Elements. Using ABSTRACT. 1. Introduction. Anish Bangalore, India. tioned devices. often consist packed. This offering both.

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1 Jornal of Alied Mathematics and Physics,, 201, 1, Pblished Online November 201 (htt:// htt://dx.doi.org/10.426/jam A Monolithic, FEM-Based Aroach for the Coled Sqeeze Film Problem of an Oscillating Elastic Micro-Plate Using D 27-Node Elements Anish Roychowdhry 1,2*, Ar Nandy 1, C. S. Jog 1, Rdra Prata 1,2 1 Deartment of Mechanical Engineering, Indian Institte of Science, Bangalore, India 2 Center for Nano Science and Engineering, Indian Institte of Science, Bangalore, India * anishroy@mecheng.iisc.ernet.in Received Agst 201 ABSRAC In this stdy we describe an FEM-based methodology to solve the coled flid-strctre roblem de to sqeeze film effects resent in vibratory MEMS devices, sch as resonators, gyroscoes, and acostic transdcers. he aforemen- tioned devices often consist of a late-like strctre that vibrates normal to a fixed sbstrate, and is generally not erfectly vacm acked. his reslts in a thin film of air being sandwiched between the moving late and the fixed sbstrate, which behaves like a sqeeze film offering both stiffness and daming. yically, sch strctres are actated electro-statically, necessitating the thin air ga for imroving the efficiency of actation and the sensitivity of detection. o accrately model these devices the sqeeze film effect mst be incororated. Extensive literatre is resent on modeling sqeeze film effects for rigid motion for both erforated as well as non-erforated lates. Stdies which model the late elasticity often se aroximate mode shaes as int to the 2D Reynolds Eqation. Recent works which try to solve the coled flid elasticity roblem, reort iterative FEM-based soltion strategies for the 2D Reynolds Eqation coled with the D elasticity Eqation. In this work we resent a FEM-based single ste soltion for the coled roblem at hand, sing only one tye of element (27 node D brick). he strctre is modeled with 27 node brick elements of which the lowest layer of nodes is also treated as the flid domain (2D) and the integrals over flid domain are evalated for these nodes only. We also aly an electrostatic loading to or model by considering an eqivalent electrostatic ressre load on the to srface of the strctre. hs we solve the coled 2D-flid-D-strctrwith both existing analytical soltions roblem in a single ste, sing only one element tye. he FEM reslts show good agreement and blished exerimental data. Keywords: Sqeeze Film Daming; Coled Problem; 27-Node Brick; Micro-Plate 1. Introdction he wide scale alication of electro-statically driven MEMS sensors, sing arallel late caacitors have led to increasing interest in the stdy of energy dissiation de to the thin film of air traed in sch devices. yi- cally sch devices consist of a late like strctre, vibrating normally to a fixed sbstrate, with a thin air film traed in-between (see Figre 1). If the lateral dimen- sions of the late haen to be mch larger than the height of the air ga, the traed air behaves both like a sring and a viscos damer, a henomenon known as sqeeze film effect. Sqeeze film daming is the domi- nant dissiation mechanismm in Si based MEMS devices oerating in the aforementioned conditions [1]. he * Corresonding athor. sqeezee film offered daming and stiffness changes the dynamic characteristics of the vibratory MEMS device [2]. In order to correctly model sch devices, it is necessary to accrately determine the stiffness and daming de to the sqeeze film. Accrate modeling of sch systems involves coling of three domains, electrostatics, strctral and flid. raditionally the sqeeze film domain effect is modeled sing the lbrication theory, via Figre 1. Schematic of an air film traed between a vibrating elastic late and a fixed sbstrate. Oen Access

