Symplectic Integration of Bar Linkage Motion Equations

Abstract

In this study, the motion and the qualitative behaviour of an unconstrained simple bar-linkage which is under the influence of different types of forces is studied. Standard non-symplectic integration methods are not suited for integrating systems of differential equations over long periods of time. Therefore, symplectic integration algorithms based on the discretization of the Hamilton variational principle are used. Unconstrained multi-body linkages obey to complex motion equations that require adequate numerical integrations. One non-symplectic integration method – the forward Euler method- as well as three different symplectic methods –the discrete midpoint Lagrangian method, the discrete trapezoidal Lagrangian method and the Newmark-β method have been implemented. The advantages of symplectic integrators over non-symplectic integrators for this class of systems including stability, proper qualitative behaviour and energy conservation. The results and the performances of both symplectic and non-symplectic integration methods are analysed and compared, in terms of energy, movement and computational costs. These numerical experiments suggest the use of symplectic methods for conservative and near-conservative multi-body mechanical systems, whereas dissipative systems in general do not require the use of symplectic methods to provide accurate computations.