Sensors and Actuators A 147 (2008) 310–323

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Sensors and Actuators A: Physical

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Optimization of clamped circular piezoelectric composite actuators

Melih Papilaa,∗, Mark Sheplakb, Louis N. Cattafesta III b

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a Sabancı University, Faculty of Natural Sciences and Engineering, Orhanlı, Tuzla, 34956b Interdisciplinary Microsystems Group, Department of Mechanical and Aerospace Engi

a r t i c l e i n f o

Article history:Received 20 December 2007Received in revised form 20 March 2008Accepted 4 May 2008Available online 18 May 2008

Keywords:Volume displacementBandwidthPareto optimal designPiezoelectric actuator

a b s t r a c t

This paper addresses the dfigurations, specifically thbandwidth. An optimizatiobimorph configurations uramic patches. A range of aand natural frequency of odesign space, Pareto optimvolume displacement is rsquare of the natural freqCharacteristic trends aremal piezoceramic patch thexample, in the design of

1. Introduction

Piezoelectric composite plates are commonly used in many

engineering applications, such as actuators for flow control applica-tions (Glezer and Amitay [1]), transducers for acoustic applications(Horowitz et al. [2]), and in locomotion of robotic systems(Yumaryanto et al. [3]). Piezoelectric circular composite plates, inparticular, are widely used for systems/applications where actu-ation and/or sensing are needed. Examples include, but are notlimited to, mechanical actuation and sound generation or receivingdevices (Chee et al. [4]), zero-net mass-flux or synthetic jets (Gallaset al. [5,6]), micropumps (Morris and Foster [7]), energy harvesting(Kim et al. [8,9], Horowitz el al. [10], and Liu et al. [11]) and activestructural health monitoring applications (Liu et al. [12]).

Optimal system performance in such applications is dictatedby the electromechanical characteristics of the piezoelectric com-posite plates, and therefore design optimization of the compositeplate configurations is of great interest. In particular, it requires thedetermination of the electromechanical response when the designparameters are varied. Morris and Foster [7] studied optimizationof a circular bimorph concept using the finite element method(FEM). However, FEM-based optimization is cumbersome becausea new mesh may be required when the geometric design variables

∗ Corresponding author. Tel.: +90 216 483 9546.E-mail address: mpapila@sabanciuniv.edu (M. Papila).

0924-4247/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.sna.2008.05.018

bul, Turkeyg, University of Florida, Gainesville, FL 32611, USA

of clamped circular piezoceramic composite unimorph and bimorph con-flicting requirements of maximum volume displacement for a prescribedblem is formulated that implements analytical solutions for unimorph and

aminated plate theory, including the use of oppositely polarized piezoce-or geometric parameters are studied, and bounds for volume displacemental designs are determined and presented via design curves. In the selectedion results for unimorph and bimorph configurations show that optimal

to the bandwidth by a universal power law such that the product of they and the displaced volume, a “gain-bandwidth” product, is a constant.escribed that are independent of the actuator radius for the Pareto opti-ess and radius versus normalized bandwidth. The results are relevant, fornet mass-flux or synthetic jet actuators used in flow control applications.

© 2008 Elsevier B.V. All rights reserved.

are varied. Furthermore, coupling with an external optimizer maybe required. Hence, analytical solutions of piezoelectric compositeplates for their electromechanical response are desirable in suchdesign optimization problems. Coorpender et al. [13], for instance,noted the ease of their model and formulation to accommodatechanging material parameters or geometry. They demonstrated the

effects of inactive plate thickness as well as piezoceramic patchradius and thickness on the displacement for a fixed input voltage.

There are a number of published analytical studies on the deter-mination of the electrically-driven transverse deflection of variouspiezoelectric composite actuators (see Fig. 1). Li et al. [14] stud-ied the electromechanical behavior of PZT-brass rectangular andcircular unimorphs where the PZT patch completely covered thebrass plate. Ha and Kim [15] developed a model for an asymmet-rical annular bimorph configuration. Prasad et al. [16] derived astatic analytical model of a clamped axisymmetric piezoelectricunimorph transducer consisting of a piezoelectric inner disc per-fectly bonded to a metal shim. They used lumped element modeling(LEM) to estimate the dynamic response.

Other configurations and boundary conditions have also beenstudied recently. In a preliminary version of the present work, Gal-las et al. [5] developed analytical models for clamped annular andinner disc axisymmetric configurations designed to maximize thevolume displacement of the actuator (see Fig. 1). Li and Chen [17]obtained a solution for both clamped and simply-supported bound-ary conditions for a valve-less micropump. Their model is valid fora disc in pure bending loaded with an axisymmetric moment at the

M. Papila et al. / Sensors and Actuators A 147 (2008) 310–323 311

ly pol

objectives of maximum volume displacement and maximum nat-ural frequency, are computed. Finally, empirical design curves as afunction of overall actuator radius are provided to determine theexpected bounds on the objective for a specified actuator size, andan optimal design algorithm with examples is provided.

2. Description of actuator and design parameters

Piezoelectric circular unimorph and bimorph composite platessubject to electrical and differential pressure loads were studiedin this work. Fig. 2 shows the unimorph configuration subject todifferential pressure loading. The clamped composite plate consists

Fig. 1. Circular composite piezoceramic unimorph and bimorph plates with oppositedirection and Ef denotes the positive upward electric field applied through a patch.

edge. Fox et al. [18] extended this approach to an annular configu-ration under clamped and simply-supported boundary conditions.Chang and Lin [19] studied a piezoelectric ring, which consists of anisotropic elastic ring laminated between two identical piezoelectricrings (termed a “trimorph” configuration in their paper but referredto here as a “bimorph” configuration). They developed an electroe-lastic laminated plate theory to analyze its dynamic behavior, suchas electric current response and resonant frequencies. They alsoconsidered several boundary conditions, namely clamped–free,free–clamped, and clamped–clamped at the inner-outer radius. Inan extension of their earlier work, Prasad et al. [20] presented atwo-port electroacoustic model which provided the solution forthe transverse static deflection field as a function of pressure andvoltage loading. Classical laminated plate theory (CLPT) was used toderive the equations of equilibrium of clamped circular laminatedplates containing a piezoelectric layer. Closed-form expressions forthe static deflection field as a function of the applied uniform pres-sure and/or the uniform electric field across the piezoelectric layerwere obtained. Another recent effort on modeling circular com-posite actuators is by Mo et al. [21]. Their solution was solely forelectrical loading but considered different boundary conditions aswell as full and partial coverage piezoelectric disc configurations.Experimental validation was also reported. Dong et al. [22], sim-ilar to Prasad et al. [20], also determined transverse deformation

shape of a circular axisymmetric piezoelectric-metal compositeunimorph actuator for both electrical and distributed mechanicalloading (such as uniform pressure). Using their analytical solution,they performed a parameteric study of the piezoelectric-to-metalthickness ratio and presented design curves. The most recent workto the authors’ knowledge is the analytical solution by Deshpandeand Saggere [23], which mainly focused on a unimorph configu-ration with a central piezoelectric layer. Their approach, however,permits an arbitrary number of layers, which makes it versatileenough to investigate other configurations (e.g., bimorph).

