Flexural–extensional behavior of composite piezoelectric circular plates

Flexural-extensional behavior of composite piezoelectric circular plates

N. T. Adelman a) and Y. Stavsky Department of Aeronautical Engineering, Technion, Israel Institute of Technology, Haifa, Israel (Received 28 August 1978; accepted for publication 10 August 1979)

A plate-type theory is developed for the flexural-extensional vibratory response and static voltage deformation of heterogeneous piezoelectric circular transducer elements. Known results for homogeneous disks and bimorphs are shown to be special cases of the theory. Application is made to the design of simply supported metal-piezoceramic unimorph disks, and thin piezoceramic bimorph benders possessing metallic electrodes of non-negligible thickness.

PACS numbers: 43.40.Dx, 85.50.Ly, 43.88.Fx

INTRODUCTION

Electroacoustic devices utilizing circular piezoelectric components include bandpass filters, headphones, microphones, and loudspeakers. • Depending upon the application, various constructions are employed such as homogeneous piezodisks, 2 metal-ceramic unimorphs, a and oppositely polarized bilaminar benders or bimorphs, 4 giving rise to either radial, thickness, or flexural responses. The purpose of this communi- cation is to describe a theory for heterogeneous piezoelectric circular plates which includes, as special cases, all of the above-mentioned constructions and responses. The localized electro-elastic governing equations are converted to globalized plate equations which directly describe the geometry of the media while indirectly accounting for any local variations in composition, polarization, etc. This has the advantage of focusing attention on the overall flexural-extensional behavior of the plate rather than on the motion of each individual particle. Known solutions for the radial vibrations of freely supported homogeneous disks, as well as for the flexural vibrations and static voltage displacements of simply supported bimorphs, are shown to be special cases of the theory. Design curves are provided for the static voltage response of metal-piezoceramic unimorph disks as functions of geometry and material properties. Numerical results are also given for the deviation in resonant frequency and static voltage displacement of simply supported, series, and parallel connected, PZT-5H bimorphs possessing silver electrodes of non-negligible thickness. Such informa- lion is relevant to the design of thin piezoelectric bimorphs for use in wide range audio loudspeakers. 5

I. COMPOSITE PLATE ANALYSIS

A. Vibratory response

Consider a thin circular plate of radius a and thickness h, composed of n axially polarized piezoceramic layers as shown in Fig. 1 for n = 2. Major faces of the plate are traction-free, are completely coated with electrodes, and are connected to an alternating voltage source of magnitude V and circular frequency co. The

a)Passed away on 14 March 1978.

field equations governing the behavior of each piezoceramic layer have been treated in Ref. 2 under the assumption that both the normal stresses T• and the shear stresses Tr• are negligibl• small. Removing the latter restriction, the axisymmetric equations of motion and •ssociated constitutive relations are

Trr. • + T•. • + (Tr• - Too)/r = 9•t• , (la)

Try, r + Tr• /r = pii z , (lb)

(2a)

Too =U•2u• • + U•u• /r + [a•(V/h)e •øJt (2b)

(2c)

In the above T• are the components of stress, u• the components of displacement, U• the modified elastic moduli, •}• the modified piezoelectric coefficients, 9 the mass density, and a comma or a dot denotes partial derivatives with respect to a spaeial coordinate or time, respectively. Assuming a steady-state response of angular frequency co, and employing the classical kine- matic relations for a thin circular plate

' o(r)]e•t (3a) u•(r,z,t)=[U•o(r)-zu, ,

u• (r,z, t) =U,o(r)e•* , (3b) Eqs. (2a, b) become

= ' -u") •'o)/r+- V/h (4a) Trr •(Uro •o + U•2(Uro--ZU es• , !

Too=U•2(Urto-ZU•t'o)+U•i(Uro-ZU•o)/r+•3•V/h , (4b)

where U.o, U.o are the displacements at the reference surface z = 0, a prime denotes a total derivative with respect to r, and the harmonic time dependence exp(icot)

I z T

FIG. 1. Cross section of composite piezoceramic circular plate.

