Dynamic Analysis of Timoshenko Beam with Arbitrary Constraints and a Further Optimization Based on Least Energy Principle

Based on Timoshenko’s beam theory, this paper adopts segmented strategy in establishing the governing equations of a multibeam system subjected to various boundary conditions, in which free, clamped, hinged, and elastic constraints are considered. Meanwhile, Galerkin method is incorporated as a competitive alternative, in which a new set of unified, efficient, and reliable trial functions are proposed. A further optimization in regard to boundary distributions under forces is implemented and established on the least absorbed energy principle. High agreement is observed between the analytical results and the FEM results, verifying the correctness of the derivations. Complete comparisons between the analytical and the numerical results indicate the Galerkin method is beneficial when slender ratio is larger than 30, in which the continuity of the deformation is proved to be a crucial influencing factor. A modified numerical strategy about optimal boundary is employed and the remarks imply the algorithm can be availably used to reduce the energy absorption of the whole system.