A Unified Approach for the Dynamics of Beams Undergoing Arbitrary Spatial Motion

A unified approach to systematically derive the dynamic equations for flexible bodies is proposed. This approach is not limited to a particular definition of the field of kinematic representation of deformation. Dynamics of flexible bodies in arbitrary spatial motion experiencing small and large elastic deflections are considered. Two test cases are analyzed via the unified approach. For the first case, linear partial differential equations based on the Euler-Bernoulli beam theory with the von Ka´rma´n geometric constraint for flexible bodies in planar motion are derived. These equations capture the centrifugal stiffening effects arising in fast rotating structures. For the second case, analytical and numerical evidence of out-of-plane vibrations of an in-plane rotating three-dimensional Timoshenko beam with cross sectional area of arbitrary shape is reported.