Despite the availability of diverse analytical solutions, reportedimplementations in design optimization are sorely lacking. Thisstudy is a continuation of the initial assessment of different configu-rations summarized in Gallas et al. [5] that used LEM in conjunctionwith analytical models of a clamped piezoelectric composite plateto optimize the performance of unimorph-driven synthetic jets forflow control applications. Here, we incorporate analytical solutionsfor both unimorph and bimorph configurations to find optimaldesigns for maximum volume displacement over a prescribedbandwidth, which is desirable in pump and flow control applica-tions. The bandwidth of the device is limited to frequencies from dc

arized inner disc and outer ring configurations. Arrow tips represent the polarization

to near the natural frequency of the composite plate, which can beeasily obtained from a static solution using LEM without having toresort to FEM-based methods (Rossi [24]). Similar to an amplifier,which has a gain-bandwidth limitation, a piezoceramic compositeplate will have a tradeoff between high dc or static displacement(i.e., related to its “gain”) and a large bandwidth. Hence, optimalunimorph and bimorph designs subject to a prescribed natural fre-quency constraint are found and compared. The tradeoffs betweenthe volume displacement and the natural frequency are investi-gated within the framework of Pareto optimization (Belegundu andChandrupatla [25]) as a function of actuator geometry. The Paretocurves and their opposite ends, which correspond to the conflicting

of PZT patch(es) perfectly bonded to a brass shim. The shim andpiezoelectric layer material properties are summarized in Table 1.Note that, in practice, clamping the PZT patch is not desirable asit may fracture due to stress concentrations. This constraint willbe addressed again when the optimized unimorph and bimorphdesigns are discussed.

Fig. 1 depicts the electric field and the appropriate oppositepolarization directions of the inner and outer piezoceramic patchesfor both unimorph and bimorph bender configurations. The geo-

Fig. 2. The three regions of a unimorph composite diaphragm.

Actua

3)

12

r2

312 M. Papila et al. / Sensors and

metric radii and thickness dimensions are also defined in Fig. 1.Arrows on the piezoceramic elements indicate the polarizationdirection. In addition, Ef denotes the electric field applied througha piezoelectric patch, which is defined as positive when the fielddirection (from ground to +) due to the applied voltage is upward.

The composite plate deforms in response to both an applied acvoltage and a differential pressure. The electrical and pressure load-ing, Ef and P, respectively, creates both transverse w(r) and radialu(r) displacements. For this study, classical laminate theory is usedto solve for the transverse static deflection. The approach makesuse of Kirchhoff’s assumption for thin plates where shear deforma-tion and rotary inertia can be omitted. This assumption requiresthe radius-to-thickness ratio to be at least 10 (Duan et al. [27]). Ingeneral, the composite plate is split into three parts: a central multi-layer circular region, a homogenous shim ring region, and an outerclamped multi-layer annular region. (Fig. 2 shows a schematic forthe unimorph case.) The details of the solution that is an extensionof that presented in Prasad et al. [20] are provided in Appendix A,and the form of the piecewise solution is⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

w(1) = Pr4

64D∗(1)11

− b(1)12

r2 + b(3)2

w(2) = Pr4

64D∗(2)11

− b(2)12

r2 − b(2)1 ln r + a(

2

w(3) = P

16D∗(3)11

(14

r4 − R43 ln r

)− b(3)

1

(

where a(j)i

, b(j)i

, c(3) are constants that are found using the bound-ary and matching conditions at the junctions of the segments. Thesolution has the same form for both unimorph and bimorph config-

urations, but the constants and stiffness (or compliance) terms D∗(j)

11depend on whether the configuration is a unimorph or bimorph.The analytical solution in Eq. (1) as well as the lateral deflection,stresses, and resultant volume displacement were coded in MAT-LAB (Mathworks [28]). The analytical results were then verified viaABAQUS [29] finite element analysis software. The model makesuse of 8-node biquadratic axisymmetric standard quadrilateral andpiezoelectric quadrilateral elements for the shim and PZT layers,respectively. Fig. 3 presents the results of an example case (mate-rial properties are in Table 1) for the lateral deflection due to a unitapplied voltage creating a positive Ef. The analytical and FEA solu-tions are in good agreement. Similar agreement (not shown) foruniform unit pressure loading are also obtained.

Note that the present analysis ignores the adhesive layer for sim-plicity here, since the primary goal is to optimize the compositeplate component dimensions. Crawley and deLuis [30] and Mor-ris and Forster [7] reported that the effect of the bonding layeris minimal, provided it is very thin relative to the other thicknessdimensions. However, the existence of other elastic layers, such asan adhesive layer, can be incorporated into the solution.

Table 1Material properties and piezoceramic diaphragm details (Gallas et al. 2003 [5])

Property Piezoceramic (PZT-5A) Shim (brass)

Elastic modulus (GPa) 63 90Max. allowable stress (MPa) 24a 200Poisson’s ratio 0.310 0.324Density (kg/m3) 7700 8700Relative dielectric constant 1750 –Coercive electric field Ef max (V/m) 1.1811 × 106 –d31 (m/V) −1.75 × 10−10 –

a Dynamic peak tensile strength—Bert and Birman [26].

tors A 147 (2008) 310–323

0 ≤ r ≤ R1

R1 ≤ r ≤ R2

− R23 ln r

)+ c(3) R2 ≤ r ≤ R3

. (1)

Next, we address the performance parameters used either asan objective function or constraint in this study: the volume dis-placement �Vol and the natural frequency fN. Both parameters aredetermined by the transverse deflection w(r) determined from thesolution to Eq. (1).

2.1. Volume displacement

The free volume displaced by the composite plate, �Vol, is dueto the application of an ac voltage Vac, with zero differential pres-sure, that results in an electric field applied across the piezoceramicpatches, Ef = Vac/tp, as shown in Fig. 1. As described in Gallas et al.[6], the output velocity of a zero-net mass-flux actuator is directlyproportional to the volume velocity of the diaphragm, which isthe time derivative of �Vol, so �Vol is an appropriate metric tomaximize for the optimization of configurations

�Vol = 2�

∫w(r)|p=0rdr. (2)

2.2. Natural frequency by lumped element modeling

As described in Prasad et al. [20], the composite plate is lumpedinto a short-circuit acoustic compliance CaS and equivalent acousticmass MaS, which represents the stored potential and kinetic energy,respectively. This approach provides a simple method to accuratelyestimate the resonance frequency of the plate structure.

The short-circuit acoustic compliance is determined by integrat-ing the transverse plate displacement generated by a unit pressureloading with the piezoelectric patches shorted [20].

CaS = �VolP

∣∣∣Vac=0

=2�∫

w(r)∣∣Vac=0

rdr

P. (3)

The acoustic mass is determined by equating the lumped kineticenergy of the electrically shorted vibrating plate expressed inacoustic conjugate power variables to the total kinetic energy [20].

MaS = 2�

∫�A(r)

(w(r)

∣∣Vac=0

�Vol

)2

rdr. (4)

where �A(r) is the radial dependent area density of the compositeplate. Finally, the first mode natural frequency of the plate fN canbe estimated as

fN = 1

2�√

CaSMaS

. (5)

Since the frequency is a very important parameter in ouractuator optimization, we also compared the natural frequencycalculated by the above LEM approach with a modal analysis per-formed using ABAQUS [29] for the same configuration describedin Fig. 3. In particular, the frequencies of the first mode are 707and 840 Hz for the unimorph and bimorph, respectively. The corre-sponding frequencies estimated by the lumped element model are709 and 841 Hz, respectively.

