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.