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Finite element modelling for a piezoelectric ultrasonic system
Measurement 43 (2010) 1387–1397
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Finite element modelling for a piezoelectric ultrasonic system Li-Hong Juang * Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia
a r t i c l e
i n f o
Article history: Received 22 December 2009 Received in revised form 15 June 2010 Accepted 4 August 2010 Available online 10 August 2010 Keywords: Piezoelectro-mechanic Discretized equation of motion Asymmetric disc Guyan reduction Householder-Bisection inverse iteration
a b s t r a c t In this paper, a three-dimension (3-D) mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, piezoelectro-mechanic and mechanical behaviors of a metal disc structure embedded with a piezoelectric actuator. In piezoelectric finite element formulation, a discretized equation of motion is developed and solved by using the integration scheme to explain why an adaptive boundary condition, a simple support condition with three non-equal-triangular (120°–90°–150°) fixed points near the edge, which is the asymmetric disc used as the stator of the studied ultrasonic motor, for the mechanical design of an asymmetric disctype piezoelectric ultrasonic stator, is defined so that a lateral elliptical motion of the contact point between stator and rotor can be realized for driving the rotor. It starts from hybrid elements with displacement and electric potential as the nodal d.o.f.s model and uses Guyan reduction and Householder-Bisection inverse iteration to find the displacement profile and displacement vector flow of the stator under frequency driving. The standing wave existence is also proven by the displacement patterns of the finite element theoretical model. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For the design of piezoelectric devices such as piezoelectric transducers and actuators, the steady state responses give useful information, which readily provides the input admittance at the electrical terminal, modal shapes and the transmission characteristic between the electrical and mechanical terminals. The numerical techniques for that purpose are well established in [1–6]. For the simulation of the steady state piezoelectric vibrations, the three-dimensional finite element has been developed by Kagawa and Gladwell [1] and Tzou and Tseng [4]. Piezoelectric materials have been indispensable for electrochemical resonators, transducers, sensors, actuators and adaptive structure. Due to the complexity of the governing equations in piezoelectricity, only a few simple problems such as simply supported beam and plates can be * Tel.: +60 75534555; fax: +60 75566159. E-mail address: [email protected] 0263-2241/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2010.08.002
solved analytically [7–9]. Bansevicius and other researchers in [10,11] employed one piezoelectric element but used different design approaches to develop many implementations of rotary drives, manipulators, mini-robots with much more degrees of freedom and with quite simple constructions. In their designs, the rotating wave is generated by the combination of two standing natural flexural waves whose phases differ by 90°, both spatially and temporally. These two standing waves raise the driving frequency of rotary-type actuators up into ultrasonic frequency range by using axle vibration. Since [12] presented their work on finite element (FE) method for piezoelectric vibration analysis, the FE method has been the dominant practical tool for design and analysis of piezoelectric devices and adaptive structures. Following Allik and Hughes’s work, all of the finite element models proposed in [13–18] include displacement and electric potential as the only assumed field variables. Other fields such as stress, electric displacement, etc., are derived from displacement and electric potential. These models and the
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L.-H. Juang / Measurement 43 (2010) 1387–1397
Table 1 The relative material parameters and dimensions of the asymmetric ultrasonic stator. Values used for stator Piezoelectric membrane Material Diameter (mm) Thickness (mm) Young’s modulus Density (kg m3) Piezoelectric strain constant d31 Poisson’s ratio
PZT-5H 24.8 0.1 65 GPa 7500 274 1012 m/V 0.34
Metal back plate Material Diameter (mm) Thickness (mm) Young’s modulus Density (kg m3) Poisson’s ratio
Ni-alloy 41 0.23 170 GPa 8800 0.35
associated formulations can be classified as irreducible in the sense that the number of field variables cannot be further reduced [19]. Similar to the irreducible or displacement elements in structural mechanics, irreduc-
ible piezoelectric elements are often too stiff, susceptible to mesh distortion and aspect ratio. To overcome these drawbacks, Tzou et al. [4,18] made use of bubble/incompatible displacement modes [19,20] to improve the eight-node hexahedral element. In addition to the bubble/incompatible displacement method, hybrid variational principles in structure mechanics have been successfully employed for enhancing the element accuracy and circumventing various locking phenomena [21]. In this light, [22] proposed a piezoelectric hybrid finite element model in which electric potential and displacement are assumed. Their model is markedly superior to the irreducible model. Finite element method has been well developed and has been widely used in solid mechanics in recent decades. For example, [17] used the finite element method to study the dynamic characteristics of a piezoceramic disc. In that research, the electrical term was considered as an extra mechanical degree of freedom in solving the eigenvalue problem. Then a modal analysis by the Lanczos method was employed to acquire steady state response functions such as vibration modes and natural frequencies in the mechanical response and electrical impedance function. In this case, r modes were the majority of types considered.
Z
Y X
120o 150o
90o
Fig. 1. Illustration of the working principle of the disc-type ultrasonic motor which generates the lateral elliptical motion at the single contact point.
L.-H. Juang / Measurement 43 (2010) 1387–1397
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Fig. 2. The displacement profile (z dir. view) under audible frequency excitation: (a) 1 k–4 kHz, (b) 5 k–8 kHz, (c) 9 k–12 kHz, (d) 13 k–16 kHz and (e) 17 k– 20 kHz by finite element simulation.
