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Analytic modeling and driving for an ultrasonic wheel system
ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 514– 522
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Analytic modeling and driving for an ultrasonic wheel system Puu-An Juang Department of Electrical Engineering, National Chung-Hsing University, Taichung, Taiwan, ROC
a r t i c l e i n f o
abstract
Article history: Received 7 September 2007 Received in revised form 8 May 2008 Accepted 12 May 2008 Available online 20 May 2008
In this paper, a 3-D mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, mechanical, and piezoelectro-mechanic behaviors of an ultrasonic piezoelectric wheel system. 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 one fixed point on the wheel central for the mechanical design of ultrasonic directly driving system, is defined so that a lateral elliptical motion of the contact point between wheel and ground can be realized for driving the wheel carrier. Guyan reduction and Householder–Bisection inverse iteration are used as an optimal solver to find the modal frequencies and associated mode shapes. In this research, a speed control scheme is implemented by using commercial DSP technique for an open-loop speed driver. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Piezoelectro-mechanic Discretized equation of motion Guyan reduction Householder–Bisection inverse iteration
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 Refs. [1–6]. For the simulation of the steady state piezoelectric vibrations, the 3-D 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 solved analytically [7–9]. Since Allik and Hughes [10] 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 Refs. [11–16] 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 associated formulations can be classified as irreducible in the sense that the number of field variables cannot be further reduced [17]. Similar to the irreducible or displacement elements in structural mechanics, irreducible piezoelectric elements are often too stiff, susceptible to mesh distortion and aspect ratio. To overcome these drawbacks, Tzou and Tseng [4] and Tzou [16] made use of bubble/ incompatible displacement modes [17,18] 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 [19]. In this light, Ghandi and Hagood
Tel.: +886 422 840 688.
E-mail address: [email protected] 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.05.005
ARTICLE IN PRESS P.-A. Juang / Mechanical Systems and Signal Processing 23 (2009) 514–522
515
[20] proposed a piezoelectric hybrid tetrahedral 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, Guo and Cawley [15] 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. 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. To express the operational principle of the piezoelectric wheel system, a simple support condition with one fixed points the wheel central was defined. When a sinusoidal input is applied to the wheel, it induces a lateral elliptical motion and generates a torque to move the wheel carrier. The mechanical design of the wheel system will enlarge the lateral elliptical motion between wheel and ground. The desired vibration modes can be concentrated at 600 Hz, 6, 45, and 65 kHz, respectively. The aim of the study is to present the operational verification of an ultrasonic wheel system by the finite element method: modal analysis and harmonic analysis. These modal shape patterns are also acquired through the verification. Meanwhile, a DSP technique for an open-loop speed driver is implemented for the real test. 2. The principle of driving mechanism In this research, we propose a piezoelectric wheel used as the wheel of system. The wheel system configuration is shown in Fig. 1(a) and (b). It consists of a carrier and two displacement members which are, respectively, pivoted on the carrier. In addition, the carrier installed the wheels must be forward–backward and laterally movable in order to transfer a
Fig. 1. The piezoelectric wheel system configuration: (a) 3-D cad design figure and (b) the real system picture.