M. Papila et al. / Sensors and Actua

(

Fig. 3. Verification of the analytical solution by FEA for a design with (R1, R2, R3, tp,ts) = (16.89, 17.58, 18.50, 0.123, 0.081) mm. (a) Unimorph and (b) bimorph.

3. Design optimization

The next step is to optimize the design of piezoelectric circularplate actuators. The specific goal of this study is to find the optimalpiezoelectric composite circular plate configuration and design formaximized volume displacement and bandwidth. These conflict-ing tradeoffs are investigated via Pareto optimization. This sectionfirst defines the design variables, objective function, and the con-

straints for the optimization. A formal problem formulation is thenprovided, followed by the results.

3.1. Design variables

The design variables specify the composite plate geometrydefined by the three radii shown in Fig. 2, as well as the thick-ness of the shim and piezoceramic patches. Note that the thicknessof the inner disc and annular ring patches can easily be imple-mented as separate variables, but here we forced them to be equalfor simplicity. In addition, among the three radii, the outer actua-tor radius R3 was considered to be fixed at several specified valuessince applications often place overall size constraints on the actu-ator. In other words, the four optimization variables for a specifiedactuator diameter (R3) were R1, R2, tS and tP. The specified R3 valueswere 5, 10, 15, 20, 25, 30, 35 and 40 mm.

3.2. Constraints

Physical bounds and operational requirements define the con-straints included in the problem formulation and are briefly

tors A 147 (2008) 310–323 313

Table 2Lower (LB) and upper (UB) limits for the design variables

Variable name Variable bounds

x xLB − xUBR1 (mm) 0.1R3–R3R2 (mm) 0.1R3–R3tS (mm) 0.100–0.250tP (mm) 0.076–0.508

described as follows:

(i) Lower (LB) and upper (UB) bounds for the variables: Alongwith the specified actuator radius R3, the bounds for thedesign variables were determined by considering dimensionsof commercially available products of piezoceramic compos-ite plate/actuator providers, such as APC International [31],Kyocera [32], and Piezo Systems Inc. [33]. The following boundsare listed in Table 2,

xLB ≤ R1, R2, tS, tP ≤ xUB. (6)

(ii) Natural frequency: The first mode resonance or natural fre-quency due to Eq. (5) was also constrained by pre-specifiedlower limits flim in the demonstration of the volume displace-ment and frequency range tradeoff,

flim − fN ≤ 0. (7)

The frequency limiting values are varied for specified plateor actuator radius and are reported in Section 4 when eachactuator radius or overall dimension of the plate are presented.

iii) Bending stresses: Mechanical stresses due to bending of thecomposite plate were also constrained. The maximum mag-nitude (compressive or tensile) radial and tangential stresseson both brass shim and PZT patches, |�(S)|max and |�(P)|max,respectively, were computed when the coercive electric fieldwas applied, and compared with the allowable stress values�(S)

all and �(P)all listed in Table 1,

|�(S)|max − �(S)all ≤ 0

|�(P)|max − �(P)all ≤ 0

. (8)

(iv) Applied electric field: The applied electric field was kept

constant at the coercive electric field (see Table 1) of the piezo-electric material. This was not implemented as a constraintin the next section because it simply defines the maximumapplicable electrical loading for the optimization study.

3.3. Problem formulation

The objective of the optimization study is to maximize is thevolume displacement �Vol subject to the constraints given in theprevious section. The mathematical representation of the optimiza-tion problem can be stated as

Maximizex=R1,R2,ts,tp

�Vol(x)

such that xLB ≤ x ≤ xUB

flim − fN ≤ 0∣∣�(S)∣∣max

− �(S)all ≤ 0∣∣�(P)

∣∣max

− �(P)all ≤ 0

. (9)

314 M. Papila et al. / Sensors and Actuators A 147 (2008) 310–323

nstra

Table 3Optimal unimorph designs for maximum volume displacement w/o a bandwidth co

R3 (mm)

5 10 15

Optimum designR2 (mm) 5.00 10.00 15.00

R1 (mm) 4.43 8.86 13.30tS (mm) 0.100 0.100 0.100tP (mm) 0.134 0.134 0.134

Performance(fN)LB (Hz) 10000 2500 1100�Vol (mm3) 0.413 6.607 33.45(dvol)UB (mm3/V) 0.0026 0.0417 0.2110

Geometric ratiosR1/R3 0.89 0.89 0.89R3/(tS + tP) 21.35 42.70 64.05tp/(tS + tP) 0.57 0.57 0.57

a Bold and italic numbers indicate an active constraint.

3.4. Pareto optimality implementation

The goal of maximizing the volume displacement is at oddswith the goal of maximizing the natural frequency or bandwidth.Without a bandwidth constraint, the optimization results in a com-pliant configuration that can achieve large volume displacementsbut has a low natural frequency. Since bandwidth is also an impor-tant quantity, we aim to also maximize the natural frequency along

Table 4Bimorph optimal designs for maximum volume displacement w/o a bandwidth constrain

R3 (mm)

5 10 15

Optimum designR2 (mm) 5.00 10.00 15.00R1 (mm) 4.42 8.84 13.27tS (mm) 0.100 0.100 0.100tP (mm) 0.100 0.100 0.100

Performance(fN)LB (Hz) 10000 2500 1100�Vol (mm3) 0.528 8.446 42.76(dvol)UB (mm3/V) 0.0045 0.0713 0.3608

Geometric ratiosR1/R3 0.88 0.88 0.88R3/(tS + 2tP) 16.63 33.26 49.892tp/(tS + 2tP) 0.67 0.67 0.67

a Bold and italic numbers indicate an active constraint.

Table 5Unimorph Pareto optimal designs for R3 = 15 mm w/bandwidth constrainta

R3 (mm)

15 15 15

Optimum designR2 (mm) 15.00 15.00 15.00R1 (mm) 13.30 13.52 13.89tS (mm) 0.100 0.100 0.100tP (mm) 0.134 0.155 0.201

PerformancefN = flimj (Hz) 1100 1200 1400�Vol (mm3) 33.45 33.19 31.51dvol (mm3/V) 0.2110 0.1814 0.1330

Geometric ratiosR1/R3 0.89 0.90 0.93R3/(tS + tP) 64.05 58.85 49.91tP/(tS + tP) 0.57 0.61 0.67

a Bold and italic numbers indicate an active constraint.

inta

20 25 30 35 40

20.00 25.00 30.00 35.00 40.00

17.73 22.16 26.59 31.02 35.450.100 0.100 0.100 0.100 0.1000.134 0.134 0.134 0.134 0.134

630 400 280 200 160105.7 258.1 5351. 9914. 16913.0.6670 1.628 3.377 6.256 10.672

0.89 0.89 0.89 0.89 0.8985.40 106.7 128.1 149.4 170.80.57 0.57 0.57 0.57 0.57

with the volume displacement and thus consider the problem asmulti-objective optimization.

