Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 Senor & Tranducer 04 by IFSA Publihing, S. L. http://www.enorportal.com Electromechanical Dynamic for Micro Film Lizhong Xu, Xiaorui Fu Mechanical engineering intitute, Yanhan Univerity, Qinhuangdao 066004, China Tel.: +86-335-805703, fax: +86-335-8074783 E-mail: xlz@yu.edu.cn Received: May 04 /Accepted: 3 July 04 /Publihed: 3 Augut 04 Abtract: In thi paper, an electromechanical coupled dynamic model of the micro film ubjected to electrotatic force i preented and it electromechanical coupled dynamic equation i given. By the equation, the natural frequencie and vibration mode of the micro film are invetigated. The forced repone of the film to voltage excitation are preented. Reult how that the natural frequency of the micro film ubjected to an electrotatic force i affected by mechanical and electric parameter uch a micro film ize, film tenion, voltage and clearance. A the film tenion grow, the deviation between the natural frequencie of the film with and without the electrotatic force decreae. Copyright 04 IFSA Publihing, S. L. Keyword: Electromechanical coupled, Micro film, MEMS, Natural frequency.. Introduction Microelectromechanical Sytem (MEMS are the integrated ytem coniting of microelectronic, microactuator and microenor []. Typical MEMS conit of micro beam or micro film. An example i a mall rectangular ilicon film with ide in the order of mm and thickne of the order of micron, that deform when ubject to electric field. Owing to it mall ize, ignificant force and deformation can be obtained with the application of low voltage. Example of device that utilize vibration of uch film are ynthetic microjet, and micropeaker, etc. []. Some device are propoed for generating electrical power from mechanical energy by reonance vibration of the micro beam or micro film [3-5]. Dynamic of the micro film i an important ubject that hould be developed. Rafael Nadal- Guardia ued one-dimenional (-D lumped model of parallel film electrotatic tranducer and tudied it dynamic behavior [6]. Lu ued a thin film model to interpret the thickne-dependent um frequency generation pectra collected from the air/ilica/poly (n-butyl methacrylate/water ytem [7]. Akai experimentally invetigated the vibration mode and tranmiion ultraonic wave by pmut uing epitaxial Pb(Zr,TiO 3 thin film on the epitaxial γ- Al O 3 /Si ubtrate to radiate ultraonic wave effectively [8]. Li preented an electrotatic film vibration voltage tranducer [9]. Gualdino fabricated a thin-film ilicon micro reonator by urface micromachining and tudied vibration mode of micromechanical dik reonator made from hydrogenated amorphou ilicon thin film [0]. Sorokin tudied the dynamic of a circular film membrane with attached current-carrying conductor in zero gravity and calculated the pectrum of natural vibration []. Sun invetigated electromechanical coupled non-linear dynamic for micro plate imply upported on four ide []. Sriniva invetigated the tatic diplacement and dynamic characteritic of microplate ubjected to electrotatic excitation [3]. In a word, a number of the tudie about dynamic of the micro film have been done. http://www.enorportal.com/html/digest/p_67.htm 99
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 However, electrotatic excitation i the mot frequently applied principle combining veratility and imple technology. In a MEMS device excited by the electrotatic force, the micro film i in an electromechanical coupled field. In thi paper, an electromechanical coupled dynamic equation of the micro film i preented which include mechanical force field and electrotatic field. The electromechanical coupled free vibration of the micro film i invetigated. The electromechanical coupled natural frequencie of the micro film and their change along with ytem parameter are dicued. A number of reult are given which can be ued in deign and manufacture of the MEMS unit.. Dynamic Equation Fig. illutrate a micro film under electrotatic force. Here uxyt (,, v Wxyt (,,, v i the initial clearance between the film and fixed plate, h i the thickne of the film, W i the tranvere diplacement of the film, a i the length of the film, b i the widene of the film, qxyt (,, i the tranvere load per unit area on the film, U i the voltage between the film and fixed plate. The dynamic equation of the micro film i W ρh