Geometrically Non-Linear Beam Element for Dynamics Simulation of Multibody Systems

Abstract

SUMMARYThe 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.

Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators

“…The adopted techniques for this purpose can be grouped into two main categories: procedures based on the approximation of the solution by means of a finite series of given functions and approaches resulting in lumped parameter systems. In particular, to model multilink flexible arms, many authors employed the assumed modes method [Bellezza et al 1990;Khorrami et al 1991;De Luca and Siciliano 1991] and the finite element formulation [Ramachandran et al 1992;Sharf 1996], both of which belong to the first group; other authors prefer the lumped-parameter approach [Rubinstein 1999;Dupac and Noroozi 2014;Giorgio and Del Vescovo 2018] for ease of use.…”

“…However, from a computational point of view, this method requires fewer mathematical operations; therefore, it is particularly suited for dynamic model-based online controller implementations [Theodore and Ghosal 1995]. Although the finite element method can be identified as a different version of the assumed modes method, it can be generalized to be used in a wider context, in particular, when nonlinear effects arise as for a multilink manipulator [Sharf 1996;Eugster et al 2014;Luongo and D'Annibale 2013]. To address some issues related to failures in convergence that are occasionally experienced, some authors have proposed a mixed formulation, based on both stress and displacement degrees of freedom, which appears very promising in this respect [Hodges 1990;Garcea et al 1998].…”

Modal Analysis for the Rotating Cantilever Beam with a Tip Mass Considering the Geometric Nonlinearity

“…Pesheck (12) 는 본-카르만 변형률(von-Karman strain)을 사용하여 축변위와 회전 면외방향(out-of-plane) 변위 만을 고려하여 비선형 운동방정식을 유도하고 섭동 법으로 정적 평형상태에서 선형화된 운동방정식을 구하였다. 이후, Sharf (13) 와 Wang 유연체 외팔보의 변형 전 임의의 위치를 P라고 하면, 변형 후 위치는 P * 로 표현 할 수 있다. 이때, …”