Multi-objective optimization is also often referred to as Paretooptimization. Mezura-Montes and Coello [34] reviewed its historyas originally proposed by Francis Ysidro Edgeworth [35] and latergeneralized by Vilfredo Pareto [36]. A design x ∈ X where X is the fea-sible set is called Edgeworth–Pareto optimal if no other design x∈Xexists such that for any criterion (objective) i Fi(x) � Fi(x) [37]. That

ta

20 25 30 35 40

20.00 25.00 30.00 35.00 40.0017.69 22.11 26.53 30.96 35.380.100 0.100 0.100 0.100 0.1000.100 0.100 0.100 0.100 0.100

630 400 280 200 160135.1 329.9 6841. 12674. 21622.1.140 2.784 5.773 10.696 18.247

0.88 0.88 0.88 0.88 0.8866.52 83.15 99.78 116.4 133.00.67 0.67 0.67 0.67 0.67

15 15 15 15

15.00 15.00 15.00 15.0014.13 14.42 14.62 14.740.100 0.100 0.100 0.1000.245 0.332 0.439 0.508

1600 2000 2500 280029.35 25.23 21.13 18.980.1014 0.0643 0.0407 0.0316

0.94 0.96 0.97 0.9843.48 34.72 27.81 24.660.71 0.77 0.81 0.84

Actua

M. Papila et al. / Sensors and

Table 6Bimorph Pareto optimal designs for R3 = 15 mm w/bandwidth constrainta

R3 (mm)

15 15 15

Optimum designR2 (mm) 15.00 15.00 15.00R1 (mm) 13.27 13.50 13.85tS (mm) 0.100 0.100 0.100tP (mm) 0.100 0.117 0.154

PerformancefN = flimj (Hz) 1100 1200 1400�Vol (mm3) 42.76 42.39 40.22dvol (mm3/V) 0.3608 0.3056 0.2217

Geometric ratiosR1/R3 0.88 0.90 0.92R3/(tS + 2tP) 49.89 44.79 36.842tP/(tS + 2tP) 0.67 0.70 0.75

a Bold and italic numbers indicate an active constraint.

is, it seeks to provide the best compromise among conflicting objec-tives. Different optimal designs are sought for which one objectivecannot be improved without deterioration in one of the other objec-tives. The collection of these designs constructs a hypersurface,called a Pareto front, in the space of the objective functions. In thepresent problem of the two conflicting objectives, the Pareto frontis a curve on which there is no point that provides an improvementcompared to any other with respect to both the volume displace-ment (increase) and the natural frequency (increase).

There are different approaches to deal with the multi-objectiveproblems and to construct the Pareto front, as described in Bele-gundu and Chandrupatla [25]. Setting one criterion as the objectivefunction F1 and adding constraints Cij on the others Fi, Eq. (10)(Papila et al. [38]), can be employed in order to evaluate the multi-objective nature of the problems and is given by

for j = 1, 2, . . . , Kmaximize F1such that Fi ≥ Cij i = 2, 3, . . .

. (10)

By using different sets of possible Cij, we obtain K Pareto optimalpoints that are designs where one objective cannot be improvedwithout deterioration in one of the other objectives, and thus con-struct a Pareto hypersurface (front) [25]. For the implementation ofPareto optimality Eq. (10), we maximize the volume displacement(F1 = �Vol) subject to constraints as in Eq. (9), while treating the

natural frequency as a conflicting objective with several limitingvalues (F2 = fN, C2j = flimj where flimj are various limiting values onbandwidth). These details are clarified in the Section 4.

The optimization problem was implemented in MATLAB [28]using its fmincon function that employs sequential quadratic pro-gramming for nonlinear constrained problems, which estimates thegradients by the finite difference method.

4. Results

Several different size unimorph and bimorph configurationswere optimized using the MATLAB optimization toolbox. Optimumpiezoelectric patch dimensions of the actuators at the selected radiiwere determined for maximum volume displacement correspond-ing to application of the coercive electric field. A discussion of theresults is delayed until the next section.

4.1. Optimum actuator designs with no-bandwidth constraint

First, solutions to Eq. (9) were obtained with a specified actuatorradius R3 in the range of 5–40 mm. Tables 3 and 4 summarize the

tors A 147 (2008) 310–323 315

15 15 15 15

15.00 15.00 15.00 15.0014.08 14.36 14.55 14.720.100 0.100 0.100 0.1000.189 0.258 0.343 0.508

1600 2000 2500 350037.47 32.24 27.04 20.450.1680 0.1059 0.0668 0.0345

0.94 0.96 0.97 0.9831.41 24.37 19.11 13.600.79 0.84 0.87 0.91

optimization results for the unimorph and bimorph configurations,respectively, when the natural frequency was not constrained. Themaximum volume displacement per coercive electric field Ef max(see Table 1) was also computed via

dvol = �VolEf maxtp

. (11)

4.2. Tradeoff between volume displacement and naturalfrequency

Pareto optimizations, Eqs. (9) and (10), were also completed foreach actuator radius in the set of R3 values (5–40 mm). Examples ofthe results are provided in Tables 5 and 6 for actuator with radiusR3 = 15 mm of unimorph and bimorph configurations, respectively.Note that the first numerical column of the tables corresponds tothe maximized volume displacement design when no-bandwidthconstraint was imposed; i.e. it is identical to the columns ofTables 3 and 4 associated with R3 = 15 mm. The performance metricsof these columns provide the upper bound of volume displace-ment and lower bound of natural frequency (dvol)UB|R3 and (fN)LB|R3 ,respectively. The last columns of Tables 5 and 6, on the other hand,correspond to the maximum natural frequency design with the useof the maximum thickness PZT material (tP is at its upper bound).The performance metrics of these columns provide lower boundof volume displacement and upper bound of natural frequency∣ ∣

(dvol)LB

∣R3

and (fN)UB∣R3

, respectively.

The results presented in Tables 5 and 6 are plotted in Fig. 4,and the metric bounds associated with bimorph design of actua-tor radius R3 = 15 mm are also indicated. Similar sets of tables andfigures were prepared for the other actuator radii, but they arenot shown due to space limitations. Instead, the Pareto optimaldesigns, bandwidth versus volume displacement, are collectivelyplotted in Fig. 5(a) and (b), associated with unimorph and bimorphconfigurations, respectively.

Next, as shown in Fig. 6, the optimal piezoceramic normalizedpatch thickness tp/tSs versus fN/(fN)LB plots for all actuator radii, R3,collapse to a single curve. The abscissa is the bandwidth fN nor-malized by its associated lower limit (fN)LB corresponding to thenatural frequency for the no-bandwidth constraint optimizationresults listed in Tables 3 and 4. For example, the raw bandwidth fNof Table 5 is normalized by (fN)LB|R3=15 = 1100 Hz. The correspond-ing linear curve fits are given in Eqs. (12) and (13) for unimorph andbimorph configurations, respectively,

tp

ts= 2.465

fN(fN)LB

− 1.125 (12)

316 M. Papila et al. / Sensors and Actuators A 147 (2008) 310–323

Fig. 4. Pareto optimal fronts for the optimized unimorph and bimorph configura-tions with R3 = 15 mm.

Fig. 5. Pareto optimal fronts combined for all actuator radii R3 = 5–40 mm withpower law predicting volume displacement dvol versus bandwidth fN for (a) uni-morph and (b) bimorph configurations.

