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Research Papers

An Eigenvalue Problem for the Analysis of Variable Topology Mechanical Systems

[+] Author and Article Information
József Kövecses

Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, QC, H3A 2K6, Canadajozsef.kovecses@mcgill.ca

Josep M. Font-Llagunes

Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montréal, QC, H3A 2K6, Canadafont@cim.mcgill.ca

We name these quantities in accordance with the tradition of continuum mechanics and tensor calculus (14-15).

Based on the interpretation of P̂c and Pc, it can be shown that PcTMPc=GTP̂cG and GTPcTMPcG1=P̂c.

It can be shown that if an n×n matrix is idempotent with rank r then r of its eigenvalues are unity and nr are zero.

If we consider the case of an ordinary particle m1=m2 then we obtain this normal direction, which corresponds to angle α=tan1(y/x).

The situation shown in Fig. 2 corresponds to the case where m2/m1 is greater than 1.

J. Comput. Nonlinear Dynam 4(3), 031006 (May 20, 2009) (9 pages) doi:10.1115/1.3124784 History: Received May 07, 2008; Revised November 07, 2008; Published May 20, 2009

Mechanical systems with time-varying topology appear frequently in natural or human-made artificial systems. The nature of topology transitions is a key characteristic in the functioning of such systems. In this paper, we discuss a concept that can offer possibilities to gain insight and analyze topology transitions. This approach relies on the use of impulsive constraints and a formulation that makes it possible to decouple the dynamics at topology change. A key point is an eigenvalue problem that characterizes several aspects of energy and momentum transfer at the discontinuous topology transition.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A generalized particle approaching and attaching to a circular block

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Figure 2

Coordinate systems at the topology change

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Figure 3

CAD model of the dual-pantograph device, and experimental setup using two devices

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Figure 5

Configuration investigated for topology transition

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Figure 6

Kinetic energy decomposition and measured normal forces, Fn, for γ=−7.58 deg

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Figure 7

Kinetic energy decomposition and measured normal forces, Fn, for γ=0 deg

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