2 A. ROYCHOWDHURY E AL. 21 the Reynolds Eqation []. With rigid late assmtion the Reynolds Eqation can be decoled from the elasticity Eqation and, frther, on linearization can be solved to obtain analytical exressions for stiffness and daming. Blech [] stdied the effect of sqeeze film indced stiffness and daming for rigid lates with trivial ressre bondary conditions. Darling et al. [4] resented analytical soltions to the linearized Reynolds Eqation for varios venting conditions, sing a Greens fnction aroach. For flexible strctres one has to accont for the variable air ga, and the elasticity Eqation has to be coled with the Reynolds Eqation for accrate modeling. Hng et al. [5] resented a redced order macro model based on basis fnctions generated from finite difference simlations. hey alied this techniqe to model a ressre sensor as a clamed-clamed microbeam to stdy the ll-in dynamics of the system sing a 1D Eler beam Eqation and the non-linear Reynolds Eqation. McCarthy et al. [6] stdied cantilever microswitches sing a transient finite difference method aroximating a arabolic ressre distribtion along the length and non variance along the width and obtained good agreement with exerimental measrements. Yonis et al. [7] stdied the effect of sqeeze film daming for an electrically actated micro-late, sing the comressible Reynolds Eqation. hey sed ertrbation methods to derive analytical exressions for ressre distribtions in terms of the strctral mode shae. Pandey et al. [8] stdied the effect of flexral mode shaes on the sqeeze film offered stiffness and daming for a cantilever resonator, they sed Green s fnction to solve the linearized comressible Reynold s Eqation and sed the modal rojection method available in ANSYS to solve the coled flid strctre roblem for several flexral modes of vibration. he analytical and nmerical vales of daming obtained were in good agreement with exerimental reslts. Li et al. [8] acconted for the static bias deflection for a fixed- fixed micro-beam and a cantilever nder electrostatic actation. hey assmed a arabolic fnction for the ressre along the beam width and a cosine series along the beam length, and solved the coled Reynolds Eqation and the Eler Bernolli beam Eqation. Hannot et al. [9] resented an aroach to solve the coled elasticity Eqation and Reynolds s Eqation for modeling a caacitive micro-switch. hey emloyed a non-linear Newmark time integration scheme for the mechanical Eqations and a traezoidal rle for the flid Eqations. he above mentioned models attemt to solve the coled roblem, thogh not in a single ste. he geometry modeled is also limited to 1D beam tye strctres. hese methods, thogh accrate, are cmbersome and involve iterative or staggered aroaches. We roose a single ste methodology to solve the coled flid- strctre sqeeze film roblem. We se the D elasticcity Eqation, ths not restricting orselves to any articlar geometry, and cole it with the 2D Reynolds Eqation for sqeeze film. A single ste monolith aroach [10] is resented to solve the coled roblem. he nmerical reslts show good agreement with blished exerimental data and existing analytical soltions. 2. Nmerical Modeling he roblem at hand involves solving for ressre on the vibrating late de to the sqeezed film, taking into accont the elasticity of the late. hs the roblem involves soltion of the Reynolds Eqation for the flid domain coled with the D elasticity Eqation for the strctral domain. In or finite element model, the air ga is treated as a 2D layer and the strctral domain is modeled in three dimensions. he element sed for modeling the strctral domain is D, 27 node brick element. he wet face of the D element is treated as the flid domain. hs the relevant integrals for the flid domain are evalated over the corresonding 9 noded wet srface of the D, 27 node brick element Variational Formlation for the Flid Domain he linearized Reynolds Eqation is given as follows [2], h 12 P P h P H, P t t x y eff a where μ eff