However, that research focused on transducer applications. Meanwhile, an adhesive layer under the piezoceramic disc such as metal back plate was not included in the finite element analysis. In our study, we refer to that analysis model for the piezoelectric membrane and consider the adhesive metal plate. The piezoceramic itself serves as an actuator in our case. Because the ultrasonic stator consists of a piezoelectric membrane and a metal back plate with the asymmetric boundary condition, the finite element model must base on a linear piezoelectric equation but it needs to define the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate using a 3-D mechanical element with an extra electrical degree of freedom. The electric boundary condition needs to fit the summation of the nodal elec-
tric charges being zero, and the mechanical boundary condition needs to fit the zero displacements at the edge fixed points. Then, because the piezoelectric membrane is embedded on the metal back plate, it needs to base on a piezoelectric–mechanic coupled equation to acquire the induced extra stiffness, the interactive force, and the extra induced charge. Finally, because the metal back plate is used as the stator to rotate it rotor, a mechanic finite element model with structure damping is needed to acquire the final rotating force for the rotor. Due to the metal back plate of the ultrasonic stator is the soft structure, the irreducible and incompatible element types are not suitable for our finite element model. But the ultrasonic stator includes electric potential and displacement, according to the paper [22] suggestion, we adopts the hybrid element
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Standard 1987, but here the contribution of the research work is on successfully defining the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate, later it will be explained. A 3-D mechanical element with an extra electrical degree of freedom is involved into the linear piezoelectric equation. Then, the coupling between the elastic field and the electric field for piezoelectric actuator is given by
T ¼ C E S eT E; D ¼ eS þ eS E;
Fig. 2 (continued)
type for the finite element analysis in order to get the accuracy data. Therefore, the contribution of the research work is mainly highlighted on the operational verification of an asymmetric disc-type piezoelectric ultrasonic stator by using the finite element theoretical model derivation to acquire the displacement profile and displacement vector flow of the stator under frequency driving. The standing wave existence is also proven by the displacement patterns of the finite element theoretical model.
2. Finite element model The finite element theoretical derivation was carried out by commercially available finite element (FEM) ANSYS5.51/Multiphysics software, which provides structure analysis with piezoelectric effect. The simulating model consists of hybrid SOLID5 structure element to model the elastic-metal-back plate and coupling element for a piezoelectric membrane. In ANSYS software, the SOLID5 structure element with four degree of freedom at each node is solved for the nodal displacement in x, y, and z axes plus electrical potential (see Section 1 to ANSYS for Revision 5.51 and Dynamics User’s Guide for Revision 5.51). Under fitting mechanical and electrical boundary conditions, ANSYS FEM code to get the relative simulation graphs of displacement, stress and electrical potential for the elastic-metal-back plate and piezoelectric membrane. The detail finite element theoretical model is separated into three parts: linear piezoelectric, piezoelectric–mechanic, and mechanical. First linear piezoelectric model is described as follows: A linear piezoelectric constitutive equation is employed to establish the dynamic model of the stator. The linear piezoelectric equation has been published earlier by IEEE
ð1Þ
where T: 6 1 stress-vector, D: 3 1 electrical displacement vector, S: 6 1 strain vector, and E: 3 1 electrical field vector respectively, CE: 6 6 elasticity matrix, e: 3 6 piezoelectric matrix and eT is its transpose, and eS: 3 3 dielectric matrix. The electric field E is related to the electric potential / by E ¼ r/. The displacement and potential for each element can be expressed, respectively, as
^; u ¼ Nu u ^ / ¼ N/ /;
ð2Þ
where u is the displacement vector, Nu and N/ the interpolation function for the variables of / and u, and ^ denotes the nodal values. To put the strain–displacement relation in terms of the ^ . Here, Bu is the product nodal displacement yields S ¼ Bu u of the differential-operating matrix relating S to the shape ^¼ function matrix Nu. Similarly, let E ¼ r/ ¼ rN / / ^ [23]. B/ / In this paper, we use the finite element method to study the different aspects of behavior in the asymmetric piezoelectric stator. In this code, a multi-field solid element and a standard isotropic solid element are involved in modelling the piezoelectric ceramic and disc base, respectively. They are both eight-node brick elements and three spatial degree nodal displacements for the degrees of freedom. In addition, each node has voltage as an extra degree of freedom in the multi-field elements. The second part of finite element derivation is for piezoelectric–mechanic coupled finite element model, as follows. The governing dynamic equation of the asymmetric piezoelectric stator in matrix form for the piezoelectric ceramic phase, which exists the electrical effects (see Table 1, piezoelectric membrane), is as follows:
€ p þ K puu up þ K pu/ / ¼ F p ; Mp u K pu/ u þ K p// / ¼ P;
ð3Þ
where Mp, K puu , K pu/ , K p// , Fp and P are mass matrix, elastic stiffness matrix, piezoelectric coupling matrix, dielectric stiffness matrix, mechanical force vector and electrical charge vector, respectively. The superscript p denotes the piezoelectric ceramic. These relative substitutive variables are given, respectively, by
L.-H. Juang / Measurement 43 (2010) 1387–1397
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Fig. 3. The displacement vector flow under audible frequency excitation: (a) 1 k–4 kHz, (b) 5 k–8 kHz, (c) 9 k–12 kHz, (d) 13 k–16 kHz and (e) 17 k–20 kHz by finite element simulation.
Mp ¼ K puu ¼ K pu/ ¼
Z
qNTu Nu dV;
Z
BTu C E Bu dV;
Z
BTu eT B/ dV; Z s K p// ¼ BT/ b B/ dV; Z Z NTb fb dV þ NTS1 fS dS1 þ NTu fc ; Fp ¼ V S1 Z P¼ N TS2 qS dS2 NT/ qc ;
ð4Þ
S2
where fb is the body force, fs is the surface force, fc is the concentrated force, qs is the surface charge, qc is the point
charge, S1 is the area where mechanical forces are applied and S2 is the area where electrical charges are applied. The contribution of the research work for the piezoelectric– mechanic coupled finite element model is on successfully deriving electrical–mechanic coupling induced the extra stiffness, that is piezoelectric coupling matrix K pu/ , the interactive force, that is mechanical force vector Fp, and the extra induced charge, that is electrical charge vector P. The third part of finite element derivation is for mechanic finite element model, as follows. For the disc base (see Table 1, metal back plate), the electrical effects do not exist and the finite element equation is given by
€s þ K suu us ¼ F s ; Ms u
ð5Þ
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Fig. 5. The displacement vector flow under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz by finite element simulation. Fig. 3 (continued)
Fig. 4. The displacement profile (z dir. view) under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz by finite element simulation.
