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Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

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Civil Engineering Topics, Volume 4

Abstract

A nonlinear structure will often respond periodically when it is excited with a sinusoidal force. Several methods are available that can compute the periodic response for various drive frequencies, which is analogous to the frequency response function for a linear system. The simplest approach would be to compute a sequence of simulations where the equations of motion are integrated until damping drives the system to steady state, but that approach suffers from a number of drawbacks. Recently, numerical methods have been proposed that use a solution branch continuation technique to find the free response of unforced, undamped nonlinear systems for different values of a control parameter. These are attractive because they are built around broadly applicable time-integration routines, so they are applicable to a wide range of systems. However, the continuation approach is not typically used to calculate the periodic response of a structural dynamic system to a harmonic force. This work adapts the numerical continuation approach to find the periodic, forced steady-state response of a nonlinear system. The method uses an adaptive procedure with a prediction step and a mode switching correction step based on Newton-Raphson methods. Once a branch of solutions has been computed, it explains how a full spectrum of harmonic forcing conditions affect the dynamic response of the nonlinear system. The approach is developed and applied to calculate nonlinear frequency response curves for a Duffing oscillator and a low order nonlinear cantilever beam.

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Authors and Affiliations

  1. Department of Engineering Physics, University of Wisconsin-Madison, 534 Engineering Research Building 1500 Engineering Drive, Madison, WI, 53706, USA

    Michael W. Sracic (Graduate Research Assistant, Ph.D Candidate) & Matthew S. Allen (Assistant Professor)

Authors
  1. Michael W. Sracic
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  2. Matthew S. Allen
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Correspondence to Michael W. Sracic .

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  1. Society for Experimental Mechanics, School Street 7, BETHEL, 06801-1405, Connecticut, USA

    Tom Proulx

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Sracic, M.W., Allen, M.S. (2011). Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems. In: Proulx, T. (eds) Civil Engineering Topics, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9316-8_5

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  • DOI: https://doi.org/10.1007/978-1-4419-9316-8_5

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