On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates

In the classical approach, it has been common to treat free vibration of rectangular Kirchhoff or thin plates in the Euclidian space using the Lagrange system such as the Timoshenko’s method or Lévy’s method and such methods are the semi-inverse methods. Because of various shortcomings of the classical approach leading to unavailability of analytical solutions in certain basic plate vibration problems, it is now proposed here a new symplectic elasticity approach based on the conservative energy principle and constructed within a new symplectic space. Employing the Hamiltonian variational principle with Legendre’s transformation, exact analytical solutions within the framework of the classical Kirchhoff plate theory are established here by eigenvalue analysis and expansion of eigenfunctions in both perpendicular in-plane directions. Unlike the classical semi-inverse methods where a trial shape function required to satisfy the geometric boundary conditions is pre-determined at the outset, this symplectic approach proceeds without any shape functions and it is rigorously rational to facilitate analytical solutions which are not completely covered by the semi-inverse counterparts. Exact frequency equations for Lévy-type thin plates are presented as a special case. Numerical results are calculated and excellent agreement with the classical solutions is presented. As derivation of the formulation is independent on the assumption of displacement field, the present method is applicable not only for other types of boundary conditions, but also for thick plates based on various higher-order plate theories, as well as buckling, wave propagation, and forced vibration, etc.