IMECE DRAFT

Transcription

1 Poceedings of IMECE8: 8 ASME Intentionl Mechnicl Engineeing Congess nd Eposition, Octobe -Novembe, 8 Boston, MA A IMECE8-8-DRAFT SENSOR DIAPHRAGM UNDER INITIAL TENSION: NONLINEAR INTERACTIONS DURING ASYMMETRIC OSCILLATIONS Xinhu Long Stte Ke Lboto of Mechnicl Sstem & Vibtion, Shnghi Jio Tong Univesit, Shnghi,, P.R. Chin, Autho fo coespondence(ph:(8-)-,, Emil: hlong@sjtu.edu.cn) Mio Yu nd Blum Blchndn Deptment of Mechnicl Engineeing, Univesit of Mlnd, College P, MD 7-. ABRACT In this ticle, investigtions into the nonline smmetic vibtions of pessue senso diphgm unde initil tension e pesented. A compehensive mechnics model bsed on plte with in-plne tension is pesented nd the effect of cubic nonlineit is studied on the nonline smmetic esponse when the ecittion fequenc is close to the ntul fequenc of n smmetic mode of the plte. The obtined esults show tht in the pesence of n intenl esonnce, depending on the initil tension, the esponse cn hve not onl the fom of stnding wve but lso the fom of tveling wve. The esults of this wo should be elevnt to diphgm-tpe stuctues used in mico-scle sensos including pessue sensos.. INTRODUCTION Thin film diphgm stuctues e fequentl used in silicon piezoesistive sensos, cpcitive sensos, nd fibe-optic sensos []-[]. One cn detect the vibtions of these diphgm stuctues though the displcements of the diphgm stuctues. The senso sensitivit, bndwidth, nd lineit e diectl elted to the stuctul behvio of the diphgm. Due to theml epnsion nd mismtch between djcent wfes, the wfe-bonding opetions m intoduce in-plne esidul stesses in the thin film diphgm stuctues. In tpicl silicon pessue senso, the diphgm is stetched thin stuctue nd the initil tension cn be s lge s GP []. The diphgm vibtions e usull nlzed b using membne equtions. Sttic membne equtions hve lso been used in othe senso designs []-[]. Howeve, s pointed out in the ecent wo of Yu nd Blchndn [], membne model is not lws the most ppopite one. Shepl nd Dugundji [] cied out sttic nlsis of clmped cicul plte unde initil tension nd studied the tnsition nge fom plte behvio to membne behvio in tems of the tension pmete. Su, Chen, Robets, nd Speing. [7] etended this wo to nlze lge deflections of pe-tensioned nnul plte bonded with igid boss unde ismmetic pessue in the pesence of in-plne loding. In the wo of Yu, Long, nd Blchndn, the wo pesented in [] is etended to the dnmic cse nd the tdeoffs between sensitivit, bndwidth, nd dnmic nge e ddessed though nonline nlsis. In most of el esech effots on the dnmic esponse of diphgm stuctue, hmonic nd smmetic ecittions e consideed. Fo studing smmetic esponses, Sidh, Moo, nd Nfeh [9] deived genel solvbilit condition fo nonline intections in the vibtions of clmped cicul plte. Yeo nd Lee [] e-emined pim esonnce stte studied b Sidh et l. nd coected the modultion equtions deived b Sidh et l. The esults

2 indicte the sted-stte esponse cn hve not onl the fom of stnding wve but lso the fom of tveling wve. In this ppe, the uthos follow the wo pesented in efeences [9-] nd build on thei elie effots [, 8], nd c out nonline nlsis of the smmetic vibtions of pessue senso diphgm unde initil tension. The est of this ticle is ognized s follows. In the second section, the model of plte with in-plne tension is povided. In the thid section, the equtions govening the nonline smmetic vibtions e deived when the sstem epeiences pim esonnce ecittion of the one-one mode. Numeicl esults fo thee diffeent diphgm stuctues e pesented in the fouth section. Finll, some ems e collected togethe nd pesented.. MODEL DEVELOPMENT AND SYEM EQUATIONS In Figue, clmped, cicul diphgm of dius of nd thicness h is illustted. The Young s modulus of elsticit nd Poisson s tio of the diphgm mteil e denoted b E nd v, espectivel. The initil tension pe unit length pplied to the diphgm is epesented b N. Figue : Illusttion of diphgm clmped long its edge. A non-dimensionl tension pmete is defined s N ( v ) T = = () D h E whee the constnt D = Eh /((- v )) nd T = N / h is the tension pe unit e. In the nlsis tht follows, it is shown the choice of plte model o membne model ctull depends on the tension pmete, not on just the initil tension pe unit length N pplied to the diphgm. Stting fom Love s equtions (Soedel [], Nfeh nd Moo []), including dmping, il in-plne foce pe unit length N, nd the tnsvese loding pe unit e f(,θ; t), the nonline ptil-diffeentil eqution govening plte with initil tension cn be obtined s w w Φ Φ Φ w w ρh + D w N w= t θ θ Φ Φ w w w μ (, ; ) + f θ t θ θ θ θ t () whee is the dil distnce fom the cente, θ is the ngul coodinte, w(,θ; t) is the tnsvese displcement, nd μ is the dmping coefficient. θ θ θ w w w w w Φ= Eh + () Fo convenience, the uthos ewite these equtions in tems of nondimensionl vibles, denoted b steiss, which e defined s follows:

