wen sheng, zhang tiemin*, zhang jiantao and yang xiuli...

Wen Sheng, Zhang Tiemin*, Zhang Jiantao and Yang Xiuli

Optimal Design of Cymbal Stack Transducer in aPiezoelectric Linear Actuator by Finite ElementMethod

Abstract: The optimal design of a piezoelectric linearactuator using parametric optimum method-based finiteelement method (FEM) was presented. First, the FEMmodel of the cymbal stack transducer was generatedwith its initial configuration. The structural parameterswere chosen as the design variables and the displace-ment on the top surface of the transducer taken as theobjective function. Second, the zero-order optimizationmethod was chosen as the basic tool of the structuralupdating. The structural optimization scheme of the cym-bal stack transducer was carried out based on ANSYSparametric design language (APDL). Finally, an exampleof dynamic response analysis was performed on the cym-bal stack transducer to verify the structural optimizationscheme. The results show that the displacement on thetop surface is increased by 32.9% compared with the caseof initial configuration.

Keywords: cymbal stack transducer, piezoelectric linearactuator, finite element method, parametric optimum

DOI 10.1515/ehs-2015-0006

Introduction

Small piezoelectric actuators have been used in the fieldsof robot technology and information devices for theiroutstanding characters such as simple functionalmechanism, compact structure, rapid response to exter-nal excitation and immunity from magnetic interference(Watson, Friend, and Yeo 2009; Chen and Liu 2013).

Different types of piezoelectric actuators, such asinchworm (Lu et al. 2009), traveling wave (Liu et al.2010) and standing wave (Szufnarowski and Schneider2011), have been developed and studied by numerousresearchers. Although the existing actuators have exhib-ited excellent performance, there are still some problemsto be overcome, for example, it is difficult for piezoelec-tric actuators to perform a linear driving system with asimple structure (Morita et al. 2012). In recent years,piezoelectric inertia actuator (PIA) has received a growingamount of attention to the fact that PIA has a simpleconstruction and is controlled by a single signal, whichallows for low production costs and simplifies miniatur-ization (Duan and Wang 2005; Lee et al. 2011). The PIAutilizes the inertia of a mass to drive a slider via frictioncontact. The principle behind the PIA is a stick-slipmotion achieved by the piezoelectric effect (Hunstig,Hemsel, and Sextro 2013; Bajuri et al. 2012). Mostly, thedisplacement directly produced by piezoceramic cannotsatisfy the need of the PIA (Chen et al. 2009). Therefore,people began to find a new way to amplify the displace-ment of the piezoceramic. The “Cymbal” transducer,which consists of a poled piezoelectric disk bonded totruncated conical metal endcaps, can produce 40 timesdisplacement of the piezoceramic with the same dimen-sion (Tsivgouli et al. 2007).

Recently, many researchers have paid increasingattention to evolution-based algorithms in the structuraloptimization field. These algorithms are robust and canprovide a more reliable result to obtain the global opti-mum. What’s more, they do not require continuity andderivative existence of the objective function. Geneticalgorithms (GAs) are applied to composite structural opti-mization problems (Chakraborti 2004; Alibeigloo,Shakeri, and Morowa 2007; Ameri, Aghdam, andShakeri 2012). Particle swarm optimization (PSO) isanother evolutionary global algorithm, which has gainedpopularity (Poli, Lee, and Kwak 2007; Kulkarni andVenayagamoorthy 2011). In general, one of the mainadvantages of PSO over GA is its algorithmic simplicity.The PSO algorithm does not involve complicated

*Corresponding author: Zhang Tiemin, College of Engineering,South China Agricultural University, Guangzhou 510642, China,E-mail: [email protected] Sheng: E-mail: [email protected], Zhang Jiantao: E-mail:[email protected], Yang Xiuli: E-mail: [email protected], Collegeof Engineering, South China Agricultural University, Guangzhou510642, China

Energy Harvesting and Systems 2015; 2(3-4): 169–176

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encoding, decoding or special genetic operators, and thusit has fewer parameters to be adjusted. Although therehas been much research concerning the optimization ofpiezoelectric structures (Araujo et al. 2013; Park, Lee, andKwak 2012; Kim, Wang, and Brennan 2011; Jayachandran,Guedes, and Rodrigues 2011), few works aimed to opti-mizing the structure of the cymbal transducer.

