MODELLING OF BEAM EXCITED BY PIEZOELECTRIC ACTUATORS IN VIEW OF TACTILE APPLICATIONS

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1 MODELLING OF BEAM EXCITED BY PIEZOELECTRIC ACTUATORS IN VIEW OF TACTILE APPLICATIONS Clément Ndl, Christophe Girud-Audine, Frédéric Girud, Michel Amberg, Betty Lemire-Semil To cite this version: Clément Ndl, Christophe Girud-Audine, Frédéric Girud, Michel Amberg, Betty Lemire-Semil. MODELLING OF BEAM EXCITED BY PIEZOELECTRIC ACTUATORS IN VIEW OF TACTILE APPLICATIONS. ELECTRIMACS 2014, My 2014, Vlenci, Spin <hl > HAL Id: hl Submitted on 1 Sep 2014 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re published or not. The documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.

2 MODELLING OF BEAM EXCITED BY PIEZOELECTRIC ACTUATORS IN VIEW OF TACTILE APPLICATIONS C. Ndl, C. Girud-Audine, F. Girud, M. Amberg nd B. Lemire-Semil L2EP-IRCICA, Université de Lille 1, 50 venue Hlley, Villeneuve d Ascq, Frnce e-mil : clement.ndl@ircic.univ-lille1.fr Abstrct - This pper dels with n one-dimensionl semi-nlyticl modelling of the vibrtory behviour of rectngulr bem excited by severl piezoelectric cermics glued on its lower fce. The estblishment of the equtions of motion is gurnteed by the ppliction of Hmilton s principle. From this pproch, it results n ccurte knowledge on the mode shpe in function of the geometry, the structure nd the positions of the piezoelectric ctutors. This pproch deployed on simple cse flls within generl process of the tctile surfces optimiztion. This rticle shows tht there is trde-off between the optimiztion of the homogeneity of the tctile stimultion nd the electromechnicl coupling fctor. Keyword - Semi-nlyticl modelling, Piezoelectricity, Tctile pplictions, Hmilton s principle. 1 INTRODUCTION During the lst decde, noticeble interest hs been demonstrted for the sense of touch in the humn-computer interction, owing to its importnce in the perception of our world by the mnipultion nd the identifiction of the objects surrounding us. In order to enhnce the interction between the user nd the communicting device, it is considered to provide to the former rendering of tctile senstions mtching the ctions he performs. In this sense, the field of tctile feedbck hs been emphsized by the emergence of severl technologies. One of them uses lterl forces which crete the illusion of fine textures nd surfce fetures 1]. Mny hptic surfces hve been relized to tke dvntge of this illusion. Wtnbe nd Fukui 2] developed the first ultrsonic vibrting plte, with resonnt frequency of 75.6 khz nd mplitude 2 µm, ble to control the roughness surfce displyed to bre finger. The min ide ws to use the reduction of the ective friction force thnks to n ir film between the vibrting plte nd the user s finger, the so-clled squeeze film ect. Biet et l. 3] used monomorph composed of n rry of piezoelectric ctutors bonded on the lower fce of metllic plte to generte out-of-plne vibrtions of 1 µm mplitude t 30.5 khz resonnt frequency. A circulr vrint of such device ws developed by Winfield et l. 4] who mounted glss disk on circulr piezoelectric ctutor. In ner future, these technologies will be destined to equip generl public systems with tctile screen (smrt phones, tblet computers, lptops, remote controls, etc.). In fct, their implementtion in limited environment gives rise to optimiztion questions in terms of, especilly, energy consumption nd mss of used piezoelectric cermics. A first ttempt hs been crried out by Sergent et l. 5] who determined the optiml dimensions of the resontor nd the cermics composing the device in order to obtin mximl deflection with miniml supply voltge. In this pper, different problem is ddressed. Indeed, it is shown 3] tht n homogeneous deformtion is