3 * ρh * h * =, t = t, w= w, D h * h * * u = u, v = v, N = N, D ( ν ) * μ = ρhdμ, () ( ν ) Dh * Eh * f = f, Φ = Φ 7 Afte substituting Eq. () into () nd () nd dopping the steiss in the esult, one cn obtin w + w N w t w Φ Φ Φ w w = ε θ θ Φ Φ w w θ θ θ θ w μ + f (, θ; t ) t () w w w w w Φ= + θ θ θ () whee ( ν )h ε = () = + + (7) θ The eltionships mongst Φ, w nd the in-plne displcements u nd ve given b u w e = + u v w e = + + θ θ u v v w w γ = + + θ θ () () () The clmped immovble bound is descibed b w w=, =, u =, nd v = t = () Ming use of equtions (8-), one cn obtin the following two conditions on Φ : Φ Φ Φ ν + = t = () θ Φ Φ Φ + ν Φ + ν Φ + + = θ θ t = (). NONLINEAR INTERACTIONS: COUPLED OSCILLATOR EQUATIONS Conside pim esonnce ecittion of the n-th mode hving the fom (, ;) ˆ i t F θ t = f()cos nθe Ω + cc (7) nd let the displcement esponse be ppoimted s ( ) w (, θ;) t η()cos t nθ + ζ ()sin t nθ φ() (8) whee J( β) φ() = J( β) I ( α) I ( α ) = Φ + Φ ν Φ e θ Φ Φ Φ e = ν + θ Φ Φ γ = ( + ν) θ θ (8) (9) () The coefficient α nd β e detemined b chcteistic equtions I ( α ) J ( β ) J ( β ) I ( α ) = (9) ' ' nd is detemined b = φ( ) φ( ) d. On substituting Eq. (8) into Eq. (), the esult is Φ= ( η + ζ ) χ() + ( ζ η ) cosnθ ηζ sin nθ χ() () n φ = () whee χ ( φ φ ) ( φ n φ)

4 n φ χ = ( φ φ) + ( φ n φ) (b) The solution of eqution () tht stisfies the bound conditions cn be Φ= ψ() ( η + ζ ) + ( ζ η ) cosnθ ηζ sin nθ ψ() () whee ψ nd ψ cn be epnded in tem of the eigenfunctions of + Ψ = λ Ψ d d n d d () Ψ νψ + νn Ψ = t = (b) Ψ +Ψ + 8n + νn Ψ + n + ν Ψ= ( ) ( ) t = (c) Ψ () < (d) The solution of eqution (), which is bounded t the oigin, cn be epessed s Ψ = cj ( λ) + ci ( λ) () n n Imposing the bound conditions (b-c) leds to c λ J n νj n + νn Jn + c λ I n νi n + νn In = () c λ J n λ J n ( 8n νn ) λj n n ( ν) J n + c λ I n λ I n ( 8n νn ) λi n n ( ν) I n = (b) Fo ψ, n =, nd in this cse, n = fo ψ. One cn detemine λ nd c c b using equtions (7, b). Then, one hs c [ J ( ) c c I ( ) ] () ψ = λ + λ = c [ J n( ) c c I n( ) ] (b) ψ = λ + λ = In ode to get the coefficients c, ψ is multiplied b c [ J( λ) + c c I( λ) ] so tht [ J( λ) + c c I( λ) ] i c[ J( λ) + c c I( λ) ] = [ J( λ) + c c I( λ) ] χ( ) d = (7) Afte substituting Eqs. (8) nd () into Eq. (), multipling the outcome with φ()cos θ nd φ()sin θ, espectivel, nd integting the esults fom θ = to θ = π nd = to =, one cn obtin the coupled oscillto equtions whee η+ ω η = εμη εαη εαη ζ εαηζ εαζ + ε f cosωt (8) ζ + ω ζ = εμζ εα ζ εα ηζ εα ζη εαη (8b) α = α = α = α = φ ψ φ ψ φ + ψ φ + ψ φ 7 8 ( ) 7 n n n ( ψφ ψφ ) ( ψφ ψφ ψφ ψφ ) (9) ψφ d α = α = α = α8 = (9b) μ = μφ d nd f = f φd (). NUMERICAL RESULTS As epesenttive cse, Ml diphgm with the Young s modulus of elsticit E =. 9

