* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download THE MODELLING OF THE ELECTROMECHANICAL MULTILAYER
Survey
Document related concepts
Hall effect wikipedia , lookup
Electric motor wikipedia , lookup
Immunity-aware programming wikipedia , lookup
Insulator (electricity) wikipedia , lookup
Electromotive force wikipedia , lookup
Induction heater wikipedia , lookup
Stepper motor wikipedia , lookup
Lorentz force wikipedia , lookup
Alternating current wikipedia , lookup
Electrical injury wikipedia , lookup
Mechanical filter wikipedia , lookup
Mains electricity wikipedia , lookup
Electromagnetism wikipedia , lookup
High voltage wikipedia , lookup
Friction-plate electromagnetic couplings wikipedia , lookup
Variable-frequency drive wikipedia , lookup
Multiferroics wikipedia , lookup
Transcript
THE MODELLING OF THE ELECTROMECHANICAL MULTILAYER PIEZOCERAMIC MICROACTUATORS Authors : Dr.Eng. Mircea IGNAT Senior Researcher INCDIE CA ,Dep.1/10 Eng. George ZARNESCU Junior Researcher, INCDIE CA ,Dep 1/10 Student Sebastian SOLDAN INCDIE CA ,Dep.1/10 Abstract: In this paper ,the modelling of a new class of microelectromechanical piezoceramic actuators (which represent the important application of the inverse piezoelectric effect) ; multilayer actuators are presented. the theoretical and experimental relationship between the piezoceramic ratio and parameters and design elements of piezoceramic micromotors and microactuators have been developed. The ultrasonic microactuators advantages include [1] : - both high torque and high efficiency at low speed; - the possibility of direct drive; -ability to maintain its own position; - good control characteristics at start and stop; - simple structure and flexibility of shape; - no restriction through induction (e.g. electromagnetic); - silent operation. Essentially a theoretical commentary on the ceramic properties (used in fabrication of piezoceramic microactuators ) is presented ; dielectric constant ; 31 , 33 , piezoelectric parameters ; d 31 , d 33 , compliance, dielectric loss and mechanical coupling ratio. The most important equation regarding electrical parameters: D T E d T (1) The most important equation regarding mechanical parameters: u s T d E (2) Where u is the displacement, T is the mechanical stress, D is the electric induction and E is the intensity of electric field. The theoretical and experimental relationship between the piezoceramic ratio and parameters and design elements of piezoceramic micromotors and microactuators have been developed. The microactuator displacement as a function of force and voltage will be therefore : hs E Fpiezo dU (3) u A where ; h -is the height of the piezoceramic element, A -the area of the piezoceramic actuator, s - the elastic compliance, d is the piezoelectric constant, U - the applied voltage. The modelling estimation of design microactuators characteristics ; displacement ,force ,torque as a function of input; voltage and piezoceramic properties has been obtained in a systematic approach from proposed method in this study. It is presented the case study of microelectromechanical multilayer actuator with the modelling of the main parameters : microforce and microdisplacent(an example in fig.1). .Fig.1.D representation of the linear actuator force function by the piezoceramic ratios; d and s References [1] S.Ueha,Y.Tomikawa,M.Kurosawa,N.Nakamura,”Ultrasonic Motors.Theory and Applications”,Clarendon Press Oxford ,1993. [2] P.Hagedorn, W.Seemann, “Problems in the Mathematical Modelling of Ultrasonic Motors “ , in “Modern methods of analytical mechanics and their applications” edited by V.V. Rumyantsev, A.I.V.Karapetyan ,Springer ,Wien,New York ,1998. [3]E.C.N.Silva ,S.Nishiwaki, N.Kikuchi,”Topology Design of Flextensional Actuators”, IEEE Transactions, Ferroelectrics and Frequency Control,vol.47,no.3, may 2000,p.657671. Proposal topics: Coupled problems. - Poster. INCDIE CA - National Research Institute of Electrical Engineering –Advanced Researche. Splaiul Unirii 313 ,sector 3,Bucharest ROMANIA Tel.346.72.31, 3467235. Email : [email protected], [email protected]. 