Symplectic elasticity approach for free vibration of rectangular thin plates

Using a new symplectic method commonly applied by theoretical physicists, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some long standing basic problems in free vibration of rectangular thin plates with any boundary conditions where exact solutions are hitherto unavailable. Employing the Hamiltonian principle with Legendre’s transformation, analytical free vibration solutions could be obtained by eigenvalue analysis and expansion of eigenfunctions in both lengthwise and widthwise directions. Unlike the classical semi-inverse approaches using trigonometric, hyperbolic and/or Bessel functions where a trial amplitude function is pre-determined, this new symplectic approach is completely rational without any guess functions and yet it renders exact solutions beyond the scope of applicability of the semi-inverse approaches. In short, the symplectic approach developed in this paper presents a new approach in an area previously unaccountable in classical mechanics and the semi-analytical approach forms a limited sub domain of this new approach. Examples for plates with selected boundary conditions are solved and the exact solution is discussed. Comparison with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only for other types of boundary conditions, but also for thick plates as well as bending, buckling, wave propagation, etc.