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stable and optimal output to the wheel system. The motion transfer from the piezodisc to the wheel is that two vibration modes, standing and traveling waves excited by an input voltage propagate on an elastic stator; they will create a lateral elliptical motion at its edge. A driving force between the wheel and the ground will be produced when pre-load (carrier weight) exists between them. It will generate a relative motion to move the wheel. It should be noted that there must exist pre-pressure between the wheel and the ground for initially driving itself. So far boundary condition, a fixed support condition with a fixed point at its central point is used for mechanism design of the piezoelectric wheel system. The lateral elliptical motion can be enlarged by adding load on the system, as shown in Fig. 1(b). The paper [21] also used the similar concept to develop a micro-robot ‘‘Mush II-P’’ system which used rectangular plate with four reformative trials of piezomotors using longitudinal and flexural vibrations for promoting ultrasonic motor application. But our wheel system is no rotor requirement. A DSP-based controller is designed for the piezoelectric wheel system in order to meet the performance specifications as reliability and practicability. The control architecture of the control system is shown in Fig. 2, which is composed of the piezoelectric wheel, a voltage-mode PWM module, and DSP-based system with PWM interface and analog to digital signal interface. The task of the DSP-based controller is to process and transfer the signal from the reference signal to output after executing required control algorithms. The DSP-based velocity control system shown in Fig. 3 has been constructed successfully and the code programming in C has been implemented systematically through the flowchart. The combination of the DSP-based control system and the control algorithm programming makes the realization of a high-performance controller become true. Based on the simplified model of the piezoelectric, the velocity control is divided into direction control and speed control. The direction control is depended on the special fixed resonance frequency of the piezoelectric. The speed control is depended on the drive current amplitude at resonance frequency of the piezoelectric. The TI DSP drives the PWM module generate a single-phase voltage source with the reference resonance frequency and the peak-to-peak voltage value.
DSP based system
POWER
Reference
TI DSP POWER SUPLLY
Load
PWM DRIVER interface A/D interface
MOS
module
i
Fig. 2. System architecture of the control system.
Fig. 3. The DSP-based control system.
piezoelectric wheel
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3. 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 eightnodes SOLID5 structure element to model the elastic–metal–back plate and coupling element for a piezoelectric wheel. 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 piezoelectric wheel. The detail finite element theoretical derivation is described as follows. A linear piezoelectric constitutive equation is employed to establish the dynamic model of the new stator. The coupling between the elastic field and the electric field for piezoelectric actuator is given by [22] T ¼ C E S eT E, D ¼ eS þ S E,
(1)
where T is the 6 1 stress vector, D is the 3 1 electrical displacement vector, S is the 6 1 strain vector, and E is the 3 1 electrical field vector, respectively; CE is the 6 6 elasticity matrix, e is the 3 6 piezoelectric matrix and eT is its transpose, and eS is the 3 3 dielectric matrix. The electric field E is related to the electric potential f by E ¼ rf. The displacement and potential for each element can be expressed, respectively, as ^ u ¼ Nu u, ^ f ¼ N f f,
(2)
where u is the displacement vector, Nu and Nf are the interpolation function for the variables of f and u, and 4 denotes the nodal values. ^ Here, Bu is the product of the To put the strain–displacement relation in terms of the nodal displacement yields S ¼ Bu u. ^ ¼ Bf f ^ [23]. differential-operating matrix relating S to the shape function matrix Nu. Similarly, let E ¼ rf ¼ rNf f In this paper, we use the finite element method to study the different aspects of behavior in the piezoelectric wheel. In this code, a multi-field solid element and a standard isotropic solid element are involved in modeling 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 multifield elements. The governing dynamic equation of the piezoelectric wheel in matrix form for the piezoelectric ceramic phase [10] is as follows: p
M p u€ þ K puu up þ K puf f ¼ F p , K puf u þ K pff f ¼ P, p
K puu ,
K puf ,
K pff ,
(3) p
where M , F , 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 Z Mp ¼ rN Tu N u dV, Z K puu ¼ BTu C E Bu dV, Z K puf ¼ BTu eT Bf dV, Z s K pff ¼ BTf b Bf dV, Z Z Fp ¼ NTb f b dV þ N TS1 f S dS1 þ NTu f c , V S1 Z P¼ N TS2 qS dS2 NTf qc , (4) 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. For the disc base, the electrical effects are ignored and the finite element equation is given by s
M s u€ þ K suu us ¼ F s ,
(5)
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where the superscript s denotes the disc. In general, all structures are slightly damped due to structure damping. Thus, Eqs. (3) and (5) can be modified to p
M p u€ þ C p u_ p þ K puu up þ K puf f ¼ F p , K puf u þ K pff f ¼ P,
(6)
s
M s u€ þ C s u_ s þ K suu us ¼ F s , p
p
lK puu
s
(7) s
lK suu
and C ¼ ZM þ [22], where Z and l are Rayleigh coefficients. It shall be noted that the where C ¼ ZM þ damping matrix in ANSYS implements [24], C ¼ bp K puu þ bs K suu , where bp and 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. 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 electric field boundary condition requires that the electrode surface is an equipotential one and the summation of the nodal electric charges on it should be zero as shown in the following: X ^i ¼ F ^ iþ1 ¼ ¼ constant or F Q i ¼ 0. (8) The adhesive layer on the piezoelectric wheel will be ignored in the finite element analysis. To analyze the steady state response characteristics of the piezoelectric wheel, first modal response analysis has to be made to determine the mode shape and the natural frequency of the piezoelectric wheel. In modal analysis, it should be
Fig. 4. The displacement profile (z direction view) under (a) 600 Hz, (b) 6 kHz, (c) 45 kHz, and (d) 65 kHz by finite element modal simulation.
