Nonlinear Three-Dimensional Beam Theory for Flexible Multibody Dynamics ∗ Shilei Han and Olivier A. Bauchau University of Michigan-Shanghai Jiao Tong University Joint Institute 800 Dong Chuan Road, Shanghai, 200240, China Abstract In flexible multibody systems, it is convenient to approximate many structural components as beams or shells. Classical beam theories, such as Euler-Bernoulli beam theory, often form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the beam’s section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such approach is capable of capturing the kinetic en- ergy of the system accurately, it cannot represent the strain energy adequately. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions. Kinematically, the problem is decom- posed into an arbitrarily large rigid-section motion and a warping field. The sectional strains associated with the rigid-section motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the equations describing geometrically exact beams and those describing local deformations. The governing equations for geometrically exact beams are nonlinear, one-dimensional equations, whereas a linear, two-dimensional analysis provides the detailed distribution of three-dimensional stress and strain fields. Within the stated assump- tions, the solutions presented here are exact solution of three-dimensional elasticity for beams undergoing arbitrarily large motions. 1 Introduction A beam is defined as a structure having one of its dimensions much larger than the other two. The generally curved axis of the beam is defined along that longer dimension and the cross-section slides along this axis. The cross-section’s geometric and physical properties are assumed to vary smoothly along the beam’s span. Numerous components found in flexible multibody systems are beam-like structures: linkages, transmission shafts, robotic arms, etc. Aeronautical structures such as aircraft wings or helicopter and wind turbine rotor blades are often treated as beams. The solid mechanics theory of beams, more commonly referred to simply as “beam theory,” plays an important role in structural analysis because it provides designers with simple tools to analyze numerous structures [1]. Several beam theories have been developed based on various assumptions and lead to different levels of accuracy. One of the simplest and most useful of these theories is due to Euler who analyzed the elastic deformation of slender beams. In many applications, beams * Multibody System Dynamics, 34(3): pp 211-242, 2015 1
Nonlinear Three-Dimensional Beam Theory for Flexible Multibody Dynamics
Jun 19, 2023
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