Symplectic integration of an accurate beam finite element in non-linear vibration

Abstract

We make two contributions in this paper: (1) to form beam finite element matrices within the large deflection and small rotation assumptions and (2) to integrate the resulting equations by symplectic schemes. The inherent approximation is introduced by the assumed shape functions only in our finite element formulation. The induced axial force is not averaged and the stiffness is defined by the first-, second- and third-order matrices. In the solution stage we use symplectic integration which does not require the linearization of stiffness. All necessary conservative laws during numerical integration are observed. Both free and forced vibrations and damped and undamped vibrations are studied.