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!

 

[sardar]

I am familiar with differential equations and their applications in various fields, including piezoelectric vibration. The problem you are working on is interesting, and it is important to find a solution to the equation of motion (equation 4) in order to understand the behavior of the piezoelectric ring.

One approach to finding a solution to this type of differential equation is to use numerical methods, such as finite element analysis or boundary element method. These methods can provide accurate solutions for complex structures like the one you are studying.

Another approach is to look for analytical solutions, which can provide insight into the behavior of the system. In this case, you may want to consider using perturbation methods or variational methods to find a solution.

There are also several textbooks and research papers available on the topic of piezoelectric vibration, which may provide useful information and references for you to explore. Some potential resources include "Piezoelectric Vibration and Acoustic Resonance Insulators" by L. E. Cross and "Piezoelectricity: An Introduction" by R. J. Prins and G. W. Taylor.

I hope this helps guide you in your research and I wish you success in finding a solution to your equations. Keep exploring and learning, and don't hesitate to reach out to other scientists or experts in the field for guidance and support.