Super Convergent Finite Beam Element for Torsional Vibration of Open Thin-Walled Vlasov Beam under Torsional Excitations

Abstract

A highly accurate finite beam element is developed for the steady state torsional-warping dynamic analysis of open thin-walled doubly symmetric Vlasov beams subjected to various harmonic torsional and warping moments. The governing dynamic equation and associated boundary conditions for torsional-warping response is derived through Hamilton variational principle. The formulation is based on Vlasov beam theory and incorporates both warping deformation and warping inertia effects. From the resulting torsional equation, the closed form solution is exactly obtained. A set of shape functions developed based on the exact solution of the field torsional equation is utilized to formulate the finite beam element. The two-noded beam element, with four degrees of freedom per element, effectively captures the quasi-static and steady-state torsional responses of open thin-walled doubly symmetric beams under harmonic torsional excitations. Additionally, it is used to extract the natural torsional frequencies and mode-shapes from the steady state dynamic response. The results obtained from the finite-element formulation are evaluated and verified by comparing them with well-established finite-element solutions and exact solutions available in the literature. The present beam element validity is demonstrated through several numerical examples, with results exhibiting excellent agreement with exact solutions available in the literature and Abaqus and Galerkin finite element models, achieved at a significantly reduced computational and modeling cost.