required in order to get n uniform tctile stimulus thus, the gol is to obtin such condition, with miniml mount of cermics nd enough mplitude. Moreover, the loction of the piezoelectric cermic ctutors is lso optimized to select specific mode shpes to relize stisfctory stimultion. Briefly put, the deformtion should be s homogeneous s possible in order to obtin n uniform tctile stimulus ll over the plte surfce while hving the lrgest conversion fctor. To chieve this, semi-nlyticl model is developed in this pper. Its min im is to study the influence of the cermics on the resonnt frequencies (the squeeze film ect being ective for frequency greter thn 25 khz 3]) nd the mode shpes, especilly in terms of promotion nd uniformity. Bsed on some wellknown ssumptions used for the study of lminted structures, such modelling is commonly crried out for the ctive vibrtion control. For instnce, Ducrne et l. 6] optimize the plcement nd the dimensions of shunted piezoelectric cermics dedicted to

3 vibrtion reduction. Numericl methods hve been lso vlidted for the understnding of the dynmic of these ctive structures (see e.g. 7]). In this pper, generlized model of free-free bem excited by n piezoelectric ctutors bonded on its lower fces, defining non-symmetric geometry, is detiled. In fct, in this rticle, the geometry nd the different ssumptions enbling its formultion will be firstly specified. Then, the equtions of motion will be deduced from the ppliction of Hmilton s principle. Finlly, exmple structures composed of two cermics will be simulted nd studied in terms of homogeneity nd promotion of mechnicl mode shpes. 2 SEMI-ANALYTICAL MODELLING 2.1 GEOMETRY AND ASSUMPTIONS The semi-nlyticl modelling developed below relies on geometry of rectngulr cross-section bem composed of N prts in the longitudinl direction nd excited on its bck, s illustrted by Fig. 1, by n piezoelectric cermics behving like ctutors. The bem is supposed to be prllelepiped of dimensions L b w b h b. The i th ctutor is prllelepiped of the sme width thn the bem w = w b, length L nd thickness h. These cermics re polrized long the z-xis nd re supplied from their lower nd upper fces in order to preferentilly use trnsverse piezoelectric coupling (vrition of the smple length long the x-xis perpendiculr to the polriztion direction). Therefter, the Crtesin coordinte system is chosen nd its origin is put t the center of the bem. Moreover, the i th cermic ctutor re locted by the set of bscisse (x, x +). The modelling of the bem dynmic is bsed on the Clssicl Lminted Plte Theory (CLPT) (see e.g. 8], Sec. 3.3, pp ) dpted to n one-dimensionl problem. The min hypothesis of this pproch re reminded below to which the electricl ssumptions hve been dded. M.1 The thicknesses of the bem h b nd the cermics (h ) i 1,n re supposed to be smll compred to the min dimension of the bem. M.2 Any plne prllel to the xz-plne is supposed to be symmetry plne for the problem geometry. Thus, ll mechnicl nd electricl quntities re independent of the spce vrible y. Furthermore, the displcement long the y-xis, u y, the trnsverse strin nd stress, S yy nd T yy respectively, nd the sher strins round x- nd z-xis, S yz nd S xy respectively, re neglected (u y = 0, S yy = S yz = S xy = 0 nd T yy = 0). M.3 All cross-sections perpendiculr to the neutrl xis remin plnr nd perpendiculr fter deformtion. The sher in the xy-plne is implicitly neglected (S xz = 0). M.4 The pure flexure cse being the min point of interest in this study, it is ssumed tht the norml extension is negligible during the deformtion. The deformed bem does not consequently stretch long the z-xis resulting in zero norml strin (S zz = 0). The norml stress is lso neglected (T zz = 0). M.5 The cermics re perfectly glued on