1.INTRODUCTION With the development of new piezoceramic materials a basis for the realization of efficient new types of piezoelectric micromotors and microactuators was established[1,2]. The functional principle of these piezoelectric microactuators and micromotors is totally different from that of conventional electromagnetic actuators and motors. In electromagnetic conventional motors and actuators the forces and mechanical moments are generated by magnetic field whereas in piezoelectric microactuators and micromotors , piezoceramic elements are used for energy transformation and to obtain high frequency vibrations for a relative motion between the different parts of the system and different working principles are possible. A classification of piezoelectric linear micromotors and microactuators by vibrational modes is [1]: -A. Non-self moving : A1. Traveling –wave using flexural wave (by Sashida). A2.Driven by layered piezoelectric ceramics. -B.Self –moving . B1. Hybrid transducer with longitudinal vibrator . B2.Hybrid transducer –type motor using flexural vibrator. B3. -shaped transducer. B4.Flexural-flexural double –mode bar. B5. Mode annular plate actuator. The ultrasonic micromotors and microactuators advantages include [1] : - both high torque and high efficiency at low speed; - the possibility of direct drive; -ability to maintain its own position; - good control characteristics at start and stop; - simple structure and flexibility of shape; - no restriction through induction (e.g. electromagnetic); - silent operation. Its disadvantages include ; - need for electrical source of high frequency and voltage; - inevitable wear and tear of the contact surfaces; - expensive electrical source and piezoelectric elements. Two important goals in the design of piezoceramic micromotors and microactuators are output parameters [2,3,4] ; - Displacement linear or angular, Force or torque (blocking force or torque). The main problems of the piezoelectric micromotors and microactuators researche and design have the following formulations ; I. The determination of the force or the torque of the piezoelectric motor or actuator from the given properties of motion (it is adapted the inverse problem of dynamics [8]) II.Given (to see Figure 1); - the topology of the active piezoelectric micromotor or microactuator ; m and boundary o , - the topology of the domains of mechanical forces or torques ; t1, t 2 , - the topology of the boundaries where is applied the voltage; t1 , t 2 , s , - the properties of the active piezomaterial ; piezoelectric, dielectric ,mechanic and electromechanic ratios and constants, to determine the micromotor or actuator torque(F,T) ,angular speed or force and linear speed. t 2 , t 2 F T 0 t1 , t1 , m F T s Fig.1 Referring to the main problem formulates of piezoelectric micromotors and microactuators. The relations between the different aspect of piezoelectric micromotors and microactuators design field are presented in Figure 2. A.Piezomaterial constant and ratios FORCES TORQUES B.Topologies of the micromotors and microactuators C.Kinematic elements of the motion . LINEAR MICRODISPLACEMENTS ANGULAR MICRODISPLACEMENTS Fig.2 Relation necessaries for piezoelectric micromotors and actuators problem formulations. Because the important dependence of the main micromotors or microactuatos parameters of the piezoceramic characteristics the last is the essential problem formulation on design. The force and torque are function of different piezoceramic parameters,by exemple : F ( , , , d , g , e, s ), (1) T ( , , , d , g , e, s) where ; mass density, Y -Young modulus, -Poisson modulus, d - piezoelectric distortion constant, g - voltage output coefficient, s - compliance, r piezoceramic material . - relative permittivity for The dependence of the piezoelectric properties material is more important comparative with the conventional electromagnetic machines and actuators. 2.PIEZOELECTRIC COEFFICIENTS [6,11,12,13] The most important equation regarding electrical parameters: D T E d T (2) The most important equation regarding mechanical parameters: u s T d E (3) Where u is the displacement, T is the mechanical stress, D is the electric induction and E is the intensity of electric field. We can consider the case in which beside the electric exterior applied field we apply an external mechanical stress, it is a cumulative effect for material polarization, we have a polarization produced by electrical field over which is added the polarization produced by external mechanical stress, in the second case we have the same cumulative effect for displacement. The relation between absolute and relative permittivity for piezoceramic material is, these are tensors because we have an anisotropic dielectric medium: T 0 ij , (4) in which ij ji , if cartesian axes (x, y, z) are the same with main polarization axes of crystal, we have only three relative dielectric constants for each axis 11 , 22 , 33 . In piezoceramic material catalogues generally it is given 11 , 33 , or the ratio 1T 3T , which , 0 0 represents the relative permittivity, each piezoceramic crystal have a specific type of symmetry therefore we’ll have less coefficients and some of these coefficients we’ll be equal. This is the equation for the direct piezoelectric effect in which we encounter the voltage output coefficient, the other equation of polarization for direct piezoelectric effect is already integrated in the general equation for electrical parameters. E g T (5) Piezoelectric tensors g and d