5 P, densit ρ =.9 g/m, nd Poisson s tio υ =. is consideed. Fo diphgm dius of.7 mm nd thicness vlues of μm. μm, nd μm, the dependence of the - mode ntul fequenc on the tension pmete is shown fo ech of these cses in Figue. The ntul fequencies incese s the tension pmete is incesed. It is noted tht it is possible to get the sme - mode ntul fequenc (.87 Hz) s tht fo diphgm with h= μm nd = b choosing the ppopite tension pmetes. As pointed out in Figue, the tension pmete vlues e =. nd =9., fo h= μm nd h= μm, espectivel. The stuctul nd sstem pmetes fo the peviousl mentioned diphgm stuctues e povided in Tble. Tble : Stuctul sstem pmetes nd chcteistics Cses h (μm) (mm) f (Hz) h= μm. -. Ntul fequenc: ω (Hz) h= μm h= μm Figue.. Vition of the ntul fequenc with espect to the tension pmete fo diphgm stuctues with diffeent thicness vlues, fo h= µm, fo h= µm, nd fo h= µm. In Figue, the fequenc esponse cuves obtined b using AO97 [] e pesented to illustte the esponse of stuctue coesponding to cse of Tble. The coesponding mode hs one nodl dimete nd no othe nodl cicle ecept the one t the bound. In Figue, the bnches lbeled,,, nd coespond to the stble stnding wve, stble tveling wve, unstble stnding wve, nd unstble tveling wve, espectivel. The stble bnches nd unstble bnches e denoted b solid lines nd dshed lines, espectivel. Fom Figues nd, one cn find the esponse of diphgm tnsfom fom stnding wve into tveling wve with the incese of ecittion fequenc. The bifuction point is locted t (97Hz,. - m). As the ecittion fequenc inceses, thee is one unstble Response Amplitude η Response Amplitude ζ Ecittion Fequenc (Hz) Ecittion Fequenc (Hz) Figue. Vitions of the diphgm esponse mplitudes with espect to the ecittion fequenc fo Cse with = nd pessue p=p, unstble bnch, stble bnch. stnding wve bnch nd one stble tveling wve bnch in the fequenc nge (97Hz, Hz). Following tht, the esponse of diphgm becomes complicted nd thee e multiple unstble bnches nd multiple stble bnches in the fequenc nge (Hz, Hz). When the

6 ecittion fequenc eches Hz, thee is one stble stnding wve bnch nd multiple unstble bnches. In ode to show the deflection of the diphgm, the esponses obseved t ecittion fequencies of Hz nd Hz e consideed. The coesponding esults obtined b time domin simultions e pesented in Figue nd to illustte the stble esponses of the diphgm ove one peiod of ecittion. A stnding wve is shown in Figue, whee one cn obseve nodl line in ech subplot. In Figue, clocwise otting tveling wve is shown. The esponses shown in Figues nd coespond to pim esonnce ecittion. - t= - t=t/ t=t/ - t=t/ t=t/ -7 t=t/ Figue. Deflections of the diphgm in Cse ove one peiod of ecittion when Hz π Ω = nd p=p.

7 - t= - t=t/ t=t/ - t=t/ t=t/ - t=t/ Figue. Deflections of the diphgm in Cse ove one peiod of ecittion when Ω π = Hz nd p=p. To loo into the initil-tension effects on the esponse of diphgm, the fequenc-esponse cuves obtined fo Cses nd of Tble e pesented in Figue nd 7. Comping Figue nd 7 with Figue, one cn find tht the bnches,,, nd e simil to those of Figue. The diffeences e in the loctions of the bifuction points. In Cses nd, the fist bifuction points e locted t (998Hz,.8 - m) nd (Hz,. - m), espectivel, s the ecittion fequenc is incesed. One unstble stnding wve bnch nd one stble tveling wve bnch e locted in the fequenc nges (Hz, Hz) nd (Hz, 8Hz) in Cses nd, espectivel. Response Amplitude (η) Ecittion Fequenc (Hz) Response Amplitude (ζ) Ecittion Fequenc (Hz) Figue. Vitions of the diphgm esponse

