S.-W. Ricky Lee, M.-L. Zhu & H.L. Wong Department of Mechanical Engineering, Hong Kong University of Science & Technology, Hong Kong

Transcription

1 Modeling for prototyping of rotary piezoelectric motors S.-W. Ricky Lee, M.-L. Zhu & H.L. Wong Department of Mechanical Engineering, Hong Kong University of Science & Technology, Hong Kong Abstract A new type of ultrasonic rotary motor driven by an anisotropic piezoelectric composite laminate is introduced in this paper. The driving element of the motor is a three-layer laminated plate. A piezoelectric layer is sandwiched between two anti-symmetric composite laminae. Due to the material anisotropy and the anti-symmetric configuration, torsional vibration can be induced through the in-plane strain actuated by the piezoelectric layer. In the present study, an analytical model is formulated. The motor is considered to be a coupled dynamic system that includes the longitudinal and torsional vibrations of the laminate and the rotating motion of the rotor under the action of contact force. The developed model is able to predict the characteristics of the motor. For numerical illustration, the motor performance curve in terms of rotating speed and output torque is generated by the present model. The analytical results are in good agreement with the experimental data obtained from the prototype of the said motor. With the developed analytical model, further parametric studies may be performed to optimize the design for future generations of rotary piezoelectric motors. 1 Introduction The development of piezoelectric motors has lasted for more than three decades [1]. Earlier efforts were focused on the design and implementation of various types of motors. More recently, the attention has been brought to the issues related to modeling and analyses. Most previous studies divided the analysis into two steps. At first, using either analytical methods [2-3] or finite element

2 22 Computational Methods for Smart Structures and Materials II (a) (b) Figure 1: (a) composite laminate with extension-twisting coupling, (b) rotary piezoelectric motor under investigation. analyses (FEA) [4-5], researchers studied the vibration of the stator to determine the mode shape, the resonant frequency and the vibration amplitude. Secondly, the contact force between the rotor and the stator is applied assuming that the vibration characteristics of the stator are not affected by the contact loading [6-8]. Strictly speaking, such kind of approach is not at all rigorous because the vibration of stator should depend on the contact force. Therefore, the existing analytical models can only partially predict the characteristics of piezoelectric motors. Besides, a majority of publications in this field was dedicated to the travelling wave type of ultrasonic motors only [9-1]. So far there is very little effort investigating the "standing wave" type of piezoelectric motors. This paper introduces an analytical model for the prototyping of a newly developed piezoelectric motor (standing wave type). This rotary actuator is driven by an anisotropic piezoelectric composite laminate. Due to material anisotropy and structural coupling, torsional vibration can be induced through in-plane strain actuation. In the present analysis, the whole motor is treated as a coupled dynamic system. The analytical formulation includes the longitudinal and torsional vibrations of the laminate and the rotating motion of the rotor together with the action of contact force. The developed model can predict characteristics such as rotating speed and output torque of the motor. Testing data are obtained as well from the prototype of the motor. The good agreement between analytical and experimental results validates the developed model. 2 Principle of actuation The driving element of the present motor is a three-layer piezoelectric laminate as shown in Figure 1. A piezo-layer is sandwiched between two graphite/epoxy composite laminae with anti-symmetric configuration (±45 ). Due to material anisotropy and structural coupling, torsional motion can be induced by the inplane strain actuated by the PZT layer. Figure 2 demonstrates a schematic diagram for the actuation principle of the present motor. One end of the driving

3 Computational Methods for Smart Structures and Materials II 23 ^ ^ _ Twisting Applied voltage Rotor lit I II yu J Stator i (\Yj; (P^ c 5 ^jp beam) ^T ^r>% J L J L Loose contact Firm contact Loose contact Separate Loose contact Figure 2: Schematic diagram for the actuation principle of the rotary element is clamped on a rigid foundation while the other end is originally in loose contact with a rotor. It should be noted that, in order to maximize the twisting motion of the laminate, the frequency of applied voltage is tuned to the torsional resonant frequency of the laminate. This driving frequency is far below the resonance of longitudinal vibration because the in-plane stiffness is much higher than the twisting stiffness. As a result, the torsional deformation of the laminate is always 9 out of phase with respect to the applied voltage, while the longitudinal deformation is in-phase. Once the piezo-layer is subjected to a rising electric field, the whole plate will extend and twist, resembling the motion of a screw driver. Similar to a clutch mechanism, the extension of "the laminate brings one end into firm contact with the rotor. Due to mechanical friction, the rotor will turn with the end of the laminate. When the electric field decreases from the peak, the laminate will contract but is still twisting in the same direction due to the phase lag. Once the applied voltage reverses its sign to negative, the tip of the laminate is separating from the rotor and the laminate starts to twist backwards. However, due to the rotary inertia, the rotor will continue to rotate in the same direction. Eventually, when the applied voltage reverses its sign back to positive, the contact will resume and the laminate begins to twist forwards again. In such a cycle, the rotor rotates a small angle. A continuous rotary motion can be achieved by applying a high frequency cyclic voltage to the piezo-layer. The detailed design and prototyping of the present motor can be found in [11]. 3 Analytical formulation and modeling Dynamic modeling of the piezoelectric motor is rather complicated. The characteristics of the motor may depend on many factors such as the geometrical and material properties of the driving element and the rotor. Also, it is very difficult to consider all aspects of concern in a single model. For the convenience of establishing a practical model of the motor, the following fundamental assumptions are introduced: 1. If there is a compressive force at the end of the laminate, the end of the laminate is considered in contact with the rotor. If the speed of the torsional vibration at the end of the laminate differs from the tangential speed of the rotor, slipping occurs. Otherwise, sticking occurs.