F W q( x, y, t, ( t where +, ρ i the ma denity of the x y film, t i the time, F i the tenion in the film. The electrotatic force per unit area i q Fe εε 0 r U e, ( A v W where ε 0 i the permittivity contant of free pace, ε r i the relative dielectric contant of the inulating layer. The diplacement W of the micro film conit of a tatic component W and a dynamic one Δ W W W +Δ W (3 The load qxyt (,, of tatic( q and dynamic ( Δ q component a well qxy (,, t q +Δ q (4 q U εε 0 0 r ( v W dq From Δ q Δ W, we know dw Δ q ΔW 0 0 r 3 ( v W (5 (6 Subtituting Eq. (6 into Eq. (, yield the following equation q W (7 F W ρ Δ Δ t h F W q 3. Static Diplacement Let q Here m n C in in 4 C q in in dxdy 0 0 ab 4q π ( co mπ( co nπ (8 (9 Let W Ain in (0 m n Subtituting Eq. (9 and (0 into Eq. (7, yield Fig.. Dynamic model of a micro film. A ( π( π 4q co m co n 4 m n Fπ + ( 00
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 Thu W Ain in m n in in 6q 4 π F m,3,5... n,3,5... m n + a b ( From Eq. (8, the natural frequencie of the film can be given ω U + ρ εε m n 0 0 r Fπ 3 h ( v W (9 5. Forced Repone to Voltage Excitation The average tatic diplacement of the film i W ab a b 0 0 m,3,5... n,3,5... W dxdy 64q 6 m n π F( + a b (3 From Eq. (5, we know the diplacement W i included in term q. Here, we take the average diplacement to replace the diplacement W, and then Eq. (3 i changed into a nonlinear equation about the average diplacement. From the equation, the average diplacement of the film can be obtained. 4. Free Vibration Subtituting Eq. (6 into Eq. (8, yield W ρh Δ F Δ W ΔW 0 0 r 3 t ( v W Let ( (4 Δ W Δ φ x, y in( ωt + ϕ, and then ubtituting it into Eq. (4, yield where Δφ Δφ α φ x y + + Δ 0 ρhω α + F v W F 0 0 r 3 (, (5 The mode function of the micro film i conidered a Δ φ Ain in (6 Subtituting Eq. (6 into (5, yield m n A π + α in in 0 From Eq. (7, we know m π a n b + α 0 (7 (8 Let the diturbing voltage Δ U Eco( ωet, which i ubjected between the micro film and the bae plate. The exciting electric field force will be caued. Conidering weak nonlinear influence of the above exciting electric field force, the dynamic force per unit area i given Δ q E ω t 0 0 r in ( v W Subtituting Eq. (0 into Eq. (8, yield W ρh F W E ωet Let 0 0 r in t ( v W m n e (0 ( W( x, y, t q ( t Δφ ( x, y ( Subtituting Eq. ( into Eq. (, yield 0 0 r ( v W Einω t e ρhq ( t Δφ ( x, y (3 m n Fq ( t Δφ ( x, y m n a b From ρh( φ 0 0 Δ, we obtain 4 Δ φ ( xy, in in (4 ρhab The olution of Eq. (3 i q ( t a inωt+ b coωt t, (5 + Q ( in ( 0 τ ω t τ dτ M ω where Q ( t Δq 0 0 Δφ( x, y dxdy ab U 0εε 0 r Einω et ρh π ( v W 0
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 From initial condition m n ( W( x, y,0 q (0 φ x, y 0, we know q a b 0, Thu ab UE 0 εε 0 r (co mπ (co nπ ρh π M ω ( v W ( t ( ω inωet ωe inωt ω ωe (6 Table a. Natural frequencie (rad/ for variou F. F(N order 0. 0.6 Mode (, 4000 3874.67 Mode (, 7635.9 4883.56 5753.60 Mode (3,3 34985.7 67660.9 8873.0 Mode (4,4 5009.99 976.96 9436.8 6. Reult and Dicuion Conider a micro film defined by the data given in Table. The material ued for the micro beam i Al. It i ubjected to an electric field force. By Eq. (3, the variation of the voltage with repect to the micro film diplacement i obtained a hown in Fig.. From Fig., it i een: a (mm Table. Parameter of the micro film. b (mm h (μm v (μm ρ (kg m -3 5.7 0 3 Fig.. Change of voltage along with diplacement. There i an extreme voltage point, after which the micro beam buckle. Under the extreme voltage, the micro film diplacement i about equal to /3 of the initial clearance between micro film and bae plate. A the tenion in the film grow, the extreme voltage grow. It i becaue larger tenion in the film will caue larger tiffne of the film. Above equation are utilized for the free vibration analyi of the micro film. The parameter of the numerical example are hown in Table. Natural frequencie of the film under variou film tenion (here, v μm, U 0 V are hown in Table a; natural frequencie of the film under variou voltage (here v μm, F N are hown in Table b. εr Table b. Natural frequencie (rad/ for variou U0. U0(V order 3 Mode (, 8809.7. 3874.67 9979.98 Mode (, 5974. 