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noted that the extra voltage degrees of freedom in the finite element equation have to be condensed using Guyan reduction. Then the modal frequencies and associated mode shapes can be found using Householder–Bisection inverse iteration. Additionally, because the electrical degrees of freedom do not 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.
4. Analysis and test Table 1 lists the ultrasonic wheel system known material parameters and dimensions. In this study, all steady state response analyses will be added the structure damping, that is, Z and l. We assumed Z ¼ 8 and l ¼ 3 105 [22], internal damping and Poisson’s ratio of the passive metal disc base influence the damping loss of the piezoelectric wheel 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 piezoelectric ultrasonic wheel as a simple support condition with one fixed point on the wheel central. Then, we used ANSYS code to run all response analyses. The analysis results show that there are four responses near 600 Hz, 6, 45, and 65 kHz. These are the driving frequencies for the piezoelectric wheel since the system can only be moved near these frequencies. We can also identify that changing the exciting frequency, amplitude, and phase would yield a better performance. The wheel can be driven into right lateral
Fig. 5. The displacement vector flow (x direction view) under (a) 600 Hz, (b) 6 kHz, (c) 45 kHz, and (d) 65 kHz by finite element modal simulation.
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movement nearby 600 Hz and left lateral movement nearby 6 kHz. Meanwhile, the wheel can be driven into backward movement nearby 45 kHz and forward movement nearby 60 kHz. The applied voltage (ac power 20 V) is fixed for all of the desired frequency ranges. Fig. 4(a)–(d) shows the z direction displacement profile under harmonic excitation. The exciting frequencies are at 600 Hz, 6, 45, and 65 kHz under a ac power 20 Vp–p input signal to match the real driving condition. The results reveal that the wheel has standing wave that exists, especially significant at the higher driving frequency. As we know, the standing wave transfers most driving energy into the wheel system. However, obviously there is much acoustic noise at an audible frequency range. Fig. 5(a)–(d) shows the displacement vector flow at 600 Hz, 6, 45, and 65 kHz. The results show why 600 Hz is right lateral movement, 6 kHz is left lateral movement, 45 kHz is backward movement, and 65 kHz is forward movement. The real driving frequency closely agreed with the predicted one. The wheel system can be oriented its motion by tuning driving frequency. The velocity trajectory of the controlled system follows a velocity reference model G(s) with an input command of a step function: GðsÞ ¼
100000 . s2 þ 195s þ 10000
(9)
The frequency reference mode is built as a reference table for the direction control and compensation of the speed gain. The first experiment is the right lateral movement velocity control by setting the frequency to 600 Hz and a speed command input to controller. When the load is zero, the speed of the wheel system is slow. Then change the load to 20 g, the speed is slow down, but the resonance frequency is not drift. At the left lateral movement velocity control experiment, the frequency is change to 6 kHz and a speed command input to controller. When the load is zero, the speed of the wheel system is fast. Then change the load to 20 g, the speed is slower, but the resonance frequency is not drift. The final is the
Fig. 6. The one-wheeled actuator driven by one piezoelectric element configuration: (a) a recent actuator and (b) with an open-loop circuit driver.
