The Electromechanical Behavior of a Micro-Beam Driven by Traveling Electrostatic Force

Transcription

1 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong The Electroechancal Behavor of a Mcro-Bea Drven by Travelng Electrostatc Force Yuh-Chung Hu We-Hsang Tu Pe-Zen Chang Chh-Kung ee Davd T.W. n and Chung-Neng Huang Abstract There s no lterature entonng the electroechancal behavor of a cro-bea one of the coon crostructure n cro devces drven by travelng electrostatc forces. Ths paper s to present the dynacs and stabltes of a cro-bea on whch an electrostatc force travels to and fro. The travelng electrostatc force ay be nduced by a row of actuated electrodes whch are arranged beneath the cro-bea and actuated sequentally. The lnearzed dstrbuted odel consderng the electroechancal couplng effect between electrostatc force and structure s derved based on the sall deflecton assupton. Accordng to the analytcal results the cro-bea wll resonate when the travelng speeds of the electrostatc force approach soe crtcal speeds. The crtcal speeds are equal to the rato of the natural frequences to the wave nuber of the correlatve natural ode of the cro-bea. Besdes the cro-bea ay be unstable at soe travelng speeds dfferng fro the crtcal speeds. Furtherore the unstable regons expand wth ncreasng drvng voltage. Ths artcle ay lead to a new research branch on electrostatc-drven cro devces. Index Ters electrostatcs MEMS crostructures stabltes T I. INTRODUCTION he electrostatc-drven devce s coonly used for actuatng and sensng n MEMS feld due to any advantages such as rapd response low power consupton copatblty wth the fabrcaton process of ntegrated crcuts (ICs). However t gves a challenge to accurately odel electrostatc crostructures because of the nonlnear electroechancal couplng behavor. Ths couplng behavor results fro the effects of non-deal boundary condtons Manuscrpt receved January 3 14; revsed January Ths work was n part supported by the Natonal Scence Councl of Tawan under Grant NSC 1-68-E MY3. Yuh-Chung Hu s wth the Natonal Ian Unversty Ian 641 Tawan (correspondng author to provde phone: ; fax: ; e-al: ychu@nu.edu.tw). We-Hsang Tu was wth the Natonal Tawan Unversty Tape 1617 Tawan. He s now wth the CADIP nternatonal corporaton (e-al: whtu@ntues.net). Pe-Zen Chang s wth the Natonal Tawan Unversty Tape 1617 Tawan (e-al: changpz@ntues.net). Chh-Kung ee s wth the Natonal Tawan Unversty Tape 1617 Tawan (e-al: cklee@ntues.net). Davd T.W. n s wth the Natonal Unversty of Tanan Tanan 7 Tawan. (e-al: davd@al.nutn.edu.tw). Chung-Neng Huang s wth the Natonal Unversty of Tanan Tanan 7 Tawan. (e-al: kosono@al.nutn.edu.tw). ISBN: ISSN: (Prnt); ISSN: (Onlne) frngng felds the pre-deforaton nduced by ntal stresses and non-hoogeneous structures. In 5 Marquès [1] presented an overvew of the exstng technques about of the odelng and dynac behavor of the electrostatc-drven cro structures. In 8 Rhoads [] provded a coprehensve survey of the fundaental research on nonlnear behavors of electrostatc-drven croresonators ncludng drect and paraetrc resonances paraetrc aplfcaton pacts self-excted oscllatons and collectve behavors such as localzaton and synchronzaton whch arse n coupled resonator arrays. In 7 other revew papers [3] [4] gave a general survey of the exstng analytcal odels for electrostatc-drven crodevces. In 1 the authors Hu and Chang publshed a revew paper [5] whch ntroduced the technques for the physcal odel of pull-n voltage dynac characterstc analyss ar dapng effect relablty nuercal odelng ethod and applcaton of electrostatc-drven MEMS devces. Besdes the effects of non-deal boundary condtons frngng felds