A Hamiltonian state space formulation of piezoelectric materials in cylindrical coordinates

A method is presented to derive the state equations of the free vibration of piezoelectric materials under cylindrical coordinates in Hamiltonian system. Based on the 3D theory of piezoelectricity, the constitutive relations and the Lagrangian function of the generalized strain energy are rewritten in terms of the generalized displacements, i.e., displacements and electric potential and their derivatives. The Legendre transform is then applied to release the Hamiltonian function and the canonical equations are obtained through the variational operation. The canonical equations, i.e., the desired state equations, in cylindrical coordinates are obtained readily and could be used in the further analysis.