## Uf f euT eydSe

PROBLEM 2

We find Condition 27.15 holds on the electrodes. Allowing for the formulas

we write equation 27.3 in the form

We solve this equation meeting the conditions 27.15 and find lieeting the cc and Then we calculate U{sh) as

The computation was performed for a shell made of PZT-4 of length 21 = 2R,h/r = 0.025. Figure 5 gives the amplitude values of forces and displacements near (a) the first resonance for a shell and (b) the second resonance as functions of the longitudinal coordinate in the interval [-/, +/] (the edges of the shell are not clamped). Each dimensionless quantity in Figure 5 is divided by

FIGURE 5. Distribution of dimensionless forces, displacements, electric strength components along the generatrix of a cylindrical shell near (a) the first and (b) the second resonances the same number that corresponds to the greatest of all the values they assume. Figure 6 shows different electro-mechanical coupling coefficients as functions of the frequency parameter A. The dashed line is for the values of k computed by 27.10; the solid line gives ke computed by 27.13; and the circles are used for k,i computed by 27.12. Since kd gives the EMCC values in the interval [A(r). A<rt)], the circle in Figure 6 is situated in the middle of this interval. We see from the figure that the ¿¿-values lie on the curve kc. The computation using formula 27.10 gives qualitatively different results. We know that kj is a reliable and well-checked characteristic widely used in applications; therefore, the obtained coincidence confirms 27.13 for computing ke. Figure 7 gives the current versus /♦(/* = //(ef| Vru>)) the frequency parameter. We use the following formula for amplitude value of current f8]:

Note that the number k,/ is computed from the current values at resonance and antiresonance frequencies corresponding to the infinite and zero current values, respectively.

The computational results for the first three natural frequencies depending on the shell length and the respective EMCC values found by different formulas are given in Table 2. The superscript (r) or («) shows that the value was obtained for the resonance or antiresonance frequency, respectively. We see from the table that kj is the averaged EMCC for the whole range of the frequency parameter A from the resonance to the antiresonance. Formula 27.13 allows us to determine the EMCC's for static and dynamic cases for each value of A.

Thus, for computing the forced vibrations, we have broken the range of the frequency parameter A into four intervals. For A lying in the first and third intervals, the computation is done using the membrane theory (the shell performs quasitransverse vibrations with small variability). For A lying in the second interval, we use the moment theory (the shell performs quasitransverse vibrations with greater variability). For A from the fourth interval, we use the

FIGURE 5. Distribution of dimensionless forces, displacements, electric strength components along the generatrix of a cylindrical shell near (a) the first and (b) the second resonances a b a b

FIGURE 6. Electromechanical coupling coefficients as functions of the frequency parameter X
FIGURE 7. Current as a function of the frequency parameter A

TABLE 2.

Resonance and anliresonance frequency parameters and (lie corresponding EMCC's as functions of llie shell length (axisymmetric problem)

TABLE 2.

Resonance and anliresonance frequency parameters and (lie corresponding EMCC's as functions of llie shell length (axisymmetric problem)

 l/r A« A<"> kd * k(r) *<»> 1.06 1.20 0.345 0.436 0.261 0.000 0.338 0.5 1.25 1.25 0.013 0.007 0.019 0.323 0.318 10.68 16.72 0.599 0.615 0.578 0.285 0.661 0.98 1.18 0.413 0.522 0.279 0.100 0.392 1.0 1.20 1.21 0.041 0.021 0.081 0.337 0.331 2.94 4.28 0.559 0.564 0.558 0.263 0.646 0.76 1.10 0.556 0.598 0.332 0.112 0.413 1.5 1.19 1.20 0.068 0.045 0.110 0.334 0.336 1.68 2.03 0.418 0.398 0.432 0.186 0.583

equations describing the quasitangential vibrations. Our computation shows that the EMCC assumes the greatest values for the quasitangential vibrations, decreasing as the shell length grows. The EMCC values for A from the second interval are close to zero.

28 FORCED VIBRATIONS OF A CIRCULAR CYNDRICAL SHELL WITH LONGITUDINAL POLARIZATION (NONAXISYMMETRIC PROBLEM)

Consider a sheii with longitudinal polarization, which performs forced vibrations caused by an electrical load acting on the shell's edges [87], The edges are covered by dissected electrodes and the electrical potential on each is given. Wc expand the electrical load and all the sought-for quantities in the Fourier series in the circular coordinate tp\

OG OO JC

//,- = ± ^ t„ sin/up, P\ - ^^ P\n sinmp, P2 = ^ Pi»cos/iy?. /t=i /i=i

Here, P\ is any of the quantities xj)t, 7",«, «„, , E\„, and D\t ; and Pj is any of the quantities 5», v,, £2„, and D2,; t„ are constants. Both P\„ and P2„ are functions of the longitudinal coordinate

We substitute expansions 28.1 into the equations of shell theory, equating the coefficients at the same trigonometric functions to zero. We find for every n the system

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