4 24 Computational Methods for Smart Structures and Materials II 2. A perfect bonding is assumed between the piezo-layer and composite laminae. The thickness of bonding is negligible. 3. The constraint force at the fixed end of the laminate in the y direction is not considered. The compact form of constitutive equations for general piezoelectric materials under plane stress condition can be written as (1) where {a } is the stress vector, [ ~Q ] is the elastic constant matrix of the piezoelectric material, {S} is the mechanical strain vector, [e] is the piezoelectric constant matrix, {E} is the electric field vector, {D} is the electric displacement vector, and [,,] is the dielectric constant matrix. Since the structure under consideration is a laminated plate, the poling direction of the piezoelectric material is arranged in the thickness direction, and the external voltage is applied along the poling direction. Therefore, the electric field {E} and the electric displacement {D} in Eq. (1) are non-zero in the z-direction only; i.e., { }=? and {D}=D,. Besides, the piezoelectric constant matrix becomes [e] -[.?/ e^ ], and the dielectric constant matrix becomes [e^ ]=?. According to the laminated plate theory, Eq. (1) can be integrated through the thickness to obtain the constitutive equations for the piezoelectric laminate. For actuation, only the first equation needs to be considered. Because the driving element of the present motor has an anti-symmetric configuration, the constitutive equation of such a piezoelectric laminate has the form of X N, = ^M, My M.n. A, A: o o o #,/,4,2 4,2 #26 o o A,, a,a a,* o o o a,, D,, D,2 o,6 #12 #22 _a,a 2,6 o o o D^ X' < c%. K\ K\ K\. - " N ; " N? yv;;, M; Mf *f.n. (2) where N, M,, and K denote force resultant, moment resultant, midplane strain, and midplane curvature, respectively. The terms A//, #,/, D,; in the matrix are the stiffnesses of the laminate. The detailed derivation of Eq. (2) is given in [12]. From this equation, it can be seen that the in-plane deformation is coupled with twisting through the terms BX, and 826- This phenomenon has been experimentally proven in [12]. Assuming the in-plane inertia is negligible, by variational principle, the governing equations of the laminate can be expressed in terms of generalized displacement pjt) and electric voltage V?( as

5 Computational Methods for Smart Structures and Materials II M,p,(t) + C,p.(t) + K,p,(t) =, V,(/) + F (t) 25 (3) where M = C,V, (f) + e,,p,,.( +Q,^,,. (t) +Q,,,,,, (f) = (4) (5) (6) (7) where the terms M,, C/ and KI represent the mode mass, the coefficient of mode damping and the mode stiffness of the laminate for the ith mode of vibration, respectively. Also, &) is the natural frequency of the laminate and, is the damping factor. It should be noted that the damping term is artificially added. In addition, the term 6>, represents the electromechanical coupling and describes the conversion of the applied voltage to an equivalent force on the laminate. It is therefore called the electromechanical coupling factor. F^ represents the generalized frictional force. C? denotes the electric capacity of the piezoelectric layer. Qdccmc and <2/>r«.v.v denote the electric charges produced in the piezoelectric layer by the contact compressive force. Note that Eqs. (3) and (4) are the equation of motion and the sensor equation of the laminate, respectively. Several contact models were evaluated in [13] for the traveling wave type of piezoelectric motors. In the present analysis, the linear spring model mentioned in [13] is adopted to study the contact behaviors of the motor driven by the piezoelectric composite laminate. The end of the laminate is originally in contact with the rotor under an initial compressive force. The contact is modeled as distributed linear springs with a stiffness k as shown in Figure 3. At the steady state, the end of the laminate pushes the rotor making its front face displace a distance from the original position by the amount A. On the other hand, assuming the initial compressive force per unit length is PQ, then the end

6 26 Computational Methods for Smart Structures and Materials II AA/yv AA/#- AA/yi^ \^/i/^ k """ %, wo(%,},r) Figure 3: Analytical model for contact. of the laminate has an initial deformation, say UQ(<Z, V,), under the action of this axial compressive force. The relation between PQ, UQ (a, _y,) and A is When the laminate is subjected to an external alternative voltage, the laminate will extend or contract in the x direction and twist about the jc-axis, and the axial compressive force will change. If the longitudinal displacement is denoted by UQ (%, y, t), the contact force becomes (8) :(,_y,r)-wo(6z,y,)] if WQ(o,)V)>-A (g) if WQ(a,y,f) <-A. It should be noted that, when UQ (a, y,t) > A, the laminate and the rotor are in contact. The model of the rotor is based on the equation of motion of the rotor under the action of contact force. The equation of angular momentum about the %-axis yields the differential equation of the rotor (1) where /,. is the polar moment of inertia of the rotor; T^t is the external load of the motor; (t)is the angular displacement of the rotor; T(f) is the torque generated by the frictional force and is the product of the frictional force and its contact radius, which can be expressed as

