A time-stepping method for multibody systems with frictional impacts based on a return map and boundary layer theory

• Published in: International journal of non-linear mechanics Vol. 131; p. 103683
• Main Authors: Natsiavas, S., Passas, P., Paraskevopoulos, E.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.05.2021; Elsevier BV
• Subjects: Augmented Lagrangian; Boundary layers; Configurations; Coordinate transformations; Equations of motion; Geometric cubic splines; Impact and friction; Mathematical analysis; Multibody dynamics; Multibody systems; Numerical integration; Return map; Stiffness; Multibody dynamics; Return map; Geometric cubic splines; Impact and friction; Augmented Lagrangian
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2021.103683

This work presents a new numerical integration method for determining dynamics of a class of multibody systems involving impact and friction. Specifically, these systems are subject to a set of equality constraints and can exhibit single frictional impact events. Such events are associated to significant numerical stiffness, appearing in the equations of motion. The new method is a time-stepping scheme, involving proper incorporation of a novel return mapping into an augmented Lagrangian formulation, developed recently for systems with bilateral constraints only. Namely, when an impact is detected during a time step, this map is applied at the end of the step in order to bring the system position back to the configuration manifold with the allowable motions. The construction of this map is based on the concept of Jacobi fields on non-flat manifolds. Moreover, once an impact event is detected, the post-impact state is determined by employing a combination of analytical and numerical tools. First, a proper coordinate transformation is performed, bringing the system into a new set of coordinates, which are suitable for describing the impact dynamics. In these coordinates, the dominant dynamics is described by a system of three equations of motion only, which are valid during the short contact interval. In addition, these equations are geometrically discretized by using appropriate cubic splines on the configuration manifold. In this way, the inherent numerical stiffness of the class of systems examined is properly addressed, since it is restricted to a space with a much smaller dimension and a much shorter time scale. Finally, the accuracy and efficiency of the new method is demonstrated by applying it to a selected set of mechanical examples. •Time stepping numerical solution to the general single contact frictionless problem in mechanical systems with bilateral constraints.•Set up of an appropriate weak form in conjunction with application of an augmented Lagrangian formulation.•Application of a novel return map based on the concept of Jacobi fields on non-flat manifolds.•Dominant dynamics is expressed by a system of three ODEs inside a boundary layer.•Geometric discretization within the boundary layer using appropriate cubic splines.