where the superscript s denotes the disc. In general, all structures are slightly damped due to structure damping. Thus, Eq. (3) can be modified to p p €
p p _
M u þC u þ
K puu up
þ
K pu/ /
K pu/ u þ K p// / ¼ P;
^ iþ1 ¼ ¼ constant or U^i ¼ U
X
Q i ¼ 0:
ð8Þ
p
¼F ;
ð6Þ
Eq. (6) still exists the electrical effects but with structure damping, in addition, Eq. (5) can be modified to
€ s þ C s u_ s þ K suu us ¼ F s ; Ms u
Eq. (7) still does not exists the electrical effects but with structure damping, where C p ¼ gM p þ kK puu and C s ¼ gM s þ kK suu , where g and k are Rayleigh coefficients. It shall be noted that the damping matrix in ANSYS implements [24], C ¼ bp K puu þ bs K suu where bpand bs are the damping coefficients associated with piezoelectric ceramic and disc material, cannot model dielectric losses but does not allow the two attenuation mechanisms to differ due to being normally insignificant to structural damping. The mechanic finite element model is often used on the most finite element analysis. The contribution of the research work for the mechanic finite element model is on successfully testing out the matched structure damping value. To evaluate the influence of damping loss, harmonic analysis was run to predict the electrical impedance of the stator as a function of frequency. Harmonic analysis (ANSYS code) allows incorporation of loss within a medium by use of damping factors for any operating frequency. The damping is usually adjusted following the experimental test until a reasonable match is found. The boundary conditions of the simulation are separated into the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate. The electric field boundary condition the piezoelectric membrane requires that the electrode surface is an equipotential one, that is unity potential value, and the summation of the nodal electric charges on it should be zero as shown in the following:
ð7Þ
The adhesive layer between the piezoelectric ceramic and the disc base will be ignored in the finite element analysis. The mechanical boundary condition for the metal back plate requires that the three non-equal-triangular (120°–90°–150°) [25] fixed points near the edge of the stator (see Fig. 1), which is the asymmetric disc, must have zero displacement.
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Fig. 6. The displacement patterns for verifying standing wave existence: (a) experimental measurement and (b) analysis simulation (? indicates fixedpoint location).
To analyze the steady state response characteristics of the asymmetric piezoelectric stator, first modal response analysis has to be made to determine the mode shape and the natural frequency of the asymmetric piezoelectric stator. In modal analysis, it should be noted that the extra voltage degrees of freedom in the finite element equation have to be condensed using Guyan reduction. Then the displacement profile and displacement vector flow of the stator under frequency driving can be found using Householder-Bisection inverse iteration. Additionally, be-
cause the electrical degrees of freedom don’t have mass associate with them, the mass matrix may not be positive definite and the stiffness matrix would be non-positive definite due to the negative dielectric.
3. Analysis and discussion To express the operational principle of the motor, a simple support condition with three non-equal-triangular
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Fig. 7. The displacement vector flow at the lateral edge under harmonic excitation (20 V input): (a) 75 kHz, (b) 83.13 kHz and (c) 98 kHz by finite element (? indicates fixed-point location).
(120°–90°–150°) [25] fixed points near the edge of the stator was defined (see Fig. 1), which is the asymmetric disc used as the stator of the studied ultrasonic motor. When a sinusoidal input is applied to the stator, it induces a lateral elliptical motion and generates a torque to rotate the rotor. The mechanical design of the stator will enlarge the lateral elliptical motion between two fixed points with a 90° included angle. The desired vibration modes can be concentrated at 75 kHz, 83.13 kHz and 98 kHz, respectively. The asymmetric piezoelectric ultrasonic stator consists of piezoelectric membranes and a circular metal disc base. Table 1 lists its known material parameters and dimensions. The analysis parameters for linear piezoelectric,
piezoelectric–mechanic, and mechanical finite element models now can be properly used in this simulation verification by Table 1. In this study, all steady state response analyses will be added the structure damping, that is, g and k. We assumed g = 7.5 and k ¼ 2 105 , internal damping and Poisson’s ratio of the passive metal disc base influence the damping loss of the ultrasonic stator considerably. The damping was maintained constant for the stator material during supplementary theoretical analysis. The soft metal disc base with almost unimodal behavior usually is predicted quite well by the finite element theory. Additionally, we defined the boundary condition of the asymmetric piezoelectric ultrasonic stator as a simple support condition with three
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Fig. 8. The one-wheeled actuator driven by one piezoelectric element configuration: (a) a recent actuator and (b) with an open-loop circuit driver.
