Nonlinear dynamics of rigid and flexible multibody systems

A unified numerical approach for dynamics modeling of multibody systems with rigid and flexible links is proposed. The dynamic equations are obtained with respect to a minimal set of generalized coordinates that describe the parameters of gross relative motion of adjacent bodies and their small elastic deformations. The procedure eliminates the constrained equations and consists of the following stages: structural decomposition of elastic links into fictitious rigid bodies connected by joints in which small force-dependent relative displacements are achieved; kinematic analysis; and deriving explicit form dynamic equations. The algorithm is developed for elastic slender beams and finite elements, achieving spatial motion with three translations and three rotations in each node. The beam elements are basic design units in mechanical devices such as space station antennae and manipulators and cranes, undergoing three-dimensional motion with large elastic deflections that cannot be neglected or linearized. Stiffness coefficients and inertia parameters of the fictitious links are calculated using well-known numerical procedures of finite element theory. Their equivalence with procedures of recently developed finite segment approaches is shown, presented as a finite element approach in relative coordinates. Elastic beam elements are presented as insubstantial links and at least six joints with linear elastic springs with coefficients that are obtained from the stiffness matrix of the beam element. Calculations for stiffness coefficients and lengths of insubstantial joints and links are presented. Inertia characteristics of flexible elements are reduced to the nodes that do not coincide with the centers of insubstantial links. Basic numerical data are obtained at the stage of kinematic analysis. Matrix methods are used for solution of nonlinear kinematic constraint equations, and a recursive numerical algorithm is applied for computing partial derivatives of motion characteristics with respect to generalized coordinates. These results are used for solution of nonlinear static problems, as well as for deriving explicit configuration space dynamic equations, and using the principle of virtual work and Euler-Lagrange equations. Inertia forces in nodes of elastic links and rigid body mass centers are defined. The problem defined allows computational methods for multibody system analysis to be successfully applied.