Dij dn

C2d31 «31

£1.1. = —TT-, £2n* = -nv„ - </*'„,, w,„ = —— +/7H,„. (2S.6)

The numbers a, A, ci2, t^i, and i/2 are taken from Section 27.

The dimensionless quantities with asterisks are introduced by formulas 27.5, to which we should add c2 D2 ¿"2* = -¿'2, £>2*

The initial equations are membrane. This means that we do not consider the vibrations described using the moment equations of the theory of piezoceramic shells. We can do this because, as it was shown in Scction 27, the frequency parameters A, for which the problem cannot be broken into the principal problem, and auxiliary problem correspond to the EMCC values close to zero.

The effect of the electrical load at the edges of the shell is like that of a longitudinal load, applied to the edges. For low frequencies of the load, the shell will perform quasitransverse vibrations, while for A >> 1 the vibrations are quasitangential. The quasitransverse vibrations with small variability are described by the membrane system 28.1 to 28.6 for q = 1. The equations for the quasitangential vibrations are obtained from the same system for q = 0.

We solve system 28.1 to 28.6 with respect to u„t,v„t, i/w

if4«22

qa 1

sf4»22

LJ44C2

"33

"33

£33i44"22

i,f4e33"22 .

Let us introduce an unknown function <1\ defined as

+ bi

ei^.+e 3

¿4¿

The quantities «„„, v,„, ?/>„, are expressed in terms of as u„t = D|4>, = D2<I>, I',,» = £>3<i>

For the nth expansion of V*» 'he following electrical conditions should hold at the edges:

For definiteness, we consider the case when there are two electrodes at each edge:

^.|ia±(/r = ±r„, r„ = — (1 -(-»)"). (28.11)

We will also assume that the shell edges are not clamped:

As in Section 27, we compute the EMCC using different formulas 27.10 to 27.13. The values of k and kt( are found directly, after the solution of the initial problem has been found. In order to find kt, we should solve two additional problems.

PROBLEM 1

Suppose that the electrodes are disconnected. Since every electrode is an exponential surface, its electrical potential is an unknown constant. In our problem with two electrodes at every edge and the same mechanical conditions at the edges £ = ±l/r, it is sufficient to introduce one unknown constant c:

In order to find V'»", we integrate equation 28.5 under the assumption that the strains are known. It is more convenient to determine ?/',','» by writing equation 28.5 in the form iUde+d\

We know functions v,„ and from the solution of the initial problem. The arbitrary integration constants in equation 28.14 are found from the conditions 28.13.

Having found all the terms of the trigonometric series for i/ij", we substitute the resultant series into the integral condition 27.14. By integrating the coordinate ip, we find c.

The energy U^ is computed by the formula

where

C2dy\

PROBLEM 2

The shell electrodes are short-circuited. In order to find ij>[sh), we must find the solution to equation 28.14; change the superscript {d) for (.s/i); and satisfy conditions 27.15 on the electrodes. The energy i/''1' is found from 28.15 and 28.16, where we put a = sh.

We calculated the first natural frequencies for different shell lengths and placed them in Table 3. The first column gives the shell's length divided by its radius. The second column gives the resonance frequency. The third gives the antiresonance frequency. The remaining columns give the EMCC values computed using different formulas. The superscripts (r) and (a) show that the values were computed for the resonance and antiresonance frequencies respectively.

TABLE 3.

Resonance and amiresonance frequency parameters and the corresponding EMCC's as functions of the shell length (nonaxisymmetric problem)

TABLE 3.

