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 = D4>, = 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 1Suppose 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 shortcircuited. 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)
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 810 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 nonaxiallysymmetric problem, the EMCC dependence on A is more complicated than in the axialiysymmetric 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 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 1Consider 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 prepolarized along the 0lines, 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

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