non-equal-triangular (120°–90°–150° fixed points near the edge). Then we used ANSYS code to run all response analyses. Fig. 2a–e shows the z direction displacement profile under harmonic excitation. The exciting frequencies are at an audible frequency range of 1 k–20 kHz. The results reveal that the stator has no significant standing wave that exists, especially at the stator edge. As we know, the standing wave transfers most driving energy into the rotor. However, obviously there is not much useful transformational energy at an audible frequency range. So, the proposed motor cannot operate at an audible frequency range. Fig. 3a–e shows the displacement vector flow under audible frequency driving at 1 k–20 kHz. The results show that there is no lateral elliptical motion that exists. It proves again that audible frequency is not adaptive for a lateral elliptical motion type of ultrasonic motor, which we simply define as LEMUM. The audible frequency means that a frequency is under 20 kHz, which is not ultrasonic frequency range. For a piezoelectric motor being operated
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at an audible frequency, it will have lots acoustic noise and energy dissipation. Figs. 2a–e and 3a–e prove that the piezoelectric motor must be operated at a non-audible frequency, that is an ultrasonic frequency. A piezoelectric motor being operated at an audible frequency will be quite and low energy dissipation. Fig. 4 shows the z direction displacement profile under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz. The results show that there are more significant standing waves that exist at lower ultrasonic frequency driving. There is a more stable and a denser standing-wave pattern at 65 kHz. Fig. 5 shows the displacement vector flow under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz. Lateral displacement motions have become very significant at 55 kHz and 65 kHz. However, the maximum lateral motions are not located in the 90° included angle section. In a real test, the proposed motor can be driven at 45 kHz, 55 kHz and 65 kHz using the isolating-resistant type driving circuit. However, the vibration patterns at 45 kHz, 55 kHz and 65 kHz can be acquired from the laser vibrometer due to input voltages being limited using the isolating-resistant type driving circuit and in contrast, input signals being too weak without the driving circuit. Fig. 6a and b shows the graphs correlating the frequency and standing waves from the experimental measurement and the analysis simulation, which will be able to give a much better insight of the standing wave existence, 75 kHz was used just for explanation. To clarify the mechanism of the asymmetric piezoelectric ultrasonic stator is adaptive for the best output performance of the ultrasonic motor, we ran an analysis of displacement vector flow at its lateral edge for a 75 kHz, 20 V input signal. Fig. 7a–c shows that lateral elliptical motions are induced at the stator edge, and that they confirm the operation principle in Fig. 1. The stator will be able to drive the rotor at 75 kHz, 83 kHz and 98 kHz. The maximal displacement around the edge occurs in a 90° included angle section. It is the optimal output location for rotating the rotor. The similar theoretic model and a real test have been also verified by ‘‘characterization of one-wheeled actuator driven by one piezoelectric element” in the accepted paper [26]. Fig. 8a and b shows the configuration of the wheel actuator with an open-loop circuit driver. Fig. 9a and b shows the displacement vector flows (y dir. view) at the driving frequencies of 24.12 Hz and 42.753 Hz. The whole wheel actuator is found to have clockwise rotation (CW) at 24.12 Hz and counter-clockwise rotation (CCW) at 42.753 Hz. As depicted in Fig. 10a and b, the experimental results confirm that the onewheeled actuator can be driven into clockwise (CW) rotation at 24 Hz and counter-clockwise (CCW) rotation at 42 Hz.
4. Conclusions By modelling the dynamic formulary of the asymmetric piezoelectric stator in the linear piezoelectric, piezoelectric–mechanic, and mechanical finite element method, the eigenvalue problem can be solved under defining the
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Fig. 9. The displacement vector flow (y dir. view) at: (a) 24.12 Hz and (b) 42.753 Hz by finite element modal simulation.
electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate. It shall be noted that the extra electrical degrees of freedom cannot have mass associate with them, and the mass matrix may not be positive definite and the stiffness matrix would be non-positive definite due to the negative dielectric. The mechanical response under con-
stant voltage excitation is acquired. The operational characteristics of the asymmetric piezoelectric stator are verified by analysis of displacement vector flow at its lateral edge. A standing wave under the higher ultrasonic frequency is the key fact for successfully driving the asymmetric piezoelectric stator. It has great potential to service as a design guideline for possible use in practical
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Fig. 10. Experimental results: (a) continuous CW rotation and (b) continuous CCW rotation.
stator design, especially in the stator of an ultrasonic motor using its lateral motion. References [1] Y. Kagawa, G.M.L. Gladwell, Finite element analysis of flexure type vibrators with electrostrictive transducers, IEEE Trans. Sonics Ultrason. 17 (1970) 41–49. [2] Y. Kagawa, T. Yamabuchi, Finite element simulation of composite piezoelectric transducer, IEEE Trans. Sonics Ultrason. 26 (1979) 81– 88. [3] M. Kuribayashi, S. Ueha, E. Mori, Excitation condition of flexural traveling waves for a reversible ultrasonic linear motor, J. Acoust. Soc. Am. 77 (1985) 1431–1435. [4] H.S. Tzou, C.I. Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: piezoelectric finite element approach, J. Sound Vib. 138 (1990) 17–34. [5] M. Kuribayashi, S. Ueha, Single-phase drive of a circular ultrasonic motor, J. Acoust. Soc. Am. 90 (1991) 1723–1728. [6] V. Chaika, Steady state response of a system of rotors with various types of flexible couplings, J. Sound Vib. 177 (1994) 521–538. [7] M.C. Ray, K.M. Rao, B. Samanta, Exact solution for static analysis of an intelligent structure under cylindrical bending, J. Comput. Struct. 47 (1993) 1031–1042. [8] J.S. Yang, R.C. Batra, X.Q. Liang, The cylindrical bending vibration of a laminated elastic plastic due to piezoelectric actuators, J. Smart Mater. Struct. 3 (1994) 485–493. [9] P.R. Heyliger, D.A. Saravanos, Exact free-vibration analysis of laminated piezoelectric layer, J. Acoust. Soc. Am. 98 (1995) 1547– 1557. [10] R.H. Bishop (Ed.), The Mechatronics Handbook, CRC Press, Boca Raton, 2002, ISBN: 0-8493-0066. [11] Responsive systems for active vibration control. NATO Science Series, II. Mathematics, Physics and Chemistry, vol. 85, Kluwer Academic Publishers, Dordrecht, 2002, p. 394, ISBN: 1-4020-0897-X (HB). [12] H. Allik, T.J. Hughes, Finite element method for piezoelectric vibration, Int. J. Numer. Methods Eng. 2 (1970) 151–157.
