Natural frequencies of submerged piezoceramic hollow spheres

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submerged in a compressible fluid is presented in this paper. A separation method is adopted to simplify the basic equations for spherically isotropic ...
ACTA MECHANICA SINICA (English Series), Voi.16, No.l, Feb. 2000 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

N A T U R A L F R E Q U E N C I E S OF S U B M E R G E D PIEZOCERAMIC HOLLOW SPHERES* Cai Jinbiao (~.~b~)

Chen Weiqiu ( ~ . ~ $ )

Ye Guiru ( ~ ' i ~ )

Ding Haojiang ( - V ~ . )

(Department of Civil Engin.eering, Zhejiang University, Hangzhou 310027, China) An exact 3D analysis of free vibration ofa piezoceramic hollow sphere submerged in a compressible fluid is presented in this paper. A separation method is adopted to simplify the basic equations for spherically isotropic piezoelasticity. It is shown that there are two independent classes of vibration. The first one is independent of the fluid medium as well as the electric field, while the second is associated with both the fluid parameter and the piezoelectric effect. Exact frequency equations are derived and numerical results are obtained. ABSTRACT:

K E Y WORDS: piezoceramic sphere, spherical isotropy, coupled free vibration

1 INTRODUCTION The study of piezoelectric plates and shells has become important due to their wide application in industry engineering [1~3]. The vibration of piezoelectric shells in fluid media is of particular importance because some electromechanical devices are working principally in fluids [4'5] . Completely three-dimensional (3D) analysis of piezoelectric cylindrical shells filled with compressible fluid was recently conducted by Ding et al. [6]. There is no corresponding 3D exact analysis of coupled vibrations of piezoelectric spherical shells. The piezoelastic vibration of an empty spherical shell was exactly considered by Shul'ga et al.[7,s]. In particular, for the general nonaxisymmetric free vibratiQn, by introducing two displacement functions and two shear stress functions, Shul'ga Is] observed two independent classes of vibrations. It is noted that Shul'ga et al. [7,s] used a polynomial expansion method (PEM) to solve the resulting ordinary differential equations so that the solution will not be valid for a solid sphere. On the other hand, Cohen et al.[9] employed the Frobenius power series method (FPSM) to solve a second-order ordinary differential equation set for vibration analysis of spherically isotropic elastic hollow spheres. It is obvious that the rate of convergence of FPSM is much faster than that of PEM. Ding et al. [1~ further developed a generalized FPSM adopting the matrix formulations, which has shown great advantages in obtaining solutions for all different cases. In this paper, the general non-axisymmetric free vibration of a piezoceramic hollow sphere submerged in a compressible fluid medium is exactly analyzed by employing the separation formula for displacements only[ TM, in connection with the spherical harmonics expansion method. In particular, the resulting coupled set of three second-order ordinary Received 19 October 1999 * The project supported by the National Natural Science Foundation of China (No.19872060)

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differential equations is solved by employing the matrix FPSM. The effect of the ambient fluid medium is considered by introducing a relationship between the normal stress and radial displacement at the outer spherical surface. Exact frequency equations are then derived with numerical results presented. 2 THE SEPARATION

METHOD

Similar to the static analysis presented in Chen and Ding [11], three displacement functions w, G and ~ are employed to rewrite the mechanical displacement components (u~,uo,u,) in spherical coordinates (r, 0, r as follows uo --

1 0r sin0 0r

OG 00

0r ur - 00

10G sin0 0r

u~ = w

(1)

Utilizing Eq.(1), through some tedious manipulations, one obtains from the basic equations of a spherically isotropic piezoelectric body [s,111 A + r2pO2/Ot2G = 0

(2)

B - r2pO2/Ot2r = 0

(3)

2 02W

L3w- r p-~

(4)

- L4V~G + L s ~ = 0

(5)

L v w - L s V ~ G - Lg~ = 0 where ~ is the electric potential, p is the mass density, and A = LlW - L2G + L s ~

B = [c44V~ - 2c44 + Cll - c12+ ( C l l -- C 1 2 ) V 2 ] ~ )