the pre-deforaton nduced by ntal stresses were also dscussed n that revew paper. Hu had also nvestegated n detal about the dynacs and nstabltes of crostructures drven by electrostatc forces [6]-[8]. Up to now there s only one lterature [8] publshed by Hu nvestgatng the behavor of a cro-rng drven by travelng electrostatc forces. Thus ths paper the extenson of [8] presents the dynacs and stabltes of a cro-bea one of the coon structures n cro electrostatc devces drven by travelng electrostatc forces. We derve the governng equaton of a cro-bea on whch a haronc electrostatc force travelng to and fro. The cro-bea s claped at both ends that s the deflecton and roraton angle of the bea are zero at both ends. The Euler-Bernoull bea odel [9] s used n odelng the cro-bea structure. The ovng feature of the travelng electrostatc force s odeled based on the author Hu s publshed work [6] naely as a Drac delta functon whose locaton vares wth the travelng speed. Frst of all a lnearzed densonless governng equaton s derved based on the aforesad assuptons whch s a lnear partal dfferental equaton wth perodcally te-varyng coeffcent. Next dscretze the governng equaton to a set of lnear ordnary dfferental equatons wth perodcally te-varyng coeffcents. Then solve the dscretzed governng equaton by the Runge-Kutta ethod whch s a nuercal ntegraton ethod avalable n the coercal software Matheatca. The dynacs and stabltes of the IMECS 14

2 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong cro-bea are both studed n ths artcle. The Floquet theory [1] [11] s used to deterne whether the cro-bea s stable or not. II. SYSTEM GOVERNING EQUATIONS Consder a cro bea beneath whch s a row of dentcal and equal-spaced electrodes as shown n Fg. 1. The cro bea s claped at both ends that s the deflectons and rotaton angle are both zero at the ends. We assue that the wdth of the electrodes and ther gaps are uch saller than the length of the bea. The dea of travelng electrostatc load can be realzed by the sequentally actuatng electrodes. Consder a cro bea subjected to the sequentally actuaton of equal-length (l) dstrbuted electrodes as shown n Fg. 1 the governng equaton s gven by ˆ wx tˆ 4 wˆ xˆ t ˆ ˆ A EI ˆ 4 t xˆ ˆ bv U xˆ x l U xˆ x l g wˆ where A s cross-sectonal area E s Young s odulus I s nerta oent of the cross-sectonal area ρ s densty of the cro bea b s the bea wdth g s the ntal gap between the flexble bea the fxed botto electrode V ˆ s the potental dfference between the flexble bea and the fxed botto electrode ε s the delectrc coeffcent of the delectrc between the flexble bea and the botto electrode respectvely whle xˆ and ˆt are the varables of space and te respectvely. x represents the central locaton of the actve electrode whle the unt step functon U s defned as U xˆ xˆ for xˆ xˆ 1 for xˆ xˆ. The central locaton x of the actve electrode at any nstant can be represented by tˆ x 1cos 1 cos ˆ tˆ. T In (3) T s the perod of the travelng pont load respectvely (1) () (3) and the te ˆt was contnuous f the length and space of the electrodes are uch less than the bea length otherwse dscrete. Now let us consder the case that the length and space of the electrodes are uch less than the bea length. Then the governng equaton can be rewrtten as ˆ wx tˆ 4 wˆ xˆ t ˆ ˆ A EI ˆ 4 t xˆ ˆ xˆ 1cos. ˆ ˆ ˆ wv t g w Here we defne the followng densonless varables: x xˆ w wˆ h t tˆ T ˆ C (5) T C where T C s a te-scale defned as T A 4 / EI. Then the governng equaton can be rewrtten n the densonless varables wx t 4 w x t 4 t x 1 x 1cos 1 ˆ V t w 4 3 where V Vˆ and s defned as ( b ) ( EIg ). Expand the electrostatc ter V (1 w) by Taylor seres