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Fig. 7. The displacement vector flow (y direction view) at (a) 24.12 Hz and (b) 42.753 Hz by finite element modal simulation.
Fig. 8. Experimental results: (a) continuous CW rotation and (b) continuous CCW rotation.
521
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turn left velocity control. The tests for backward movement velocity control at 45 kHz and forward movement velocity control at 60 kHz are the same results. The similar theoretic model and a real test have been also verified by ‘‘characterization of one-wheeled actuator driven by one piezoelectric element’’ in a recent accepted paper [25]. Fig. 6(a) and (b) shows the configuration of the wheel actuator with an open-loop circuit driver. Fig. 7(a) and (b) shows the displacement vector flows (y direction view) at the driving frequencies of 24.12 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. 8(a) and (b), the experimental results confirm that the one-wheeled actuator can be driven into clockwise (CW) rotation at 24 Hz and counter-clockwise (CCW) rotation at 42 Hz. 5. Conclusions By modeling the dynamic formulary of the ultrasonic wheel in the finite element method, especially when the electrical term is treated as an extra mechanical degree of freedom, the eigenvalue problem can be solved. The mechanical response under constant voltage excitation is acquired. The operational characteristics of the ultrasonic wheel are verified by finite element analysis and a real DSP speed control driver. It has been shown that the theoretic model can exactly verify the operational model of an ultrasonic wheel system. It has great potential to service as a design guideline for possible use in practical design, especially in the ultrasonic actuator used as a direct-driving motion. Boundary condition design is the successful key because it induces a standing wave but let a traveling wave be partly reflected and mixed. It is helpful to drive the carrier system.
Acknowledgment The authors deeply acknowledge the financial support from the Ministry of Education, Taiwan, ROC, under the ATU plan. References [1] Y. Kagawa, G.M.L. Gladwell, Finite element analysis of flexure-type vibrators with electrostrictive transducers, IEEE Transactions on Sonics and Ultrasonics 17 (1970) 41–49. [2] Y. Kagawa, T. Yamabuchi, Finite element simulation of composite piezoelectric transducer, IEEE Transactions on Sonics and Ultrasonics 26 (1979) 81–88. [3] M. Kuribayashi, S. Ueha, E. Mori, Excitation condition of flexural traveling waves for a reversible ultrasonic linear motor, Journal of the Acoustical Society of America 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, Journal of Sound and Vibration 138 (1990) 17–34. [5] M. Kuribayashi, S. Ueha, Single-phase drive of a circular ultrasonic motor, Journal of the Acoustical Society of America 90 (1991) 1723–1728. [6] V. Chaika, Steady state response of a system of rotors with various types of flexible couplings, Journal of Sound and Vibration 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, Journal of Computer Structures 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, Journal of Smart Materials and Structures 3 (1994) 485–493. [9] P.R. Heyliger, D.A. Saravanos, Exact free-vibration analysis of laminated piezoelectric layer, Journal of the Acoustical Society of America 98 (1995) 1547–1557. [10] H. Allik, T.J. Hughes, Finite element method for piezoelectric Vibration, International Journal for Numerical Methods in Engineering 2 (1970) 151–157. [11] M. Naillon, R.H. Coursant, F. Besnier, Vibration response of sonar transducers using piezoelectric finite elements, Acta Electronica 25 (1983) 341–362. [12] 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, Proceedings of the IEEE Ultrasonics Symposium (1988) 685–688. [13] R. Lerch, Simulation of piezoelectric devices by two- and three-dimesional finite elements, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 37 (1990) 233–247. [14] J.A. Hossack, G. Hayward, Finite element