Research Papers

A Boundary Layer Approach to Multibody Systems Involving Single Frictional Impacts

[+] Author and Article Information
S. Natsiavas

Department of Mechanical Engineering,
Aristotle University,
Thessaloniki 541 24, Greece
e-mail: natsiava@auth.gr

E. Paraskevopoulos

Department of Mechanical Engineering,
Aristotle University,
Thessaloniki 541 24, Greece

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 17, 2018; final manuscript received October 5, 2018; published online November 19, 2018. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 14(1), 011002 (Nov 19, 2018) (16 pages) Paper No: CND-18-1316; doi: 10.1115/1.4041775 History: Received July 17, 2018; Revised October 05, 2018

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

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Grahic Jump Location
Fig. 1

(a) Original configuration manifold M and constrained manifold X, for a system subject to a unilateral constraint. (b) Magnification around a boundary point p.

Grahic Jump Location
Fig. 2

An x-basis and horizontal part HbS* of cotangent space Tbp*X at a boundary point p

Grahic Jump Location
Fig. 3

Collision of a four-particle system with a rigid wall

Grahic Jump Location
Fig. 4

Collision of a pendulum with a rigid plane

Grahic Jump Location
Fig. 5

Slip trajectories of a spherical pendulum for (a) θ=π/4 and k=1; (b) θ=π/4 and k=0.1; (c) θ=π/4 and k=10; and (d) θ=π/3 and μ=μc

Grahic Jump Location
Fig. 6

Effect of friction coefficient on the rebound speed of a spherical pendulum for (a) k=0.1 and k=1, and (b) k=10 and comparison with results of the Darboux–Keller approach

Grahic Jump Location
Fig. 7

Collision of a double planar pendulum with a rigid wall

Grahic Jump Location
Fig. 8

Energy loss during impact of a double pendulum against a rough half space for (a) c=0 and (b) c>0



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