[13] M. Naillon, R.H. Coursant, F. Besnier, Vibration response of sonar transducers using piezoelectric finite elements, Acta Electron. 25 (1983) 341–362. [14] J.H. Jeng, X. Bao, V.V. Varadan, V.K. Varadan, Complete finite element eigenmode analysis for 1–3 type of piezoelectric composite transducer including the effect of fluid loading and internal losses, Proc. IEEE Ultrason. Symp. (1988) 685–688. [15] R. Lerch, Simulation of piezoelectric devices by two- and threedimensional finite elements, IEEE Trans. Ultrason. Ferroelectr. Frequency Control 37 (1990) 233–247. [16] J.A. Hossack, G. Hayward, Finite element analysis of 1–3 composite transducers, IEEE Trans. Ultrason. Ferroelectr. Frequency Control 38 (1991) 618–627. [17] N. Guo, P. Cawley, The finite element analysis of the vibration characteristics of piezoelectric discs, J. Sound Vib. 159 (1992) 115–137. [18] H.S. Tzou, Piezoelectric Shells: Distributed Sensing and Control of Continua, Kluwer Press, 1993. [19] R.L. Taylor, P.J. Beresford, E.L. Wilson, A non-conforming element for stress analysis, Int. J. Numer. Methods Eng. 10 (1976) 1211–1220. [20] J.C. Simo, M.S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, Int. J. Numer. Methods Eng. 29 (1990) 1595–1638. [21] T.H. Pian, P. Tong, Basis of finite elements for solids continua, Int. J. Numer. Methods Eng. 29 (1969) 3–28. [22] K. Ghandi, N.W. Hagood, A hybrid finite element model for phase transitions in nonlinear electro-mechanically coupled material, SPIE Proc. 3039 (1997) 97–112. [23] Y.H. Lim, V.V. Varadan, V.K. Varadan, Closed loop finite element modeling of active structural damping in the time domain, J. Smart Mater. Struct. 8 (1999) 390–400. [24] G. Hayward, Assessing the influence of pillar aspect ration on the behavior of 1–3 connectivity composite transducers, IEEE Trans. Ultrason. Ferroelectr. Frequency Control 43 (1996) 98–108. [25] M. Ouyang, A thin-disc actuating type ultrasonic motor, Taiwan Patent 502488, September 11, 2002 (in Chinese). [26] P.A. Juang, C.C. Tsai, Characterization of one-wheeled actuator driven by one piezoelectric element, Measurement 42 (1) (2009) 112–118.
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Finite element modelling for a piezoelectric ultrasonic system Li-Hong Juang * Department of Applied Mechanics, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia
a r t i c l e
i n f o
Article history: Received 22 December 2009 Received in revised form 15 June 2010 Accepted 4 August 2010 Available online 10 August 2010 Keywords: Piezoelectro-mechanic Discretized equation of motion Asymmetric disc Guyan reduction Householder-Bisection inverse iteration
a b s t r a c t In this paper, a three-dimension (3-D) mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, piezoelectro-mechanic and mechanical behaviors of a metal disc structure embedded with a piezoelectric actuator. In piezoelectric finite element formulation, a discretized equation of motion is developed and solved by using the integration scheme to explain why an adaptive boundary condition, a simple support condition with three non-equal-triangular (120°–90°–150°) fixed points near the edge, which is the asymmetric disc used as the stator of the studied ultrasonic motor, for the mechanical design of an asymmetric disctype piezoelectric ultrasonic stator, is defined so that a lateral elliptical motion of the contact point between stator and rotor can be realized for driving the rotor. It starts from hybrid elements with displacement and electric potential as the nodal d.o.f.s model and uses Guyan reduction and Householder-Bisection inverse iteration to find the displacement profile and displacement vector flow of the stator under frequency driving. The standing wave existence is also proven by the displacement patterns of the finite element theoretical model. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For the design of piezoelectric devices such as piezoelectric transducers and actuators, the steady state responses give useful information, which readily provides the input admittance at the electrical terminal, modal shapes and the transmission characteristic between the electrical and mechanical terminals. The numerical techniques for that purpose are well established in [1–6]. For the simulation of the steady state piezoelectric vibrations, the three-dimensional finite element has been developed by Kagawa and Gladwell [1] and Tzou and Tseng [4]. Piezoelectric materials have been indispensable for electrochemical resonators, transducers, sensors, actuators and adaptive structure. Due to the complexity of the governing equations in piezoelectricity, only a few simple problems such as simply supported beam and plates can be * Tel.: +60 75534555; fax: +60 75566159. E-mail address: [email protected] 0263-2241/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2010.08.002
solved analytically [7–9]. Bansevicius and other researchers in [10,11] employed one piezoelectric element but used different design approaches to develop many implementations of rotary drives, manipulators, mini-robots with much more degrees of freedom and with quite simple constructions. In their designs, the rotating wave is generated by the combination of two standing natural flexural waves whose phases differ by 90°, both spatially and temporally. These two standing waves raise the driving frequency of rotary-type actuators up into ultrasonic frequency range by using axle vibration. Since [12] presented their work on finite element (FE) method for piezoelectric vibration analysis, the FE method has been the dominant practical tool for design and analysis of piezoelectric devices and adaptive structures. Following Allik and Hughes’s work, all of the finite element models proposed in [13–18] include displacement and electric potential as the only assumed field variables. Other fields such as stress, electric displacement, etc., are derived from displacement and electric potential. These models and the
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L.-H. Juang / Measurement 43 (2010) 1387–1397
Table 1 The relative material parameters and dimensions of the asymmetric ultrasonic stator. Values used for stator Piezoelectric membrane Material Diameter (mm) Thickness (mm) Young’s modulus Density (kg m3) Piezoelectric strain constant d31 Poisson’s ratio
PZT-5H 24.8 0.1 65 GPa 7500 274 1012 m/V 0.34
Metal back plate Material Diameter (mm) Thickness (mm) Young’s modulus Density (kg m3) Poisson’s ratio
Ni-alloy 41 0.23 170 GPa 8800 0.35
associated formulations can be classified as irreducible in the sense that the number of field variables cannot be further reduced [19]. Similar to the irreducible or displacement elements in structural mechanics, irreduc-
ible piezoelectric elements are often too stiff, susceptible to mesh distortion and aspect ratio. To overcome these drawbacks, Tzou et al. [4,18] made use of bubble/incompatible displacement modes [19,20] to improve the eight-node hexahedral element. In addition to the bubble/incompatible displacement method, hybrid variational principles in structure mechanics have been successfully employed for enhancing the element accuracy and circumventing various locking phenomena [21]. In this light, [22] proposed a piezoelectric hybrid finite element model in which electric potential and displacement are assumed. Their model is markedly superior to the irreducible model. Finite element method has been well developed and has been widely used in solid mechanics in recent decades. For example, [17] used the finite element method to study the dynamic characteristics of a piezoceramic disc. In that research, the electrical term was considered as an extra mechanical degree of freedom in solving the eigenvalue problem. Then a modal analysis by the Lanczos method was employed to acquire steady state response functions such as vibration modes and natural frequencies in the mechanical response and electrical impedance function. In this case, r modes were the majority of types considered.
