# Conservative rigid body dynamics by convected base vectors with implicit constraints

Steen Krenk, Martin Bjerre Nielsen

Research output: Contribution to journalJournal articleResearchpeer-review

## Abstract

A conservative time integration formulation is developed for rigid bodies based on a convected set of orthonormal base vectors. The base vectors are represented in terms of their absolute coordinates, and thus the formulation makes use of three translation components, plus nine components of the base vectors. Orthogonality and unit length of the base vectors are imposed by constraining the equivalent Green strain components, and the kinetic energy is represented corresponding to rigid body motion. The equations of motion are obtained via Hamilton’s equations including the zero-strain conditions as well as external constraints via Lagrange multipliers. Subsequently, the Lagrange multipliers associated with the internal zero-strain constraints are eliminated by use of a set of orthogonality conditions between the generalized displacements and the momentum vector, leaving a set of differential equations without additional algebraic constraints on the base vectors. A discretized form of the equations of motion is obtained by starting from a finite time increment of the Hamiltonian, and retracing the steps of the continuous formulation in discrete form in terms of increments and mean values over each integration time increment. In this discrete form the Lagrange multipliers are given in terms of a representative value within the integration time interval, and the equations of motion are recast into a conservative mean-value and finite difference format. The Lagrange multipliers are eliminated explicitly within each integration interval leaving a projection operator expressed in terms of displacement component mean values. Hereby the number of variables is reduced by six for each rigid body in the problem, and the difference equations lead to conservation of the orthonormality conditions for the local base vectors. Examples demonstrate the efficiency and accuracy of the procedure.
Original language English Computer Methods in Applied Mechanics and Engineering 269 437-453 0045-7825 https://doi.org/10.1016/j.cma.2013.10.028 Published - 2014

## Keywords

• Rigid body dynamics
• Conservative time integration
• Implicit constraints
• Hamiltonian dynamics