8 mplitudes with espect to the ecittion fequenc fo Cse with =. nd pessue p=p, unstble, stble bnch. 7 - Response Amplitude η (m) Response Amplitude ζ (m) Ecittion Fequenc (Hz) Ecittion Fequenc (Hz) Figue 7. Vitions of the diphgm esponse mplitudes with espect to the ecittion fequenc fo Cse with =9. nd pessue p=p, unstble bnch, stble bnch, Response Amplitude (η) (m) Ecittion Foce (P) Response Amplitude (ζ) (m) Ecittion Foce (P) Figue 8. Vitions of the diphgm esponse mplitudes with espect to the ecittion pessue fo Cse with =9. nd Ω=. Hz, -----, unstble bnch, stble bnch. To investigte nonline intections with espect to the ecittion pessue level, the esponse cuves obtined in Cse e shown in Figue 8 when the ecittion fequenc is close to the - mode ntul fequenc. Fom this figue, one cn find tht stble stnding wves occu in the smll nge (, P). With incese of ecittion pessue, the esponses of diphgm become complicted nd thee e multiple stble bnches nd multiple unstble bnches in the nge ( P, P)..CLOSURE In this effot, the nonline smmetic vibtions of pessue senso diphgm unde initil tension e investigted. A compehensive mechnics model bsed on plte with in-plne tension is pesented nd the effect of cubic nonlineit is studied on the nonline smmetic esponse when the ecittion fequenc is close to the ntul fequenc of n smmetic mode of the plte. The obtined esults show tht in the pesence of n intenl esonnce, the bifuction loctions depending on the initil tension. The esponse cn hve not onl the fom of stnding wve but lso the fom of tveling wve. REFERENCES. Mtsuo, Y., Ymmoto, Y., Tnbe, M., Shimd, S., Ymd, K., Ysuw, A., nd H. Mtsuz, (99), Low-pessue mesuement limits fo silicon

9 piezoesistive cicul diphgm sensos, Jounl of Micomechnics nd Micoengineeing, Vol., pp. -.. Pedesen, M., Meijein, M. G. H., Olthuis, W., nd Begveld, P., (997), A cpcitive diffeentil pessue senso with polimide diphgm, Jounl of Micomechnics nd Micoengineeing, Vol. 7, pp... Zeng, N., Shi, C., Wng, D., Zhng, M., nd Y. Lio, (), Diphgm-tpe fibe-optic intefeometic coustic senso, Opticl Engineeing, Vol. (9), pp. 8.. Cho, S. T. Njfi, K, nd Wise, K. D., (99), Intenl Stess Compenstion nd Scling in Ultsensitive Silicon Pessue Senso, IEEE Tnsction on Electon Devices, Vol. 9, pp Yu, M. nd Blchndn, B. (), Senso diphgm unde initil tension: Line nlsis, Epeimentl Mechnics, Vol., pp.-9.. Shepl, M. nd Dugundji, J.. (998), Lge deflections of clmped cicul pltes unde initil tension nd tnsitions to membne behvio, ASME Jounl of Applied Mechnics, Vol., pp Su, Y..H., Chen, K.S., Robets, D.C., nd Speing S. M., (), Lge deflection nlsis of pe-stessed nnul plte with igid boss unde ismmetic loding, Jounl of Micomechnics nd Micoengineeing, Vol., pp Yu, M., Long, X.-H., nd Blchndn, B., (8), Senso diphgm unde initil tension: Nonline esponses nd desigh implictions, Jounl of Sound nd Vibtion, Vol., pp Sidh, S, Moo, D. T., nd Nfeh, A. H., (978), Nonline esonnces in the foced esponses of pltes. Pt II: smmetic esponses of cicul pltes, Jounl of Sound nd Vibtion, Vol. 9, pp Yeo, M. H. nd Lee, W. K., (), Coected solvbilit conditions fo non-line smmetic vibtions of cicl plte, Jounl of Sound nd Vibtion, Vol. 7, pp. -.. Nfeh, A. H. nd Pi, P. F., (), Line nd Nonline Stuctul Mechnics, Wile, New Yo.. Nfeh, A. H. nd Moo, D. T., (979), Nonline Oscilltions, Wile, New Yo.. Soedel, W., (99), Vibtions of Shells nd Pltes, Mcel Dee, New Yo.. Doedel, E. J., Chmpnes, A. R., Figieve, T. F., Kuznetsov, Y. A., Sndstede, B., nd Wng, X., (997), AO97: Continution nd Bifuction Softwe fo Odin Diffeentil Equtions (with HOMCONT), Technicl Repot, Concodi Univesit.