Evaluation of Non Linear Response of Beams by Symplectic Integration Method

摘要

The non linear free vibration of beams obey the law of conservation of energy. The application of symplectic integration (SI) scheme for solution of non linear beam problems ensures conservation of energy, momentum and volume of the system. In this paper the Langrangian of the continuous system is discretized by Ritz-Galerkin's method. The discretized Lagrangian is transformed to Hamiltonian. The resulting Hamilton's equation which is a system of ordinary differential equations is solved by second order and eighth order SI scheme. A comparison is made among the results obtained by the present symplectic method, FEM displacement method, analytical method, and Differential Quadrature (DQ) method as reported in literature. It is observed that eighth order symplectic method is better than the FEM displacement method and is not far from DQ method except for the case when the ratio of amplitude to radius of gyration becomes more than five. A system of n differential equations of second order is transformed to Hamiltonian system of 2n canonical equation of first order in the phase space and is solved by the SI scheme. The effectiveness of the SI scheme is illustrated using a numerical example of a pinned-pinned beam under three different conditions. The first mode is chosen for analysis. The motion is found to be periodic and the period is related to the initial conditions. It becomes dissipative if the time step is more than ten micro seconds. The (SI) method is the simplest of the methods mentioned in the literature.