Abstract
SUMMARYMechanical systems are usually subjected not only to bilateral constraints, but also to unilateral constraints. Inspired by the generalized‐ α time integration method for smooth flexible multibody dynamics, this paper presents a nonsmooth generalized‐ α method, which allows a consistent treatment of the nonsmooth phenomena induced by unilateral constraints and an accurate description of the structural vibrations during free motions. Both the algorithm and the implementation are illustrated in detail. Numerical example tests are given in the scope of both rigid and flexible body models, taking account for both linear and nonlinear systems and comprising both unilateral and bilateral constraints. The extended nonsmooth generalized‐ α method is verified through comparison to the traditional Moreau–Jean method and the fully implicit Newmark method. Results show that the nonsmooth generalized‐ α method benefits from the accuracy and stability property of the classical generalized‐ α method with controllable numerical damping. In particular, when it comes to the analysis of flexible systems, the nonsmooth generalized‐ α method shows much better accuracy property than the other two methods. Copyright © 2013 John Wiley & Sons, Ltd.
Highlights
The equations of motion for a mechanical system with bilateral constraints can be characterized by a mixed set of second order differential and algebraic equations
Considering that many mechanical systems are subjected to both bilateral constraints and unilateral constraints, an extension of the generalized- ̨ method is proposed to account for the nonsmooth phenomena induced by unilateral constraints
Numerical tests are studied and results show that the Moreau–Jean, the fully implicit Newmark and the nonsmooth generalized- ̨ methods are valid when unilateral constraints are expressed at the velocity level
Summary
The equations of motion for a mechanical system with bilateral constraints can be characterized by a mixed set of second order differential and algebraic equations. Within the framework of structural dynamics with rigid and flexible bodies, this paper proposes to take the best of the generalized- ̨ method and the Moreau–Jean scheme by keeping the integration rule of the generalized- ̨ scheme for the nonlinear smooth dynamics; the aim is to enjoy the stability, the robustness and the controlled numerical dissipation at high frequencies; dealing, as in the Moreau–Jean scheme, with the contact forces and impacts through their associated impulses; in this way, the consistency of the evaluation of the contact reactions is ensured when the time-step vanishes and an impact occurs; treating, as in the Moreau–Jean scheme, the unilateral constraint at the velocity level together with the impact law; this yields an energetically consistent treatment of the jumps in velocity together with a correct stabilization of the velocity at contact.
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