wth respect to the equlbru poston.e. w = V 1 w 3 V 1 w 3w 4 w. Our nterest les n sall deflectons about the equlbru poston.e. otons confned to sall neghborhoods of the equlbru poston. Based on the sall deflecton assupton the second and hgher order ters of the electrostatc expanson can be neglected thus the governng equaton can be lnearzed as wx t 4 w x t 1 x 4 1 cos t V w t x 1 1 cos. x t V C (4) (6) (7) (8) Fg. 1. The scheatc dagra of a cro claped-claped bea drven by a travelng electrostatc force. III. DYNAMIC ANAYSIS The dynacs of a claped-claped cro-bea subjected to a travelng electrostatc load s analyzed n ths secton. It s known that the dynac characterstcs of a syste depend on the geoetrcal densons and ateral propertes. However for an electrostatc-drven syste the electrostatc forces wll alter the dynac characterstcs of structures and as the structure defors the electrostatc forces redstrbute thereby odfyng the echancal loads. The so-called electroechancal couplng effect ust be taken ISBN: ISSN: (Prnt); ISSN: (Onlne) IMECS 14

3 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong nto account on the desgn of electrostatc-drven cro-systes. For a cro claped-claped bea the deflecton functon w( x t ) can be expanded as the lnear cobnaton of the bea s natural odes naely w x t t x. (9) 1 where ( x) s the -th ode and the coeffcent (t) to be solved s the assocated odal partcpaton factor. Snce the natural ode s the exact soluton to the free vbraton of structures t essentally satsfes the boundary condtons and the hoogeneous part of the governng equaton of a dynacs syste. Thus the natural odes for the foundaton for forced-response calculatons n structural dynacs. Ths s why we can expand the deflecton functon w(xt) as the lnear cobnaton of the bea s natural odes. The natural odes of a bea claped at both ends are gven by ( x) cosh x cos x snh x sn x (1) where (cosh cos ) (snh sn ) and λ satsfes cos cosh 1. Substtutng (9) nto (8) gves 4 t x t x x 1 cos t V t x cos x t V (11) where s the densonless natural frequency of the -th ode. Multplyng each ter by the n-th ode and then ntegratng over the doan of the bea gves 4 t x x dx t x x dx n n (1) 1 cos x t V t x n x dx cos x t V n xdx. Snce the ode shape functons posses the followng orthogonaltes TABE I GEOMETRIC AND MATERIA PARAMETER OF THE CAMPED-CAMPED MICRO-BEAM. Paraeters 4 1 n t nn t V t 1 cos 1 t (14) 1 1 n 1 cos t V n 1 cos t. By the use of 13-ters odal expanson (n = 1-13) (14) can be rewrtten as n t nn t V t 1 cos 1 t (15) 1 1 n 1 cos t V n 1 cos t. Substtutng the soluton of (15) nto (9) gves the dynac response of the syste. To obtan the dynac response we have to solve (15) by the coercal software Matheatca. Set zero ntal condtons naely () (). Table I shows the geoetrc and ateral paraeters of the cro-bea. The frst three densonless crtcal speeds (Ω) of the claped-claped cro-bea are and 19.1 respectvely. Fg. shows the frequency response of the claped-claped cro-bea at the locaton of travelng electrostatc force. There are crtcal speeds for the odd-nuber odes / (the subscrpt odd n odd k odd k represent the postve odd ntegers and k = ) for exaple Ω 11 = ω 1 / Ω 1 = ω 1 /4... Ω 31 = ω 3 / Ω 3 = ω 3 /4 Ω 33 = ω 3 /6...etc. and the crtcal speeds for the even-nuber odes are deterned as / ( 1) (the subscrpt even r even r n Values Young s odulus (E) (GPa) 168 Densty (ρ) (kg/ 3 ) 33 ength () () Wdth (b) () 5 Thckness (h) (μ) 1 Gap (g) (μ) 1 Perttvty of free space (k) (F/) Delectrc constant (ε) 1 Mode 1 Mode Mode 3 x 1 for n n xdx for n. (13) Consequently Fg.. The frequency response of the claped-claped cro-bea at the locaton of the travelng electrostatc force. ISBN: ISSN: (Prnt); ISSN: (Onlne) IMECS 14

