«Modelling of piezoelectric actuators : an EMR approach»

Transcription

1 EMR 15 Lille Sept. 215 Summer School EMR 15 Energetic Macroscopic Representation «Modelling of piezoelectric actuators : an EMR approach» Dr. C.Giraud-Audine (1), Dr. F. Giraud (2) L2EP, (1) Arts et Métiers Paris-Tech, (2) University Lille1 christophe.giraud-audine@ensam.eu

2 - Outline EMR 15, Lille, Sept Introduction to piezoelectricity 2. EMR of a piezoelectric actuator 3. Case study : a speed/displacement estimator used in forging processes 2

3 EMR 15 Lille Sept. 215 Summer School EMR 15 Energetic Macroscopic Representation «PART 1 Introduction to Piezoelectricity»

4 The piezoelectric effect EMR 15, Lille, Sept Non piezoelectric Polarization Piezoelectric Piezoelectric effect occurs in some crystals with special lattices symmetries [Ikeda 96] 4

5 The piezoelectric effect EMR 15, Lille, Sept. 215 In some materials strain and polarization are linked This is a reversible effect, thus an electric field induces stress Piezoelectricity is a linear coupling between : Mechanical variables : stress (T), strain (S) Electrical variables : electric field (E) and polarization (P) or equivalently electrical displacement (D) Note : electrostriction involves the same variables, but it is a quadratic relationship 5

6 Piezoelectric laws EMR 15, Lille, Sept Van-Voigt notation : E t S =s T d E ( T D=dT +ε E X 11 X 12 X 31 Example : PZT ceramics d 15 d= d 15 d 31 d 31 d 33 s= X 12 X 22 X 23 ( X 31 X 23 X 33 ) X 11 X1 X 22 X2 X X 33 = 3 X 23 X4 X 31 X5 X 12 X6 ( )( ) s 11 s12 s 13 s12 s 11 s 13 s 13 s13 s 33 s 44 s 44 s 66 ) ε 11 ε= ε 11 ε 33

7 Piezoelectric laws EMR 15, Lille, Sept. 215 The equations can be expressed equivalently according to the set of electrical and mechanical variables chosen E S =s D T + g t D t S =s T d E T D=dT +ε E E T E =-gt +β D t D T =c S e E t T =c S h D S S D=eS + ε E E =-hs +β E Properties tensors can be obtained by matrix algebra e.g : 1 T g =( ε ) d 7

8 Ferroelectric ceramics EMR 15, Lille, Sept. 215 Industrial ceramics used in actuator are sintered ferroelectrics powders, which are then polarized E P Before polarization P During polarization After polarization This allows to realize many shapes, and a preferential coupling direction 8

9 Ferroelectric ceramics EMR 15, Lille, Sept. 215 The polarization will define the way the ceramic will be used P P P mode (longitudinal) 2 31 mode (transversal) 15 mode (shear) 9

10 Ferroelectric ceramics EMR 15, Lille, Sept. 215 Ferroelectric ceramic have a strong non linear behavior. Piezoelectricity is an approximation 1

11 Pro and cons EMR 15, Lille, Sept. 215 Very large stress can be produced High dynamic Compacity : high energy/mass ratio Reversible effect (actuation and sensing) Very small strains : very small displacements Low efficiency Hysteresis 11

12 EMR 15, Lille, Sept. 215 «PART 2 EMR of a piezoelectric actuator» 12

13 Problem EMR 15, Lille, Sept. 215 F Slender piezoelectric rod : Quasi static motion 33 mode Mechanical load applied on the horizontal surfaces of the rod Voltage is imposed on the electrodes a a V P 3 6 F Find : The displacement field (u3 at the tip of the rod) The electrical charge : Q 13

14 Assumptions : stress EMR 15, Lille, Sept. 215 The mechanical loading is along the direction 3 and the vertical surfaces are unloaded. Therefore, the ceramic being thin we can assume that only T3 is non-zero, due to the loading F : T 1=T 2=T 4 =T 5=T 6= F T 3= 2 at x 3= L a Thus : T = T 3 () 14

15 Assumption : electric potential EMR 15, Lille, Sept. 215 The electrical potential φ is supposed to vary only in the direction 3 : ϕ = x1 ϕ E 2 = = x2 ϕ E 3= x3 E 1= Assuming a linear dependency of φ on x3 : ϕ( x 3)=V x3 V + ϕ E 3 = l l 15

16 Mechanical solution EMR 15, Lille, Sept. 215 E t 16 Recall that : S =s T d E thus using the previous assumptions { E S 1 =s13 T 3 +d 31 E 3 S 2 =s E13 T 3 + d 31 E 3 E S 3 =s33 T 3 + d 33 E 3 and { S 4 = S 5 = S 6 = F a2 Using the last boundary condition : T 3= { F V d 31 l a2 V E F S 2 =s 13 2 d 31 l a F V S 3 =s E33 2 d 33 l a E S 1 =s 13 and { u1 x1 u S2= 2 x2 u S 3= 3 x3 { F l d 31 V 2 a E F l u 2= s13 d 31 V 2 a E F l u 3=s 33 2 d 33 V a E u 1=s 13 S 1= hence