Z
Y X
120o 150o
90o
Fig. 1. Illustration of the working principle of the disc-type ultrasonic motor which generates the lateral elliptical motion at the single contact point.
L.-H. Juang / Measurement 43 (2010) 1387–1397
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Fig. 2. The displacement profile (z dir. view) under audible frequency excitation: (a) 1 k–4 kHz, (b) 5 k–8 kHz, (c) 9 k–12 kHz, (d) 13 k–16 kHz and (e) 17 k– 20 kHz by finite element simulation.
However, that research focused on transducer applications. Meanwhile, an adhesive layer under the piezoceramic disc such as metal back plate was not included in the finite element analysis. In our study, we refer to that analysis model for the piezoelectric membrane and consider the adhesive metal plate. The piezoceramic itself serves as an actuator in our case. Because the ultrasonic stator consists of a piezoelectric membrane and a metal back plate with the asymmetric boundary condition, the finite element model must base on a linear piezoelectric equation but it needs to define the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate using a 3-D mechanical element with an extra electrical degree of freedom. The electric boundary condition needs to fit the summation of the nodal elec-
tric charges being zero, and the mechanical boundary condition needs to fit the zero displacements at the edge fixed points. Then, because the piezoelectric membrane is embedded on the metal back plate, it needs to base on a piezoelectric–mechanic coupled equation to acquire the induced extra stiffness, the interactive force, and the extra induced charge. Finally, because the metal back plate is used as the stator to rotate it rotor, a mechanic finite element model with structure damping is needed to acquire the final rotating force for the rotor. Due to the metal back plate of the ultrasonic stator is the soft structure, the irreducible and incompatible element types are not suitable for our finite element model. But the ultrasonic stator includes electric potential and displacement, according to the paper [22] suggestion, we adopts the hybrid element
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Standard 1987, but here the contribution of the research work is on successfully defining the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate, later it will be explained. A 3-D mechanical element with an extra electrical degree of freedom is involved into the linear piezoelectric equation. Then, the coupling between the elastic field and the electric field for piezoelectric actuator is given by
T ¼ C E S eT E; D ¼ eS þ eS E;
Fig. 2 (continued)
type for the finite element analysis in order to get the accuracy data. Therefore, the contribution of the research work is mainly highlighted on the operational verification of an asymmetric disc-type piezoelectric ultrasonic stator by using the finite element theoretical model derivation to acquire the displacement profile and displacement vector flow of the stator under frequency driving. The standing wave existence is also proven by the displacement patterns of the finite element theoretical model.
2. Finite element model The finite element theoretical derivation was carried out by commercially available finite element (FEM) ANSYS5.51/Multiphysics software, which provides structure analysis with piezoelectric effect. The simulating model consists of hybrid SOLID5 structure element to model the elastic-metal-back plate and coupling element for a piezoelectric membrane. In ANSYS software, the SOLID5 structure element with four degree of freedom at each node is solved for the nodal displacement in x, y, and z axes plus electrical potential (see Section 1 to ANSYS for Revision 5.51 and Dynamics User’s Guide for Revision 5.51). Under fitting mechanical and electrical boundary conditions, ANSYS FEM code to get the relative simulation graphs of displacement, stress and electrical potential for the elastic-metal-back plate and piezoelectric membrane. The detail finite element theoretical model is separated into three parts: linear piezoelectric, piezoelectric–mechanic, and mechanical. First linear piezoelectric model is described as follows: A linear piezoelectric constitutive equation is employed to establish the dynamic model of the stator. The linear piezoelectric equation has been published earlier by IEEE
ð1Þ
where T: 6 1 stress-vector, D: 3 1 electrical displacement vector, S: 6 1 strain vector, and E: 3 1 electrical field vector respectively, CE: 6 6 elasticity matrix, e: 3 6 piezoelectric matrix and eT is its transpose, and eS: 3 3 dielectric matrix. The electric field E is related to the electric potential / by E ¼ r/. The displacement and potential for each element can be expressed, respectively, as
^; u ¼ Nu u ^ / ¼ N/ /;
ð2Þ
where u is the displacement vector, Nu and N/ the interpolation function for the variables of / and u, and ^ denotes the nodal values. To put the strain–displacement relation in terms of the ^ . Here, Bu is the product nodal displacement yields S ¼ Bu u of the differential-operating matrix relating S to the shape ^¼ function matrix Nu. Similarly, let E ¼ r/ ¼ rN / / ^ [23]. B/ / In this paper, we use the finite element method to study the different aspects of behavior in the asymmetric piezoelectric stator. In this code, a multi-field solid element and a standard isotropic solid element are involved in modelling the piezoelectric ceramic and disc base, respectively. They are both eight-node brick elements and three spatial degree nodal displacements for the degrees of freedom. In addition, each node has voltage as an extra degree of freedom in the multi-field elements. The second part of finite element derivation is for piezoelectric–mechanic coupled finite element model, as follows. The governing dynamic equation of the asymmetric piezoelectric stator in matrix form for the piezoelectric ceramic phase, which exists the electrical effects (see Table 1, piezoelectric membrane), is as follows:
€ p þ K puu up þ K pu/ / ¼ F p ; Mp u K pu/ u þ K p// / ¼ P;
ð3Þ
where Mp, K puu , K pu/ , K p// , Fp and P are mass matrix, elastic stiffness matrix, piezoelectric coupling matrix, dielectric stiffness matrix, mechanical force vector and electrical charge vector, respectively. The superscript p denotes the piezoelectric ceramic. These relative substitutive variables are given, respectively, by
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Fig. 3. The displacement vector flow under audible frequency excitation: (a) 1 k–4 kHz, (b) 5 k–8 kHz, (c) 9 k–12 kHz, (d) 13 k–16 kHz and (e) 17 k–20 kHz by finite element simulation.
