thermo actuation, use of shape memory alloys, piezo actuation, magneto static actuation an electrostatic actuation. Out of all these, electrostatic actuation is wiely use. The popularity of electrostatic actuation is ue to ease of fabrication, low power consumption an higher energy ensity. * Corresponing author. Tel.: ; fax: aress:.sharma@nirmauni.ac.in The Authors. Publishe by Elsevier Lt. Open access uner CC BY-NC-ND license. Selection an peer-review uner responsibility of Institute of Technology, Nirma University, Ahmeaba. oi: 1.116/j.proeng

2 Anchit J. Kaneria et. al / Proceia Engineering 51 ( 13 ) Various analytical an numerical stuies have been carrie out on the pull-in parameters of the MEMS evices [1-5]. The biggest isavantage of using electrostatically riven actuators is the well-known pull-in instability in which one of the movable capacitor plates strikes its fixe counterpart after travelling a certain istance. This causes the snapping of two electroes, which can cause a short circuit an it will make the evice non- functional [3]. The root cause of the pull-in instability lies in the interaction of nonlinear electrostatic force an the linear mechanical restoring force. The pull-in isplacements as percentages of the original gaps in many structural moels of these actuators are 33.33% for parallel plates, 45% for cantilevers, 44.4% for torsional actuators an 35.8% for fixe-fixe beams [5]. Looking to the results quote in literature, energy techniques has been wiely use for the analysis of micro actuator. The stuy of literature inicates that more than 5% of the initial gap is unutilise for the travel range. The motivation behin this article is to stuy the total isplacement in micro cantilever beam using Galerkin approach an to compare pull-in parameters with existing literature an commercial finite element package (COMSOL). Figure 1 Electrostatic actuator use in MEMS [6]. Static Analysis of Electrostatically Actuate Micro cantilever beam The micro-beam requires mechanical energy to obtain a vibrating or translating motion base on the application of MEMS. An electric charge is create aroun the electric fiel ue to potential ifference between two conuctors (Refer Figure ). When the electrostatic force of attraction is in action, the stiffness of the movable electroe offers the require restoring force to maintain the system in equilibrium. The electrostatic force being a surface force, varies as per the inverse square law, i.e. the force of attraction between the two electroes is inversely proportional to the square of the istance between them. The mechanical restoring force offere by movable plate is irectly proportional to its isplacement. The interaction of this linear an non-linear pair of force leas to pull-in instability [7]. Figure Principle of electrostatic actuation For analyzing the static behavior of micro cantilever beam, Galerkin s approach is use to arrive at the precise values of the static pull-in parameters. It also explains the epenence of static pull-in parameters on various non-imensional quantities, which woul enhance the overall unerstaning of the static pull-in behaviour of prismatic an non-prismatic micro-beam. The governing ifferential equation of a electrostatically actuate micro-beam is nonlinear in nature an is expresse by, u AV EI (1) x ( g u )

3 778 Anchit J. Kaneria et. al / Proceia Engineering 51 ( 13 ) Figure 3: Schematic of an Electrostatically Actuate Micro cantilever beam An electrostatically actuate micro-cantilever beam is schematically shown in Figure 3. When the voltage V is applie across the two electroes, electrostatic force is generate. Electrostatic force acting on a beam is proportional to the eflection of the micro-beam. Since the eflection varies along the length, the intensity of the electrostatic force also varies proportionately. In case of the micro-cantilever beam, the maximum electrostatic force is experience by its tip, since it unergoes the maximum eflection. The eflection function is given as uner, ux ( ) a ax ax ax ax () 1 Satisfying following bounary conitions u, u ' (3) x x EIu '', EIu ''' (4) x L x L Equation for eflection can be written as Normalising the Eq.(5) ux ( ) a4 6xL 4xL x (5) ˆ ˆ ˆ ˆ ˆ where, ˆ ˆ ˆ Governing ifferential Eq. (1) can be rewritten in the normalise form as uner F Vˆ a x 4 1 af ux ( ) a6x 4x x (6) ux ˆ( ˆ) af (7) F 6x 4x x (8) (9) Non imensional parameters are efine as The omain resiual can be written as xˆ x, uˆ u, Vˆ bv L L g EIg R ( x) 4a 4 3 Vˆ 1 af (1) (11) By using Taylor series expansion, the above Eq. (11) can be rewritten as