4 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong even represent the postve even ntegers and r = ) for exaple Ω 1 = ω 1 /1 Ω = ω /3 Ω 3 = ω /5...etc. ω s the densonless natural frequency of the th ode. Fg. 3 shows the frequency response of the claped-claped cro-bea at the locaton of the travelng electrostatc forces wth dfferent drvng voltages. The stffness of the claped-claped cro-bea wll softened by the drvng voltage and the varaton ncreases wth ncreasng the agntude of drvng voltage. IV. STABIITY ANAYSIS Accordng to the results of dynac response the bea drven by a travelng electrostatc force ay be unstable or resonant at the crtcal speeds of the travelng electrostatc force. Therefore we nvestgate the dynac nstablty usng Floquet theory n ths secton. Frst of all (14) ust be transfored nto the state space to reduce the order. Thus defne the state vector as dagonal atrces whose eleents are M n 4 1 for n n for n Kn for n for n. (1) The equvalent electrostatc-stffness atrx [H n ] s a syetrc atrx whose eleents are gven by 1 Hn V t 1cos t 1 1 n 1 cos t () The generalzed force Q n s defned as 1 Q 1 cos. n V n t (3) Y () t () t (16) Consder the hoogeneous part of (17) naely the rght-hand sde of the equal sgn s zero and set n lnear-ndependent ntal condtons naely and transfor (14) nto state space as M ˆ Y K ˆ Y F ˆ where Mn K Mˆ H n n Kn Hn K H Kˆ n n F Q n ˆ. (17) (18) (19) () The ass atrx [M n ] and the stffness atrx [K n ] are 1 1 (4) () () (). 1 n 1 Y Y Y n1 n1 n1 By the Runge-Kutta nuercal ntegraton ethod we obtan n lnear-ndependent hoogeneous solutons of (17) n one perod T naely {Y(T)} 1 {Y(T)} {Y(T)} n. The lnear-ndependent hoogeneous solutons construct the onodroy atrx [C] also known as the state transton atrx.e. C Y T Y T Y T nn 1 n (5) The stablty can be deterned by the nature of the egenvalues (μ ) of the onodroy atrx [C] [1] [11]. The syste s stable f all the egenvalues have agntudes less than unty.e. 1 unstable f at least one egenvalue greater than unty.e. 1 and argnally stable f at least one egenvalue wth unt agntude and ultplcty less than unty. Fg. 4 shows the nuercal results of stablty analyss of the frst three odes n travelng speed Ω over the regon of 1.6 to 1. The nstable regons near Ω nspq = ω p /q say Ω ns11 = ω 1 Ω ns1 = ω 1 /... Ω ns = ω / Ω ns3 = ω /3 Ω ns4 = ω /4... Ω ns34 = ω 3 /4 Ω ns35 = ω 3 /5 Ω ns36 = ω 3 /6 Ω ns37 = ω 3 /7...etc. Fg. 5 shows that the dynac nstable regons expand wth the ncreasng of the drvng voltage and toward lower frequences. Fg. 3. The frequency responses of the claped-claped cro-bea at the locaton of the travelng electrostatc forces wth dfferent drvng voltages. ISBN: ISSN: (Prnt); ISSN: (Onlne) V. CONCUSION An analytcal approach to the dynac and stablty analyss of the crostructures subjected to-and-fro travelng IMECS 14

5 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong Fg. 4. The nstablty of the claped-claped cro-bea drven by a travelng electrostatc force. Fg. 5. The enlarged dynac nstable regon n the parts of Ω ns1 (= ω 1) and Ω ns311 (= ω 1/5.5). electrostatc forces s presented n ths paper. A lnearzed dstrbuted syste odel consderng the structure-electrostatc couplng effect s derved based on the sall deflecton assupton. Accordng to the analytcal results the cro-bea wll resonate when the travelng speeds of the electrostatc force approach soe crtcal speeds. The crtcal speeds are equal to the rato of the natural frequences to the wave nuber of the correlatve natural ode of the cro-bea. Accordng to the results of stablty analyss the nstable regons appear not only near the