Symplectic Galerkin method for non-linear vibration of beams and plates

Non-linear free vibration of beams and plates is conservative in the sense that the total energy of the system remains constant. Symplectic numerical integration aims to preserve the energy, momenta and area (volume) of the phase space, and is ideal in the study of non-linear free vibration. In this paper, the Lagrangian of continuous systems is discretized by Galerkin's method to obtain the discrete Hamiltonian and Hamilton's equations. Symplectic numerical integration schemes are applied to the resulting ordinary differential equations to construct the phase diagram. It is shown that, due to modal coupling, bounded quasi-periodic and non-periodic free vibrations are very common for continuous systems. Comparison is made with the Runge-Kutta method. It is shown that the Runge-Kutta integration reduces the total energy of an undamped system while symplectic integration almost preserves it. The symplectic schemes are extended to ordinary differential equations by means of the Lagrangian and Hamiltonian, and the non-linear vibration of beams and plates is studied by the schemes for the first time. Non-linear modal coupling is emphasized. The extension to shell structures involves the complication of curvature effects. Research is in progress.