Resonance and amiresonance frequency parameters and the corresponding EMCC's as functions of the shell length (nonaxisymmetric problem)

l/r

A<r>

A<">

*<■>

*<«>

2.07

2.24

0.282

0.370

0.206

0.060

0.275

o.so

7.00

7.04

0.071

0.118

0.072

0.267

0.282

9.86

10.80

0.310

0.401

0.200

0.268

0.359

12.34

12.44

0.086

0.106

0.069

0.422

0.424

1.91

2.10

0.300

0.394

0.218

0.110

0.305

0.75

4.63

4.76

0.166

0.238

0.108

0.278

0.319

5.87

5.27

0.135

0.163

0.106

0.420

0.431

7.87

8.88

0.336

0.402

0.270

0.154

0.399

1.64

1.82

0.312

0.410

0.232

0.265

0.388

1.00

3.13

3.19

0.042

0.078

0.056

0.264

0.271

4.48

4.53

0.110

0.168

0.060

0.471

0.475

5.33

5.93

0.317

0.377

0.263

0.128

0.382

1.37

1.52

0.325

0.398

0.258

0.283

0.426

1.25

2.68

2.70

0.086

0.089

0.074

0.229

0.230

4.07

4.20

0.171

0.341

0.050

0.111

0.428

4.26

4.34

0.250

0.165

0.235

0.502

0.453

As in Section 27, the ¿^-values are in good agreement with k,i averaged over the interval [A(r), A'"']; k assumes quite different values in the same interval. Figures 8 to 10 show the needed quantities as functions of coordinates for a shell with length 2r. Figure 8 gives the displacements, forces, and electrical quantities for the first resonance; Figure 9 gives the same quantities for Ihe second resonance; and Figure 10 gives ihe quantities for the third resonance. Each dimensionless quantity in Figures 8-10 is divided by the same number that corresponds to the greatest of all the values they assume. The solid line in Figure 11 is for ke versus A; the dashed line is for k versus. A; and the circles denote k¡¡. We see that all the circles lie on the ke curve. Figure 12 presents the current /,(I, = I j(£Tu Vrco)) as a function of the frequency parameter A.

For the nonaxially-symmetric problem, the EMCC dependence on A is more complicated than in the axialiy-symmetric case. But there is the same tendency for the EMCC to assume the greatest values for the quasitangential vibrations.

FIGURE 8. Distribution of dimensionlcss (a) forces, (b) displacements, and (c) electric strength components along the generatrix of a cylindrical shell near the first resonance (f = tt/4); (tl) displacements along the edge line £ = I as functions of the (^-coordinate (the first resonance)

FIGURE 9. Distribution of dimensionless (a) forces, (b) displacements, and (c) electric strength components along (lie generatrix of a cylindrical shell near the (us! resonance (y~ = n/AY. (tl)

FIGURE 10. Distribution of dimensionless (a) forces, (b) displacements, and (c) electric strength components along the generatrix of a cylindrical shell near ihe first resonance (<p = 7t/4); M) displacements along the edge line £ = 1 as functions of the «¿»-coordinate (the third resonance)

FIGURE 11. Electromechanical coupling coefliciem as a function of the frequency parameter for a shell with cutted electrodes

FIGURE 12. Current as a function of the frequency parameter for a shell with cutted electrodes

29 TWO NONCLASSICAL PROBLEMS FOR SHELLS WITH TANGENTIAL POLARIZATION

PROBLEM 1

Consider a spherical shell. We place its middle surface in a geographic coordinate system where the position of a point is specified by the polar distance 6 and longitude </?. In parametric form, the equations of the spherical surface will be x = rsmflcosip, y = rsinfl sintp, We change the independent variables

Then the coefficients of the first quadratic form have the form [21]:

FIGURE 13. Pari of a spherical shell polarized in meridional direction k2 =

_1 dB AB da sinha

Let the shell be pre-polarized along the 0-lines, and its surfaces be covered with electrodes on which we are given the potential

Our shell is part of a sphere with two rigidly fixed edges: 0 = 9\ and 6 = < 6 < $2 (see Figure 13). We restrict ourselves to calculating the electroelastic state in the roughest approximation.

