Solution to differential equations of piezoelectric vibration

[athosanian]

dear all, I am working on a problem about the vibration of a piezoelectric ring with electrodes on the upper and lower surface. the coordinate system is cylinder coordinate system. the structure is shown below. This is an axisymmetry problem.

the polarization of the piezoelectric ring is along thickness direction.

constitutive equations are

the equation of motion are

The geometric relations are

substituting constitutive relation and geometric relation into the equation of motion , I obtain

I want to find a solustion to the above equ. (4). Any one who is familiar with such equations could provide some references for me to read ? Thanks a lot.

 

[UltrafastPED]

Do you expect an analytic solution?

You might determine some appropriate boundary conditions and carry out a series of numerical solutions; this will help you locate the most sensitive parameters, and guide you in some appropriate approximations which will simplify the equations.

 

[athosanian]

I just want to find a form for the solution. For example, in Cartesian coordiantes, the vibration equation is

the solution for the equations is

Then I can continue the discussions.

 

[GregoryS]

I guess you want to say that there is one of the possibile solutions for the equations you wrote!