Fig. 6. Pareto optimal piezoceramic patch thickness versus normalized bandwidthtp versus fN/(fN)LB (R3 = 5–40 mm). (a) Unimorph and (b) bimorph, where (fN)LB isprovided in Tables 3 and 4, respectively.

and

tp = 1.923fN − 0.923. (13)

ts (fN)LB

Similarly, optimal R1/R3 versus fN/(fN)LB data collapse in Fig. 7using polynomial fits. The curve fit polynomials are given in Eqs.(14) and (15) for unimorph and bimorph configurations, respec-tively,

R1

R3= −0.029

[fN

(fN)LB

]4

+ 0.239[

fN(fN)LB

]3

−0.753[

fN(fN)LB

]2

+ 1.105[

fN(fN)LB

]+ 0.326 (14)

and

R1

R3= −0.013

[fN

(fN)LB

]4

+ 0.128[

fN(fN)LB

]3

−0.469[

fN(fN)LB

]2

+ 0.784[

fN(fN)LB

]+ 0.455. (15)

M. Papila et al. / Sensors and Actuators A 147 (2008) 310–323 317

Fig. 8. Unimorph configuration Pareto optimal front ends as a function of actu-ator radius R3: (a) yUB = (ln dvol)UB and yLB = (ln dvol)LB ;(b) yLB = (ln fN)UB andyLB = (ln fN)LB. Open symbols indicate optimization results for R3 = 12.5 mm that wasnot used in the curve fit.

Table 7Coefficients ˇi of bounding curves for optimal unimorph designs

Fig. 7. Pareto optimal piezoceramic patch radius ratio versus normalized badwidth:R1/R3 versus fN/(fN)LB (R3 = 5–40 mm). (a) Unimorph and (b) bimorph, where (fN)LBis provided in Tables 3 and 4, respectively.

4.3. Volume displacement–natural frequency bounds

The Pareto curves and their opposite ends, which correspond to

the conflicting objectives of maximum volume displacement andmaximum natural frequency, were computed and are shown inFigs. 8 and 9 for unimorph and bimorph configurations, respec-tively. Note the data are transformed as y = ln dvol and y = ln fN tofacilitate 4th-order polynomial fits

y = ˇ4x4 + ˇ3x3 + ˇ2x2 + ˇ1x + ˇ0. (16)

The coefficients ˇi for bounding curves of ln dvol and ln fN cor-responding to unimorph and bimorph optimal Pareto designs arereported in Tables 7 and 8, respectively. A potential use of thesebounding curves is suggested in the next section.

5. Discussion

First, optimal unimorph versus bimorph configurations are com-pared. Tables 3 and 4 indicate that, as expected, the bimorphconfiguration provides larger volume displacement. The naturalfrequencies of optimal unimorph and bimorph designs coincidewhen they are not constrained. For a fair comparison of the volumedisplacement, however, one might consider the percentage gain in

Fig. 8(a) Fig. 8(b)

yUB = (ln dvol)UB yLB = (ln dvol)LB yUB = (ln fN)UB yLB = (ln fN)LB

ˇ4 −8.856 × 10−6 −8.796 × 10−6 4.388 × 10−6 4.428 × 10−6

ˇ3 1.021 × 10−3 1.012 × 10−3 −5.046 × 10−4 −5.105 × 10−4

ˇ2 −4.533 × 10−2 −4.495 × 10−2 2.242 × 10−2 2.266 × 10−2

ˇ1 1.061 1.056 −5.269 × 10−1 −5.306 × 10−1

ˇ0 −10.23 −12.11 12.27 11.35

Curves are fit with R2 = 0.9999.

Table 8Coefficients ˇi of bounding curves for optimal bimorph designs

Fig. 9(a) Fig. 9(b)

yUB = (ln dvol)UB yLB = (ln dvol)LB yUB = (ln fN)UB yLB = (ln fN)LB

ˇ4 −8.856 × 10−6 −8.431 × 10−6 4.176 × 10−6 4.428 × 10−6

ˇ3 1.021 × 10−3 9.789 × 10−4 −4.856 × 10−4 −5.105 × 10−4

ˇ2 −4.533 × 10−2 −4.393 × 10−2 2.184 × 10−2 2.266 × 10−2

ˇ1 1.061 1.044 −5.205 × 10−1 −5.306 × 10−1

ˇ0 −9.696 −12.02 12.47 11.35

Curves are fit with R2 = 0.9999.

318 M. Papila et al. / Sensors and Actua

Fig. 9. Bimorph configuration Pareto optimal front ends as function of actu-ator radius R3: (a) yUB = (ln dvol)UB and yLB = (ln dvol)LB; (b) yUB = (ln fN)UB andyLB = (ln fN)LB. Open symbols indicate optimization results for R3 = 12.5 mm that wasnot used in the curve fit.

volume displacement or volume displacement/voltage versus thepercentage increase of PZT material use. When no-bandwidth con-straint is imposed, the bimorph configuration provides 71% gain involume displacement per volt at a cost of approximately 49% morePZT material as indicated in Table 9.

Similar comparisons can also be made along the Pareto front foreach actuator radius when a bandwidth constraint is imposed. Forexample, for R3 = 15 mm, Pareto solutions indicate the benefit, interms of dvol, of the bimorph configuration decreases for increas-ingly higher bandwidths (Table 10).

The following important points are noted concerning observedtrends in the optimal designs. Optimal solutions invariably push theshim thickness ts to its lower bound, essentially rendering it a con-stant. Second, both unimorph and bimorph configurations resultin optimal designs with an inner PZT disc alone. This is attributedto the fact that, compared to when the shim alone is clamped, the

Table 9Performance comparison of optimal designs w/o bandwidth constraint for all radiiR3: bimorph/unimorph

PZT material volume ratio 1.49�Vol ratio 1.28dvol ratio 1.71

tors A 147 (2008) 310–323

Table 10Performance comparison of optimal designs w/bandwidth constraint for R3 = 15mm:bimorph/unimorph

fN (Hz)

1100 1200 1400 1600 2000 2500

PZT materialvolume ratio

1.49 1.51 1.52 1.53 1.54 1.54

�Vol ratio 1.28 1.28 1.28 1.28 1.28 1.28dvol ratio 1.71 1.68 1.67 1.66 1.65 1.64

existence of an outer PZT ring stiffens the plate in the vicinity ofthe clamp where there is a transition between the zero-to-finiteslope. Third, the resulting designs tend towards full but not com-plete coverage of the shim by the inner piezoceramic layer, that isR1, R2 → R3, as the bandwidth constraint is increased. This trend canbe explained by examining the slope of the transverse deflection,from which one finds that the curvature changes the sign at theend of the inner disc (i.e., an inflection point exists – see Fig. 3). Theoptimization tries to push the inflection point towards the clampededge for maximum volume displacement. Fourth, when the band-width is not constrained, the ratio R1/R3 ≈ 0.88 − 0.89. However,when it is constrained, the bandwidth constraint is enforced viaan increase in the PZT layer thickness, which decreases the result-ing penalty on the volume displacement. Note that the practicalconstraint of avoiding a clamped PZT ring mentioned earlier is cir-cumvented because the optimal designs do not possess a clampedouter PZT patch.

Similarly, the following points are noted concerning the trade-off between the volume displacement and bandwidth. Without abandwidth constraint, the optimization results in a very compli-ant diaphragm that can achieve large volume displacements witha concomitant lower natural frequency. Table 5, Table 6 and Fig. 4demonstrate the expected tradeoff between the two performancemetrics for R3 = 15 mm. When the Pareto optimal points of all actu-ators with radii R3 = 5–40 mm are compiled, the overall tradeoffbetween bandwidth and volume displacement reveals an intrigu-ing result. Fig. 5 shows that there exist a power law relationshipbetween the two conflicting metrics as given by Eqs. (17) and (18),for unimorph and bimorph, respectively, by which one can estimatethe optimal volume displacement for a specified bandwidth,

dvolf2N = 2.60 × 105 (17)

dvolf2N = 4.35 × 105 (18)

Noting that fNdvol is proportional to the volume velocity per voltor the “gain” of the actuator at resonance, and fN is the bandwidth,we see that the optimization has essentially found the “constant”in the gain-bandwidth product. Furthermore, the constant is ∼1.67times larger for a bimorph than a unimorph.