819 J. Acoust. Soc. Am. 67(3), Mar. 1980 0001-4966/80/030819-04500.80 ¸ 1980 Acoustical Society of America 819

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has been suppressedø

The localized equations of motion and constitutive equations may be converted to globalized plate equations by integrating over the thickness in accordance with Boussinesq's method. 6 Defining the stress resultants, elastic moduli, and piezoelectric coefficients, by the following integrals, respectively:

(Nr ,No, Qr ) = (T,.,. , Too, T,.•) dz, (5a)

•o h (M, ,Me)= (%,, Too)z az, (5b)

from the boundary conditions, and the •'s are obtained from the roots of the cubic equation

aota+a•t2+a2t +aa=0 , (9a)

where

t = (/•a) •- , (9b)

ao= 1 - bd, a• = - (eta) 2, a 2 = - (/Ja) 4, a a = (ota)a(/ja) 4 , (9c)

b=B•/A•, d=B•/D• , (9d)

•4 = RoCO2/D• •2: Roco2/A l• , . (9e)

•o h (no, n,n3 = z, z-)p (5c)

(A• ,B• ,Dis )= (1,z,z•')U• dz (i,j = 1, 2) , (5d)

•o h (E •, E•.) = (1, z)•a• dz . (5e)

Choosing the reference plane for the integrals in Eqs. (5) at a distance

h h he = pz dz / p dz , (6)

measured from the lower face of the plate, R•, vanishes identically. Multiplying the localized constitutive relations, Eqs. (4), alternately by (1,z), and integrating over the plate thickness, leads to globalized plate constitutive relations -

E

V (7)

It is noted that the piezoelectric terms in the above relations are of significance when treating nonresonant phenomena. Performing similar operations of the equations of motion, Eqs. (1), the resulting expressions were found to be independent of E•, E•. and therefore identical to the corresponding equations derived by Staysky and Loewy ? for nonpiezoelectric media. The solutions for the mechanical displacements, in terms of the Bessel function d and modified Bessel functions

I, are

2 •z G•.J• (t•. r) giro---- --b 2 • 2 alJl(•l •') +'0/2 -- •2 2

L(m r) , (8a) + 2 2 3 3

U,o= - G•Jo(•r)/• - G•.Jo(iZ•.r)/Iz•. + Galo(rnar)/rn a . (8b)

For homogeneous or symmetrically layered plates, the appropriate solutions of mid-plate displacements are

U•o= G•.3 •(ar) , (10a)

o = - - o.Jo(ar)/a + . (0b)

Resonant eigenfrequeney equations may be obtained by applying the relevant boundary conditions at r = a and re- quiring that the determinant of the coefficients of the G• constants vanish.

B. Static behavior

Mechanical displacements caused by an applied static voltage may be obtained from the governing equations derived in Ref. 7 by setting co =0. For this case, any reference may be conveniently chosen since the governing equations are independent of the mass density. Solutions which are nonsingular at the origin were found to be

Uro=H • r, U•o= H•.r •' + Ha ß (11)

II. HOMOGENEOUS DISKS AND BIMORPHS

The general theory developed above is here shown to include, as special cases, the known results for homogeneous disks and bimorphs. The plate constants and parameters evaluated from Eqs. (5), (6), and (9) for homogeneous plates are

A•i=U•h, D• =U•h3/12 (i=1,2), (12a)

B•=B•.=0, R o=h, (12b)

E• E•.={ e-a• h, O, for disks ' O, Fa•h•'/4, for bimorphs,

ot = (p/Uzz)z/2co, [j= (12pco2/Uzzh2) •/4

(12c)

(12d)

Treating first traction-free disks, the boundary condition

N• (a) = 0, (13)

applied to Eq. (10a), leads to the radial mode resonant frequency condition

aaJo(aa)/J•(aa) = 1 - (U•./Un) , (14)

In the above, the G's are constants to be determined in agreement with Meitzler, O'Bryan, and Tiersten. •- In

820 J. Acoust. Soc. Am., Vol. 67, No. 3, March 1980 N. T. Adelman and Y. Stavsky: Piezoelectric circular plates 820


addition to the radial mode of vibration there also exists a thickness mode [as indicated by Eq. (10b)] which, however, will not be treated here. The flexural vibratory response of simply supported bimorphs may be found by applying the boundary conditions

U,.o(a)=Ueo(a)=M,.(a)=O , (15)

to the governing Eqs. (10). This results in the resonant condition

d•(tJa)/do(i•)+I • (13a)/Io(13a)= 2/3a/(1 - •'•./•'•0, (16)

which was given by Leissa 8 for nonpiezoelectric plates and was found by McElroy et al. 9 to apply equally well for piezoelectric bimorphs.