the bem nd the thickness of the bonding lyer is negligible so it is not tken into ccount. Moreover, the displcements re supposed to be common to the bem nd the cermics. M.6 The bem is supposed to be free on its ends which implies tht the xil force N, the bending moment M nd the sher force Q = M,x (the symbol (.),x denotes prtil derivtive with respect to the vrible x) re null t x = ±L b /2. E.1 The cermic ctutors being ssumed resonbly thin nd supplied from its lower nd upper fces, the electric field E, deriving from the electric potentil ϕ, is supposed to be only orientted long the z- xis so much tht : E = 0 0 ϕ,z ] T (1) where the superscripted T indictes mtrix trnsposition. E.2 The i th cermic ctutor is supposed to be respectively connected to the potentil v (t) nd the ground on its lower nd upper fces so tht : { ϕ(x, z = hb /2 h, t) = v (t) (2) ϕ(x, z = h b /2, t) = CONSTITUTIVE RELATIONSHIPS The mteril of the bem is supposed to be liner, isotropic nd homogeneous, therefore the strin S nd stress T vectors re linked by the Hooke s lw. From the ssumptions (M.2), (M.3) nd (M.4), the constitutive reltionship reduces to the following eqution : T = c]s T xx = E b S xx (3) where E b is the Young s modulus of the mteril constituting the bem. Due to the ssumptions (M.2), (M.3), (M.4) nd (E.1), the equtions describing the piezoelectric behviour (see e.g. 9], Sec , pp. 11, eq ) reduce to : T xx = c E 11S xx e 31 E z D z = e 31 S xx + ε S 33E z (4)

4 x (1) x (1) + x x + y x (n 1) x + (n 1) x (n) x (n) + w b x () (Γ 1 ) (Γ 2 ) (Γ 3 ) (Γ k 1 ) (Γ k ) (Γ k+1 ) (Γ N 2 ) (Γ N 1 ) (Γ N ) x 1 x 2 x 3 x 4 x k 1 x k x k+1 x k+2 x N 2 x N 1 x N x N+1 z h b x (b) v (1) (t) h (1) v (k) (t) h (k) h (n 1) v (n 1) (t) v (n) (t) h (n) L b Fig. 1. Bottom view () nd cross view (b) of bem excited by n piezoelectric ctutors where c E 11, e 31 nd ε S 33 re respectively the trnsverse stiffness t constnt electric field, the piezoelectric coicient nd the longitudinl permittivity t constnt strin. The br symbol on these structurl coicients denotes the specific vlue of the trnsverse coupling mode expressed s follows : c E 11 = 1 s E ; e 31 = d s E ; ε S 33 = ε T 33(1 k31) 2 (5) 11 where k 31 is the trnsverse piezoelectric coupling fctor. 2.3 KINEMATIC AND ELECTRICAL QUANTITIES The displcement corresponding to the mechnicl ssumptions re (see e.g. 8], Sec , p. 114, eq ) : u = u(x, t) zw,x(x, t) 0 (6) w(x, t) where u nd w chrcterize the displcement fields in longitudinl extension nd in flexure, respectively. Note tht the coupling between u nd w implicitly enbles to tke into ccount the vrition of the neutrl xis position rising from symmetry introduced by the gluing of the cermics of one side of the bem. The strin vector S is besides deduced from the previous reltionship by pplying the displcement grdient in the frmework of liner elsticity so much tht the only nonzero component of this vector is written s follows : S xx (x, z, t) = u,x (x, t) zw,xx (x, t) (7) Concerning the electricl quntities, for ech cermic ctutor, it follows from (1), (7) nd (4) tht the electric displcement field is given by D z = e 31 (u,x zw,xx ) ε S 33ϕ,z (8) Besides, since D is conservtive, D z,z = 0. Using this condition nd (2), the electric potentil in the i th ctutor cn be deduced nd the z-xis component of the electric field is consequently written s follows : E z (x, z, t) = v (t) + e ( ) 31 h ε S z h m w,xx (x, t) 33 (9) where h m = (h b + h )/2 is the height of the center of the i th cermic ctutor. 