Dynamics of flexible multibody systems using cartesian co‐ordinates and large displacement theory

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Abstract
A formulation for the dynamic analysis of flexible systems, composed of slender bodies that can be accurately modelled by beams is presented in this paper. A new set of state variables, composed of Cartesian co‐ordinates of points and unit vectors, is introduced to define the beam with respect to an inertial frame. A non‐linear Timoshenko beam finite element capable of handling finite displacements with small linear elastic strains is developed. This allows relative displacements between material points of a single beam to be arbitrarily large. Since deformations are not explicit variables, there is no need to define a moving reference frame attached to each flexible body. Instead, deformations are obtained through a displacement‐deformation relation based on finite‐displacement beam theory. The differential equations of motion are obtained using the Lagrange equations. A symmetric, constant and sparse mass matrix is obtained in the inertial frame. Constraints are introduced with a penalty formulation and the resulting set of ordinary differential equations is integrated with Newmark's family of methods. The whole formulation is extremely simple and the results demonstrate the capabilities and efficiency of the proposed method for dynamic simulation, even when relative displacements are finite in a single beam or coupling effects are significant.

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