Cr J i

On the electrodes covering the bar ends fi, the boundary conditions for the electrical quantities have the form for


Short-circuited electrodes

The potential difference on the electrodes is given as rl>, =±V,. (31.7)

In equations 31.1 to 31.4, we use dimensionless quantities, dimensionless variable and dimensionless parameter A:

^ = v. = d-fv A = pw2/2sf3, where 21 is the bar length.

Let us find the solution to the system of equations 31.1 to 31.4 for the different boundary conditions 31.5 to 31.7. When working with the conditions 31.7, we assume that the potential difference Vm on the electrodes is an unknown quantity. We will determine it under the condition that the bar ends are not displaced. The solution of the problem has the form u, = Cj sinkx, a, = —^—(¿C2 cosfcr + C|), k2-

l+o ' 1 + a where C\ and C2 are determined by the following formulas

kcosk for the conditions 31.5;

- -r~.—:—r I w k cos k + a sink kcosk + as'mk for the conditions 31.6;

for the conditions 31.7.

The computational results for a bar made of PZT-4 are given in Figure 23. The solution for the boundary condition 31.5 is marked by 1, that for condition 31.6 by 2, and that for condition 31.7 by 3. We see that by properly choosing the potential difference V, = 1 +a, we can eliminate any displacement. In this case, the stresses will be identically equal to one.

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0.0 0.5

FIGURE 23. Distribution of displacements along the bar length for different boundary conditions; (1),(2) with only mechanical load and (3) with an additional specially chosen electrical load

FIGURE 23. Distribution of displacements along the bar length for different boundary conditions; (1),(2) with only mechanical load and (3) with an additional specially chosen electrical load


Consider the bending vibrations of a two-layer beam withbimorphic properties. One end of the beam is clamped and the other is loaded by a periodically varying bending moment with transverse forces. The beam is subject to a normal load uniformly distributed over its length. The faces of the beam are covered by 2n electrodes with cuts (see Figure 24) that are symmetric relative to the middle surface. The dimensionless length of every beam part covered with a solid electrode is = 1 /n. The beam's dimensionless length is equal to one. For brevity, we will use only dimensionless quantities. The starting system of the differential equations has the form dN* 7 », dG*

d2w* dwt

Here, the dimensionless quantities with asterisks were introduced by the formulas

Here, Bu,i/, are given by 12.7 in Section 12, and / is the beam length.

We will solve two problems. In the first problem, all the electrodes are treated as short-circuited. In the second problem, a potential difference is applied to each pair of the electrodes symmetric with respect to the middle surface so that the deflections at the ends of every segment are zero. We will consider each beam segment separately.


2 h

FIGURE 24. The geometry of a bimorphic bar with cutted electrodes and the mechanical load acting on it

For the /th segment covered with solid electrodes, the solution has the form

Wj = Cue~iX + C2le(X + Cv sin A£ + C4/ cos A£ + W, (31.9)

where IV, is a particular solution. The constants will be found from the boundary conditions at the beam's ends

The conditions at the interface and the condition that the deflections at the junction points of all the segments = i/= 1,2,...,«- 1 are zero:

The computational results for a bar of PZT-4 are plotted in Figure 25. Curve (a) in Figure 25 shows the deflection w* as a function of E, when the bar is subjected to only mechanical load (Nq = 1, Go = 1, Z„ = 1). Curve (b) in Figure 25 depicts the deflection due to the electrical and mechanical loads. It turns out that, for/) = 3, A = 1, the deflection due to the electrical load decreases

104 times.


Suppose that a circular two-layer plate (bimorph) with an opening in the center performs forced bending axisymmetrical vibrations under the action of bending moments and transverse forces distributed over the outer edge. The inner edge is rigidly clamped. The faces are completely covered by circular electrodes with cuts. As for the bar, we solve two problems: the one for short-circuited electrodes and the other where on every pair of electrodes symmetric to the

FIGURE 25. Distribution of displacements along the length of a bimorphic bar (a) under the effect of a pure mechanical load and (b) after suppressing the vibrations by an electrical load

middle surface the potential difference is chosen so that the deflections at the electrode cuts are zero.

The initial system to the dimensionless quantities consists of the Equilibrium equations

Electroelasticity relations

GI. = -Ow +v,K2.) - V» G2* = -(«2. + l'.Kl,) - V,

Strain-displacement formulas dw»

d 7i

The dimensionless quantities with asterisks were introduced by it'

where /? is the radius and 2h is the thickness of the plate. The numbers Ru and are given by formulas 12.7. The resultant equation

je t<is

y/ -


-1 - - - ^



FIGURE 26. Distribution of displacements along the radius of a bimorphic plate (a) under the effect of a pure mechanical load and (b) after suppressing the vibrations by an electrical load has a solution

Wi* = C,,Jo(AO + C2,Ko(AÇ) + C3i/0(AO + C4iK0(\O + VV,.

Here, W, is particular solution of the equation 31.12.

We satisfy conditions 31.10 and 31.11 and find the integration constants and the unknown potential differences. The computational results are given in Figure 26. Curve (a) in Figure 26 shows the deflection as a function of the radial coordinate when the plate is subject to only mechanical load (/Vo = l,Go = 1,Z, = 1). Curve (b) in Figure 26 depicts the plate's deflection after suppressing the vibrations by a specially chosen electrical load (n = 6). All the solutions show that piezoeffect can be used to diminish the vibration amplitude.


It was shown in [86] that the problem of the vibration of an elastic body immersed in a viscous fluid can be essentially simplified. A simultaneous asymptotic analysis of the equations describing the elastic body and the Navier-Stoke's equations describing the viscous fluid show that the Navier-Stoke's system can be approximately replaced by simpler systems. One describes the fluid boundary layer quickly decreasing with the distance from the elastic body, and the other describes the penetrating perturbation in the fluid. The latter can be reduced to an equation similar to the Helmholtz equation.

The system for the fluid boundary layer can easily be integrated. We use the solution to express the tangential stresses at the interface between the body and the fluid in terms of the tangential displacements of points on the surface of the body.

It is shown in [92] that for shells and plates performing tangential vibrations in a fluid, the effect of the viscous fluid is allowed for by introducing a load A", into the equations for the motion of the surface load, so that

T7T + T-^-+kj(T>~Tj)+ lk'S ~ 77 + Upu2ii; + X< = 0 A, da, Ajdctj R/


ps is the density of the material of the shell, pf is the density of the fluid, and p is the viscosity of the fluid.

We use ujv to denote the resonance frequency of the tangential vibrations of the plate (shell) in a vacuum or air and w/ to denote its resonance frequency in the viscous fluid. The system describing the vibrations in a vacuum differs from that for fluid by an inertial term: 2hpsu2,u, for a vacuum and

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