2.4 APPLICATION OF HAMILTON S PRINCIPLE According to 10] (Sec. 4.8, pp ), generl form of Hmilton s principle pplied to piezoelectric system is t2 t 1 (δl + δw ext )dt = 0 (10) where δl nd δw ext re respectively the vritions on the time intervl t 1, t 2 ] of the modified Lgrngin L nd the work done by externl forces nd sources which is expressed s follows : n δw ext = nδudγ + pδwdγ + i δλ Γ b Γ b i=1 (11) where n(x, t) nd p(x, t) re respectively the liner density of xil nd norml forces. The modified Lgrngin L is the sum of the Lgrngin L of the globl structure to which Lgrnge multipliers (for more detils see e.g. 12], Sec. II.5, pp ) hve been dded in order to respect the 3(N 1) kinemtic conditions t the interfces between ech section Γ k which re expressed s follows : u(x = x k, t) = u(x = x+ k, t) w(x = x k, t) = w(x = x+ k, t) (12) w,x (x = x k, t) = w,x(x = x + k, t)

5 The Lgrngin L of the studied system is the sum of n + 1 contributions corresponding to the ctutors nd the bem so much tht : n L = L b + L (13) i=1 where L b is the Lgrngin of the bem which is clssiclly written s follows : L b = 1 Mb ( u 2 + ẇ 2 ) K b u 2,x D b w 2 ],xx dγ 2 Γ b (14) where Γ b = N k=1 Γ k symbolises the neutrl xis of the bre bem nd M b, K b nd D b re respectively the liner mss, xil stiffness nd flexurl rigidity of the bem defined by h 3 b M b = ρ b w b h b ; K b = E b w b h b ; D b = E b w b 12 (15) nd L is the Lgrngin of the i th cermic ctutor defined s follows : L = 1 2 Γ M ( u 2 + ẇ 2 ) K u 2,x 2J u,x w,xx D w,xx 2 ] +2 ψ u,x + φ w,xx v + C (v ) 2] dγ (16) where M = ρ w h is the liner mss of the i th cermic ctutor nd K, J, D re respectively the liner xil stiffness, elstic coupling fctor, flexurl rigidity defined by K D = c E 11w h = c E 11w h = c E 11w h h m (17) ; J ] (h ) 2 12(1 k31 2 ) + (h m) 2 (18) ψ, φ nd C re the liner xil nd flexurl electromechnicl coupling fctors nd the clmped cpcitnce whose expression re respectively: ψ = ē 31 w ; φ = ē 31 w h m ; C w = ε S 33 h (19) Applying the rules of vritionl clculus (see e.g. 10], Sec. 4.8, pp ), the sttionry of the definite integrl (10) leds to the equtions governing the system dynmic nd dditionlly gives the 3(N 1) extr continuity reltionships by removing the Lgrnge multipliers. These foregoing equtions ffect the xil force N, the bending moment M nd the sher force Q which my be expressed in generl wy on the section Γ k = Γ s follows : Continuity reltionships t x = x k N k 1 (x k, t) = N k (x k, t) N (t) M k 1 (x k, t) = M k (x k, t) M (t) Q k 1 (x k, t) = Q k (x k, t) (20) Continuity reltionships t x = x k+1 N k (x k+1, t) N (t) = N k+1 (x k+1, t) M k (x k+1, t) M (t) = M k+1 (x k+1, t) Q k (x k+1, t) = Q k+1 (x k+1, t) (21) It should be noted tht the bem vibrtion is loclly induced t the ends of ech cermic ctutor by mens of n xil force N (t) = ψ v (t) nd bending moment M (t) = φ v (t). In pure flexure cse, where n uncoupling of the xil u nd flexurl w motions hs been cted (for more detils see 6], Sec , p. 3290), the resulting mechnicl equtions re written on ech section (Γ k ) k 1,N s follows : x Γ k, M k ẅ(x, t) + D k w,xxxx (x, t) = p(x, t) (22) where M k nd D k re the liner mss nd flexurl rigidity of the section Γ k given by nd D k = M k = { Mb + M { D b M b J D b ] 2 /K, if Γ k = Γ, otherwise b, if Γ k = Γ, otherwise (23) (24) with K b = K b + K nd D b = D b + D. From the eqution (22), the determintion of eigenmodes is crried out for free bem with short circuited cermic ctutors. Assuming tht the mechnicl nd electricl quntities hrmoniclly evolve, which mens for the deflection tht w(x, t) = w(x) exp (jωt) with j 2 = 1 nd w(x) its mplitude t the ngulr frequency ω, it leds to : x Γ k, w,xxxx (x) (α k k b ) 4 w(x) = 0 (25) where k b = (ω 2 M b /D b ) 1/4 is the wve vector relted to the flexurl modes of the bre bem nd α k = (M k /M b )/(D k /D b )] 1/4. The generl solution of this system of equtions is written s liner combintion of Duncn s function defined for i = 1, 2 by c i, s i ] = cosh +( 1) i+1 cos, sinh +( 1) i+1 sin]. The use of the boundry conditions (M.6) nd the continuity reltionships (12), (20) nd (21) leds to system of 4N equtions which degenertes for prticulr frequencies defining the resonnces. Furthermore, it enbles to find the mode shpes ϕ (n) up to multiplictive constnt which cn be set using the modl mss normliztion criterion given by the following reltionship : n N, N k=1 Γ k M k ϕ (n) (x)] 2 dx = 1 (26)