analysis of 1–3 composite transducers, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 38 (1991) 618–627. [15] N. Guo, P. Cawley, The finite element analysis of the vibration characteristics of piezoelectric discs, Journal of Sound and Vibration 159 (1992) 115–137. [16] H.S. Tzou, Piezoelectric Shells: Distributed Sensing and Control of Continua, Kluwer Press, 1993. [17] R.L. Taylor, P.J. Beresford, E.L. Wilson, A non-conforming element for stress analysis, International Journal for Numerical Methods in Engineering 10 (1976) 1211–1220. [18] J.C. Simo, M.S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29 (1990) 1595–1638. [19] T.H. Pian, P. Tong, Basis of finite elements for solids continua, International Journal for Numerical Methods in Engineering 29 (1969) 3–28. [20] K. Ghandi, N.W. Hagood, A hybrid finite element model for phase transitions in nonlinear electro-mechanically coupled material, SPIE Proceedings 3039 (1997) 97–112. [21] T. Tomikawa, T. Funakubo, O. Miyazawa, M. Aoyagi, T. Takano, Some reformative trials of piezo-motors using longitudinal and flexural vibrations, World Congress on Ultrasonics (WCU 2003) (2003) 1569–1576. [22] Y.H. Lim, V.V. Varadan, V.K. Varadan, Finite-element modeling of the transient response of MEMS sensor, Journal of Smart Materials and Structures 6 (1997) 53–61. [23] Y.H. Lim, V.V. Varadan, V.K. Varadan, Closed loop finite element modeling of active structural damping in the time domain, Journal of Smart Materials and Structures 8 (1999) 390–400. [24] G. Hayward, Assessing the influence of pillar aspect ration on the behavior of 1–3 connectivity composite transducers, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 43 (1996) 98–108. [25] P.A. Juang, C.C. Tsai, Characterization of one-wheeled actuator driven by one piezoelectric element, Measurement (2008), in press.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Analytic modeling and driving for an ultrasonic wheel system Puu-An Juang Department of Electrical Engineering, National Chung-Hsing University, Taichung, Taiwan, ROC
a r t i c l e i n f o
abstract
Article history: Received 7 September 2007 Received in revised form 8 May 2008 Accepted 12 May 2008 Available online 20 May 2008
In this paper, a 3-D mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, mechanical, and piezoelectro-mechanic behaviors of an ultrasonic piezoelectric wheel system. 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 one fixed point on the wheel central for the mechanical design of ultrasonic directly driving system, is defined so that a lateral elliptical motion of the contact point between wheel and ground can be realized for driving the wheel carrier. Guyan reduction and Householder–Bisection inverse iteration are used as an optimal solver to find the modal frequencies and associated mode shapes. In this research, a speed control scheme is implemented by using commercial DSP technique for an open-loop speed driver. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Piezoelectro-mechanic Discretized equation of motion Guyan reduction Householder–Bisection inverse iteration
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 Refs. [1–6]. For the simulation of the steady state piezoelectric vibrations, the 3-D 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 solved analytically [7–9]. Since Allik and Hughes [10] 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 Refs. [11–16] 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 associated formulations can be classified as irreducible in the sense that the number of field variables cannot be further reduced [17]. Similar to the irreducible or displacement elements in structural mechanics, irreducible piezoelectric elements are often too stiff, susceptible to mesh distortion and aspect ratio. To overcome these drawbacks, Tzou and Tseng [4] and Tzou [16] made use of bubble/ incompatible displacement modes [17,18] 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 [19]. In this light, Ghandi and Hagood
Tel.: +886 422 840 688.