is the effective viscosity, h is the initial air 0 ga, P is the ambient air ressre, P is the flid resa sre (ertrbed abot P ) and H is the air ga (ertrbed a abot h ). he last term on the right hand side of the 0 above Eqation coles the strctre and the flid doj t main. Sbstitting for harmonic soltion P e and j t H ze in Eqation (1) and considering as variation of in a weighted integral sense we have, h0 2 h j 0 jz d 0. (2) 12 P eff a For the flid domain we have 0 on the oen borders and 0 on the closed borders. After doing nˆ integration by arts and imlementing the above bondary conditions, we get the governing Eqation for flid domain as, h h j 0 0. d d 12 P eff a j d0 z (1) () Oen Access

3 22 A. ROYCHOWDHURY E AL FEM Formlation for the Flid Domain For the FEM formlation we se 9 noded qadrilateral elements for 2D flid domain. Pressre, its variation and at any oint are obtained from interolation of the z corresonding nodal vales sing the following relationshis. 1 2 N ˆ N1 N2 N9, (4) 9 N ˆ, (5) 1 v 1 w 1 N ˆ 0 0 N 0 0 N 1 9. z z 9 v9 w Pressre gradient can be exressed as where 9 (6) B, ˆ (7) N1 N2 N9 x x x B. (8) N1 N2 N9 y y y Similarly we have, B ˆ. (9) where N is (1 9), ˆ is (9 1), ˆ is (9 1) B δ is (2 9), N z is (1 27) and û is (27 1). Sbstitting Eqations (4), (5), (6), (7) and (9) in Eqation () and sing arbitrariness of ˆ we get, δ 12 0 jh0 d h eff a BB NNdˆ P j NNd z ˆ FEM Formlation for the Strctre (10) We have the variational statement for dynamic strctral roblem withot anybody force as: : ( ) d d.td, (11) t where is density, is dislacement, is its variation, τ is stress, t is rescribed traction over the bondary and ε is given by Here, t 1 ( ). 2 : ( ) : ( ), ij with the smmation convention over reeated indices. For coled sqeeze film daming roblem with strctral interaction, the wet srface (the srface which constittes the 2D flid domain) is sbjected to rescribed traction t n ˆ, then Eqation (11) can be written as : ( ) d d.nˆ d wet t.td, ij (12) where the last term on the left hand side signifies coling effect of the flid over the strctral domain. he standard FEM discretization for dislacement and other qantities are N ˆ, N ˆ, ( ) CB ˆ, ( ) B ˆ, ( ) B ˆ. Sbstitting above relations in Eqation (12) and sing arbitrariness of we have the discretized Eqation for the strctral domain as BCBd ˆ d ˆ NN N d ˆ nn Ntd. wet t 2.4. Coled FEM Formlation (1) For the coled roblem at hand the 2D flid domain corresonds to the wet srface of the strctral domain. hs coling the flid and strctre domains we have (combining Eqations (10) and (1)) K K ˆ f, (14) K K ˆ 0 Oen Access

4 A. ROYCHOWDHURY E AL. 2 K M K 2 M N N, d, K BCB d, K N nn ˆ wet f N t t d, d, h jh K B B N N d, 0 0 d 12 P eff wet a wet K N N j wet d. We have sed 27 noded brick element for the strctral domain, whose wet srface (consisting of 9 noded sqare face) is modelled as the 2D flid domain. ressre and h 0 is the initial air ga. Figre 2 shows the lots for F s and F d for the OOOO case and Figre shows the same for the OCOC case. We see from these lots that the nmerical stiffness and daming forces are in close agreement with the analytical reslts for both the flow bondary cases stdied. We also note that the two methods show good agreement at both high and low vales (ths high and low freqencies). he deviation between the nmerical and analytical reslts have been fond to be less than 2% for both the flow bondary conditions stdied here..2. Modeling a Cantilever In order to comare or reslts with exerimental data we have modeled a cantilever beam as er dimensions. Reslts and Discssion For validation of or FEM reslts we have comared or nmerical reslts with analytical soltions reorted by Siddartha et al. [11,12], as well as exerimental reslts from work by Pandey et al. [8]..1. Modeling a Varying Flow Bondary Elastic Microlate Siddartha et al. [11,12] stdied the effect of varying flow bondary conditions on the