A Consistent Dissipative Time Integration Scheme for Structural Dynamics: Application to Rotordynamics

Abstract

In order to integrate the equations of evolution of dynamical systems, one can resort to two families of algorithms: the implicit family and the explicit family. In this paper, we focus on the implicit family. The most widely used implicit algorithm is the Newmark algorithm. For linear models, this algorithm is unconditionally stable if some conditions on the parameters are verified. For non-linear models, Belytschko and Schoeberle, and Hughes proved that the discrete energy is bounded if it remains positive. Nevertheless, Hughes et al. have proved that, in the non-linear range, the Newmark algorithm remains physically consistent only for small time step sizes. The contradiction between these two observations results from the fact that the discrete energy is computed from the work of the internal forces, which is different from the internal potential when the Newmark algorithm is used in the non-linear range. To avoid divergence due to numerical instabilities, numerical damping was introduced, leading to the generalized-α methods. Nevertheless, the unconditional stability of these methods occurs only for linear systems or asymptotically for the high frequency in the non-linear range. Therefore a new kind of dynamics integration algorithms has appeared that verifies the mechanical laws of conservation (i.e. conservation of linear momentum, angular momentum and total energy) and that remains stable in the nonlinear range. The first algorithm (EMCA or Energy Momentum Conserving Algorithm) verifying these properties was described by Simo and Tarnow. It consists in a mid-point scheme with an adequate evaluation of the internal forces. This adequate evaluation (i.e. that leads to a consistent algorithm) was given by Simo and Tarnow for a Saint Venant-Kirchhoff hyperelastic material. A generalization to other hyperelastic models was given by Gonzalez. The extension