Simulation of a Moving, Elastic Beam Using Hamilton's Weak Principle

Abstract

A proof-of-concept algorithm for calculating the general motion of elastic beams in the context of a multibody system analysis has been developed and validated. The governing equations for moving elastic beams with closed cross sections have been formulated using Hamilton’s law of varying action. They are then discretized with a mixed space–time finite element scheme, which results in a system of nonlinear algebraic equations. These equations are solved using unconstrained optimization techniques, thereby obtaining steady-state and time-accurate solutions for linear and nonlinear structural dynamics problems. Solutions obtained for a variety of static, steady-state, and transient test cases have demonstrated the accuracy of the formulation. Illustrative examples, including the cantilever elastica problem, rotation of a beam at a constant angular velocity, and prescribed flapping motion of a beam, are presented.