sqeeze film stiffness and daming for an all sides clamed micro-late. he late is considered to vibrate in its fndamental mode which imarts the flexibility effect. Analytical exressions have been derived for stiffness and daming forces on the late (clamed at all sides) de to the traed air film (sbjected to different flow bondaries). We se 4 4 mesh for FEM modeling of the late. We design two reresentative flow bondary conditions with or nmerical scheme, namely the all for sides oen ( OOOO ) and the two oosite sides closed ( OCOC ) cases. he late is sbjected to harmonic dislacement bondary condition corresonding to its first mode shae. he reslting ressre distribtion is integrated over the wet srface to get the force on the moving late. he sqeeze film sring (F s ) and daming (F d ) forces are obtained from the real and imaginary comonent of the resltant force resectively. he forces are non dimensionalised (see [11]) and lotted against a non dimensional arameter, directly related to the freqency of harmonic exci- 2 2 eff 0 0 tation as follows, 12 L / h, where μ eff is the effective viscosity [1], ω is the harmonic excitation freqency, L is the late side dimension, 0 is the ambient Figre 2. Sring and daming forces vs sigma for OOOO configration. Figre. Sring and daming forces vs sigma for OCOC configration. Oen Access

5 24 A. ROYCHOWDHURY E AL. mentioned in the work of Pandey et al. [8]. We comare the nmerically obtained Qality factors (Q) for the first three modes of vibration as well as the effect of asect ratio on the qality factor of the beam for the first mode of vibration. he beam modeled is of length 50 μm, width 22 μm, thickness 4 μm, with an initial air ga of 1.4 μm. he beam material is olysilicon with density 20 Kg/m, Yong s Modls 160 GPa and Poisson s ratio Air is considered to be the flid medim with the relevant roerty vales (nder standard temeratre and ressre ) as follows: density ρ air = 1.2 Kg/m, dynamic viscosity µ air = N.s/m 2, and ambient ressre a = Pa. In or simlations we have sbjected the cantilever to a sinsoidal voltage of 1.5 V, which is well below the ll-in voltage of 6 V for the given cantilever. he int voltage has been alied as an electrostatic ressre load of magnitde 5.08 N/m 2 to or FEM model for the cantilever beam, considering a arallel late caacitor with small dislacement aroximation [14]. he beam ti velocities have been obtained from the simlations and normalized with resect to the alied voltage and lotted against freqency. he freqency resonse so obtained is shown in Figre 4. he lot shows three distinct eaks corresonding to the first three modes, and is in close agreement with a similar lot reorted by Pandey et al. [8]. Q factors for different modes are obtained sing half ower method from the freqency resonse lot. Convergence stdy of the Q factor (able 1) has been done sing three levels of mesh refinement considering a very fine mesh ( ) reslt as or benchmark. Q factors for sfficiently fine mesh ( ) are comared with blished reslts from Pandey et al. [8] in able 2. We see that the data from the nmerical simlations are in good agreement with blished exerimental and nmerical reslts. We frther stdied beams with varying asect ratios. We considered beams with lengths varying from 150 µm to 50 µm, having a constant width of 22 µm and thickness 4 µm. Only the first mode of vibration is considered in this stdy. Comarative vales of Q factors for the different beams ( mesh) are resented in able. Or simlation reslts show a deviation of less than 10% from the reorted exerimental data. 4. Conclsion We have discssed an FEM formlation to solve the coled flid-strctre sqeeze film roblem withot resorting to iterative soltions. Or reslts show good agreement with exerimental data available from the literatre. Or nmerical scheme is seen to give good reslts for varying asect ratio strctres with dimensions below 100 μm. he roosed scheme can