5753.60 537.88 Mode (3,3 9088.69 8873.0 86040.69 Mode (4,4 055.56 9436.8 7430.83 From Table, the following obervation are worth noting: Without conidering electrotatic force, the natural frequencie of the film vibration are larger than thoe of conidering the electrotatic force. It i becaue the electrotatic force ytem i equivalent to a oft pring ytem. The electrotatic force can caue decreae of the natural frequencie of the micro film. The deviation between the natural frequencie of the film with and without the electrotatic force decreae with increaing the order number of the mode. For mode (, and FN, the relative error between the natural frequencie i 0.98 %. For mode (4,4 and FN, the relative error between the natural frequencie i.8 %. It how that influence of the electrotatic force on the natural frequencie decreae with increaing the order number of the mode. 3 At a contant voltage, the natural frequency of the film increae ignificantly with increaing the film tenion. A the film tenion grow, the deviation between the natural frequencie of the film with and without the electrotatic force decreae. At F0.N, the relative error between the natural frequencie with and without the electrotatic force i 58.6 % for mode (4,4. At F0.6N, the relative error between the natural frequencie with and without the electrotatic force i 4. %. At FN, the relative error between the natural frequencie with and without the electrotatic force i.8 %. By ubtituting thee natural frequencie into Eq. (4, the vibration mode of the micro film can be obtained (ee Fig. 3. The variation of the natural frequencie with repect to the variou parameter are hown in Fig. 4 (here, only reult of the firt mode are given. 0
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 (a mode (, (b mode (, and y0.5 mm (c mode (, (d mode (, and y0.5 mm (e mode (3,3 (f mode (3,3 and y0.5 mm Fig. 3. Vibration mode of the micro film. (a a change (b b change (c v change (d h change Fig. 4. Variation of the natural frequencie with repect to the variou parameter. 03
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 From Fig. 3 and 4, we know: In mode (,, the maximum dynamic diplacement occur at poition xa/ and yb/. In mode (,, the maximum dynamic diplacement occur at poition xa/4 and yb/4 and poition x3a/4 and y3b/4; the negative peak diplacement occur at poition xa/4 and y3b/4 and poition x3a/4 and yb/4. In mode (3,3, five poitive peak diplacement and four negative peak diplacement occur. The mode function are only dependent on mechanical parameter of the micro film and independent from electric parameter of the electromechanical ytem. A the length of the micro film increae, it natural frequency decreae firt rapidly, and then decreae lowly. For a higher order mode, the natural frequency decreae more rapidly with increaing the length of the micro film. 3 A the width of the micro film increae, it natural frequency decreae firt rapidly, and then decreae lowly a well. For a higher order mode, the natural frequency decreae more rapidly with increaing the width of the micro film a well. 4 A the clearance between micro film and bae plate increae, the natural frequency of the micro film increae. Under a maller clearance, the natural frequency of the micro film increae ignificantly while under a larger clearance, the natural frequency increae gradually. 5 A the thickne of the micro film increae, it natural frequency decreae gradually. For a higher order mode, the natural frequency decreae more rapidly with increaing the thickne of the micro film a well. Eq. ( and (6 are utilized for the analyi of the forced repone of the micro film to an exciting voltage. The forced repone correponding to each mode and the total forced repone are given a hown in Fig. 5 where only the firt five mode are conidered, exciting voltage E0.5 V, and exciting frequency ωe 00 πrad /. Fig. 6 how forced repone at center point (xa/ and yb/ on the micro film. The frequency repone of the micro film to voltage excitation are hown in Fig. 7. From Fig. 5 and 7, the following obervation were made: For different mode, though vibrating magnitude at different point of the micro film change along with time, the relative magnitude relationhip between the different point of the film are invariable. For given imple harmonic voltage excitation, the forced repone are alo imple harmonic vibration. A the exciting voltage frequency i near to the natural frequency for the mode (,, the forced repone magnitude correponding to mode (, i the larget. 3 When the exciting frequency ω e of the voltage i near ome natural frequency, reonance will occur. Under reonance, the magnitude and phae relation between different point on the micro film do not change. 4 A the order of the mode increae, the peak dynamic diplacement decreae obviouly. When the order number of the mode increae, the number of the peak dynamic diplacement increae. (a mode (, and yb/ (b mode (, and xa/ (c mode (3,3 and yb/ (d mode (3,3 and xa/ Fig. 5. Forced repone to voltage excitation. 04