The mechanical displacement of a simply supported plate, due to an applied static voltage, can be obtained by applying the boundary conditions, Eqs. (15), to the governing Eqs. (11) with the result that

=0, u =[VE2(r2-a2)]/[2h(D• +D•2)]. (17) /gr 0 ZO 1

Substituting the appropriate values from Eqs. (12) for a bimorph, and noting that the piezoelectric coefficient

d31 = •31/(•11 q- •12) , (18)

the maximum flexural displacement at r = 0 is of magnitude

(ggZo)max - d31 a2V/2h 2 , (19) as previously found by Germano. •ø

• •.s

'-

o

o 0.2 o.i. 0.6 0.8

hllh lO

o N

:D

5

3

2

1

o o

• C 11 + C 12

I

0'.4 0'.6 0.8 1.0 h11h

FIG. 2. Static voltage displacement of metal-piezoceramic unimorphs as a function of geometry and material properties- (a) base simply supported, (b) junction simply supported.

III. NUMERICAL EXAMPLES

A. Metal-piezoceramic unimorphs-static behavior

One of the basic static voltage control elements useful for the generation of large mechanical displacements and low forces, is the metal-piezoceramic unimorph described by Berlincourt. • These devices are construc- ted in the form of bilaminar plates (Fig. 1), such that expansion of the ceramic component, due to an applied field, results in a primary flexural response. Efficient design of these elements requires determination of the optimal thickness ratio h•/h which will provide the maximum displacement for a given material combination. This problem is here analyzed by application of the composite plate theory developed above.

The ceramic and metal portions of the plate are as- sociated with regions I and 2, respectively, and will be referenced by employing the appropriate subscript or superscript on all material constants and field quanti- ties. Two types of support conditions are considered' (a) base simply supported, and (b) junction simply supported. The solution for the static voltage flexural displacements may then be obtained from Eq. (17) by pro- per evaluation of the plate constants from Eqs. (5d, e). Defining the nondimensional parameter

--(2) --(2) --(t) --(t) ) (c• + c•,. )/(c• + c•. , (20)

the results of these calculations are plotted in Fig. 2. It

is seen that the response is highly dependent on the lo- cation of the support, being nominally twice as large for the junction-supported case as for the base support. The maximum displacement of a symmetric (h•/h = 0.5), homogeneous (r/= 1), junction-supported unimorph is identical to that of a bimorph as given by Eq. (19). Much larger displacements can be obtained with the unimorph, however, by choosing a material combination having r/< I and selecting the optimal thickness ratio from Fig. 2. For example, a junction-supported unimorph of r/= 0.1 and h/h = 0.35 will provide a response approximately two and one half times the value given by Eq. (19).

B. PZT-5H biomorphs possessing silver electrodes

Hugo Schafft s has recently discussed the use of circular piezoelectric bending transducers for wide range audio loudspeakers. He notes that for useful sound out- puts, transducer capacities in the microfarad range are required which can be achieved by bimorphs ranging from 2 to 4 in. in diameter and from 2 to 6 mil in

thickness. The response of such thin bimorphs, however, will be influenced to some degree by the pres- ence of electrodes whose thicknesses are non-negligible. To investigate the extent of this influence, the response of simply supported, series, and parallel connected, PZT-5H bimorphs possessing silver electrodes is here treated. The central electrode of the parallel unit was

821 J. Acoust. Soc. Am., Vol. 67, No. 3, March 1980 N.T. Adelman and Y. Stavsky' Piezoelectric circular plates 821


TABLE I. Selected material properties of PZT-5H and silver.

Physical property PZ T- 5H Silver

C•l(N/m 2 x10 lø) 6.52 7.24 C•2(N/m 2 x10 lø) 1.82 2.68 e31(C/m 2) 23.4 0 p(kg/m 3 x 103) 7.5 10.5

chosen to have twice the thickness of either outer electrodes; thus, for a given percentage of PZT-5H, the electrode thickness of the parallel unit is half that of the series type. Selected material properties of PZT- 5H and silver are given in Table I.

