Cr J i
On the electrodes covering the bar ends fi, the boundary conditions for the electrical quantities have the form for
DYNAMIC PROBLEMS IN THEORYOEPIEZOCERAMIC PIATF.S & SHELLS 131 Disconnected electrodes
Shortcircuited 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. = dfv 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 PZT4 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.
4  
Mr1 
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
PROBLEM 2
Consider the bending vibrations of a twolayer 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 shortcircuited. 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.
DYNAMIC PROBLEMS IN THEORY OF PIEZOCERAMIC PLATES & SHELLS 133 2 No
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 PZT4 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. PROBLEM 3Suppose that a circular twolayer 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 shortcircuited electrodes and the other where on every pair of electrodes symmetric to the 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, Straindisplacement 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

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