GEOMETRICALLY NON‐LINEAR BEAM ELEMENT FOR DYNAMICS SIMULATION OF MULTIBODY SYSTEMS

SUMMARY The main objective of the present paper is to derive an explicit expression for the stiffness term in the dynamics equations of an elastic beam based on the non-linear kinematics of deformation. In particular, we develop a closed-form expression for the non-linear stiffness matrix which operates on the total elastic deformations to give the vector of internal forces. The derivation begins with the description of the deformation field and the exact non-linear strain4isplacement relations for a conventional beam element. The strain energy is formulated in accordance with the slender beam theory and various non-linear terms are identified. The non-linear stiffness matrix is then obtained in two stages. The first step involves application of Castigliano’s theorem to generate the vector of nodal internal forces S from the strain energy of the element. In the second step, we expand S in the nodal co-ordinates q using the Taylor formula. Because of the particular form of S, this expansion is exact and consists of only three terms: linear, quadratic and cubic. It allows us to directly evaluate the respective three terms of the non-linear stiffness matrix by setting: The closed-form expressions for the elemental matrices XG and X,, are provided in terms of the nodal co-ordinates. These were determined using the Maple symbolic computation program, which was also employed to generate the LATEX format of the results and the corresponding Fortran code. The paper is concluded with two numerical examples demonstrating the performance of the dynamics models incorporating the proposed non-linear stiffness term. The first example illustrates the response of a cantilever beam to a prescribed moment at the base. The second example is modelled after an experimental planar manipulator driven by two direct-drive actuators. The results illustrate the differences between three dynamics models formulated with the linear, second-order and third-order approximations for the stiffness term.

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