7 Computational Methods for Smart Structures and Materials II 27 if v,_. (y, f) > v, (y, f) and w<, (11) In order to predict the overall characteristics of the motor, the equations of motion for the motor can be obtained by assembling the aforementioned equations. Let,^w (,8(, (12) then (13) where _ " 1 -C,. /M, -AT,/M, " 1 (14) o (15) Eq. (13) is a nonlinear inter-linkage differential equation. After the geometry dimensions and material properties of the motor are given, the input parameters of the model are the applied voltage, V?(, which include the amplitude, the frequency and the wave form of the applied voltage, the initial compressive force, po, and the external load,7^.,. It should be noted that the generalized displacement and the generalized velocity, p^(t) andp^( ), and the angular displacement and the angular speed of the rotor, 6(t) and 9(t), are the unknowns to be solved for. 4 Numerical results and discussion Based on the aforementioned analytical model, the overall characteristics of the motor such as the rotating speed of the rotor, the output torque, and the efficiency of the motor can be predicted. Although many other parameters can be investigated as well using the developed model, the present study is focused on the effects of initial compressive force on the performance of the motor.

8 g Computational Methods for Smart Structures and Materials II In order to use the proposed analytical model to obtain numerical results, it is necessary to introduce a shape function, O^Cx, y). For the torsional vibration, the following shape function is considered in a separate-variable form: where (16) (, (x) = sin rjx - sinh TJX + D (cos 77% - cosh 77*), cos 77 + cosh 77 D = sin 77*2 - sinh 77^ 77 = 1.875, (17) These functions satisfy the geometric boundary conditions. The fourth and fifth order Runge-Kutta formulas are used to solve Eq. (13) for the initial value problem X={ }^. An analysis program is compiled using MATLAB. All of the geometrical and material properties of the motor must be incorporated into the model. The values adopted in the present analysis are given in Table 1. The moment of inertia of the rotor, J#, is measured from the prototype in [11] as kgmnt. In addition, the dynamic friction coefficient, ju, is assumed to be.2; the damping factor,, is 5%; the contact stiffness, &, is 3xl(fN/m. Furthermore, the amplitude of applied voltage (sinusoidal wave form) to the PZT is 15 V and the driving frequency is the torsional resonant frequency of the laminate (1622 Hz). The performance of the motor in terms of rotating speed and output torque is shown in Figure 4. From this figure, it is observed that the rotating speed of the rotor monotonically reduces as the torque increases. The experimental data obtained from the prototype is also given for comparison. The agreement between analytical and experimental results validates the developed model. In addition to the results shown in Figure 4, the present model is able to investigate other characteristics of the motor such as output power and efficiency. It is also very useful for the parametric study on design factors such as geometries and material properties. With this analytical tool, prototyping of new generations of piezoelectric motors becomes much easier. Currently a mini-piezoelectric motor with the drivng element of the size of 15 (L) x 1 (W) x 3 (T) mm (see Figure 5) is under development. More parametric studies using the present analytical model is still on-going for further miniaturization. Table 1. Mechanical properties of the driving element of the motor. PropertiesVMaterials Elastic Modulus (GPa) Poisson's Ratio Density (kg/m ) Graphite/Epoxy Eu=138, E?2=8.96,2= PZT(d3i=-171xlO'^m/V) EH =61, 22=6! G%2= Dimensions: 9 (L) x 25 (W) x 3 (T) mm.

9 Computational Methods for Smart Structures and Materials II 8 29 & 6- % 4 GO C 2 - Analytical Modeling Experimental Data Torque (Nmm) Figure 4: Comparison between experimental data and modelling results. Figure 5: New generation of piezoelectric motor with further miniaturization 5 Concluding remarks This paper presents an analytical model for a rotary motor driven by an anisotropic piezoelectric composite laminate. This model considered the motor as a coupled dynamic system. The governing equations of the longitudinal and torsional vibrations of the laminate and the equation of motion of the rotor under the action of compressive contact forces were derived. The developed model is able to evaluate the characteristics of the motor, including the modal frequency and motion response of the laminate, the rotating speed of the rotor, the input power, the output power and the efficiency of the motor. For numerical illustration, the present model is used to evaluate the performance curve of a newly developed piezoelectric motor in terms of rotating speed and output torque of the motor. Testing data are obtained as well from the prototype of the motor for comparison. The good agreement between analytical and experimental results validates the developed model. With this analytical tool, further parametric studies may be performed to optimize the design for future generations of rotary piezoelectric motors.

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