L1 = (c13 + c44)V2 + Cll + c12 + 2c44

L2 = c44V] - 2c44 + Cll - c12 + cHV~

L3 = c33V ] - 2(cll + c12 - c13) + c44V~

L4

L5 = e33V 2 - 2e31V2 + e15~712

L6 -- (e15 + e31)V2 -t- 2e15

L7 = e33V~ +2e31V2 + 2e31 + e15V~

L8 = (e31 + els)V2 + e31 - e15

L9 = r

0 V2 = r-~r

+ elxV~

---- (C13 "~- C 4 4 ) V 2

0 0 V'~ = r~rr-~r

02 ~ V~ = ~ + cot0

V32 = V 2 + V2

-- E44 -- C l l -- C12 q-

1 + sin2 ~

02 0r 2

C13

/ (6)

where cij, eij and eij are the elastic, dielectric and piezoelectric constants, respectively. As in the static case, it is shown that the function r is uncoupled with the other two displacement functions w and G, and the electric potential 4f. 3 FREE VIBRATION 3.1

ANALYSIS

T h e G e n e r a l Free V i b r a t i o n P r o b l e m It is known that a closed spherical shell vibrates in a general nonaxisymmetric way. We thus assume

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Cai Jinbiao et al.: Coupled Vibration of Spheres = RU,~(~)Yn(O, r exp(iwt)

w = RWn(~)Y~(O, r exp(iwt)

G = RV,(~)Y,~(O, r exp(iwt)

57

]

4~ = (Re33/e33)Xn(~)Yn(O, r exp(iwt) ~

(7)

where Y~(O, r is the spherical harmonic function, w is the circular frequency, ~ = r / R is the nondimensional radial coordinate and R is the mean radius of the spherical shell. Substituting Eq.(7) into Eqs.(2)~(5), yields ~2U~[ + 2~U[~ + {/22~ 2 - [2 + (n - 1)(n + 2)(fl - f2)/2]}Un = 0

~2WnU -]- 2~WnI -[- (,(~2~2//4

(8)

J'2 ~ l l

l

- - k p l ) W n -- p 2 ~ V n -- p 3 V n q- q l ~ A n na q2~ X " "{- q 3 X n : 0

(9)

~2V.nt' -}- 2~V7~-{- (/22{ 2 q- p4)Vn -- p5~W'n -- p6Wn -b q4~X~ -k q s X n -- 0

(10)

~2 --n X " + 2~X~n + q6Xn - -, f:2W,, . n -pT~Wtn _ psWn -pg~V~ -p,oVn = 0

(11)

where a prime denotes differentiation with respect to ~, and p, = [2(fa - fl - f2) - n ( n + 1)]/f4 P3 = n ( n + 1)(fl + f2 + 1 - f 3 ) / f 4

P5 = f3 + 1

P4 = f l - f2 - n ( n + 1)fl - 2

P6 -- fl + f2 -b 2

P9 = n ( n + 1)(f5 + f6)

q2 = 2fs(1 - f 6 ) / f 4

P2 = - n ( n + 1)(f3 + 1)/f4

P7 = 2(f6 + 1)

Ps = 2f6 - n ( n + 1)f5

Plo = n ( n + 1)(f6 - f5) q3 = - n ( n + 1)fsfs/f4

ql = f s / f 4

(12)

q4 = - ( f 5 + f6)f8

q5 = - 2 f 5 f s

q6 = - n ( n + 1)f7

/2 = w R / v 2

f l = C11/C44

f2 = C12/C44

f3 "~ C13/C44

f4 :" C33/C44

f5 = e15/e33

f6 ----e31/e33

f7 = Ell/~33

f8 = e23/(E33c44)

Here v2 = ~ is the elastic wave velocity. Thus, for the free vibration problem, the basic equations have been turned to Eqs.(8).~(ll) in a dimensionless form. Equation (8) is an independent, second-order, ordinary differential equation in Un, while Eqs.(9)~(ll) are coupled with V,~,W,~ and X~, and each equation involved is a second-order ordinary differential one. 3.2 Solution to Eq.(8) Equation (8) is in fact identical to that for pure elasticity[9,12]; its solution is Un({) = ~-1/2[Bn14($2{) + B~2Yn(/2{)]

(n _> 1)