The fundamental resonant frequency and static voltage displacements of a PZT-5H bimorph as obtained

100 (a)

•- 98 •.• z

z

z 9• 0

•2

•OI • 0

SERIES

CONNECTED

•i SILVER

• P ZT -5H

PARALLEL

CONNECTED

2'0 • ' ' 6'0 40 80 100

% PZ T- 5H

lOO

"' 80

60

o

• 20

(b) _

SILVER

PZT - 5H

PARALLEL

CONNECTED

SERIES

CONNECTED

0 20 40 60 80 100

*/, PZT-5H

FIG. 3. Deviation of (a) resonant frequency and (b) static voltage displacement with composition of PZT-5H/silver bimorphs.

from Eqs. (16) and (19) are

w, =4.17 h/a •' (kHz) , (21)

and

•g)max -' 0

Deviations from these values, with composition for considered PZT-5H/silver bimorphs, are depicted in Fig. 3. It is observed that a decrease in the percentage of piezoceramic material generally leads to a corresponding decrease in both resonant frequency and static voltage displacement. For the series-type connection in the region above 85% PZT-5H composition, however, the resonant frequency is actually larger than the value indicated by Eq. (21). As a specific example, consider bimorphs of 5 rail overall thickness whose

1

electrodes are z-mil thick. For the parallel-connected unit (80% PZT-5H), the resonant frequency is found to drop 1.7% with a significant 24% reduction in the static voltage displacement. The series connected bimorph (90% PZT-5H) shows a 0.2% increase in resonant frequency and a 23% reduction in static voltage displacement.

ACKNOWLEDGMENTS

This study was supported by the Technion Research Funds.

4.2 a•'V/h"'(10 -'øm), for series connection,

(22a)

8.4 a2V/h2(lO-•øm), for parallel connections.

(22b)

1j. van Randeraat, Piezoelectric Ceramics (Philips Gloeilampenfabrieken,Eindhoven, The Netherlands, 1968).

2A. H. Meitzler, H. M. O'Bryan, Jr., and H. F. Tiersten, "Definition and Measurement of Raidal Mode Coupling Fac- tors in Piezoelectric Ceramic Materials with Large Varia- tions in Poisson's Ratio," IEEE Trans. Sonics Ultrason. SU-20, 233-239 (1973).

3W. J. Denkmann, R. E. Nickell, and D.C. Strickler, "Analy- sis of Structural-Acoustic Interactions in Metal-Ceramic Transducers," IEEE Trans. Audio Electroacoust. AU-21, 317-324 (1973).

4C. P. Germano, "Flexure Mode Piezoelectric Transducers," IEEE Trans. Audio Electroacoust. AU-19, 6-12 (1971).

•I-I. Schafft, "Wide Range Audio Transducer Using Piezoelec- tric Ceramic," Ferroelectrics 10, 121-124 (1976). 6j. Boussinesq, "•2tude Nouvelle sur l'l•quilibre et la Mouve-

ment des Corps Solides Elastiquest dont Certaines Dimen- sions sont Tr$s-Petites par Rapport • d'Autres," J. Math. Paris Ser. 2, 16, 125-274 (1871); Ser. 3, 5, 329-344 (1879).

?Y. Stavsky and R. Loewy, "Axisymmetric Vibrations of Iso- tropic Composite Circular Plates," J. Acoust. Soc. Am. 49, 1542-1550 (1971).

aA. W. Leissa, "Vibration of Plates," NASA SP-160, 9 (1969). 9j. H. McElroy, P.E. Thompson, H. E. Walker, E. H. John-

son, D. J. Radecki, and R. S. Reynolds, "Laser Tuners Us- ing Circular Piezoelectric Benders," Appl. Opt. 14, 1297- i302 (i975).

løC. P. Germano, "Useful Relationships for Circular Bender Bimorphs," Vernitron Tech. Paper TP-230 (1965).

riD. Berlincourt, "Current Developments in Piezoelectric Applications of Ferroelectrics," Ferroelectrics10, 111- 119 (1976).

822 J. Acoust. Soc. Am., Vol. 67, No. 3, March 1980 N.T. Adelman and Y. Stavsky' Piezoelectric circular plates 822


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Flexural–extensional behavior of composite piezoelectric circular plates