We saw in Chapter 4 that the internal electroelastic state can be represented as a sum of a slowly varying electroelastic state and a quickly varying auxiliary electroelastic state. The greatest stresses of the principal electroelastic state are computed from the moments that in the first approximation can be determined without solving the problem by using the formulas

«22

Gßt -

«22

Gjr =

36',

2hJll22(i\

sinha sinha

The resultant equation for the auxiliary electroelastic state was obtained in Section 24 and has the form

DYNAMIC PROBLEMS INTHEORYOE l'IEZOCERAMIC PLATES & SHELLS 117 where

hhhl

To be concrete, we will only count the values of the frequency a> such that u>2r2

and the solution to equation 29.4 will be written as w* = c\eka +cje~ka + cj cos ka + C4 sin to (29.6)

The conditions at the rigidly clamped edges 0 = 0\ and 8 = Oi for the auxiliary problem will be written as (see Section 10)

The computation was made for h/R = 0.01,6\ = 7r/4,02 = 37r/4,and d - —1.5 for a shell made of PZT-4.

Figure 14 depicts the bending moments Ggt. The dashed line gives the moment computed from the principal problem, and the solid line gives the moment computed as a sum of the principal problem and auxiliary problem. We see in Figure 14 that the bending moments of the auxiliary electroelastic state are yjRji 1 times greater than the respective moments of the principal electroelastic state. The transverse forces of the auxiliary problem are R/li times greater than the transverse forces of the principal electroelastic state.

Thus, the greatest stresses generated by the smooth surface electrical load are found from the equations of the auxiliary problem. This situation is impossible in the theory of nonelectrical shells because the greatest stresses of the auxiliary stressed-strained state do not exceed the greatest stresses of the principal stressed-strained state for any smooth surface load.

PROBLEM 2

Consider a shell that is a part of an ellipsoid whose equations have the form x = a sin 6 cos tp, y = b sin 0 sin tp, z = c cos 0.

FIGURE 14. Distribution of the bending moment along the meridian of a spherical shell (the dashed line presents the principal problem solution, the solid line combines solutions of the principal and auxiliary problems)

FIGURE 14. Distribution of the bending moment along the meridian of a spherical shell (the dashed line presents the principal problem solution, the solid line combines solutions of the principal and auxiliary problems)

FIGURE 15. Element of an ellipsoidal shell polarized along parallels

Both edges 0 = 0t and 0 = 02 of the shell are rigidly clamped (Figure 15).

The coefficients A and B of the first quadratic form and the geodesic curvatures of the 6- and ip-lines are defined by the formulas

A2 = fi2(cos2 9 cos2 tp + sin2 0) + b2 cos2 0 sin2 tp

We will assume that the shell is pre-polarized along (he i/>Iines, and its faces are covered with electrodes fot which we are given an electrical potential 29.2.

B2 = a2 sin2 9 sin2 ip + b2 sin2 0 cos2 <p 1 °A ' dB

We saw in Section 10 that the internal electroelaslic state of such a shell can be partitioned into the principal and auxiliary electroelaslic states. As in the first problem, we will perform the computation in the first approximation.

In order to find the greatest stresses, we need the bending and transverse forces. For the principal electroelaslic state, we have r Ir w 1 dk*

A o0

For the auxiliary electroelastic state we keep equation 29.4 where

and rv is the curvature radius for the t/?-lines.

Nonclassical boundary conditions for the considered case were discussed in Section 10 and were obtained in Section 45 using the asymptotic method. For the auxiliary problem, the boundary conditions for the rigidly clamped edges 6-61 and 6 = 62 have the form

The constant p\ was calculated for PZT-4 (see Table 1) using the formulas given in Section 50. It turned out to be equal to —2.03 for c = a, b = 2a, 0, ~ n/4, 02 = 3tt/4 h/r<p(0i) = hh'ipiQi) - 0.01, d=- 1.5.