Mp ¼ K puu ¼ K pu/ ¼
Z
qNTu Nu dV;
Z
BTu C E Bu dV;
Z
BTu eT B/ dV; Z s K p// ¼ BT/ b B/ dV; Z Z NTb fb dV þ NTS1 fS dS1 þ NTu fc ; Fp ¼ V S1 Z P¼ N TS2 qS dS2 NT/ qc ;
ð4Þ
S2
where fb is the body force, fs is the surface force, fc is the concentrated force, qs is the surface charge, qc is the point
charge, S1 is the area where mechanical forces are applied and S2 is the area where electrical charges are applied. The contribution of the research work for the piezoelectric– mechanic coupled finite element model is on successfully deriving electrical–mechanic coupling induced the extra stiffness, that is piezoelectric coupling matrix K pu/ , the interactive force, that is mechanical force vector Fp, and the extra induced charge, that is electrical charge vector P. The third part of finite element derivation is for mechanic finite element model, as follows. For the disc base (see Table 1, metal back plate), the electrical effects do not exist and the finite element equation is given by
€s þ K suu us ¼ F s ; Ms u
ð5Þ
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Fig. 5. The displacement vector flow under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz by finite element simulation. Fig. 3 (continued)
Fig. 4. The displacement profile (z dir. view) under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz by finite element simulation.
where the superscript s denotes the disc. In general, all structures are slightly damped due to structure damping. Thus, Eq. (3) can be modified to p p €
p p _
M u þC u þ
K puu up
þ
K pu/ /
K pu/ u þ K p// / ¼ P;
^ iþ1 ¼ ¼ constant or U^i ¼ U
X
Q i ¼ 0:
ð8Þ
p
¼F ;
ð6Þ
Eq. (6) still exists the electrical effects but with structure damping, in addition, Eq. (5) can be modified to
€ s þ C s u_ s þ K suu us ¼ F s ; Ms u
Eq. (7) still does not exists the electrical effects but with structure damping, where C p ¼ gM p þ kK puu and C s ¼ gM s þ kK suu , where g and k are Rayleigh coefficients. It shall be noted that the damping matrix in ANSYS implements [24], C ¼ bp K puu þ bs K suu where bpand bs are the damping coefficients associated with piezoelectric ceramic and disc material, cannot model dielectric losses but does not allow the two attenuation mechanisms to differ due to being normally insignificant to structural damping. The mechanic finite element model is often used on the most finite element analysis. The contribution of the research work for the mechanic finite element model is on successfully testing out the matched structure damping value. To evaluate the influence of damping loss, harmonic analysis was run to predict the electrical impedance of the stator as a function of frequency. Harmonic analysis (ANSYS code) allows incorporation of loss within a medium by use of damping factors for any operating frequency. The damping is usually adjusted following the experimental test until a reasonable match is found. The boundary conditions of the simulation are separated into the electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate. The electric field boundary condition the piezoelectric membrane requires that the electrode surface is an equipotential one, that is unity potential value, and the summation of the nodal electric charges on it should be zero as shown in the following:
ð7Þ
The adhesive layer between the piezoelectric ceramic and the disc base will be ignored in the finite element analysis. The mechanical boundary condition for the metal back plate requires that the three non-equal-triangular (120°–90°–150°) [25] fixed points near the edge of the stator (see Fig. 1), which is the asymmetric disc, must have zero displacement.
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Fig. 6. The displacement patterns for verifying standing wave existence: (a) experimental measurement and (b) analysis simulation (? indicates fixedpoint location).
To analyze the steady state response characteristics of the asymmetric piezoelectric stator, first modal response analysis has to be made to determine the mode shape and the natural frequency of the asymmetric piezoelectric stator. In modal analysis, it should be noted that the extra voltage degrees of freedom in the finite element equation have to be condensed using Guyan reduction. Then the displacement profile and displacement vector flow of the stator under frequency driving can be found using Householder-Bisection inverse iteration. Additionally, be-
cause the electrical degrees of freedom don’t have mass associate with them, the mass matrix may not be positive definite and the stiffness matrix would be non-positive definite due to the negative dielectric.
3. Analysis and discussion To express the operational principle of the motor, a simple support condition with three non-equal-triangular
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Fig. 7. The displacement vector flow at the lateral edge under harmonic excitation (20 V input): (a) 75 kHz, (b) 83.13 kHz and (c) 98 kHz by finite element (? indicates fixed-point location).
