Multibody motion in implicitly constrained director format with links via explicit constraints

A conservative time integration algorithm is developed for constrained mechanical systems of kinematically linked rigid bodies based on convected base vectors. The base vectors are represented in terms of their absolute coordinates, hence the formulation makes use of three translation components, plus nine base vector components for each rigid body.

Both internal and external constraints are considered. Internal constraints are used to enforce orthonormality of the three base vectors by constraining the equivalent Green strain components, while the external constraints are associated with the presence of kinematic joints for linking bodies together.

The equations of motion are derived from Hamilton’s equations with an augmented Hamiltonian in which internal and external constraints initially are included via Lagrange multipliers. Subsequently the Lagrange multipliers associated with internal constraints are eliminated by use of a set of displacementmomentum orthogonality conditions, leaving a set of differential equations in which additional algebraic constraints are needed only for imposing external constraints.

The equations ofmotion are recast into a conservativemean-value and finite difference format based on the finite increment of the Hamiltonian. Examples dealing with a hanging chain represented by a four body linkage serve to demonstrate the efficiency and accuracy of the algorithm.