4 Anchit J. Kaneria et. al / Proceia Engineering 51 ( 13 ) ˆ R( x) 4a V 1 af 3aF 4aF 5aF 6aF 7 af... (1) The weighte resiue form of the Eq. (9) can be written as follows Here, L R ( x) w( x ) (13) ˆ ˆ ˆ ˆ wx ( ) Fx ( ) 6x 4x x (14) After inserting the omain resiue an weighte function into Eq. (13) 1 ˆ F 4a V 1 af 3a F 4a F 5a F 6a F 7a F x (15) The above equation will be resulte in V a 636.5V a 17.5V a 4.83Va 1.835V a 3.815V 4 a V (16) 3. Results an iscussion The pull-in voltage an pull-in isplacement are to be obtaine from the governing ifferential equation subjecte to suitable bounary conitions. Using Galerkin s approach an Taylor series, the governing ifferential equation is converte into algebraic form (Eq. 16). Since, there is one equation an two unknown (pull-in voltage an pull-in isplacement), the solution can not be obtaine irectly. Initially a voltage ranges from 1 to 1 with the step size of 1 is taken an roots are checke. The voltage corresponing to which roots are changing nature (real to complex), is the approximate value of the pull in voltage. This value then can be improve by reucing step size in the range near the voltage value obtaine in first iteration. After three to four iterations very precise value of the pull in voltage can be obtaine. Base on this normalize pull in voltage the constant a can be foun. Now using Eq. (6), pull-in isplacement is foun. The normalize pull-in parameters an their comparison with existing literature can be seen from Table 1. Table 1 Static pull-in parameters of prismatic beam Microbeam type u ps V ps Present Approach Joglekar [8] % iff. Present Approach Joglekar [8] % iff. Cantilever Beam Figure FEA moel using COMSOL Figure 3 The eflection curve of beam uner application of

5 78 Anchit J. Kaneria et. al / Proceia Engineering 51 ( 13 ) electrostatic force To valiate the results through Finite Element Analysis, COMSOL software is use in multi physics environment. Moelling of cantilever beam is one in multi-physics environment. Figure shows finite element moel of movable electroe an fixe electroe. The eflection curve of beam uner application of electrostatic force is shown Figure 3. Table Comparison of pull-in parameters with COMSOL Pull-in Parameters Analytical Joglekar [8] COMSOL %iff. u ps % V ps % Here from Table the value of pull-in parameter foun by using COMSOL is much close to the analytical one with the ifference as 1.15% in pull-in isplacement an 1.7% in pull-in voltage. 4. Conclusion In this work, the results of static analysis of electrostatically exite micro-cantilever beam have been presente. Pull-in analysis of the continuous mechanical systems is carrie out. Authors have evise a numerical scheme base on the Galerkin metho an voltage iteration scheme is use to solve the polynomial equation. Results are valiate with the article publishe in the literature an also compare the results with COMSOL software. Acknowlegement Authors woul like to thank Prof D N Pawaskar, Prof. R P Shimpi, Inian Institute of Technology, Bombay, Powai, Mumbai an Prof. M M Joglekar, Inian Institute of Technology Roorkee, Roorkee, Inia for many fruitful suggetions. References [1] X. L. Jia, J. Yang, S. Kitipornchai an C. W. Lim, "Pull-in instability an free vibration of electrically actuate poly-sige grae micro-beams with a curve groun electroe", Appl. Math. Moeling, 11. [] M. M. Aballa, C. K. Rey, W. F. Faris an Z. Gural, "Optimal esign of an electrostatically actuate micro-beam for maximum pull-in voltage", Computers an Structures 83, 5, pp [3] M. Z. Moghimi an M. T. Ahmaian, "Application of homotopy analysis metho in stuying ynamic pull-in instability of Microsystems", Mechanics Research Communications 36, 9, pp [4] S. Chaterjee an G. Pohit, "A large eflection moel for the pull-in analysis of electrostatically actuate micro-cantilever beams", Journal of Soun an Vibration 3, 9, pp [5] S. Krylov an Y. Bernstein, "Large isplacement parallel plate electrostatic actuator with saturation type characteristic", Sensors an Actuators, , 6, pp [6] D. Y. Qiao, W. Z. Yuan an X. Y. Li, A two beam metho for extening the working travel range of electrostatic parallel plate micro-actuators, Journal of Electrostatics, 65, 7, pp [7] O. B. Degani an Y. Nemirovsky, "Moeling the pull in parameter of electrostatic actuators with novel lumpe two egree of freeom pill-in moel", sensors an actuators, 97-98,, pp [8] M. M. Joglekar an D. N. Pawaskar, "Estimation of oscillation perio/switching time for electrostatically actuate micro-beam type switches", International Journal of Mechanical Sciences 53, 11, pp [9] R. R. Trivei, M. M. Joglekar, R. P. Shimpi an D. N. Pawaskar, "Shape optimization of electrostatically actuate micro cantilever beam with extene travel range using simulate annealing", Proceeings of the Worl Congress on Engineering 11 Vol III WCE 11, July 6-8, 11, Lonon, U.K, pp.1-6. [1] M. M. Joglekar an D. N. Pawaskar,"Variable With Electrostatic Micro-actuators with Extene Travel Range". Proceeings of 7 MEMS Innovation Fest (MIF 7), Bangalore, Inia, pp