In the present paper, a miniaturized piezoelectriclinear actuator based on cymbal stack transducer is pro-posed here. An effective and automated design strategy isused to design the cymbal stack transducer. In this strat-egy, finite element analysis, parametrical model and anumerical optimization algorithm are integrated to createan automated design tool. Using this approach, shapedesign of the cymbal stack transducer is formulated inthe form of an optimization problem that can be solvedeasily by a conventional numerical optimizationalgorithm.

Description of Configuration andDriving Principle

The stator of the piezoelectric linear actuator is composedof a shaft, a metal block and the cymbal stack transducer,as shown in Figure 1. The linear slider is preloaded onthe shaft, which consists of two materials, as shown inFigure 2. The outer material is stainless steel for the basicinertial weight, and the inner material is resin with highfriction factor, such as break material of automobile.

The slider is driven by two successive motions, namely,slow forward movement and rapid backward movement.As shown in Figure 3, these motions may correspond to asawtooth displacement. The driving procedure is as follows.

From (a) to (b): During the slow forward movement,the voltage signal is applied as shown in Figure 4. Thecymbal stack transducer will produce extension

1

3 4 5 6 7 98

10

2

Figure 1: Schematic of the piezoelectric linear actuator.Note: 1, pedestal; 2, rubber pad; 3, sleeve; 4, cymbal stack trans-ducer; 5, metal block; 6, spring; 7, screw cap; 8, linear bearing; 9,linear slider; 10, shaft.

Figure 2: Structure of the slider.

Figure 3: The principle of inertia displacement.

c

b

aTime

Displacem

ent

Slider

Piezo

Figure 4: The sawtooth electrical potential.

170 W. Sheng et al.: Optimal Design of a Piezoelectric Linear Actuator by FEM



displacement in horizontal direction and push the shaftto move forward. The slider slowly moves with the statorby the frictional force. This movement is called stickmotion.

From (b) to (c): Subsequently, when the shaftmoves quickly backward to its original position, aslip will be produced between the linear slider and theshaft because the inertial force is larger than the staticfrictional force. Therefore, the linear slider remains at thelocation where it was placed during sticking motion.

By repeating these two stages, the slider can bedriven a long distance. Reversible slider movement ispossible by changing the order of slow and rapid move-ments of the piezoelectric linear actuator.

Parameters Optimization Design

The driving capability of the piezoelectric linear actuatordepends on the energy conversion efficiency of the cym-bal stack transducer. Because of small size, the structureparameters of the cymbal stack transducer have impor-tant influence on converting electric energy into mechan-ical energy. So, it is necessary to optimize the design ofthe cymbal stack transducer.

In this section, a novel design method, using finiteelement analysis and the zero-order optimizationmethod, is proposed and applied to the design of thecymbal stack transducer. The finite element method(FEM) is applied to modal analysis and harmonic analysisof the cymbal stack transducer. The zero-order optimiza-tion method is used to locate optimum design of thedesign variables.

Finite Element Model

The aim of numerical modeling is to calculate naturalfrequencies and modal shapes of the cymbal stack trans-ducer and to perform harmonic or transient responseanalysis. Basic equations for the motion of the cymbalstack can be written in matrix form (Narayanan andSchwartz 2007):

½M� @2fug@2t

þ ½C� @fug@t

þ ½K1�fug þ ½K2�fΦg ¼ fFNg ½1�

½K2�Tfug þ ½K3�fΦg ¼ fQg ½2�where {u} is the nodal displacement vector, {Φ} isthe electrical potential vector, [M] is the mass matrix,

