Modeling And Dynamic Simulation Of Large Deformation Flexible Beams In Multibody Systems

The slender beam is not only the most used flexible component in mechanical systems, but also one of the most popular research objects in flexible multibody system dynamics. With the wide application of light materials and the improvement of the operating speed in modern mechanical systems, the dynamic characteristics of the flexible beam become extremely complex, and the coupling between large rigid body motions and large deformations is becoming more and more significant. Previous researches have indicated that the traditional modeling method in flexible multibody system dynamics based on small deformations and rotations assumption has been unable to provide reliable numerical simulation results for this kind of mechanical system. This paper presents an in-depth study on the modeling theory of large deformation beams and the numerical methods for dynamic equations of rigid-flexible coupling systems.With the assumption of rigid cross-section, the strain measures defined in the geometrically exact beam theory are of objectivity, which can be used to analyze the deformation of the beam with large displacements and finite rotations. Based on the internal virtual power of the geometrically exact plane Euler-Bernoulli beam, a strain-interpolation beam element, which is suitable for molding of large deformation plane slender beam in flexible multibody systems, is proposed. It is the strain rather than the displacement directly related to the beam strain energy. So, when structuring the finite element, the axial strain and the curvature of the beam element are selected to disperse as the basic variables. This is not affected by the rigid-body displacement of the element, but also can get simple element nodal forces and stiffness matrix. The centroid displacement and angle of the cross-section at two ends are obtained by integrating the geometry equation, which automatically captures the dynamic stiffening terms. The proposed beam element with strain interpolation is not only suitable for the small deformation problems of plane beams, but also the large deformation rigid-flexible coupling dynamics.The independent interpolation of finite rotations is widely adopted in the existing geometrically exact spatial beam element, which causes the problem of shear locking when modeling the flexible slender beam. On considering the deformation coupling relationships of the slender beam, a spatial Euler-Bernoulli beam element is proposed, which is suitable for the modeling of large deformation spatial slender beam in flexible multibody systems. The global displacement and rotation vectors are selected as the nodal coordinates and an element deformation field is constructed, in which the beam cross-sections can keep perpendicular to the current neutral axes by employing a special coupled interpolation of the centroid position and the cross-section orientation. On this basis, the beam element nodal force, tangent stiffness matrix and consistent mass matrix are derived according to the geometrically exact beam theory. The proposed spatial Euler Bernoulli beam element is not only suitable for the large deformation flexible beam's modeling in multibody system dynamics, but also can be used to the geometric nonlinear problem of large deformation slender structure.The dynamic equations of flexible multibody systems are often a set of stiff equations, which are very difficult to solve. At present, stiff equation solvers are generally adopted, of which the basic idea is to filter out the high frequency response by numerical damping. Although many problems have been solved successfully, the computational efficiency of the stiff equation solvers is still unsatisfactory. By replacing the original instantaneous strains in the system internal force expression with an average strains over a small time interval, an average strain method is proposed, which can filter out the high frequencies of the system equation. The strain damping and inertia terms are introduced to the system equations through this method, and generally the modified system dynamic equations can be solved by using nonstiff solvers. Finally, the average strain method is applied to the dynamic modeling of the large deformation beams, and the efficiency of the proposed method is verified by the numerical simulations.