From a design standpoint, the tradeoff curves obtained for differ-ent actuator radii (e.g., Fig. 4) can be used to estimate the achievablevolume displacement–natural frequency bounds empirically. Theseempirical expressions can be used to estimate the potential perfor-mance for a given actuator radius if the radius and available PZTpatch and shim thickness are within the range studied here.

For instance, consider a R3 = 12.5 mm actuator as a test case.This radius lies within the bounds of the radii considered but wasnot used to construct the fits. As a test, the Pareto curve limitswere estimated via Eq. (16), as shown in Tables 7 and 8, for uni-morph and bimorph configurations, respectively, and compared tooptimization results. Note that Figs. 8 and 9 include the optimiza-tion results as open symbols, which qualitatively agree with theempirical curves. A quantitative comparison between the predic-tion by the curves and the Pareto optimal design data via MATLAB isshown in Table 11. This example illustrates how customized designs

Actua

M. Papila et al. / Sensors and

Table 11Comparison of estimated versus Pareto optimal designs for R3 = 12.5 mm dvol(mm3/V) and fN (Hz)

(dvol)UB (fN)LB (dvol)LB (fN)UB

UnimorphPareto optimization 0.1018 1602 0.0152 4079Predictions by Eq. (16), Fig. 8, Table 7 0.1031 1586 0.0154 4058

BimorphPareto optimization 0.1740 1596 0.0159 5090Predictions by Eq. (16), Fig. 9, Table 8 0.1758 1586 0.0161 5063

for targeted volume displacement and natural frequency may berequested from a vendor using the empirical Pareto bounds formu-las.

A potential use of the bounding curves is now described. Sup-pose a designer seeks a unimorph actuator with a specified volumedisplacement dvol = 1 mm3/V. This corresponds to y = 0 in Fig. 8(a).The bounding curves in this figure show that the actuator radiusR3 lies between 22.2 and 35.2 mm, while Fig. 8(b) indicates thecorresponding natural frequencies for these limits will almostbe the same and approximately 500 Hz. A similar evaluation fordvol = 0.1 mm3/V (corresponding to y = −2.3 in Fig. 8(a) shows theactuator radius R3 lies between 12.4 and 19.9 mm, while the corre-sponding natural frequencies are approximately the same around1600 Hz. Note that the actuator radius R3 = 15 mm summarized in

Fig. 10. Pareto optimal design sequence.

(

tors A 147 (2008) 310–323 319

Table 5 is in the above range and dvol = 0.1 mm3/V is obtained whenthe natural frequency is also 1600 Hz (4th numerical column ofTable 5). These results suggest, for a specified volume displacementdvol, the designer can use the upper bounding curve in Fig. 8(a)because the cost of the smaller actuator is expected to be less. Analternative design method is to use Fig. 5(a) with Eq. (15), whichpredicts that when fN = 500 Hz, dvol = 1.034 mm3/V (3% error) andwhen fN = 1600 Hz, dvol = 0.101 mm3/V (1% error).

Finally, in the light of the above discussions, the design flowchartin Fig. 10 may be proposed, for which the following example isprovided:

(i) Select a unimorph configuration with fN = 1600 Hz(ii) dvol = 0.101 mm3/Viii) ts = (ts)LB = 0.100 mm

(iv) (R3)min − (R3)max = 12.4–19.9 mm1. (R3)min = 12.4 mm, tp = 0.134 mm, R1 = 10.9 mm2. (R3)max = 19.9 mm tp = 0.508 mm, R1 = 19.5 mm3. R3 = 15 mm, (fN)LB|R3 = 1087 Hz, fN/(fN)LB = 1.472,

tp = 0.250 mm, R1 = 14.2 mm

6. Concluding remarks

Clamped circular composite piezoceramic actuators were opti-mized for volume displacement at a prescribed bandwidth usingclassical laminated plate theory. Unimorph and bimorph configura-tions, including oppositely polarized PZT patches, were considered.The results indicate that the optimized bimorph configuration is anideal choice to maximize volume displacement when no constrainton bandwidth is imposed. As the bandwidth of the piezoceramicincreases, the performance advantage of the bimorph diminishes,and its added cost and complexity may not be warranted.

This research studied a range of actuator radii representative ofcommercially available products, and bounds for volume displace-ment and natural frequency of optimal designs were determined.The optimal volume displacement was found to be related to thebandwidth by a power law to provide a constant gain-bandwidthproduct, from which one can estimate the maximum static volumedisplacement for a specified bandwidth or vice versa. Design curvesand procedures to determine the required geometric dimensions ofthe prescribed actuator were provided.

Acknowledgment

The authors gratefully acknowledge financial support fromNASA Grant NNX07AD94A, monitored by Brian G. Allan and TheScientific and Technological Research Council of Turkey – TUBITAKGrant 106M364.

Appendix A

Analytical solutions for piezoelectric circular composite platesimplemented in the present optimization framework are sum-marized in the appendix. It is reporting supplementary effort ofon different circular piezoelectric plate configurations presentedearlier in Gallas et al. [5,6] and Prasad et al. [20]. The analyticalsolutions are now generalized for annular and inner disc unimorphand bimorph configurations, and study the combination of them byoppositely polarizing the inner and outer PZT patches.

The equilibrium equations (Timoshenko [39]) of the axisymmet-ric plates are

dNr

dr+ Nr − N�

r= 0, (A1)

Actua

320 M. Papila et al. / Sensors and

Qr = dMr

dr+ Mr − M�

r, (A2)

and

dQr

dr+ P + Qr

r= 0, (A3)

where Nr and N� are the force resultants in the radial and circumfer-ential directions, respectively. Similarly, Nr and N� are the momentresultants, and Qr is the transverse shear force resultant. The num-ber of equilibrium equations can be reduced to two by substitutingfor Qr from Eq. (A2) into Eq. (A3) to obtain

1r

ddr

(r

dMr

dr+ Mr − M�

)+ P = 0. (A4)

The radial and circumferential strain–displacement relation-ships from Kirchoff’s plate theory are

εrr = ε0rr + z�r (A5)

and

ε�� = ε0�� + z��, (A6)

where �r = −d2w/dr2 = −d�/dr and �� = −(1/r)(dw/dr) = −(�/r)are the radial and circumferential curvatures, respectively, and � isthe transverse deflection slope. The strains in the reference plane(z = 0) are

ε0rr = du0

dr(A7)

and

ε0�� = u0

r. (A8)

The constitutive equations for a transversely isotropic, linearelastic axisymmetric piezoelectric plate are{

�rr

���

}= [Q ]

({ε0

rr

ε0��

}+ z

{�r

��

}− Ef

{d31

d31

}), (A9)

where

[Q ] = E

1 − 2

{1

1

}, (A10)

E is the Young’s modulus, and is Poisson’s ratio. The last termin Eq. (A9) is due to the piezoelectric effect, where Ef = Vac/hp is theelectric field strength for an electrically conductive shim, and d31 is

the piezoelectric modulus. The explicit z-dependence of the mate-rial properties is understood and is omitted here for convenience.