Figure 16 shows Ggt and Ngt as functions of the variable 0 for <p = 7r/4. The dashed line gives the computation results obtained by using 29.7 (the principal electroelastic state) without allowing for the boundary layer corrections. The solid line gives the transverse forces and moments computed allowing for the boundary layer corrections (the transverse forces and moments are found as sums of the principal and auxiliary electroelastic slates). Comparing the solutions where the boundary layer corrections are and are not allowed for, we see that the latter variant gives absolutely the wrong results.

FIGURE 16. Distribution of (a) the bending moment, (b) transverse force along the meridian of a ellipsoidal shell with allowing for the boundary layer corrections (solid line) and without the corrections (dashed line)

FIGURE 16. Distribution of (a) the bending moment, (b) transverse force along the meridian of a ellipsoidal shell with allowing for the boundary layer corrections (solid line) and without the corrections (dashed line)

30 AXISYMMETRIC TANGENTIAL VIBRATIONS OF CIRCULAR PLATES WITH THICKNESS POLARIZATION

Some devices employ elements whose surfaces are only partially covered with electrodes that may either be solid or have cuts.

Let us investigate the effect produced by different electrical conditions given for the plate faces on the dynamic electroelastic state. We will consider the free and forced vibrations of plates whose faces are (a) completely covered with electrodes, (b) have no electrodes, and (c) are partially covered with electrodes. We will solve the problems using the constitutive relations of the theory of piezoceramic plates given in Section 13.

Let us consider a circular plate with thickness polarization. Its radius is R and its thickness is 2h. The numerical computation will be carried out for the PZT-5 piezoceramic. Let us write the equilibrium equations, strain-displacement formulas, and electrostatics equations in the polar coordinates, which have the same form for all our problems.

The equilibrium equations are dT, T, - Te

The strain-displacement formulas are du u

The electroelaslicily relations can have different forms depending on (he electrical conditions on the faces. We will write them separately for each case.

PROBLEM 1 A. Plate with Electrodes.

Consider the free and forced vibrations of a plate with electrode-covered faces due to an electrical load. The edge of the plate is not fixed and is free from the edge forces:

The electroelasticity relations are

2h sUl-v2)

J 31

33"

Here, we have taken into account that the electrical potential on the electrodes is given as

We express the stresses in terms of the displacements and substitute them into the equilibrium equations to get the equation

drl dr L J

whose solution is a first-kind Bessel function

We find the arbitrary integration constant from the condition in 30.3:

XRJ0{XR) (1 - v)Ji(XR) h The forces, strains, and electrical quantities are found from the formulas

2hC\

2d3,

2d2,

In order to determine the amplitude of the current, we integrate the component Dj of the electric induction vector over the surface Q of the electrode on one of the plate faces:

Here we have used the formula rR fR

Now we can use the solution obtained for a numerical computation. The resonance frequencies are found from

The first four roots of equation 30.11 for a plate made of PZT-5 are 2.08, 5.40, 8.58, 11.34.

The antiresonance frequencies for which the current through the piezoce-ramic plate vanishes are important characteristics. Equation 30.10 implies that the antiresonances are defined by the roots of the equation

The first four roots of this equation for a plate made of PZT-5 are 2.46, 5.54, 8.67, 11.80.

The obtained solution is used in Figure 17, showing the dependence of the quantities

on the radial coordinate. Each curve was constructed for a fixed frequency, and the dimensionless quantities T,»(h») were found by dividing the force T, (u) by, the plate force (displacement) with the greatest absolute value for the given frequency. We carried out the computations for frequencies close to the first three resonances. The curves marked by 1,2, and 3 give the quantities T,,,u, as functions of a = r/R near the first, second, and third resonance frequencies, respectively.

The EMCC is of special interest in designing electroelastic elements. We will compute it using the Mason formula 27.12 and the energy formula 27.13.

A

i y

-k

FIGURE 17. Distribution of dimensionless (a) force 7>,and (b) displacement it, along the plate radius near the first three resonances

The Mason formula has the form k2 -Kà -

and gives the following values of k2t near the first four resonances: 0.282,0.051, 0.021, 0.011. We compute k] by the formula kl =

where

0 0

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