[C] is the damping matrix, [K1] is the stiffness matrix,[K2] is the piezoelectric matrix, [K3] is the dielectricmatrix, {Q} is the nodal electrical charge vector and{FN} is the nodal mechanical force vector. The solutionfor nodal displacement is obtained with ANSYS. Figure 5(a) shows the three-dimensional finite element model ofthe cymbal stack transducer. The endcaps material isbrass, and the piezoelectric material is PZT-5A. Theendcaps and piezoceramics were modeled with SOLID5which is an element type in the library of ANSYS,defined by eight nodes each having three degrees offreedom. The complete model consists of 64,320elements. The structural parameters needed for fabrica-tion of a cymbal transducer are shown in Figure 5(b),where tp and tm are the thickness of piezoelectricdisc and metal cap respectively, tc is the height ofthe endcaps. D, D1 and D2 are the diameters of thepiezoelectric disc, bottom and top of cavities, respec-tively. The initial values of the structural parameters areshown in Table 1.

Figure 5: (a) Finite element model of the cymbal stack transducer.(b) Structure parameters of the cymbal transducer.

Table 1: The initial values of the structural parameters.

Structuralparameter

Initialvalue/mm

Structuralparameter

Initialvalue/mm

D D

D tp .tm . tc .

W. Sheng et al.: Optimal Design of a Piezoelectric Linear Actuator by FEM 171



Optimization Method

There are two optimization methods provided by ANSYS,i.e. the zero-order method and the first-order method. Thezero-order method, due to its independency from usingderivatives of the problem variables, is the first candidatefor the optimization subroutine. It can be efficientlyapplied to most engineering problems (Zhang, Zhu, andZhao 2010; Bang et al. 2008). The shape or materialdesign optimization problem can generally be formulatedas a constrained minimization problem as following(Zhang et al. 2012):

Min F ¼ FðxÞ ½3�subjected to:

gjðxÞ � 0 ðj ¼ 1; . . . ; ncÞ ½4�with the design space:

xil � xi � xiu ði ¼ 1; . . . ;NÞ ½5�

where F(x) is the objective function, gj(x) (j¼ 1,..., nc) arethe constraint functions and x ¼ [x1, x2,..., xN] is thevector of design variables. xil and xiu describe physicalupper and lower bounds on design variables. nc and Nare the number of constraints and number of designvariables, respectively. Solution of eqs [3]–[5] for shapeoptimization problems can be efficiently done by repla-cing objective and constraint functions with theirresponse surface (RS) approximations. For this method,the dependent variables are replaced with the RS approx-imations by means of least squares fitting, and minimiza-tion is performed every iteration on the penalizedfunction. The approximate optimization method imple-mented in ANSYS optimum module for the objective orconstraint function as follows:

~F ¼ a0 þXNi¼1

aixi þXNi¼1

XNj¼1

bijxixj ½6�

where ai and bij are coefficients to be determined. ~F is theapproximation of the objective function. For the zero-order method, convergence is assumed if any one of thefollowing conditions is satisfied:

jFðjÞ � Fðj�1Þj � τ ½7�

jFðjÞ � FðbÞj � τ ½8�

jxiðjÞ � xiðj�1Þj � ρi ði ¼ 1; 2; 3; . . . ; nÞ ½9�

jxiðjÞ � xiðbÞj � ρi ði ¼ 1; 2; 3; :::; nÞ ½10�

where F(b) is the best objective function at the currentiteration. τ and ρi are the objective function and designvariable tolerances, respectively.

Optimization Procedures

Objective Function

Large vibration amplitude can improve the performanceof the piezoelectric linear actuator. That is to say, it isneeded to increase the vibration amplitudes of the cym-bal stack transducer’s axial stretching vibration.Arranging the saw electrical potential cycle by FFT idea,the mathematical model of optimization design can beexpressed as follows:

Max: UðPiÞs:t: Pimin � Pi � Pimax

�½11�

where U is the top displacement of the cymbal stacktransducer， Pi is the structural parameters of the cymbaltransducer, Pimin and Pimax are the minimum and max-imum value of variable Pi, respectively.

The ANSYS program always minimizes the objectivefunction, so the objective function is transformed to thefollowing minimum problem:

Min: F ¼ C � U ½12�where C is a constant, which is greater than the topdisplacement U.

