Keywords: symplectic method, piezoelectricity, cantilever.

Since the phenomenon of piezoelectricity in natural crystals was discovered in 1880 by Pierre and Jacques Curie, it has attracted great interest for its special property which combines electricity and mechanical characteristics. For this property, piezoelectric materials are developed for application to many fields, such as industrial, aviation and navigation. Many important results have been obtained.

Ding and Chen [1] used a new state-space method to study transversely isotropic piezoelectric problems that have a strong engineering background. Heyliger and Saravanos [2] developed exact solutions for predicting the coupled electromechanical vibration characteristics of simply supported laminated piezoelectric plates composed of orthorhombic layers. Mitchell and Reddy [3] developed a refined theory of laminated composite plates with piezoelectric lamina. Pai [4] presented a fully nonlinear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large-rotation and small-strain vibration.

Zhong [5] developed a symplectic phase-space for conservative Hamiltonian systems to solve the elastic problem in contrast to the traditional semi-inverse solution method. It is an analytical process without the assumptions of the potential functions required in the semi-inverse method. This is a new way to solve the elastic problem. Lagrange systems are transferred to Hamiltonian systems by the Legendre transformations. After that, the second order partial equations become first order partial equations and the method of separable variables can be used to solve problems easily. Leung and Mao [6,7] combined the finite element and symplectic procedures to investigate non-linear vibration problems with all necessary conservative laws observed.

In this paper, the symplectic method for the analytical separation of independent variables is introduced. This paper deals with transversely isotropic beams considering the piezoelectric effects. The beam is investigated as a two dimensional (2D in x and z) problem. One of the space dimensions, say x, takes the place of the time variable in the usual Hamiltonian mechanics and the Hamiltonian has derivatives in the z coordinate only. Separation of variables is therefore achieved. To solve the resulting first order Hamiltonian equation, the eigenvalue problem of the Hamiltonian matrix is required analytically. Two kinds of eigensolutions solutions corresponding to the zero eigenvalue (zero-eigensolutions) and non-zero eigenvalues (nonzero-eigensolutions) need separate consideration.

The complete zero-eigensolutions space can be obtained from the Hamiltonian equations. Each of the eigensolutions has its own physical meaning, including translation, bending and uniform electric potential. Nonzero-eigensolutions represent the localized boundary effects which are usually neglected by the Saint-Venant principle and which decay exponentially with respect to the distance away from the boundaries. For verifying the correctness and the advantages of the symplectic method, some results about the cantilever beam with the piezoelectric effects are compared with those of the polynomial solutions. Other results for the "sandwich" beam are also compared with those of the finite element.

1

H.J. Ding, W.Q. Chen, R.Q. Xu, "New state space formulations for transversely isotropic piezoelasticity with application", Mechanics Research Communications, 27(3), 319-326, 2000. doi:10.1016/S0093-6413(00)00098-7

2

P.R. Heyliger, D.A. Saravanos, "A. Exact free-vibration analysis of laminated plates with embedded piezoelectric layers", Journal of Acoustical Society of America, 98(3), 1547-1557, 1995. doi:10.1121/1.413420

3

J.A. Mitchell, J.N. Reddy, "A refined hybrid plate theory for composite laminates with piezoelectric laminae", International Journal of Solids and Structures, 32(16), 2345-2367, 1995. doi:10.1016/0020-7683(94)00229-P

4

P.F. Pai, A.H. Nayfeh, "A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors", International Journal of Solids and Structures, 30(12), 1603-1630, 1993. doi:10.1016/0020-7683(93)90193-B

5

W.X. Zhong, "Plane elastic problem in strip domain and the Hamiltonian system", Journal of Dalian University of technology, 31, 373-384, 1991.

6

A.Y.T. Leung, S.G. Mao, "Symplectic integration of an accurate beam finite element in non-linear vibration", Computers & structures, 54, 1135-1147, 1995. doi:10.1016/0045-7949(94)00388-J

7

A.Y.T. Leung, S.G. Mao, "A Symplectic galerkin method for non-linear vibration of beams and plates", Journal of Sound and Vibration, 183, 475-491, 1995. doi:10.1006/jsvi.1995.0266

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