be frther sed as a design tool for modeling and simlation of the dynamic resonse of vibratory MEMS devices sch as caacitive microhones, RF (Radio Freqency) MEMS switches, etc., for which accrate knowledge of the Q able 1. Convergence stdy for the first two modes of a cantilever beam of length 50 μm, width 22 μm. Modes Q factors for FEM Mesh (length width thickness) able 2. Q factor comarison for the first three modes of a cantilever beam of length 50 μm, width 22 μm. Modes Q factors comarison Q EXP Q ANSYS Q FEM able. Q factor comarison for beam of width 22 μm with varying lengths. Length (μm) Q EXP Q factors Q FEM % Deviation Figre 4. Freqency resonse of a cantilever beam of length 50 μm, width 22 μm and thickness 4 μm Oen Access

6 A. ROYCHOWDHURY E AL. 25 factor is critical for design. With this methodology one can directly cole the elasticity effect of the strctre with the flid flow and need not limit oneself to 1D geometries. 5. Acknowledgements his work is artially sorted by NPMASS grant for comtational micro-systems. he athors acknowledge the sort from CoNE Lab of the Center for Nano Science and Engineering at IISC. REFERENCES [1] M. Bao and H Yang, Sqeeze Film Air Daming in MEMS, Sensors and Actators A, Vol. 16, No. 1, 2007, htt://dx.doi.org/ /j.sna [2] R. Prata, S. Mohite and A. K. Pandey, Sqeeze Film Effects in MEMS Devices, Jornal of the Indian Institte of Science, Vol. 87, No. 1, 2007, htt://erints.iisc.ernet.in/id/erint/1228 [] J. J. Blech, On Isothermal Sqeeze Films, Jornal of Lbrication echnology, Vol. 105 No. 4, 198, htt://dx.doi.org/ / [4] R. B. Darling, C. Hivick and J. X, Comact Analytical Models for Sqeeze Film Daming with Abitrary Venting Conditions, International Conference on Solid State Sensors and Actators, Vol. 2, 1997, htt://dx.doi.org/ /sensor [5] E. S. Hng and S. D. Sentria, Generating Efficient Dynamical Models for Microelectromechanical Systems from a Few Finite-Element Simlation Rns, Jornal of Microelectromechanical Systems, Vol. 8, No., 1999, htt://dx.doi.org/ / [6] B. McCarthy, G. G. Adams, N. E. McGrer and D. Potter, A Dynamic Model, Inclding Contact Bonce, of an Electrostatically Actated Microswitch, Jornal of Microelectromechanical Systems, Vol. 11, No., 2002, htt://dx.doi.org/ /jmems [7] M. I. Yonis and A. H. Nayfeh, Simlation of Sqeeze- Film Daming of Microlates Actated by Large Electrostatic Load, Jornal of Comtational and Nonlinear Dynamics, Vol. 2, No., 2007, htt://dx.doi.org/ / [8] A. K. Pandey and R. Prata, Effect of Flexral Modes on Sqeeze Film Daming in MEMS Cantilever Resonators, Jornal of Micromechanics and Microengineering, Vol. 17 No. 12, 2007, tt://dx.doi.org/ / /17/12/01 [9] S. D. A. Hannot and D. J. Rixen, Coling Plate Deformation, Electrostatic Actation and Sqeeze Film Daming in a FEM Model of a Micro Switch, International Conference on Comtational Methods for Coled Problems in Science and Engineering, Barcelona, [10] C. S. Jog, An Otward-Wave-Favoring Finite Element- Based Strategy for Exterior Acostical Problems, International Jornal of Acostics and Vibration, Vol. 18, No. 1, 201, htt://erints.iisc.ernet.in/id/erint/4646 [11] S. Patra, A. Roychowdhry and R. Prata, Effect of Flid Flow Bondary Conditions on Sqeeze Film Parameters of Planar MEMS Strctres, Alied Mathematics Modelling, AMM 1457, Unblished. [12] S. Patra, Effect of Rarefaction, Flexibility and Restrictive Flow Bondary Conditions on Sqeeze Film Flow of MEMS Devices, Master s hesis, Indian Institte of Science, Bangalore, [1]. Veijola, H. Kisma, J. Lahdenerä and. Ryhänen, Eqivalent-Circit Model of the Sqeezed Gas Film in a Silicon Accelerometer, Sensors and Actators A: Physical, Vol. 48, No. 4, 1995, , htt://dx.doi.org/ / (95) [14] E. Sacedo-Flores, R. Relas, M. Florer and I. C. Chiao, Stdy of the Pll in Voltage for MEMS Parallel Plate Caacitor Actators, Materials Research Society Fall Meeting, Boston, 1-5 December 200. Oen Access