ultples of resonant frequences but also near fractons of resonant frequences. Furtherore the nstable regons expand wth ncreasng drvng voltage. [5] W. C. Chuang H.. ee P. Z. Chang and Y.C. Hu Revew on the odelng of electrostatc MEMS Sensors vol. 1 no.6 pp Jun. 1. [6] Y. C. Hu S. C. Huang Forced response of sandwcth rng wth vscoelastc core subjected to travelng loads J. Acoust. Soc. A. vol. 16 pp. 1 Mar [7] Y. C. Hu C. M. Chang and S. C. Huang Soe desgn consderatons on the electrostatcally actuated crostructures Sens. Actuat. A vol. 11 no. 1 pp Apr. 4. [8] X. Ye Y. Chen D. C. Chen K. Y. Huang and Y. C. Hu The Electroechancal Behavor of a Mcro-Rng Drven by Travelng Electrostatc Force Sensors vol. 1 no. pp Jan. 1. [9] W. Soedel n Vbratons of Shells and Plates nd ed. New York: Marcel Dekker Inc pp [1] J. A. Rchards n Analyss of Perodcally Te-Varyng Systes Berln: Sprnger-Verlag 1983 pp [11] K. Farhang and A. Mdha Steady-state response of perodcally te-varyng lnear systes wth applcaton to an elastc echans J. Mech. Des. vol. 117 no. 4 pp Dec Yuh-Chung Hu receved hs Ph.D. fro the Departent of Mechancal Engneerng Natonal Tawan Unversty of Scence and Technology Tape Tawan n He s now wth the Departent of Mechancal and Electroechancal Engneerng Natonal Ian Unversty Ian Tawan. Hs current research nterests are n the areas of cro transducers and appled echancs for cro-systes. REFERENCES [1] A. F. Marquès R. C. Castelló and A. M. Shkel Modelng the Electrostatc Actuaton of MEMS: State of the Art Unverstat Poltècnca de Catalunya Insttut d Organtzacó Control de Sstees Industals: Barcelona Span 5 IOC-DT-P [] J. Rhoads S. W. Shaw and K.. Turner Nonlnear Dynacs and Its Applcatons n Mcro- and Nano-Resonators n 8 Proc. ASME Dynac Systes and Control Conf. pp.. [3] R. C. Batra M. Porfr and D. Spnello Revew of odelng electrostatcally actuated croelectroechancal systes Sart Mater. Struct. vol. 16 no. 6 pp. R3 R31 Oct. 7. [4] W. M. Zhang G. Meng and D. Chen Stablty nonlnearty and relablty of electrostatcally actuated MEMS devces Sensors vol. 7 no. 5 pp May. 7. We-Hsang Tu receved hs Ph.D. fro the Departent of Engneerng Scence and Ocean Engneerng Natonal Tawan Unversty Tape Tawan n 13. He s now wth the CADIP nternatonal corporaton. ISBN: ISSN: (Prnt); ISSN: (Onlne) IMECS 14

6 Proceedngs of the Internatonal MultConference of Engneers and Coputer Scentsts 14 Vol II IMECS 14 March Hong Kong Pe-Zen Chang receved hs Ph.D. fro the Departent of Theoretcal and Appled Mechancs Cornell Unversty Ithaca NY n 199. He s now wth the Insttute of Appled Mechancs Natonal Tawan Unversty Tape Tawan. Hs current research nterests are n the areas of cro-systes agnetc echancal systes and appled echancs of superconductors. Chh-Kung ee receved hs Ph.D. fro the Departent of Theoretcal and Appled Mechancs Cornell Unversty Ithaca NY n He s now wth the Insttute of Appled Mechancs Natonal Tawan Unversty Tape Tawan. Hs current research nterests are n the areas of cro-systes nanotechnologes photonc systes and bochp systes. Davd T. W. n receved hs Ph.D. fro Natonal Cheng Kung Unversty Tanan Tawan n 4. He s now wth the Graduate Insttute of Mechatronc Syste Engneerng Natonal Unversty of Tanan Tanan Tawan. Hs current research nterests are n the area of optzaton electronc devce coolng and nverse proble. Chung-Neng Huang receved hs Ph.D. fro the Group of ECEI Graduate School of Engneerng Tohoku Unversty Senda Japan n. He s now wth the Graduate Insttute of Mechatronc Syste Engneerng Natonal Unversty of Tanan Tanan Tawan. Hs current research nterests are n the area of electroechancal ntegraton power systes. ISBN: ISSN: (Prnt); ISSN: (Onlne) IMECS 14