Improved stability and transient behaviour of generalized-alpha time integrators for constrained flexible systems

MARTIN ARNOLD; OLIVIER BRULS; ALBERTO CARDONA

Lieja

Conferencia; Fifth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2011); 2011

The dynamics of flexible multibody systems with large rotations may be studied convenientlyin a Lie group setting, see [6, 3]. For this problem class, the equations of motion form a secondorder differential-algebraic equation (DAE) on a k - m dimensional submanifold  {q in  G : Phi(q) = 0} of a k-dimensional manifold G with Lie group structure. In a finite element approach, q in G may represent the set of nodal translations and rotations of the system, as discussed in [3].For time integration, a generalized-alpha method with numerical damping parameter rho_inf in [0; 1) is used [5]. These methods may be extended to the Lie group formulation such that parameterizations of the Lie group G are not only avoided in the formulation of the equations of motion but also in time integration [3]. A potential drawback of the method are spurious transient oscillations of the constrained forces that are already well known from the application to constrained systems in (linear) vector spaces.In the present paper, we give a detailed mathematical analysis of these oscillations that are illustrated in Fig. 1 by simulation results for a heavy top [6] being integrated by the generalized-alpha method with rho_inf = 0.9 and rho_inf = 0.6, respectively. We apply the method to flexible multibody systems and use the results of the theoretical analysis to avoid the spurious oscillations by perturbed initial values or by analytical transformations of the equations of motion before time discretization (index reduction).