Closed Form Constraint Equations Used to Express Frictionless Slip of Multibody Systems Attached to Finite Elements—Application to a Contact between a Double Pendulum and a Beam

This paper focuses on the numerical modeling of the dynamics of mechanical systems. Robots that can inspect high-voltage lines inspired this research. Their control systems must anticipate potential grab positions appropriately. We intend to formulate equations dedicated to the numerical description of the robot/cable contact. The investigated problem is not straightforward, since parts of the modeled systems are numerically inhomogeneous. They consist of multibody and finite element components. These components interact with each other only through frictionless point contact. We limit the present investigation to the mathematical modeling of these frictionless point connections. According to the model-adopted assumption, the location of the contact point is invariant in the multibody structure, but it is variable in the finite elements part. Unlike the classically used models (i.e., spring/damper models of elastic contacts), we focus on constraint equations. We present and discuss their details in this paper. Following the presence of the constraint equations, their associated Lagrange multipliers appear in the dynamics equations of the two sub-models. The main feature/result of the presently proposed method is the closed form of the coordinate-portioning formulae, proposed in this paper, employed to eliminate the dependent coordinates and the constraint-associated Lagrange multipliers. To verify the applicability of the proposed elimination methodology, we test its use in a dedicated numerical example. During the test, we limit the investigation to a frictionless connection between a double pendulum and a beam. The results confirm that the proposed methodology allows us to model the investigated frictionless contact. We shall underline a vital property, that the proposed elimination method is universal, and thus one can easily extend/modify the above methodology to operate with other multibody/finite element contacts.