Multi ultrasonic actuators with some degree of freedom

Transcription

1 005 WA Int. Conf. on DYNAMICAL YM and CONROL, enice, Italy, Noveer -4, 005 (pp609-63) Multi ultrasonic actuators with soe degree of freedo G. KULII, D. MAZIKA, I. UMAONIN 3 ilnius Gediinas echnical University auleteio av., 040 ilnius, Lithuania Astract: Usually nuerical odelling and siulation of ulticoponent piezoelectric actuators lead to the large nuer of recurred calculations with different geoetrical paraeters of the actuator and to the proles concerning the changes of odal shape sequences during recurred calculations. he exchanges in the odal shape sequence prevents autoation of solving different proles. For this reason an iproved algorith for solving the proles related to the changes in the odal shape sequences has een proposed in the paper, also, the dependence of eigenfrequencies on various paraeters of ulticoponent piezoelectric actuators and proles related with the exchanges in the odal shapes sequence has een analyzed. Key-Words: piezoelectric actuator, odal shape, ulticoponent oscillations Introduction Whole odern technical areas (radiotechnology, acoustic, virotechnology, wave-technologies) are ased on the application of different viration processes. he developent of new device groups that use piezoactive aterials for high frequency oscillation transforation into a continuos ultidirectional otion provides an opportunity to extend the field of creating tie constant positioning drivers, icroanipulators, icropups and other drives [7]. he perforance of these devices strongly depends on the features of the actuator, which is the ain part of the piezoechanical syste. he synthesis of needful oscillation fields of the actuator can e otained optiizing the geoetrical paraeters, the vector of poliarization and the topology of excitation zones of the actuator. he piezoelectric effect and the hysteresis effect play an iportant role in the dynaical ehavior of these actuators [6]. o it is very iportant to now what odal shape will e excited when odelling piezoelectric actuators. he functioning principle of ost piezoelectric actuators is ased on the excitation of higher resonance frequencies. he nuerical analysis of such piezoelectric actuator is usually tied to a particular odal shape. While perforing nuerical analysis, when only the geoetrical paraeters of a piezoelectric actuator change, a prole arises that is related to the change in the odal shape sequence. ince viration devices usually function at one of their odal frequencies, as the odal shape sequence changes, the prole solution usually does not converge, and the nuerical analysis ecoes eaningless. Construction of Piezo Actuators he perforance of ultrasonic otors strongly depends on the features of the actuator, the ain part of any piezoechanical syste. here are soe asic shapes of piezo actuators such as ea, plate, cylinder, disc, ring and etc. arious constructions of actuators are used in order to achieve a particular law of oveent of the actuator and the final lin of the ineatics pair Fig. [7]. he actuators shaped lie eas and plates are ostly used in ultrasonic otors. he characteristics and types of the excited ulticoponent oscillations of the piezoelectric actuator depend on its geoetrical paraeters, oundary conditions and direction of the polarization vector. he topology of electrodes and geoetrical paraeters of the actuator define the direction of the excited oscillations. In order to achieve suitale characteristics of the oscillations, particular geoetrical paraeters of the actuator ust e calculated. Fig. Constructions of the piezo actuators. a) =l, n=l; ) =l, n= [9, 0]. Many different types of ulticoponent oscillations can e excited using these actuators: longitudinal-flexural, longitudinal - torsional and etc. Using the variale vector of polarization, three and four coponent oscillations of the ea shaped actuator can e achieved [].

2 005 WA Int. Conf. on DYNAMICAL YM and CONROL, enice, Italy, Noveer -4, 005 (pp609-63) 3 Prole Definition ince the analysis of ultidiensional piezoelectric actuator cannot e perfored without considering the viration device, ost often the proles of piezoelectric actuator research are solved in an integral fashion taing into account the whole device. he foral algorith for solving the prole loos as follows: Input Optiization loop Changing paraeters igenvalues Integrating & δ + K δ [ M ]{} && δ + [ C]{} [ ]{} [ ]{} ϕ = R( ω t) [ ] {} δ + [ ]{} ϕ = {} Q Calculation criteria of optiization 4 nd of optiization loop Output { } Fig. he structure of the general calculation algorith. In the case under consideration, at the first stage (changing paraeters) the geoetrical paraeters of a piezoelectric actuator are changed. At the second stage (eigenvalues) a atrix of eigenvalues is fored, whose every colun descries a corresponding odal shape; the first colun descries the first odal shape, the second colun the second shape and so on. While changing the geoetrical paraeters of a piezoelectric actuator the change in the odal shape sequence has een oserved. xchanges in the odal shape sequence could e deterined analyzing various constructions of piezoelectric actuators. For exaple, let s consider longitudinal flexural oscillations of the ea actuator. Using this type of actuator, changes of the odal shape sequence could e found analytically [8]. Using the technical oscillation theory of the ea the longitudinal oscillations are found y solving the second order differential equation [, 4]: ξ ξ () x s = 0 () t x i x Longitudinal oscillations of the ea can e expressed as follows [, 4]: ω = () l ρ - the Jung odulus; - the ode nuer of the longitudinal oscillations; l - the length of the ea; ρ ass density. Flexural oscillations of the ea are found y solving the second order differential equation [, 4]: x ξ len ξ I + = 0 ρ len (3) x t Flexural oscillations of the ea are descried y the expression [, 4]: πh ( n + 0.5) ωn = (4) 4l 3 ρ h - the height of the ea; n the ode nuer of the flexural oscillations; If certain values of and n are defined, then h/l ratio of the ea could e calculated. Fro the equations () and (4) following equation could e otained: h l 3 = π ( n + 0.5) As an exaple: (5) h = ; n = = (6) l But h/l ratio could e changed, for exaple, increasing or reducing the height of the ea. In this case value reains the sae, ut n value changes. his eans that the sequence of odal shapes changes when the geoetrical paraeters of the ea vary. For exaple, when the length and height ratio is 0.5<l/h<3 we have an ordinary odal shape sequence and in other case second and third