have the same 3 rank. Most important coefficients for the piezoelectric effect are: d - piezoelectric distortion constant measured in [C/N] or [m/V], it is the distortion resulting from the application of an uniform electric field without mechanical stress. g - voltage output coefficient [V*m/N], referring to the electrical field strength for an uniform applied stress with no electrical effect for displacement. s - compliance is the inverse of Young’s modulus Y, it is an elasticity constant measured in [m2/N]. We can establish a scalar link between these coefficients: d T g (6) Equations above can be written in a matrix form for the principal axes of polarization, P represents the polarization for all three axes. Px 11 0 P 0 22 y Pz 0 0 0 Ex 0 E y 33 E z (7) For example Barium Titanate (BaTiO3) has a tetragonal crystalline structure, this category of materials are most utilized. We can observe the matrix of piezoelectric distortion coefficient for the direct piezoelectric effect in BaTiO3 case. Px 0 P y 0 d P z 31 0 0 0 d15 0 0 d15 0 d31 d33 0 0 T 1 T 2 0 T3 0 T 4 0 T 5 T 6 (8) The general relationship between mechanical stress and deformation deduced from Hook’s law, from the theory of elasticity: u s T (9) The matrix of compliances or the compliance tensor for BaTiO3 can be written: s11 s12 s13 0 0 0 s12 s11 s13 0 0 0 s13 s13 s 33 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 66 (10) 3. THEORETICAL ASPECTS The determination problem of optimum dimension of torque and force of piezoelectric micromotors and microactuator can be a problem of maximum (extremum) determination of a multi – valued function [9,10]. Either force function ; F ( x1 , x 2 ,....x p ) and torque function ; T T ( x1 , x 2 ,....x p ) with X pm R p ( X pm is the multitude of the material properties with the following identifications; x1 , x2 Y , x3 , x4 d , x5 s,...) , the force and torque function have three partial derivative on X pm and either (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) the solution of the equation systems: F F F 0, 0,..., 0 x1 x2 x p T T T 0, 0,..., 0 x1 x2 x p (a1 , a 2 ,...a p ) (11) (b1 , b2 ,...b p ) (12) a.If the numbers : 1 A11 0, 2 where ; Aij A11 A12 A21 A22 0,..., p F (a1 , a 2 ,...a p ) xi x j or Aij A11 A12 ... A21 A22 ... A2 p ... ... ... A p1 Ap 2 ... A pp T (b1 , b2 ,...b p ) xi x j A1 p ... 0 ,then (13) the function F, T have a minimum in the points ; (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) . b.If the numbers : *1 A11 0, *2 A11 A12 A21 A22 0,..., *p (1) p A11 A12 ... A1 p A21 A22 ... A2 p ... ... ... A p1 Ap 2 ... A pp ... 0 (14) then the functions F, T have a maximum in the points; (a1 , a 2 ,...a p ) , (b1 , b2 ,...b p ) . An important observation ; in majority the application which involve the piezoelectric micromotors or microactuators impose a maximum torque or force (b. condition ). But between the piezoceramic constant and ratios are bond relations (to see relation (3) between d and s) and the number of variables are reduced. An other determination properties piezomaterial( constant and ratios) procedure is to explicit the constant piezomaterial or piezomaterial function the force (or torque) imposed by the design theme formulation . Either the force or torque function of a piezoelectric actuator or motor, with the following representation : F K s f cp ( x)U T K s f cp ( x)U (15) where; K s ratio of the dimensional parameters of the motor or actuator, U supply voltage and f cp (x) - function of piezomaterial ratio or constants.Then we explicite and analyze this function ; f cp ( x) F K sU T f cp ( x) K sU (16) 4.RESULTS AND DISCUSSION 1.The displacement and force of a linear microactuator (Fig. 3 ). Fig. 3 The main element ; displacement and force of a linear actuation The microactuator displacement as a function of force and voltage will be therefore : hs E Fpiezo dU (17) u A where ; h -is the height of the piezoceramic element, A -the area of the piezoceramic actuator, s the elastic compliance, d is the piezoelectric constant, U - the applied voltage. The force as a function of displacement and voltage will be : F piezo A u dU hs E (18) Fig.4.3D representation of the linear actuator force function by the piezoceramic ratios; d and s If in relation (18) we introduce relation (3) between the compliance and distorsion coefficient and explicite the distorsion function by compliance the force relation becomes : Fpiezo AT (u dU ) h(u dE ) (19) and Fpiezo AT (Uu u dU UdE) d h(u dE ) 2 Fpiezo and by 0 results an extremum value for distorsion coefficient; d u (1 U ) d U (1 E ) (20) (21) In Fig.4 is a 3D representation where is evidentied for design maximum force the sensible subdomain of the compliance ; distorsion s [0...1] 10 11[m 2 / N ] and ; d [0,5...5] 10 11[m / V ] with an interesting inflexion point placed on 11 d [1...2]10 [m / V ] . 