3D solutions for static/vibration of FGPM plate/pipe in Hamiltonian system

The 3-dimensional couple equations of piezoelectric-mechanic were derived into Hamilton system by the principle of Hamilton theorem. The problem of single sort of variables was converted to double sorts of variables, and the Hamilton canonical equations were established. The dynamic characteristics of the simply supported functionally graded piezoelectric material (FGPM) plate and pipe are investigated in different coordinate systems. Finally, the problem was solved by the symplectic algorithm. The results show that the complex electromechanical problems of FGPM structures can be solved in the Hamiltonian system. The general displacement and stress of the medium are divided into so-called out-of-plane variables and in-plane variables. The former is continuous while the latter is discontinuous along the depth.