(13)

where Jn and Y~ are the first and second kinds of Bessel functions, respectively, Bnl and Bn2 are arbitrary constants, and ~/2__ 4[ 9 + 2 ( n 2 + n _ 2 ) ( f l _ f 2 ) ] > 0

(14)

3.3 Solution to Eqs.(9),--(11) It is obvious that ~ = 0 is a regular singular point of the coupled system (9)--~(11). To obtain the solution to this ordinary differential equation system, the matrix FPSM developed

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by Ding et al. [1~ is employed. Details are, however, omitted here and tile general solution finally can be expressed as the linear combination of six independent solutions as follows 6

wn(

) =

6

CnjWnj(

)

6

Vn( ) =

j=l

= j=l

j=l

where Cnj are arbitrary constants and W,,j, Vnj and X,,j are convergent, infinite series about the variable ~. It should be noted that n = 0 is a special case for which the function I~(~) contributes nothing to the piezoelastic field. In fact, Eqs.(9),~(ll) will become {2Wg' + 2~W; + (1/f4)(/22{ 2 + 2fa - 2/1 - 2f2)Wo+

(fs/f4)[(eX~'+

(2 - 2f6){X~] = 0

2 X.,,o + 2 { X ~ - { 2 W~. - ( 2 f 6 +

(16)

2){W~ - 2 / 6 W 0 = 0

(17)

We can" then use the matrix FPSM to obtain the following solution of Eqs.(16) and (17) 4

Wo({) = Z

4

Co, Wo~({)

Xo({) = E Co,Xo,({)

i=1

(18)

i----1

Having obtained the solutions, the stresses, electric displacements, mechanical displacements and electric potential can be expressed in terms of b~(~), Vn((), W~(~) and X,~(~). These are omitted for the sake of simplicity. 4 EXACT COUPLED

FREQUENCY

EQUATIONS

Now let's consider the free vibration of a piezoceramic spherical shell submerged in a compressible fluid medium. Supposing its inner and outer radii to be a and b, respectively, then we have the following boundary conditions at the two spherical surfaces

ar = 0

(r=a)

=

r,e=r~r au./at

=

(r=a,b) (r = b)

(19) (20)

where PI and uI are the dynamic pressure and velocity of the fluid. Chen [12] has shown that condition (20) could be transformed into the following one

ar = -pfcocfh,~(kr)ur

(r = a; n = 0, 1, 2 , . . . )

(21)

where k = W/cl,p] and cf are respectively the density and sound velocity of the fluid, ha(x) = xh~2)(x)/[nh~ ) (x) - ;Ct~nq_ _~(2) 1 (x)], and h(2)(x) is the second kind of spherical Hankel function. It is noted that the electric condition at the spherical surfaces can also be treated as the shorted electrode-covered one as employed by Shul'ga [81. In a similar manner as in elasticity [9J2], one can find that the coupled free vibration of a submerged piezoceramic spherical shell is also separated into two independent classes. The first class is identical to the corresponding one for an empty, elastic spherical shell[ 9a2],

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while the second changes because of the piezoelectric effect and the coupling effect due to the surrounding fluid. Consequently, from the boundary conditions (19) and (21), one can derive two sets of linear homogeneous algebraic equations of undetermined constants Bni and C,u, respectively. For nontrivial solutions to exist, the coefficient determinants of the two systems should vanish so that the corresponding.frequency equations are obtained. Since the first class of vibration has been discussed in detail previously[ 9,12], we will thus focus on the second class only, of which the frequency equations are given in the following. When n >__1, one obtains six homogeneous linear algebraic equations and the following frequency equation can be derived

lEVI =

0

(i,j = 1 , 2 , . . . ,6; n > 1)

(22)

where

E~i = n(n + 1)Vni(~l)/~l + 2Wn~(~1)/~1 + (Y41/3)W'~ff~) + (fs/f3)Xlni(~l) l E~,

V~.~(~)/~ + V,.(~)/~I - Vg~(~) + AY~X,.(~I)/~i

E~ ~(~ + 1)y6V,.(~)/~I + 2y6W,.(~l)/~I + W:.(~I) - X,.(~)' E~ n(n + 1)V~(~)/~2 + 2W,.(~)/~2 + (Y4/A) W'.~(~2)+ (fs/f3)

ni(~2) + x'