The force and moment resultants are obtained by integratingthe constitutive equations through the thickness of the compositeplate, resulting in{

Nr

N�

}= [A]

{ε0

rr

ε0��

}+ [B]

{�r

��

}−{

NPr

NP�

}, (A11)

and{Mr

M�

}= [B]

{ε0

rr

ε0��

}+ [D]

{�r

��

}−{

MPr

MP�

}, (A12)

where

[A] =∫ z2

z1

[Q ]dz (A13)

is the extensional stiffness matrix,

[B] =∫ z2

z1

[Q ]zdz (A14)

tors A 147 (2008) 310–323

is the flexural-extensional coupling matrix, and

[D] =∫ z2

z1

[Q ]z2dz (A15)

is the flexural stiffness matrix. In the above expressions, z1 is the zlocation of the bottom and z2 represents the z location of the topof the plate. The reference plane was chosen to be at the middle ofthe shim material. We can rewrite Eqs. (A11) and (A12),

Nr = A11du0(r)

dr+ A12

u0(r)r

+ B11d�(r)

dr+ B12

1r

�(r) − NPr

N� = A21du0(r)

dr+ A22

u0(r)r

+ B21d�(r)

dr+ B22

1r

�(r) − NP�

(A16)

Mr = B11du0(r)

dr+ B12

u0(r)r

+ D11d�(r)

dr+ D12

1r

�(r) − MPr

M� = B21du0(r)

dr+ B22

u0(r)r

+ D21d�(r)

dr+ D22

1r

�(r) − MP�

(A17)

where the piezoelectric coupling generated force and momentresultants are given by[

NPr

NP�

]=∫ z2

z1

Ef[Q ]

[d31

d31

]dz, (A18)

and[MP

r

MP�

]=∫ z2

z1

Ef[Q ]

[d31

d31

]zdz. (A19)

The governing equations for a piezoelectric composite plate arederived by substituting for the force and moment resultants fromEqs. (A16) and (A17) into the equilibrium equations, Eqs. (A1) and(A4). Then the strain and curvature terms are replaced by displace-ment u0(r) and slope �(r) to obtain the two equilibrium equationsin terms of u0(r) and �(r) as

d2�(r)dr2

+ 1r

d�(r)dr

− �(r)r2

= − Pr

2D∗11

(A20)

and

d2u0(r)dr2

+ 1r

du0(r)dr

− u0(r)r2

= − Pr˛

2D∗11

, (A21)

where (˛ = B11/A11) and the reduced bending stiffness isD∗

11 = (D11 − B211/A11).

The general solutions to Eqs. (A20) and (A21) are obtained by

integration twice in radius r,

�(r) = b1r + b2

r− 1

D∗11

(Pr3

16

)(A22)

and

u0 = a1r + a2

r− ˛

D∗11

(Pr3

16

)(A23)

where a1, a2, b1 and b2 are constants to be determined by theboundary conditions of the plate problem.

The specific composite plate problem is solved by partition-ing the plate into three regions as in Fig. 2: central multi-layercomposite region, single layer homogenous ring region and outerfixed-boundary multi-layer composite annular region. The deter-mination of 12 constants (four for each region: a(j)

i, b(j)

i, i = 1, 2

and j = 1, 2, 3) requires boundary and interface matching condi-tions listed in Table A1 (Superscripts j = 1, 2, 3 denote the centralmulti-layer composite region, single layer homogenous ring regionand outer fixed-boundary multi-layer composite annular region,respectively):

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ions 1

M. Papila et al. / Sensors and

Table A1Boundary and interface matching conditions for composite plate

No. Boundary conditions, BC(m) Compatibility conditions, between reg

1 �(1)|r=0 < ∞ u(1)0 |r=R1 = u(2)

0 |r=R1

2 u(1)0 |r=0 < ∞ �(1)|r=R1 = �(2)|r=R1

3 u(3)0 |r=R3 = 0 N(1)

r |r=R1 = N(2)r |r=R1

4 �(3)|r=R3 = 0 M(1)r |r=R1 = M(2)

r |r=R1

The piezoelectric coupling is introduced by the piezoelectriccontributions to the integrated force, Eq. (A18), and moment, Eq.(A19), resultants in the matching or compatibility conditions atthe interfaces CC12(3) and CC12(4) for central piezoelectric regionand CC23(3) and CC23(4) for central piezoelectric region presentedin Table A1. Referring to general form of solutions in Eqs. (A22)and (A23), the rotation and radial displacements at the parti-tions of the plate structure can be found for each region byimplementing associated boundary conditions. Piecewise solutionsare

A.1. Central multi-layer composite region, i = 1 (0 ≤ r ≤ R1)

According to the boundary conditions of central plate, BC(1) andBC(2) of Table A1, b(1)

2 = a(1)2 = 0, then

�(1) = − Pr3

16D∗(1)11

+ b(1)1 r (A24)

u(1)0 = ˛(1) Pr3

16D∗(1)11

+ a(1)1 r (A25)

The expressions of Nr and Mr of the central plate become.

N(1)r = a(1)

1 (A(1)11 + A(1)

12 ) + b(1)1 (B(1)

11 + B(1)12 )

+ Pr2

16D∗(1)11

[˛(1)A(1)12 − B(1)

12 ] − NP(1)r (A26)

M(1)r = a(1)

1 (B(1)11 + B(1)

12 ) + b(1)1 (D(1)

11 + D(1)12 )

+ Pr2

16D∗(1)11

[˛(1)B(1)12 − D(1)

12 − 3D∗(1)11 ] − MP(1)

r (A27)

A.2. Ring region without fixed boundary, i = 2 (R1 ≤ r ≤ R2)

�(2) = − Pr3

16D∗(2)11

+ b(2)1 r + b(2)

21r

(A28)

u(2)0 = ˛(2) Pr3

16D∗(2)11

+ a(2)1 r + a(2)

21r

(A29)

and the expressions of Nr and Mr

N(2)r = a(2)

1 (A(2)11 + A(2)

12 ) + b(2)1 (B(2)

11 + B(2)12 ) + Pr2

16D∗(2)11

(3˛(2)A(2)11

+˛(2)A(2)12 − 3B(2)

11 − B(2)12 ) + a(2)

2

r2[A(2)

12 − A(2)11 ]

+b(2)2

r2(B(2)

12 − B(2)11 ) − NP(2)

r (A30)

tors A 147 (2008) 310–323 321

and 2: CC12(m) Compatibility conditions, between regions 1 and 2: CC23(m)

u(2)0 |r=R2 = u(3)

0 |r=R2

�(2)|r=R2 = �(3)|r=R2

N(2)r |r=R2 = N(3)

r |r=R2

M(2)r |r=R2 = M(3)

r |r=R2

M(2)r = a(2)

1 (B(2)11 + B(2)

12 ) + b(2)1 (D(2)

11 + D(2)12 ) + Pr2

16D∗(2)11

(3˛(2)B(2)11

+˛(2)B(2)12 − 3D(2)

11 − D(2)12 ) + a(2)

2

r2(B(2)

12 − B(2)11 )

+b(2)2

r2(D(2)