Design Variables

The structure parameters of piezoceramic (D and tp) remainunchanged, because the limits of the piezoceramic proces-sing technique and the design requirement. So, the otherdesign parameters are chosen as design variables as follows:

x ¼ ½D1 D2 tm tc� ½13�The variation ranges of the design variables as shown inTable 2. The variation ranges of the design variables aredetermined according to the following reasons: thedimension constraint of different components of thepiezoelectric linear actuator, and the previous experi-ences in cymbal transducer research.

Optimization Process

The optimization process of cymbal stack transducerdesign is shown in Figure 6. FEM is applied to the




modal analysis and harmonic analysis of the cymbalstack. Zero-order method is used to search optimumdesign of the design variables. The design process isprogrammed by ANSYS parametric design language(APDL). The iterative design process is as follows.

Step 1: Parametric finite element model

First, the input quantities such as dimensions, materialproperties and loads should be varied parametrically. So inthe process of optimization, the three-dimensional modelcan be constructed in different dimensions. Meanwhile,according to the actual working conditions, the bottomsurface of the cymbal stack transducer is clamped.

Step 2: Modal analysis

The modal analysis is used to determine the cymbalstack transducer’s natural frequencies. Figure 7 showsthe calculated mode shapes of the first four axial stretch-ing vibrations. The first mode is known as the fundamen-tal vibration mode of the cymbal stack transducer.

Step 3: Harmonic analysis

The frequency of the vibration modes depends on thegeometry of the cymbal transducer. With the change of

the design variables, the frequency of the mode shapehas also changed during the iterative process. In additionto the axial stretching vibration modes, the cymbalstack transducer has other torsional modes, as shown inFigure 8. Therefore, how to distinguish the axial stretch-ing vibration modes from the other modes is one of theimportant problems to be solved.

Figure 6: Flowchart of cymbal stack transducer.

Table 2: Variation ranges of the design variables.

Design variable Variation range/mm

D [; ]D [; ]tm [.; .]tc [.; ]

Figure 7: The axial stretching vibration mode shapes of the cymbalstack transducer: (a) the first-order mode shape at 720 Hz, (b) thesecond-order mode shape at 1,969 Hz, (c) the third-order modeshape at 2,778 Hz and (d) the fourth-order mode shape at 3,145 Hz.




When the axial stretching vibration is generated, the dis-placement along the Y-axis of the top surface is muchlarger than the other modes, as shown in Figure 9. Thus,the desired axial stretching vibration can be obtainedaccurately by using the harmonic analysis.

Step 4: Optimization historyThe iterative history of the design variables is shown

in Figure 10. After 24 iterations, the objective functionand design variables reach a steady state and iterationtermination. The design variables before and after theoptimization are shown in Table 3.

Step 5: Transient analysis

Transient analysis is used to determine the dynamicresponse of the cymbal stack transducer under electricload. According to the Fourier analysis, a sawtooth-shaped waveform can be decomposed into multiple sinu-soidal waves:

f ðtÞ ¼ A sinðωtÞ þ A2sinð2ωtÞ þ A

3sinð3ωtÞ þ � � � ½14�

where f(t) is a sawtooth-shaped wave function, and Ais the amplitude of the first sinusoidal signal. A saw-tooth-shaped waveform is composed of numerousintegers multiple harmonics, the second harmonic is

0 5 10 15 20 25

10

15

20

25

30

Des

ign

vari

able

s D

1D2/

mm

Iterations

D1

D2

(a)

0 5 10 15 20 25

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Des

ign

vari

able

s t c

t m/m

m

Iterations

tctm

(b)

Figure 10: Iteration histories of design variables.

(a ) 352 Hz (b) 1,101 Hz

(c ) 1,757 Hz (d) 2,132 Hz

Figure 8: The other torsional vibration modes.

200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000−2.00 E−008

−1.00 E−008

0.00 E+000

1.00 E−008

2.00 E−008

3.00 E−008

4.00 E−008

5.00 E−008

6.00 E−008

Dis

plac

emen

t /m

m

Frequency/Hz

352 Hz

720 Hz

1,101 Hz1,757 Hz

Figure 9: The diagram of harmonic analysis. Table 3: Initial and final values of design variables and naturalfrequencies.