3 005 WA Int. Conf. on DYNAMICAL YM and CONROL, enice, Italy, Noveer -4, 005 (pp609-63) odal shape of two diensional actuator changes. Calculations of ultidiensional piezoelectric actuators, of course, have the sae prole, ut to solve it analytically is very difficult or even ipossile [8]. Modal shape sequence exchanges often causes fatal errors in the calculation process well as unexpected errors in results [5]. And it eans that at igenvalues stage of solving the prole an incorrect value of w (natural frequency) can e chosen. hen other stages of solving the prole ecoe eaningless ecause the actuator ade to the paraeters otained is defective. Hence the ain prole is to choose a suitale value of w. In ost cases piezoelectric actuators are resonance systes that operate in the first or higher resonance frequency. he synthesis of the needful field of oscillations ust e otained y using a particular shape and diensions of actuator and also certain geoetry of the locations of excitation zones. quations (7) fully define the piezoeffect [4]: { σ } = [ c ]{} ε []{ e } { D} = []{} e ε + [ ]{ } (7) where [c ], [e], [ s ] the atrix of stiffness for a constant electric field; the atrix of the piezoelectric constant; the atrix of dielectric constant evaluated at the constant strain, respectively; {σ},{ε},{d}, {} the vectors of stress, strain, electric induction and electric field, respectively. arious inds of resonance oscillations of actuators - longitudinal, flexural, rotational, shear and so on - could e otained using a different geoetry of electrodes []. In order to achieve required resonance oscillations of the actuator, particular electrodes ust e excited. Analysis of the piezoelectric actuator ust e carried out appreciating the electric occurrence in the syste. Based on FM, every node of the eleent has one additional DOF used for electric potentials in FM odeling. he solution applied for the equations of otion, suitale for the actuator, can e derived fro the principle of iniu potential energy y eans of variation functional [4]. he asic dynaic FM equation of otion for piezoelectric transducers that are fully covered with electrodes can e expressed as[]: [ M ]{} && δ + [ C]{} & δ + [ K]{} δ [ ]{ ϕ} = { R( ωt) } (8) δ + ϕ = Q [ ] {} [ ]{} {} where [M], [K], [], [], [C] - the atrices of ass, stiffness, electro elasticity, capacity and daping, δ - the vectors of nodes displaceents, potentials and external echanical forces, respectively. Here: respectively; { }, { ϕ }, { R} [ K ] [ B] [ c ] [ B]d = (9) [ ] [ B] [ e] [ B ]d = (0) s [ ] [ B ] [ ] [ B ]d = () [ M ] ρ [ N ] [ N ]d [ C] α [ M ] + β[ K ] = () = (3) where [B], [B ] the atrices of geoetry used for evaluation of displaceents and potential, respectively; [N] the function of the shape used for evaluation of the ass atrix. he daping atrix [C] is derived using ass and stiffness atrices y assigning constants α and β. Usually only the first equation fro syste (8) is used in the odeling process, ecause it is considering that the current source is powerful enough and ensures defined values of the electric potentials. olving dynaic equations using noralized coordinates, the vector of nodes displaceents {u} is expressed as a superposition of odal shapes with weight coefficients [3]: { u } = [ Y ]{ z} = { y } z { y q } z q (4) Based on the equation (4) the following expression of dynaic equation of the actuator could e otained [3]: && z + ω c z& + ω z =,,..., q = { y } { F } (5) Usually piezoelectric actuators operate a in certain resonance ode and the syste of the equations (5) could e solved using the dynaical reduction ethod. his eans that only one equation, which corresponds to a certain odal shape, could e solved. But in this case the calculation algorith couldn t e statically tied with the definite equation nuer, ecause its value ecoes indefinite when geoetrical paraeters or oundary conditions change. his is the ain reason why errors appear hindering autoation of the nuerical experients. o it s necessary to identify the odal shape and find the corresponding nuer