E-mail address: [email protected] 0888-3270/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2008.05.005
ARTICLE IN PRESS P.-A. Juang / Mechanical Systems and Signal Processing 23 (2009) 514–522
515
[20] proposed a piezoelectric hybrid tetrahedral 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, Guo and Cawley [15] 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. 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. To express the operational principle of the piezoelectric wheel system, a simple support condition with one fixed points the wheel central was defined. When a sinusoidal input is applied to the wheel, it induces a lateral elliptical motion and generates a torque to move the wheel carrier. The mechanical design of the wheel system will enlarge the lateral elliptical motion between wheel and ground. The desired vibration modes can be concentrated at 600 Hz, 6, 45, and 65 kHz, respectively. The aim of the study is to present the operational verification of an ultrasonic wheel system by the finite element method: modal analysis and harmonic analysis. These modal shape patterns are also acquired through the verification. Meanwhile, a DSP technique for an open-loop speed driver is implemented for the real test. 2. The principle of driving mechanism In this research, we propose a piezoelectric wheel used as the wheel of system. The wheel system configuration is shown in Fig. 1(a) and (b). It consists of a carrier and two displacement members which are, respectively, pivoted on the carrier. In addition, the carrier installed the wheels must be forward–backward and laterally movable in order to transfer a
Fig. 1. The piezoelectric wheel system configuration: (a) 3-D cad design figure and (b) the real system picture.
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stable and optimal output to the wheel system. The motion transfer from the piezodisc to the wheel is that two vibration modes, standing and traveling waves excited by an input voltage propagate on an elastic stator; they will create a lateral elliptical motion at its edge. A driving force between the wheel and the ground will be produced when pre-load (carrier weight) exists between them. It will generate a relative motion to move the wheel. It should be noted that there must exist pre-pressure between the wheel and the ground for initially driving itself. So far boundary condition, a fixed support condition with a fixed point at its central point is used for mechanism design of the piezoelectric wheel system. The lateral elliptical motion can be enlarged by adding load on the system, as shown in Fig. 1(b). The paper [21] also used the similar concept to develop a micro-robot ‘‘Mush II-P’’ system which used rectangular plate with four reformative trials of piezomotors using longitudinal and flexural vibrations for promoting ultrasonic motor application. But our wheel system is no rotor requirement. A DSP-based controller is designed for the piezoelectric wheel system in order to meet the performance specifications as reliability and practicability. The control architecture of the control system is shown in Fig. 2, which is composed of the piezoelectric wheel, a voltage-mode PWM module, and DSP-based system with PWM interface and analog to digital signal interface. The task of the DSP-based controller is to process and transfer the signal from the reference signal to output after executing required control algorithms. The DSP-based velocity control system shown in Fig. 3 has been constructed successfully and the code programming in C has been implemented systematically through the flowchart. The combination of the DSP-based control system and the control algorithm programming makes the realization of a high-performance controller become true. Based on the simplified model of the piezoelectric, the velocity control is divided into direction control and speed control. The direction control is depended on the special fixed resonance frequency of the piezoelectric. The speed control is depended on the drive current amplitude at resonance frequency of the piezoelectric. The TI DSP drives the PWM module generate a single-phase voltage source with the reference resonance frequency and the peak-to-peak voltage value.
DSP based system
POWER
Reference
TI DSP POWER SUPLLY
Load
PWM DRIVER interface A/D interface
MOS
module
i
Fig. 2. System architecture of the control system.
Fig. 3. The DSP-based control system.