2.Bimorph actuators. Bimorph actuation (Fig. 5) consist of two independent flat piezoelectric, stacked on top of each other.By driving one element to expand while contracting the other one, the actuator is forced to bend, creating an out of plane motion. 2 3 L d 31 U 4 t 2tw d 31 Fpiez U L s11E u (22) (23) Fig.5 The structure of a piezoceramic bimorph actuator with dimensional representations. Either the ratio of the dimensional parameters and function of piezomaterial ratio or constants; Ks d 2tw , f cp 31 L s11E (24) with the explicit function of piezomaterial : f cp ( x) F K sU (25) If the dimensional initial parameters are t 0,5mm, w 5mm, L 20 and the imposed design parameters are ; ,U 100V , F 100 N we find; 3 d 31 4 10 3 s11 and extremum relations.But for compliance –the imposed distorsion domain E maxim ratio is : d 31 / / s11 5 10 10 50 with the possible d 31 50 s11E 4000s11E ideal 11 1 10 domain. How the force and voltage are imposed parameters result the design condition : Ks F 1 100 1 0,02 and is necessary an other iterative dimensional calculation. U f cp 100 50 Fig.6 3D representation of the bimorph actuator force function by the ratios; d and s . piezoceramic Fig.7 3D representation of the bimorph actuator displacement function by the piezoceramic coefficient d and voltage U. In Fig.6 is shown a 3D representation of the specific force bimorph actuator on an imposed 2D domain d s. 3.The force source of a cantilever beam piezoceramic actuator [7] has the value : Fpiezo 2d13YwU (26) cm ] is the piezoelectric constant , Y - is the Young modulus, w the width of the V piezoceramic element (lamellar geometry) , U - the applied voltage . The displacement, x a , of the microactuator can be estimated imposing conditions of static where ; d 13 [ equilibrium on the model shown in Fig. 8. Balancing the force at x a yields : FA k bd ( x a x piezo ) (27) with x a : k bd k pz F piezo x a Fa (28) k bd k pz k pz where k pz -the spring constant of the piezoceramic element and k bd - the micromechanical connection with : k pz 2Ywt L pz (29) where , t thickness and L pz length of piezoceramic element. FA k bd xa F piezo k pz x piezo Fixed rigid support Fig 8. Electromechanical model of a cantilever microactuator. The force function in the case of cantilever beam actuator is a linear function without distinguished commentaries ; this function has not three partial derivative and has not the maximum or minimum point. A numerical representation is represented in Fig.9. Fig.9 3D representation of the linear actuator displacement function by the ratios; d and s . piezoceramic 5.CONCLUSIONS The authors propose an new approached procedure to find the optimum relation between the piezomaterial properties, imposed parameters and dimensional parameters.This procedure represents an initial segment in the general piezoceramic micromotor or microactuator design algorithm. References [1] S.Ueha,Y.Tomikawa,M.Kurosawa,N.Nakamura,”Ultrasonic Motors.Theory and Applications”,Clarendon Press Oxford ,1993. [2] P.Hagedorn, W.Seemann, “Problems in the Mathematical Modelling of Ultrasonic Motors “ , in “Modern methods of analytical mechanics and their applications” edited by V.V. Rumyantsev, A.I.V.Karapetyan ,Springer ,Wien,New York ,1998. [3]E.C.N.Silva ,S.Nishiwaki, N.Kikuchi,”Topology Design of Flextensional Actuators”, IEEE Transactions, Ferroelectrics and Frequency Control,vol.47,no.3, may 2000,p.657-671. [4] K.Ragulskis,R.Bansevicius,R.Barauskas,G.Kulveitis,”Vibromotors for precision microrobots” ,Hemisphere Publishing Corporation,1988. [5] M.P.Mason “Physical Acoustics,Principles and Methods” vol.I,Part A,New York,Academic Press,1964. [6] W.G.Cady “ Piezoelectricity.An introduction to the theory and applications of electromechanical phenomena in crystals” vol.I,II, Dover Publications,New York 1964. [7] W.B.Robbins,D.L.Polla, D.E.Glumac,”High –Displacement Piezoelectric Actuator Utilizing a Meander-Line Geometry- Part I; Experimental Characterization,Part II- Theory” IEEE Transactions on Ultrasonics,Ferroelectrics and Frequency Control,” vol.38, no.5,September 1991, pp.454-467. [8] A.S.Galiulin ,“Inverse problems of dynamics”,MIR Publishers Moscow,1984. [9] M.Rosculet , “ Analiza matematica” ,Ed.Didactica si pedagogica ,Buc.1967. [10] M.Nicolescu, N.Dinculeanu, S.Marcus ,”Analiza matematica “vol.I ,II, Ed.Didactica si pedagogica , Buc.1967. [11] MURATA (Japonia) ,Piezoceramic catalogue -2005. [12]FERROPERM(Danemarca), Piezoceramic catalogue , 2003 [13] PI (Germania), Piezoceramic catalogue ,2005. [14] A.E.Glazounov,S.Wang,Q.M.Zhang,”Piezoelectric Stepper motor with Direct Coupling Mechanism to Achieve High Efficiency and Precise Control of Motion” IEEE Transactions, Ferroelectrics and Frequency Control,vol.17,no., July 2000,p.1059-1067.