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 (a mode (, (b mode (3,3 Fig. 6. Forced repone of the center point to voltage excitation. (a mode (, and yb/ (b mode (, and xa/ (c mode (3,3 and yb/ (d mode (3,3 and xa/ Fig. 7. Frequency repone of the micro film to voltage excitation. 7. Concluion In thi paper, an electromechanical coupled dynamic model of the micro film ubjected to electrotatic force i preented and it electromechanical coupled dynamic equation i given. By the equation, the natural frequencie and vibration mode of the micro film are invetigated. The forced repone of the film to voltage excitation are preented. Reult how: The natural frequency of the micro film ubjected to an electrotatic force i affected by mechanical and electric parameter uch a micro film length, width, thickne, tenion, voltage and clearance. At a contant voltage, the natural frequency of the film increae ignificantly with increaing the film tenion. A the film tenion grow, the deviation between the natural frequencie of the film with and without the electrotatic force decreae. 05
Senor & Tranducer, Vol. 76, Iue 8, Augut 04, pp. 99-06 Acknowledgment Thi project i upported by Key Baic Reearch Foundation in Hebei Province of China (39670D. Reference []. Thielicke E., Obermeier E., Microactuator and their technologie, Mechatronic, 0, 000, pp. 43-455. []. Zhongping Bao, Subrata Mukherjee, Electrotatic BEM for MEMS with thin conducting plate and hell, Engineering Analyi with Boundary Element, 8, 004, pp. 47-435. [3]. El-Hami M., Glynne-Jone P., White N. M., Hill M., Beeby S., Jame E., Brown A. D., Ro J. N., Deign and fabrication of a new vibration-baed electromechanical power generator, Senor and Actuator, A: Phyical, 9, -3, 00, pp. 335-34. [4]. Wang Pei-Hong, Dai Xu-Han, Fang Dong-Ming, Deign, fabrication and performance of a new vibration-baed electromagnetic micro power generator, Microelectronic Journal, 38,, 007, pp. 75-80. [5]. Kuehne Ingo, Frey Alexander, Marinkovic Djordje, Power MEMS-A capacitive vibration-to-electrical energy converter with built-in voltage, Senor and Actuator, A: Phyical, 4,, 008, pp. 63-69. [6]. Rafael Nadal-Guardia, Anna Maria Broa, Alfon Dehe, AC tranfer function of electrotatic capacitive enor baed on the -D equivalent model: application to ilicon microphone, Journal of Microelectromechanical Sytem,, 6, 003, pp. 97-978. [7]. Lu Xiaolin, Clarke Matthew L., Li Dawei, A um frequency generation vibrational tudy of the interference effect in poly (n-butyl methacrylate thin film andwiched between ilica and water, Journal of Phyical Chemitry, C, 5, 8, pp. 3759-3767. [8]. Akai Daiuke, Ozaki Katuya, Numata Yauyuki, Vibration analyi and tranmiion characteritic of piezoelectric micromachined ultraonic tranducer uing epitaxial Pb(Zr,TiO3 thin film on γ-alo3/si ubtrate, Japanee Journal of Applied Phyic, 5,, 0, PA04. [9]. Li Jin Bo, Xu Qi Feng, An electrotatic film vibration voltage tranducer, Applied Mechanic and Material, 3-34, 03, pp. 443-448. [0]. Gualdino A., Chu V., Conde J. P., Study of the out-of-plane vibrational mode in thin-film amorphou ilicon micromechanical dik reonator, Journal of Applied Phyic, 3, 7, 03, pp. 74904. []. Sorokin V. M., Yahchenko A. K., Vibration of a framele film membrane tabilized by the Ampère force in zero gravity, Journal of Applied Mechanic and Technical Phyic, 54, 6, 03, pp. 885-893. []. Sun L., Xu L., Electromechanical Coupled Non-linear Dynamic for Micro Plate Simply upported on four Side, in Proceeding of the International Conference on Mechanical Engineering and Material Science (MEMS 0, 0, pp.0-3. [3]. Sriniva D. J., Electromechanical Dynamic of imply-upported micro-plate, International Journal of Computational Engineering Reearch, 0,, 5, 0, pp.388-395. 04 Copyright, International Frequency Senor Aociation (IFSA Publihing, S. L. All right reerved. (http://www.enorportal.com 06