piezoelectric wheel
ARTICLE IN PRESS P.-A. Juang / Mechanical Systems and Signal Processing 23 (2009) 514–522
517
3. 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 eightnodes SOLID5 structure element to model the elastic–metal–back plate and coupling element for a piezoelectric wheel. 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 piezoelectric wheel. The detail finite element theoretical derivation is described as follows. A linear piezoelectric constitutive equation is employed to establish the dynamic model of the new stator. The coupling between the elastic field and the electric field for piezoelectric actuator is given by [22] T ¼ C E S eT E, D ¼ eS þ S E,
(1)
where T is the 6 1 stress vector, D is the 3 1 electrical displacement vector, S is the 6 1 strain vector, and E is the 3 1 electrical field vector, respectively; CE is the 6 6 elasticity matrix, e is the 3 6 piezoelectric matrix and eT is its transpose, and eS is the 3 3 dielectric matrix. The electric field E is related to the electric potential f by E ¼ rf. The displacement and potential for each element can be expressed, respectively, as ^ u ¼ Nu u, ^ f ¼ N f f,
(2)
where u is the displacement vector, Nu and Nf are the interpolation function for the variables of f and u, and 4 denotes the nodal values. ^ Here, Bu is the product of the To put the strain–displacement relation in terms of the nodal displacement yields S ¼ Bu u. ^ ¼ Bf f ^ [23]. differential-operating matrix relating S to the shape function matrix Nu. Similarly, let E ¼ rf ¼ rNf f In this paper, we use the finite element method to study the different aspects of behavior in the piezoelectric wheel. In this code, a multi-field solid element and a standard isotropic solid element are involved in modeling 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 multifield elements. The governing dynamic equation of the piezoelectric wheel in matrix form for the piezoelectric ceramic phase [10] is as follows: p
M p u€ þ K puu up þ K puf f ¼ F p , K puf u þ K pff f ¼ P, p
K puu ,
K puf ,
K pff ,
(3) p
where M , F , 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 Z Mp ¼ rN Tu N u dV, Z K puu ¼ BTu C E Bu dV, Z K puf ¼ BTu eT Bf dV, Z s K pff ¼ BTf b Bf dV, Z Z Fp ¼ NTb f b dV þ N TS1 f S dS1 þ NTu f c , V S1 Z P¼ N TS2 qS dS2 NTf qc , (4) 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. For the disc base, the electrical effects are ignored and the finite element equation is given by s
M s u€ þ K suu us ¼ F s ,
(5)
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where the superscript s denotes the disc. In general, all structures are slightly damped due to structure damping. Thus, Eqs. (3) and (5) can be modified to p
M p u€ þ C p u_ p þ K puu up þ K puf f ¼ F p , K puf u þ K pff f ¼ P,
(6)
s
M s u€ þ C s u_ s þ K suu us ¼ F s , p
p
lK puu
s
(7) s
lK suu
and C ¼ ZM þ [22], where Z and l are Rayleigh coefficients. It shall be noted that the where C ¼ ZM þ damping matrix in ANSYS implements [24], C ¼ bp K puu þ bs K suu , where bp and 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. 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 electric field boundary condition requires that the electrode surface is an equipotential one and the summation of the nodal electric charges on it should be zero as shown in the following: X ^i ¼ F ^ iþ1 ¼ ¼ constant or F Q i ¼ 0. (8) The adhesive layer on the piezoelectric wheel will be ignored in the finite element analysis. To analyze the steady state response characteristics of the piezoelectric wheel, first modal response analysis has to be made to determine the mode shape and the natural frequency of the piezoelectric wheel. In modal analysis, it should be
Fig. 4. The displacement profile (z direction view) under (a) 600 Hz, (b) 6 kHz, (c) 45 kHz, and (d) 65 kHz by finite element modal simulation.
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noted that the extra voltage degrees of freedom in the finite element equation have to be condensed using Guyan reduction. Then the modal frequencies and associated mode shapes can be found using Householder–Bisection inverse iteration. Additionally, because the electrical degrees of freedom do not 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.
4. Analysis and test Table 1 lists the ultrasonic wheel system known material parameters and dimensions. In this study, all steady state response analyses will be added the structure damping, that is, Z and l. We assumed Z ¼ 8 and l ¼ 3 105 [22], internal damping and Poisson’s ratio of the passive metal disc base influence the damping loss of the piezoelectric wheel 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 piezoelectric ultrasonic wheel as a simple support condition with one fixed point on the wheel central. Then, we used ANSYS code to run all response analyses. The analysis results show that there are four responses near 600 Hz, 6, 45, and 65 kHz. These are the driving frequencies for the piezoelectric wheel since the system can only be moved near these frequencies. We can also identify that changing the exciting frequency, amplitude, and phase would yield a better performance. The wheel can be driven into right lateral
Fig. 5. The displacement vector flow (x direction view) under (a) 600 Hz, (b) 6 kHz, (c) 45 kHz, and (d) 65 kHz by finite element modal simulation.