Initialvalue

Finalvalue

Design variable D/mm .D/mm .tm/mm . .tc/mm .

Natural frequency ofaxial stretchingvibration

First order/Hz

Second order/Hz , ,Third order/Hz , ,Fourth order/Hz , ,




half the level of the first, the third harmonic is one-third the level of the first and so on. Because thecoefficients of the third and higher harmonics aremuch smaller than the first two harmonics, the thirdand higher harmonics can be neglected in practice. Tosynthesize the axial stretching vibration, the ratio ofthe cymbal stack transducer’s second and fundamen-tal resonant frequencies can be designed to 4:1, asshown in Figure 11.

To verify the optimization results, the dynamicresponse of the cymbal stack transducer at the same

voltage amplitude is numerically analyzed by using thetransient analysis. The voltage amplitude applied on thesurface of the piezoceramics is 200 V. The top surfacedisplacement curves of the cymbal stack transducerbefore and after optimization are shown in Figure 12. Itcan be seen from the figure that the maximum displace-ment generated by the cymbal stack transducer beforeoptimization is 1.46 μm and the maximum displacementis 1.94 μm after optimization. The displacement valueafter optimization is increased by 32.9% compared withthe value before optimization.

Conclusion

This research investigates on the conceptual design andperformance analysis of a novel piezoelectric linearactuator. Based on the parametric finite elementmodel, the optimization design of the cymbal stacktransducer is realized by using the zero-order method.After optimization design, the displacement on the topsurface is increased by 32.9% compared with the case ofinitial configuration. The parametric optimizationmethod can quickly search for the optimal solution tomeet the design requirements, which provides a newway for the optimization design of the piezoelectriclinear actuator.Figure 11: Two axial stretching vibration modes.

2 3 4 5 6 7 8 9 10

−2

−1

0

1

2

Dis

plac

emen

t /10

−6

m

Time /10−3 s

Before optimization

After optimization

Figure 12: Displacement curve of cymbal stack transducer.




Funding: This work was funded by National NaturalScience Foundation of China (grant number: 51177053),Guangdong Province Science and Technology InnovationProject (grant number: 2012CXZD0016) and SpecializedResearch Fund for the Doctoral Program of HigherEducation of China (grant number: 20124404110003).

References

Alibeigloo, A., M. Shakeri, and A. Morowa. 2007. “Optimal StackingSequence of Laminated Anisotropic Cylindrical Panel UsingGenetic Algorithm.” Structural Engineering and Mechanics25 (6): 637–52.

Ameri, E., M. M. Aghdam, and M. Shakeri. 2012. “Global Optimizationof Laminated Cylindrical Panels Based on Fundamental NaturalFrequency.” Composite Structures 94 (9): 2697–705.

Araujo, A. L., J. F. A. Madeira, C. M. M. Soares, et al. 2013. “OptimalDesign for Active Damping in Sandwich Structures Using theDirect Multisearch Method.” Composite Structures 105: 29–34.

Bajuri, M. N., M. R. A. Kadir, M. M. Raman, et al. 2012. “Mechanicaland Functional Assessment of the Wrist Affected byRheumatoid Arthritis: A Finite Element Analysis.” MedicalEngineering & Physics 34 (9): 1294–302.

Bang, I. K., D. S. Han, G. J. Han, et al. 2008. “Structural Optimizationfor a Jaw Using Iterative Kriging Metamodels.” Journal ofMechanical Science and Technology 22 (9): 1651–9.

Chakraborti, N. 2004. “Genetic Algorithms in Materials Design andProcessing.” International Materials Reviews 49 (3–4): 246–60.

Chen, W. M., and T. S. Liu. 2013. “Modeling and Experiment ofThree-Degree-of-Freedom Actuators Using PiezoelectricBuzzers.” Smart Materials and Structures 22 (10): 105006.