4 005 WA Int. Conf. on DYNAMICAL YM and CONROL, enice, Italy, Noveer -4, 005 (pp609-63) of the colun in atrix [Y] during recurred calculations [8]. 4 An Algorith of Modal hape Identification Doinating coefficients is the way to define the type of oscillations of the actuator and to sort odal shapes y the doinating type of the oscillations for exaple longitudinal, flexural or torsion. When the odal frequencies analysis of ulticoponent actuators is done using FM, doinating coponents of the oscillations can e found referring to the energetic ethod of the oscillation analysis, ecause aplitudes raised to the second power are proportional to the energy of the oscillations [8]. In that way the ratios (doinating coefficients) of the coponents of aplitudes in all directions can e found, and the direction with the axiu of aplitudes can e defined. Let s calculate the following su [7]: ( A ) r p = i p, r = i= q p (6) he doinating coefficients of the odel can e expressed as follows [7]: j j =, j (7) he physical eaning of doinating coefficients is the ratios etween different oscillation energy coponents in the directions of coordinate axes. he su defines the energy of the oscillation of the natural frequency in direction and the doinating coefficient defines the relation of the oscillation energies in i and j directions of natural frequency. Doinating coefficients have the following characteristic [7]: j =, j j j (8) Based on doinating coefficients we could deterinate the type of doinating oscillations and also define the level of correlation of ulticoponent oscillations as follows: = τ lg τ =,, 3... (9) j Doinating coefficients is the ethod to define the type of oscillations of the actuator and to sort odal shapes y doinating type of the oscillations for exaple longitudinal, flexural or torsional. In order to finally identify the odal shape additional characteristic criteria ust e applied, ecause the values of doinating coefficients vary when the geoetrical paraeters change. hese criteria are the nodes points or lines nuer of the odal shape. During calculations the nuer of node points or lines could e found referring to the sign of the oscillations aplitude alternating around the equiliriu attitude (Fig. 3). Fig. 3 Modal shape identification of piezo actuators: a) ea, ) plate. he exchanges in the odal shape sequence are a general case prole concerning not only piezoelectric actuators, ut also with all echanical structures [8]. Input Optiization loop Changing paraeters igenvalues Modal shape identification Integrating M && δ + C & δ + K δ [ ] {} δ + [ ]{} ϕ = {} Q [ ]{ } [ ]{ } [ ]{} [ ]{} ϕ = R( ω t) Calculation criteria of optiization nd of optiization loop Output { } Fig. 4 he iproved structure of the general calculation algorith. a) )

5 005 WA Int. Conf. on DYNAMICAL YM and CONROL, enice, Italy, Noveer -4, 005 (pp609-63) o, when solution of the echanical prole includes dynaic equations solving during recurred calculation, it is proposed to ae calculations using the iproved structure of the general calculation algorith and to add the stage of odal shape sequence identification as is shown in Fig Conclusions While changing the geoetrical paraeters of piezoelectric actuators the change in the odal shape sequence has een oserved. Identification of odal shapes sequence is necessary step in order to autoate nuerical experients of ulticoponent piezoelectric actuators. An algorith of odal shape identification has een proposed that could e applied to all echanical structures. his algorith ust e used as an additional stage in FM software. References: [] R. Bansevicius, R. Barausas, G. Kulvietis, K. Ragulsis, irootors for precision Microroots, Heisphere Pulishing Corp., UA, 988, pp 87. [] Под редакцией В. В. Болотина, Вибрации в технике,. oм. - Москва: Машиностроение, 978, pp 35. [3] R.Bansevicius, K.Ragulsis, irootors, ilnius, 98. [4] zou H., Piezoelectric hells (Distriuted sensing and control of continua), Kluwer Acadeic Pulishers, 993, pp30. [5] A.Gleus, D.Mazeia, G.Kulvietis, R.Bansevicius, Analysis of the natural frequencies of piezoactive cylinder, Mechanical technology XXI. Conference. Kaunas, 996. [6] R. Bansevicius, G. Kulvietis, D. Mazeia, Optial electrodes configuration of two diensional piezodrives, Mechanics 97. Conference. ilnius, 997. [7] R. Bansevicius, G. Kulvietis, D. Mazeia. Design of the Multicoponent Ultrasonic Drives Based on Mode - Frequency Analysis. Mechania N5 (0), Kaunas: echnologija, 999, pp 3-7. [8] R. Bansevicius, G. Kulvietis, D. Mazeia, Nuerical Identification of Modal hapes equence of Multicoponent Piezoelectric Actuators, Journal of iroengineering, ol., No. 4, 000. [9] R.Bansevicius, G.Kulvietis, D.Mazeia, Optiisation Of lectrodes Dislocation Of Multicoponent Piezoelectric Actuators, enth World Congress On he heory Of Machine And Mechaniss. Oulu, Finland, 999 June 0 4. [0] R. Bansevicius, G. Kulvietis, D. Mazeia. Optial electrodes configuration of piezodrives. IUAM/IFoMM ypoziu on ynthesis of Nonlinear Dynaical ystes, Riga, 4-8 August 998. pp. 5-6 [] Han J. M.. A. Adriaens, Wille L. de Koning, and Reinder Banning. Modeling Piezoelectric Actuators. Ieee/ase transactions on echatronics, ol. 5, No. 4, Deceer 000 pp. 33.