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Modeling for Nonlinear Vibrational Response of Mechanical Systems

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The Art of Modeling Mechanical Systems

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 570))

Abstract

This chapter describes the modeling of mechanical systems in order to describe their nonlinear vibratory response, focusing on free vibration and frequency response near resonance. Some general thoughts about modeling are offered first, along with some archetypical models for nonlinear vibration, motivated by physical examples. We focus on considerations related to the inclusion of nonlinearities in system models and how nonlinearity affects the ability of a model to describe system response. Methods for analyzing nonlinear vibration systems are not emphasized, and only the minimum tools required to demonstrate the results of interest are introduced. Models with one and two modes with polynomial nonlinearities are considered, since these offer descriptions of generic behaviors that are quite common and qualitatively distinct from linear system response. Sample physical systems that are used to demonstrate the main ideas include the simple pendulum, transverse vibrations of beams, capacitively driven micro-systems, the spherical pendulum, and a spring-pendulum system. The chapter closes with some general thoughts about nonlinear system modeling and its use in design.

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Acknowledgments

The lead author is grateful to the organizers of the CISM course on The Art of Modeling Mechanical Systems, in particular, Professor Freidrich Pfieffer, for inviting him to contribute to the lectures and to this volume. The authors are grateful to Professor Mark Dykman of Michigan State University for discussions that were particularly helpful in the preparation of this chapter. The authors’ current related work on nonlinear vibrations is supported by the US National Science Foundation under grants 1234067 and 1100260, by US ARO grant W911NF-12-1-0235, managed by Dr. Samuel Stanton, and by DARPA grant FA8650-13-1- 7301, Mesodynamic Architectures (MESO), managed by Dr. Daniel Green.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Michigan State University, East Lansing, Mi, 48824, USA

    Steven W. Shaw, Oriel Shoshani & Pavel M. Polunin

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  1. Steven W. Shaw
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  2. Oriel Shoshani
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  3. Pavel M. Polunin
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Correspondence to Steven W. Shaw .

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Editors and Affiliations

  1. Institute for Applied Mechanics, Technical University of Munich Institute for Applied Mechanics, Garching, Bayern, Germany

    Friedrich Pfeiffer

  2. Institute of Robotics, Johannes-Kepler-University Linz Institute of Robotics, Linz, Austria

    Hartmut Bremer

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Shaw, S.W., Shoshani, O., Polunin, P.M. (2017). Modeling for Nonlinear Vibrational Response of Mechanical Systems. In: Pfeiffer, F., Bremer, H. (eds) The Art of Modeling Mechanical Systems. CISM International Centre for Mechanical Sciences, vol 570. Springer, Cham. https://doi.org/10.1007/978-3-319-40256-7_5

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  • DOI: https://doi.org/10.1007/978-3-319-40256-7_5

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