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Nonsmooth Modal Analysis: Investigation of a 2-DOF Spring-Mass System Subject to an Elastic Impact Law

[+] Author Affiliations
Anders Thorin, Mathias Legrand

McGill University, Montréal, QC, Canada

Stéphane Junca

Université de Nice Sophia-Antipolis, Nice, France

Paper No. DETC2015-46796, pp. V006T10A021; 10 pages
  • ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control
  • Boston, Massachusetts, USA, August 2–5, 2015
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5716-8
  • Copyright © 2015 by ASME


The well-known concept of normal mode for linear systems has been extended to the framework of nonlinear dynamics over the course of the 20th century, initially by Lyapunov, and later by Rosenberg and a growing community of researchers in modal and vibration analysis. This effort has mainly targeted nonlinear smooth systems — the velocity is continuous and differentiable in time — even though systems presenting nonsmooth occurrences have been increasingly studied in the last decades to face the growing industrial need of unilateral contact and friction simulations. Yet, these systems have nearly never been explored from the standpoint of modal analysis.

This contribution addresses the notion of modal analysis of nonsmooth systems. Developments are illustrated on a seemingly simple 2-dof autonomous system, subject to unilateral constraints reflected by a perfectly elastic impact law. Even though friction is ignored, its dynamics appears to be extremely rich. Periodic solutions are sought for given numbers of impacts per period and nonsmooth modes are illustrated for one and two impacts per period in the form of two-dimensional manifolds in the phase space. Also, an unexpected bridge between these modes in the frequency-energy plots is observed.

Copyright © 2015 by ASME



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