6 3 SIMULATION The initil im of such study is to quntify the influence of the cermics on the flexurl mode shpes, both the impct of their dimensions, their constitution nd one of their position, even if no dmping hs been considered in the semi-nlyticl modelling. The cse of bem excited by two cermics glued with the sme polriztion direction is hold nd the prmeters used for the simultion re listed in Tble I. In light of these considertions, preliminry prmetric study hs been crried out in order to determine the cermic positions on the host structure optimizing the energy conversion (ll others things being equl). The ide ws to study, for ech mode shpe, the evolution of the ective electromechnicl coupling coicient (EEMCC) k (n) ginst the set of prmeters (χ (1), χ (2) ) = (x (1), x (2) )/L b defining the cermic positions. In piezoelectricity, k (n) is the clssicl relevnt prmeter enbling to chrcterize the energy conversion, cting in the present cse between the cermics nd the host structure. It is defined by the IEEE stndrds 13] s follows : k (n) 1 = f (n) sc /f oc (n) ] 2 (27) where f sc (n) nd f oc (n) re the resonnt frequencies of the vibrting system while the cermic ctutors re respectively in short-circuit nd open circuit. For exmple, k (n) is depicted in Fig. 2 s function of χ(1) nd χ (2) for the fifth mode in flexure. The centrl bnd correspond with the overlpping of the cermics which hs no physicl relity. The symmetry of the pttern is due to the equivlent chrcteristics of both cermics. I. Geometricl nd structurl prmeters Definition Vlue Unit L b Length 100 mm w b Width 7 mm h b Thickness 1.94 mm ρ b Mss density 2700 kg/m 3 E b Young s modulus 69.0 GP L Length 11 mm w Width 7 mm h Thickness 1 mm ρ Mss density 8000 kg/m 3 s E 11 Complince 17 µm 2 /N d 31 Piezo. coef. 260 pm/v ε T 33/ε 0 Reltive Permittivity 5000 k 31 Coupling fctor This figure is used to deduce the loction of the cermics promoting prticulr mode. For instnce, the χ (2) (S1) k (5) (S2) χ (1) Fig. 2. EEMCC for the fifth flexurl mode shpe geometry defined by (χ (1), χ (2) ) = ( 0.250, 0.150), nmed therefter (S1), specificlly promotes the flexurl mode of rnk 5. For purpose of comprison, nother geometry nmed (S2) nd defined by (χ (1), χ (2) ) = ( 0.115, 0.005) globlly promotes the modes of rnk 1, 3 nd 5. These geometries were mnufctured nd the tble II shows the good greement between predicted nd mesured resonnt frequencies. II. Theoreticl nd experimentl resonnt frequencies of both structures (S1) nd (S2). f r (n) (Hz) Structure (S1) Structure (S2) Mode rnk Th. Exp. Th. Exp The modelling developed in the pper cn lso be used to compre the mode shpes of both geometries (see Fig. 3). This figure shows the fifth mode in flexure for ech structure nd highlights the fct tht the mode shpe is modified by the cermics. However, in tctile pplictions, the user cn be sensible to modl deformtion. Indeed, the stimultion depends on the vibrtion mplitude under the fingertip 3]. This influence cn be quntified in terms of homogeneity of the vibrtion using the minimiztion of the following criterion, defined for n N\{0, 1} by σ (n) H = n i=1 j<i ϕ (n) (χ i ) ϕ (n) (χ j ) ( ) n mx ϕ 2 (n) (χ k ) k 1,n 0 (28) where ( n k) is the binomil coicient nd (χ k ) k 1,n is the vector of dimensionless bscisse mtching the locl mode shpe extrem.

