Mechanical integrators for nonlinear flexible multibody dynamics

The modeling of flexible multibody systems in nonlinear structural dynamics as finite dimensional Hamiltonian system subject to holonomic constraints constitutes a general framework for a unified treatment of rigid and elastic components. Internal constraints, which are associated with the kinematic assumptions of the underlying continuous theory, as well as external constraints, representing the interconnection of different bodies by joints, can be accounted for in a likewise systematic way.

The formalism provides the possibility to use different methods for the constraint enforcement (cf. [

1

]), which differ significantly in the following categories: dimension of the system of nonlinear equations, condition number of the iteration matrix during the iterative solution procedure, exactness of the constraint fulfilment and computational costs. The discrete null space method developed in [

2

] provides an integration scheme for the constrained equations of motion, which has proven to perform excellently in the mentioned categories. It relies on the elimination of the constraint forces (comprising the Lagrange multipliers) from the energy-momentum conserving time stepping scheme, emanating from the underlying DAEs by direct temporal discretisation. This is accomplished via the premultiplication of the discrete scheme by a null space matrix, which spans a basis of the null space of the discrete constraint Jacobian.

A six-body-linkage possessing a single degree of freedom, addressed in [

3

], is analysed as an example. The treatment of this closed loop structure by the discrete null space method results in the solution of only one scalar equation of motion. Furthermore it circumvents the involved investigation of dependent constraints, which lead to rank-deficiency of the constraint Jacobian occurring in the time stepping scheme pertaining to the direct temporal discretisation of the DAEs.