Chen, H., W. Peng, R. Ge, et al. 2009. “Optimal Design of CompositeLaminates for Minimizing Delamination Stresses by ParticleSwarm Optimization Combined with FEM.” StructuralEngineering and Mechanics 31 (4): 407–21.

Duan, Z., and Q. Wang. 2005. “Development of a Novel HighPrecision Piezoelectric Linear Stepper Actuator.” Sensors andActuators A: Physical 118 (2): 285–91.

Hunstig, M., T. Hemsel, and W. Sextro. 2013. “Stick–Slip and Slip–Slip Operation of Piezoelectric Inertia Drives. Part I: IdealExcitation.” Sensors and Actuators A: Physical 200: 90–100.

Jayachandran, K. P., J. M. Guedes, and H. C. Rodrigues. 2011.“Ferroelectric Materials for Piezoelectric Actuators by OptimalDesign.” Acta Materialia 59 (10): 3770–8.

Kim, S. M., S. Wang, and M. J. Brennan. 2011. “Dynamic Analysis andOptimal Design of a Passive and an Active Piezo-electricalDynamic Vibration Absorber.” Journal of Sound and Vibration330 (4): 603–14.

Kulkarni, R. V., and G. K. Venayagamoorthy. 2011. “Particle SwarmOptimization in Wireless-Sensor Networks: A Brief Survey.”IEEE Transactions on Systems, Man, and Cybernetics, Part C:Applications and Reviews 41 (2): 262–7.

Lee, J., W. S. Kwon, K. S. Kim, et al. 2011. “A Novel Smooth ImpactDrive Mechanism Actuation Method with Dual-Slider for aCompact Zoom Lens System.” Review of Scientific Instruments82 (8): 085105.

Liu, Y., J. Liu, W. Chen, et al. 2010. “A Cylindrical Traveling WaveUltrasonic Motor Using Longitudinal Vibration Transducers.”Ferroelectrics 409 (1): 117–27.

Lu, H., J. Zhu, Z. Lin, et al. 2009. “An Inchworm Mobile Robot UsingElectromagnetic Linear Actuator.” Mechatronics 19 (7): 1116–25.

Morita, T., H. Murakami, T. Yokose, et al. 2012. “A MiniaturizedResonant-Type Smooth Impact Drive Mechanism Actuator.”Sensors and Actuators A: Physical 178: 188–92.

Narayanan, M., and R. W. Schwartz. 2007. “Finite Element Modelingof a Donut Flextensional Transducer.” Journal of the AmericanCeramic Society 90 (3): 850–7.

Park, J., S. Lee, and B. M. Kwak. 2012. “Design Optimization ofPiezoelectric Energy Harvester Subject to Tip Excitation.”Journal of Mechanical Science and Technology 26 (1): 137–43.

Poli, R., J. Kennedy, and T. Blackwell. 2007. “Particle SwarmOptimization.” Swarm Intelligence 1 (1): 33–57.

Szufnarowski, F., and A. Schneider. 2011. “Force Control of aPiezoelectric Actuator Based on a Statistical System Model andDynamic Compensation.” Mechanism and Machine Theory46 (10): 1507–21.

Tsivgouli, A. J., M. A. Tsili, A. G. Kladas, et al. 2007. “GeometryOptimization of Electric Shielding in Power Transformers Basedon Finite Element Method.” Journal of Materials ProcessingTechnology 181 (1): 159–64.

Watson, B., J. Friend, and L. Yeo. 2009. “Piezoelectric UltrasonicMicro/Milli-scale Actuators.” Sensors and Actuators A: Physical152 (2): 219–33.

Zhang, J., H. Zhu, and C. Zhao. 2010. “Combined Finite ElementAnalysis and Subproblem Approximation Method for theDesign of Ultrasonic Motors.” Sensors and Actuators A:Physical 163 (2): 510–15.

Zhang, J. T., H. Zhu, S. Q. Zhou, et al. 2012. “Optimal Design of a RodShape Ultrasonic Motor Using Sequential QuadraticProgramming and Finite Element Method.” Finite Elements inAnalysis and Design 59: 11–17.




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