17 Electrical solution EMR 15, Lille, Sept. 215 Since D=dT +ε T E replacing gives : D 3 =d 33 F2 εt33 V l a This is the electrical displacement at the upper surface of the rod. The electrode charge is thus : a2 V Q= d 33 F +ε l T 33 Q= D 3 dx 1 dx 2 finally { E u 3= s 33 l a 2 F d 33 V εt33 a2 Q= d 33 F + V l 17

18 Introducing causality EMR 15, Lille, Sept. 215 Introducing the relationships : { u 3 dt= E s 33 l a 2 { u 3= u 3 dt Q= i dt u 3 : speed i : current F d 33 V T 2 ε33 a i dt= d F + V 33 l Define : { a2 K s= E s 33 l N = d 33 K s T 2 ε33 a C = l 2 C b =C d 33 K s the rigidity of the rod (N/m) conversion (N/V or C/m) «free» capacitance (F) «blocked» capacitance (F) 18

19 EMR EMR 15, Lille, Sept. 215 We rewrite : { F =K s u 3 dt F p i i m dt V= Cb with { F p =N V i m =N u 3 conversion storages F p =NV im=n u 3 F=F p F k i V Fp V im u 3 ES V= 1 (i im ) dt Cb Electrical F MS u 3 u 3 Fk Mechanical F k =K s u 3 dt 19

20 Discussion EMR 15, Lille, Sept. 215 Thanks to the EMR one can analyse the causality that is not revealed by the equation of piezoelectricity : To respect causality the piezoelectric actuator should be supplied by a current source The piezoelectric devices are often considered as displacement generators, however, it is clear from the EMR that the voltage generate the «inner» force, so it is rather an imperfect force generator. 2

21 EMR 15, Lille, Sept «PART 3 Case study : a speed/displacement estimator used in forging processes»

22 Context : vibrations in forming processes EMR 15, Lille, Sept. 215 Ultrasonic vibrations superimposed during tensile tests reduce the forging load [Blaha 1959] Many research have applied this to several forming processes : Results are promising Questions are still open Small displacements, high forces and fast dynamic are required therefore piezoelectric actuators are interesting 22

23 Context : vibration in forging processes EMR 15, Lille, Sept. 215 Forging in literature : only simulation Current research : Effect of waveforms, at low frequencies on aluminium, copper or plasticine during upsetting Experimental set-up 23

24 Context : vibration in forging processes EMR 15, Lille, Sept. 215 Recent results show [Ly 9, Nguyen 12]: Similar effect are obtained at low frequencies Key parameter : waveforms 24

25 Speed estimator EMR 15, Lille, Sept The waveforms must be controlled, but the sensors are too expensive : use an estimator instead 4 flexible link (negligible) workpiece Fw F 1 piezo 3 body 5+6 tool

26 Speed Estimator EMR 15, Lille, Sept. 215 The path from the speed toward a measurable electric variable is found using the EMR of the system [Giraud-Audine 11] u 3 = F p =NV im=n u 3 F=F p F k i V Fp V im u 3 ES V= 1 (i im ) dt Cb f 1 F F acc ) dt ( m u 3 MS u 3 u 3 Fk facc F k =K s u 3 dt 26

27 Speed Estimator EMR 15, Lille, Sept. 215 The motional current is then estimated by matching the voltage of model of the blocked capacitance (purple block) to the actual voltage using a closed loop 27

28 Experimental results EMR 15, Lille, Sept. 215 The estimator is able to track the speed accurately. 28

29 Conclusion EMR 15, Lille, Sept. 215 The EMR is useful to understand the operation of a piezoelectric devices by introducing the causality in the piezoelectric equations Using this tool, it is possible to analyse the physic of the system to deduce useful properties for the control design A direct application has been presented 3

30 EMR 15, Lille, Sept. 215 «BIOGRAPHIES AND REFERENCES» 31

31 - Authors EMR 15, Lille, Sept. 215 Dr. Christophe GIRAUD-Audine graduated from the Arts et Métiers (MsC in mechanical engineering, 1992), and received a PhD in Electrical Engineering ENSEEIHT (1998).He is Associate Professor since 21 at Arts et Métiers Paristech. He is a member of L2EP, France. His research topics are modelling and control of piezoelectric devices. Dr Frédéric Giraud graduated from the Ecole Normale Supérieure de Cachan, France in 1996 in electrical engineering, and received his PhD from the University Lille1, in 22. He is a member of the electrical engineering and power electronics laboratory of Lille where he works as an Associate Professeur. His research deals with the modelling and the control of piezo-electric actuators. 32

32 - References EMR 15, Lille, Sept [Ikeda 96] Takuro Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1996 [Giraud-Audine 11] C. Giraud-Audine and F. Giraud, Preliminary feasibility study of a speed estimator for piezoelectric actuators used in forging processes, in Proceedings of the 21114th European Conference on Power Electronics and Applications (EPE 211), 211, pp [Ly 9] R. Ly, C. Giraud-Audine, G. Abba, and R. Bigot, Experimentally valided approach for the simulation of the forging process using mechanical vibration, International Journal of Material Forming, vol. 2, no. S1, pp , Dec. 29. [Nguyen 12] T. H. Nguyen, C. Giraud-Audine, B. Lemaire-S , G. Abba, and R. Bigot, Modelling of piezoelectric actuators used in forging processes: Principles and experimental validation, in 212 XXth International Conference on Electrical Machines (ICEM), 212, pp