Interaction of local and global nonlinearities of elastic rotating structures

Abstract

Geometrically nonlinear effects like buckling due to large acceleration of lightweight and thin walled structures often play an important role in flexible multibody dynamics. To account for the interaction between local and global nonlinearities, a generalized method to model the motion of large flexible elastic structures is applied. The structure is decomposed into substructures, which are discretized using the finite element method. To reduce the number of degrees of freedom, a set of well-suited modes for the description of the displacement field on substructure level is used. The formulation introduced here combines the advantages of the corotational finite element formulation and the component mode synthesis method, resulting in a reduced set of degrees of freedom including local nonlinearities on substructure level. NVESTIGATIONS into the dynamic behavior of large structures undergoing finite displacements are expensive due to the large amount of computing time required. To reduce the numerical effort, several methods based on substructuring and natural mode decom- position of the displacement field have been developed.1'6'9'10) 12 Cur- rent applications are primarily limited to linear oscillation behavior of general elastic structures. In most cases nonlinearities are incor- porated only for beam-like structures having rigid cross sections. To consider both global (finite rotations) and local (for example, dynamic buckling) nonlinearities, the proposed procedure is based on the following steps: 1) Decomposition of the structure into elastic substructures and discretization of the substructures (for example, a robot manipula- tor) is accomplished using a finite element formulation applied to a shell theory of moderately large (linearized) rotations. To simplify the nonlinearity with respect to the deformation, the derivation of the Euler-Lagrange equations and their consistent linearization is performed by using a mixed Hellinger—Reissner formulation, where the nonlinearities are of second order. 2) Reduction of the number of degrees of freedom is achieved by means of component modes and the subsequent transformation of the inertia and nonlinear stiffness matrices into the reduced sub- space on substructure level. The component modes for the conden- sation satisfy the continuity between adjacent substructures using unit displacement modes and describe the dynamic behavior within the substructures by hand of internal vibration modes. To account for the local buckling phenomena, additional static modes are used, which are adapted to the nonlinear deformation behavior. 3) The reduced substructure matrices are embedded in a coro- tational formulation for handling the finite rotations of the overall structure.