Galerkin finite element derivation for vibration of a thermopiezoelectric structure

A solution for the equations governing dynamic thermopiezoelectric behavior is assumed in the form of an exponential ei(kz+mθ+ωt)ei(kz+mθ+ωt). The resulting formulation is reduced to one-dimension and will have both real and imaginary parts. The Galerkin method and assumed shape functions are used to illustrate the formulation of the resulting eigenvalue problem.