F,~(~2)W,.(~2)/f3 E~ = W,,i(~)/~2 + V,.(G)/~ - V'~(~2)+ hfsZ,.(~2)/~2 E3~

I

(23)

I

n(,~ + 1)f6V,.(~)/~ + 2 f 6 W , . ( ~ ) / ~ + W'~(~:) - Z'~(~:) (i = 1,2,...,6)

and ~1 = a i r = 1 - e/2, ~2 = b/R = 1 + el2, R = (a + b)/2, where e = (b - a ) / R is the thickness-to-mean radius ratio of the shell, and

Fn(x) = pocof2hn( ~ x /co)

(24)

where Po = Pf/P and co = cf/v2 are density and velocity ratios between the fluid and shell, respectively. For n = 0, further investigation is needed. It is first noted the four indicials appearing in the matrix FPSM [1~ can be obtained precisely as follows

81,2

~ . / f 4 - 8(f3 - fl - f2) + fs(4f6 - 1) 2 4(f4 + Ys)

V

s3.4 = ~:1/2

(25)

We then can verify that the solution corresponding to s4 will give no contribution to the stresses and electric displacements so that we have Co4 = 0 in Eq.(18). One can further demonstrate that the radial electric displacement component Dr corresponding to either sl or s2 will be zero. It makes the two boundary conditions Dr(a) = 0 and Dr(b) = 0 identical and both give C03 = 0. Two linear homogeneous equations in unknowns Col and C02 are finally obtained from the remaining two boundary conditions. The vanishing of the coefficient determinant gives the following frequency equation

IE~ = 0

(i,j = 1,2; n = 0)

(26)

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where

E~

=

x'

2W0i(~2)/~2 + (fdS)W~i(~2) + (Is~5) o~(~2)+ Fo(~2)Wo~(~D//3

~ (i = 1, 2) (27)

Note that frequency Eq.(26) corresponds to the purely radial vibration (breathing mode). It is noted here that, if the fluid is incompressible, the above frequency equations keep unchanged, but Eq.(24) will read

(28)

F,~(x) = -po92X/(n + 1)

It is also noteworthy to point out that the analysis for shorted electrode-covered condition is similar and two classes of vibration can also be observed. In fact the first class will be exactly the same as that for the condition of zero normal electric displacement that we have adopted above. Nevertheless, the second class will be different. 5 NUMERICAL

CALCULATION

When the outside fluid is compressible, the frequency equations will be of a complex form due to the inclusion of the second kind of spherical Hankel ~function, h(~2) (x). Such complex transcendental equations should be solved using special techniques. For the sake of simplicity, only the case that the fluid is incompressible will be considered in the following. In this case, the frequency equations will be real and any conventional iterative algorithm can be adopted to obtain the frequency. Since the frequency Eqs.(22) and (26) are three dimensional, there are an infinite number of frequencies. In what follows, only the smallest positive natural frequency that is of practical significance will be given in the calculation. The piezoelectric material of the spherical shell is taken to be PZT4, of which the nondimensional material constants fi can be found in Ref.[13]. The frequency spectra for the breathing mode (n = 0) as well as for the non-breathing mode (n = 2), are displayed in Figs.1 and 2, respectively, for several values of the density ratio Po. 4.0

2.0

3.5~ ~ - ~ ~ 3.of j ~ _ -~-~--~-~--~

1.8 ~ 1.6 1.4 1.2 1.0

1.5~//~/x~1.0 /

0.2 - - ~ 0.6 [] 0.4 - v - 1

0.1 0.3 0.5 0.7 0.9

1.1 1.3 1.5

e

Fig.1 Curves of the nondimensional frequency ~2 versus the thickness-tomean radius ratio e for the breathing mode (n = 0)

~/'~

[] 0.4 ~ 0.6 1

0.8

0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 e

Fig.2 Curves of the nondimensional frequency /2 versus the thickness-tomean radius ratio e for the nonbreathing mode (n = 2)

It can be seen that the lowest natural frequency of an empty spherical shell (po = 0) is always greater than the corresponding one of a submerged spherical shell. It is known