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movement nearby 600 Hz and left lateral movement nearby 6 kHz. Meanwhile, the wheel can be driven into backward movement nearby 45 kHz and forward movement nearby 60 kHz. The applied voltage (ac power 20 V) is fixed for all of the desired frequency ranges. Fig. 4(a)–(d) shows the z direction displacement profile under harmonic excitation. The exciting frequencies are at 600 Hz, 6, 45, and 65 kHz under a ac power 20 Vp–p input signal to match the real driving condition. The results reveal that the wheel has standing wave that exists, especially significant at the higher driving frequency. As we know, the standing wave transfers most driving energy into the wheel system. However, obviously there is much acoustic noise at an audible frequency range. Fig. 5(a)–(d) shows the displacement vector flow at 600 Hz, 6, 45, and 65 kHz. The results show why 600 Hz is right lateral movement, 6 kHz is left lateral movement, 45 kHz is backward movement, and 65 kHz is forward movement. The real driving frequency closely agreed with the predicted one. The wheel system can be oriented its motion by tuning driving frequency. The velocity trajectory of the controlled system follows a velocity reference model G(s) with an input command of a step function: GðsÞ ¼
100000 . s2 þ 195s þ 10000
(9)
The frequency reference mode is built as a reference table for the direction control and compensation of the speed gain. The first experiment is the right lateral movement velocity control by setting the frequency to 600 Hz and a speed command input to controller. When the load is zero, the speed of the wheel system is slow. Then change the load to 20 g, the speed is slow down, but the resonance frequency is not drift. At the left lateral movement velocity control experiment, the frequency is change to 6 kHz and a speed command input to controller. When the load is zero, the speed of the wheel system is fast. Then change the load to 20 g, the speed is slower, but the resonance frequency is not drift. The final is the
Fig. 6. The one-wheeled actuator driven by one piezoelectric element configuration: (a) a recent actuator and (b) with an open-loop circuit driver.
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Fig. 7. The displacement vector flow (y direction view) at (a) 24.12 Hz and (b) 42.753 Hz by finite element modal simulation.
Fig. 8. Experimental results: (a) continuous CW rotation and (b) continuous CCW rotation.
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turn left velocity control. The tests for backward movement velocity control at 45 kHz and forward movement velocity control at 60 kHz are the same results. The similar theoretic model and a real test have been also verified by ‘‘characterization of one-wheeled actuator driven by one piezoelectric element’’ in a recent accepted paper [25]. Fig. 6(a) and (b) shows the configuration of the wheel actuator with an open-loop circuit driver. Fig. 7(a) and (b) shows the displacement vector flows (y direction view) at the driving frequencies of 24.12 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. 8(a) and (b), the experimental results confirm that the one-wheeled actuator can be driven into clockwise (CW) rotation at 24 Hz and counter-clockwise (CCW) rotation at 42 Hz. 5. Conclusions By modeling the dynamic formulary of the ultrasonic wheel in the finite element method, especially when the electrical term is treated as an extra mechanical degree of freedom, the eigenvalue problem can be solved. The mechanical response under constant voltage excitation is acquired. The operational characteristics of the ultrasonic wheel are verified by finite element analysis and a real DSP speed control driver. It has been shown that the theoretic model can exactly verify the operational model of an ultrasonic wheel system. It has great potential to service as a design guideline for possible use in practical design, especially in the ultrasonic actuator used as a direct-driving motion. Boundary condition design is the successful key because it induces a standing wave but let a traveling wave be partly reflected and mixed. It is helpful to drive the carrier system.
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