7 φ (5) (x) x Lb/2 0 Lb/2 Fig. 3. Fifth mode shpe for the structures (S1) (in dshed line), (S2) (in dsh-dotted line) nd the bre bem (solid line). For both structures, σ (5) H is respectively equl to 0.27 for (S1) nd 0.15 for (S2) with n EEMCC pretty much equl (k (5) 0.10). Hence, the design which optimizes the energy conversion for mode 5 is not for the homogeneity. This conclusion cn be generlized for flexurl mode shpes with rnk n 3. 4 CONCLUSION This pper hs presented n one-dimensionl seminlyticl modelling of bem excited by cermic ctutors of ny number. This pproch is first step in view of n optimiztion of tctile devices using the squeeze film ect. This phenomenon is generted between vibrting plte, ctivted by piezoelectric cermics, nd n user s finger. It is thus importnt to precisely know the ctutors impct on the dynmic of such structure. From the study of two structures composed of two cermic ctutors, this rticle hs highlighted trde-off to ct between the promotion of mechnicl mode shpe nd its homogeneity which my be uncceptble for tctile feedbck devices if it is not provided. For future works, the influence of the function sensor ccomplished by extr cermics will be studied. An extension to 2D model is lso envisged. ACKNOWLEDGEMENTS The finncil support for this study ws provided by grnt from the Europen Union which invested in the Nord-Ps-de-Clis Region (FEDER funds) within the frmework of the FUI TOUCHIT (Tctile Open Usge with Customized Hptic Interfce Technology). REFERENCES 1] M. Wiertlewski, J. Lozd nd V. Hywrd, The Sptil Spectrum Of Tngentil Skin Displcement Cn Encode Tctul Texture, IEEE Trnsctions on Robotics, Vol. 27, No. 3, pp , June ] T. Wtnbe nd S. Fukui, A Method for Controlling Tctile Senstion of Surfce Roughness Using Ultrsonic Vibrtion, 1995 IEEE Interntionl Conference on Robotics nd Automtion, Ngoy, Jpn, Vol. 1, pp , My 21-27, ] M. Biet, F. Girud nd B. Lemire-Semil, Implementtion of Tctile Feedbck by Modifying the Perceived Friction, The Europen Physicl Journl Applied Physics, Vol. 43, No. 1, pp , July ] L. Winfield, J. Glssmire, J.E. Colgte nd M. Peshkin, T-PD: Tctile Pttern Disply through Vrible Friction Reduction, 2007 World Hptics Conference, Tsukub, Jpn, pp , Mrch 22-24, ] P. Sergent, F. Girud nd B. Lemire-Semil, Geometricl optimiztion of n ultrsonic tctile plte, Sensors nd Actutors A: Physicl, Vol. 161, No. 1-2, pp , June ] J. Ducrne, O. Thoms nd J.-F. Deü, Plcement nd dimension optimiztion of shunted piezoelectric ptches for vibrtion reduction, Journl of Sound nd Vibrtion, Vol. 331, No. 14, pp , July ] C. Murini, M. Porfiri nd J. Pouget, Numericl methods for modl nlysis of stepped piezoelectric bems, Journl of Sound nd Vibrtion, Vol. 298, No. 4-5, pp , December ] J.N. Reddy, Mechnics of Lminted Composite Pltes nd Shells: Theory nd Anlysis - Second Edition, CRC Press, ] J. Yng, Anlysis of Piezoelectric Devices, World Scientific Publishing, ] A. Preumont, Mechtronics: Dynmics of Electromechnicl nd Piezoelectric Systems, Springer, ] T. Iked, Fundmentls of Piezoelectricity, Oxford University Press, ] C. Lnczos, The Vritionl Principles of Mechnics - Fourth Edition, Dover Publiction, ] IEEE Stndrd on Piezoelectricity ANSI/IEEE Std , 1988.