A geometric solution to the general single contact frictionless problem by combining concepts of analytical dynamics and b-calculus. (October 2017)

• Record Type: Journal Article
• Authors: Paraskevopoulos, E.; Natsiavas, S.
• Is Part Of: International journal of non-linear mechanics. Volume 95(2017)
• Page Start: 117
• Page End: 131
• Publication Date: 2017-10
• Subjects: Analytical dynamics -- Unilateral constraint -- Frictionless contact -- Manifold with boundary; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2017.05.007
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: This study presents a systematic approach, leading to a new set of equations of motion for a class of mechanical systems subject to a single frictionless contact constraint. To achieve this goal, some fundamental concepts of b-geometry are utilized and adapted to the general framework of Analytical Dynamics. These concepts refer to the theory of manifolds with boundary and provide a suitable and strong theoretical foundation. First, the boundary is defined within the original configuration manifold of the system by the equality in the unilateral constraint. Then, an appropriate vector bundle is considered, involving only smooth vector fields, even at the boundary. After determining the essential geometric properties (i.e., the metric and the connection) near the boundary, Newton's law of motion is applied. In this way, the equations of motion during the contact phase are derived as a system of ordinary differential equations. These equations possess a special form inside a thin boundary layer. In particular, the essential dynamics of the systems examined is found to be governed by a single second order ordinary differential equation, which is investigated fully. Moreover, a critical comparison of the present formulation with the classical formulations applied to systems with a frictionless contact is performed. Finally, the effect of the dominant parameters on the dynamics during the contact phase and the steps for the application process to mechanical systems are illustrated by two selected examples, referring to contact of a particle and a rigid body with a plane. Highlights: Geometric solution to the general single contact frictionless problem in mechanical systems. Relation between mechanical systems involving contact and manifolds with boundary. Evaluation of geometric properties (i.e., metric and connection) for manifolds with boundary. Dominant dynamics is expressed by a single ODE inside a boundary layer. Solution varies rapidly along the normal to the boundary of the configuration space. Exact solution permits a thorough examination of the effect of the dominant parameters.