12 − D(2)11 ) − MP(2)

r (A31)

A.3. Annular plate with fixed boundary, i = 3 (R2 ≤ r ≤ R3)

�(3) = − Pr3

16D∗(3)11

+ b(3)1 r + b(3)

21r

(A32)

u(3)0 = ˛(3) Pr3

16D∗(3)11

+ a(3)1 r + a(3)

21r

(A33)

Substituting the boundary conditions BC(3) and BC(4) ofTable A1 into Eqs. (A32) and (A33) we obtain

b(3)2 = PR4

3

16D∗(3)11

− b(3)1 R2

3 (A34)

a(3)2 = −˛(3) PR4

3

16D∗(3)11

− a(3)1 R2

3 (A35)

Then, Eqs. (A32) and (A33) become

�(3) = − P

16D∗(3)11

(r3 − R4

3r

)+ b(3)

1

(r − R2

3r

)(A36)

(3) ˛(3)P(

3 R43

)(3)

(R2

3

)

u0 =

16D∗(3)11

r −r

+ a1 r −r

(A37)

From Eqs. (A36) and (A37), we can obtain the expressions for Nr

and Mr in terms of r,

N(3)r = a(3)

1

[A(3)

11

(1 + R2

3

r2

)+ A(3)

12

(1 − R2

3

r2

)]

+b(3)1

[B(3)

11

(1 + R2

3

r2

)+ B(3)

12

(1 − R2

3

r2

)]

+ Pr2

16D∗(3)11

[˛(3)A(3)12 −B(3)

12 ]− P

16D∗(3)11

R43

r2(˛(3)A(3)

12 −B(3)12 )−NP(3)

r

(A38)

M(3)r = a(3)

1

[B(3)

11

(1 + R2

3

r2

)+ B(3)

12

(1 − R2

3

r2

)]

+b(3)1

[D(3)

11

(1 + R2

3

r2

)+ D(3)

12

(1 − R2

3

r2

)]− MP(3)

r

Actua

[

[

[

[

[

[

322 M. Papila et al. / Sensors and

+ Pr2

16D∗(3)11

[˛(3)B(3)12 − D(3)

12 − 3D∗(3)11 ]

− P

16D∗(3)11

R43

r2(˛(3)B(3)

12 − D(3)12 + D∗(3)

11 ) (A39)

As a result, there remains eight constantsb(1)

1 , a(1)1 , b(2)

1 , b(2)2 , a(2)

1 , a(2)2 , b(3)

1 , a(3)1 . Next, we integrate the

expressions of these slopes to get the expressions of the deflectionw = −

∫�dr

w(1) = Pr4

64D∗(1)11

− b(1)12

r2 + b(3)2 (A40)

w(2) = Pr4

64D∗(2)11

− b(2)12

r2 − b(2)2 ln r + a(3)

2 (A41)

w(3) = P

16D∗(3)11

(14

r4 − R43 ln r

)− b(3)

1

(12

r2 − R23 ln r

)+ c(3)

(A42)

Using the boundary and matching conditions in terms oftransverse displacement (w(3)|R3 = 0, w(2)|R2 = w(3)|R2 , w(1)|R1 =w(2)∣∣R1

), we can determine the three constants: b(3)2 , a(3)

2 and c(3)

c(3) = − PR43

16D∗(3)11

(14

− ln R3

)+b(3)

1 R23

(12

− ln R3

)(A43)

a(3)2 = − PR4

2

64D∗(2)11

+b(2)12

R22+b(2)

2 ln R2+ P

16D∗(3)11

(14

R42−R4

3 ln R2

)

−b(3)1

(12

R22 − R2

3 ln R2

)+ c(3) (A44)

b(3)2 = − PR4

1

64D∗(1)11

+ b(1)12

R21 + PR4

1

64D∗(2)11

− b(2)12

R21 − b(2)

2 ln R1 + a(3)2

(A45)

Finally incorporating compatibility conditions due to CC12(m)

and CC12(m) for m = 1, . . ., 4 in Table A1 and Eqs. (A43)–(A45), theconstants can be determined and generalized the final expressionsof the transverse slope for the three parts are given as.⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

w(1) = Pr4

64D∗(1)11

−b(1)

12

r2 + b(3)2 0 ≤ r ≤ R1

w(2) = Pr4

64D∗(2)11

−b(2)

12

r2 − b(2)1 ln r + a(3)

2 R1 ≤ r ≤ R2

w(3) = P

16D∗(3)11

(14

r4 − R43 ln r

)− b(3)

1

(12

r2 − R23 ln r

)+ c(3) R2 ≤ r ≤ R3

(A46)

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Biographies

Melih Papila received the B.S. and M.S. degrees in aeronautical engineering from theMiddle East Technical University, Ankara, Turkey, in 1990 and 1995, respectively. Hereceived the Ph.D. degree in aerospace engineering from the University of Florida,Gainesville, in 2001, as a member of Multidisciplinary and Structural Optimizationgroup. He is an assistant professor in the Materials Science and Engineering Pro-gram at the Sabancı University (SU), Istanbul, Turkey. Prior to joining SU in 2004,he was a Postdoctoral Associate jointly in Interdisciplinary Microsystems and Mul-tidisciplinary and Structural Optimization Groups at the Department of Aerospaceand Mechanical Engineering of University of Florida, from 2002 to 2004. His cur-rent research focuses on the electroactive polymers and composites for sensorsand actuators, design and optimization of smart/composite materials and structures(http://people.sabanciuniv.edu/∼mpapila).

Mark Sheplak received the B.S., M.S., and Ph.D. degrees in mechanical engineer-ing from Syracuse University, Syracuse, NY, in 1989, 1992, and 1995, respectively.During his Ph.D. studies, he was a GSRP Fellow at NASA-Langley Research Center,Hampton, VA, from 1992 to 1995. He is an associate professor in the Departmentof Aerospace and Mechanical Engineering and an affiliate associate professor of

tors A 147 (2008) 310–323 323

Electrical and Computer Engineering at the University of Florida (UF). Prior to join-ing UF in 1998, he was a postdoctoral associate at the Massachusetts Instituteof Technology’s Microsystems Technology Laboratories, Cambridge, from 1995 to1998. His current research focuses on the design, fabrication, and characterizationof high-performance, instrumentation-grade, MEMS-based sensors and actuatorsthat enable the measurement, modeling, and control of various physical properties(http://www.img.ufl.edu).

Louis N. Cattafesta III is currently an associate professor in the Department ofMechanical and Aerospace Engineering at the University of Florida. His primaryresearch interests are experimental fluid dynamics, particularly active flow con-trol, and aeroacoustics, particularly airframe noise. Prior to joining UF in April of1999, he was a senior research scientist at High Technology Corporation in Hamp-ton, VA, where he was the group leader of the Experimental and InstrumentationGroup. He received a B.S. degree in Mechanical Engineering in 1986 from Penn StateUniversity, a M.S. degree in Aeronautics from MIT in 1988, and a Ph.D. degree inMechanical Engineering in 1992 from Penn State University. In 1992, he joinedHigh Technology Corporation as a Research Scientist at NASA Langley ResearchCenter. His research at NASA Langley focused on supersonic laminar flow con-trol and pressure- and temperature-sensitive paint measurement techniques. Atthat time, he became involved in active control of flow-induced cavity oscilla-tions, which provoked his current research interests in active flow control andaeroacoustics.