(120°–90°–150°) [25] fixed points near the edge of the stator was defined (see Fig. 1), which is the asymmetric disc used as the stator of the studied ultrasonic motor. When a sinusoidal input is applied to the stator, it induces a lateral elliptical motion and generates a torque to rotate the rotor. The mechanical design of the stator will enlarge the lateral elliptical motion between two fixed points with a 90° included angle. The desired vibration modes can be concentrated at 75 kHz, 83.13 kHz and 98 kHz, respectively. The asymmetric piezoelectric ultrasonic stator consists of piezoelectric membranes and a circular metal disc base. Table 1 lists its known material parameters and dimensions. The analysis parameters for linear piezoelectric,
piezoelectric–mechanic, and mechanical finite element models now can be properly used in this simulation verification by Table 1. In this study, all steady state response analyses will be added the structure damping, that is, g and k. We assumed g = 7.5 and k ¼ 2 105 , internal damping and Poisson’s ratio of the passive metal disc base influence the damping loss of the ultrasonic stator considerably. The damping was maintained constant for the stator material during supplementary theoretical analysis. The soft metal disc base with almost unimodal behavior usually is predicted quite well by the finite element theory. Additionally, we defined the boundary condition of the asymmetric piezoelectric ultrasonic stator as a simple support condition with three
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Fig. 8. The one-wheeled actuator driven by one piezoelectric element configuration: (a) a recent actuator and (b) with an open-loop circuit driver.
non-equal-triangular (120°–90°–150° fixed points near the edge). Then we used ANSYS code to run all response analyses. Fig. 2a–e shows the z direction displacement profile under harmonic excitation. The exciting frequencies are at an audible frequency range of 1 k–20 kHz. The results reveal that the stator has no significant standing wave that exists, especially at the stator edge. As we know, the standing wave transfers most driving energy into the rotor. However, obviously there is not much useful transformational energy at an audible frequency range. So, the proposed motor cannot operate at an audible frequency range. Fig. 3a–e shows the displacement vector flow under audible frequency driving at 1 k–20 kHz. The results show that there is no lateral elliptical motion that exists. It proves again that audible frequency is not adaptive for a lateral elliptical motion type of ultrasonic motor, which we simply define as LEMUM. The audible frequency means that a frequency is under 20 kHz, which is not ultrasonic frequency range. For a piezoelectric motor being operated
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at an audible frequency, it will have lots acoustic noise and energy dissipation. Figs. 2a–e and 3a–e prove that the piezoelectric motor must be operated at a non-audible frequency, that is an ultrasonic frequency. A piezoelectric motor being operated at an audible frequency will be quite and low energy dissipation. Fig. 4 shows the z direction displacement profile under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz. The results show that there are more significant standing waves that exist at lower ultrasonic frequency driving. There is a more stable and a denser standing-wave pattern at 65 kHz. Fig. 5 shows the displacement vector flow under lower ultrasonic frequency excitation: 35 kHz, 45 kHz, 55 kHz and 65 kHz. Lateral displacement motions have become very significant at 55 kHz and 65 kHz. However, the maximum lateral motions are not located in the 90° included angle section. In a real test, the proposed motor can be driven at 45 kHz, 55 kHz and 65 kHz using the isolating-resistant type driving circuit. However, the vibration patterns at 45 kHz, 55 kHz and 65 kHz can be acquired from the laser vibrometer due to input voltages being limited using the isolating-resistant type driving circuit and in contrast, input signals being too weak without the driving circuit. Fig. 6a and b shows the graphs correlating the frequency and standing waves from the experimental measurement and the analysis simulation, which will be able to give a much better insight of the standing wave existence, 75 kHz was used just for explanation. To clarify the mechanism of the asymmetric piezoelectric ultrasonic stator is adaptive for the best output performance of the ultrasonic motor, we ran an analysis of displacement vector flow at its lateral edge for a 75 kHz, 20 V input signal. Fig. 7a–c shows that lateral elliptical motions are induced at the stator edge, and that they confirm the operation principle in Fig. 1. The stator will be able to drive the rotor at 75 kHz, 83 kHz and 98 kHz. The maximal displacement around the edge occurs in a 90° included angle section. It is the optimal output location for rotating the rotor. The similar theoretic model and a real test have been also verified by ‘‘characterization of one-wheeled actuator driven by one piezoelectric element” in the accepted paper [26]. Fig. 8a and b shows the configuration of the wheel actuator with an open-loop circuit driver. Fig. 9a and b shows the displacement vector flows (y dir. view) at the driving frequencies of 24.12 Hz and 42.753 Hz. The whole wheel actuator is found to have clockwise rotation (CW) at 24.12 Hz and counter-clockwise rotation (CCW) at 42.753 Hz. As depicted in Fig. 10a and b, the experimental results confirm that the onewheeled actuator can be driven into clockwise (CW) rotation at 24 Hz and counter-clockwise (CCW) rotation at 42 Hz.
4. Conclusions By modelling the dynamic formulary of the asymmetric piezoelectric stator in the linear piezoelectric, piezoelectric–mechanic, and mechanical finite element method, the eigenvalue problem can be solved under defining the
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Fig. 9. The displacement vector flow (y dir. view) at: (a) 24.12 Hz and (b) 42.753 Hz by finite element modal simulation.
electric boundary condition for the piezoelectric membrane and the mechanical boundary condition for the metal back plate. It shall be noted that the extra electrical degrees of freedom cannot have mass associate with them, and the mass matrix may not be positive definite and the stiffness matrix would be non-positive definite due to the negative dielectric. The mechanical response under con-
stant voltage excitation is acquired. The operational characteristics of the asymmetric piezoelectric stator are verified by analysis of displacement vector flow at its lateral edge. A standing wave under the higher ultrasonic frequency is the key fact for successfully driving the asymmetric piezoelectric stator. It has great potential to service as a design guideline for possible use in practical
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Fig. 10. Experimental results: (a) continuous CW rotation and (b) continuous CCW rotation.
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