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as the "added mass effect" that has been reported extensively[ 14]. From Figs.1 and 2, one finds that the nondimensional frequency ~ decreases with P0. In addition, the variation of the nondimensional frequency is not monotonic with the thickness-to-mean radius ratio e. The piezoelectric effect on the natural frequencies of a submerged spherical shell is investigated by comparing results for PZT-4 material with those for the corresponding elastic material, PZT-4(E). The elastic constants of PZT-4(E) are the same as those of PZT-4, but the piezoelectric effect is ignored. The density ratio and the thickness-to-mean radius ratio are taken to be P0 = 0.5 and e = 0.5, respectively. The lowest natural frequencies for both cases are given in Table 1. It can be seen t h a t the lowest natural frequencies of a P Z T - 4 spherical shell are usually higher than those of the corresponding elastic shell except when n = 2. This observation is of particular importance because the lowest frequency of a spherical shell usually occurs at the mode number n = 2. T a b l e 1 T h e piezoelectric effect on natural frequencies of a s u b m e r g e d spherical shell n PZT-4 PZT-4(E)

0 2.37180 2.12741

1 3.73904 3.34975

2 1.34427 1.39125

3 2.29145 2.19279

4 3.40865 3.09347

5 4.58415 4.03974

6 5.73596 4.98553

6 CONCLUSION The basic equations of a vibrating spherically isotropic piezoelectric medium were simplified through the introduction of three displacement functions in the paper. For the general nonaxisymmetric free vibration problem, the equations were further reduced to an uncoupled second-order ordinary differential equation, and a coupled system of three second-order ordinary differential equations. Solution to the coupled differential system was obtained by means of the matrix FPSM. Exact frequency equations were presented for a piezoceramic spherical shell submerged in a compressible fluid medium. Numerical calculation was then carried out for the second class of vibration when the outside fluid is incompressible. Discussion on the effects of some involved parameters was also presented in the paper. The present method can also be used to analyze piezoelectric solid spheres and laminated spherical shells. The procedure is similar to that for the elastic case that has been described in Ref.[]2] in detail. REFERENCES

1 Tiersten HF. Linear Piezoelectric Plate Vibrations. New York: Plenum Press, 1969 2 Knarouf N, Heyliger PR. Axisymmetric free vibrations of homogeneous and laminated piezoelectric cylinders. J Sound Vib, 1994, 174:539~561 3 Chen WQ, Xu RQ, Ding HJ. On free vibration of a piezoelectric composite rectangular plate. J Sound Vib, 1998, 218:741~748 4 Borisyuk AI, Kirichok IF. Steady-state radial vibrations of piezoceramic spheres in compressible fluid. Soy Appl Mech, 1979, 15:936,,~940 5 Babaev AE, But LM, Savin VG. Transient vibrations of a thin-walled cylindrical piezoelectric vibrator driven by a nonaxisymmetric electric signal in a liquid. Soy Appl Mech, 1990, 26: 1167,~1174

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6 Ding H J, Chen WQ, Guo YM, Yang QD. Free vibrations of piezoelectric cylindrical shells filled with compressible fluid. Int J Solids Struct, 1997, 34:2025~2034 7 Loza IA, Shul'ga NA. Axisymmetric vibration of a hollow piezoceramic sphere with radial polaxization. Soy Appl Mech, 1984, 20:113,-,117 8 Shul'ga NA. Harmonic electroelastic oscillations of spherical bodies. Soy Appl Mech, 1993, 29, 812~817 9 Cohen H, Shah AH, Ramakrishnan CV. Free vibrations of a spherically isotropic hollow sphere. Acustica, 1972, 26:329~333 10 Ding H J, Chen WQ, Liu Z. Solutions to equations of vibrations of spherical and cylindrical shells. Applied Mathematics and Mechanics, 1995, 16:1,-,15 11 Chen WQ, Ding HJ. Exact static analysis of a rotating piezoelectric spherical shell. Acta Mechanica Sinica, 1998, 14:257,-,265 12 Chen WQ. Coupled free vibrations of spherically isotropic hollow spheres. [Ph.D. dissertation], Zhejiang University, 1996 (in Chinese) 13 Chen WQ. Problems of radially polarized piezoelastic bodies. Int J Solids Struct, 1999, 36: 4317~4332 14 Junger MC, Feit D. Sound, Structures and Their Interaction. Cambridge: MIT Press, 1972