Abstract
To understand and analyze the behavior of realistic nonlinear structures it is desirable to reduce the dimensionality of the system, as well as simplify the equation of motion. Reduction to Spectral submanifolds (SSMs) has recently been shown to provide such a dimension reduction, yielding exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time-dependent SSMs. We demonstrate our analytical formulae on numerical examples and compare them to results obtained from alternative methods.
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Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.: Nonlinear normal modes, Part I: a useful framework for the structural dynamicest. Mech. Syst. Signal Process. 23, 170–194 (2009)
Jiang, D., Pierre, C., Shaw, S.: Nonlinear normal modes for vibratory systems under harmonic excitation. J. Sound Vib. 288, 791–812 (2005)
Gabale, A., Sinha, S.: Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds. J. Sound Vib. 330, 2596–2607 (2011)
Jezequel, L., Lamarque, C.H.: Analysis of non-linear dynamical systems by the normal form theory. J. Sound Vib. 149(3), 429–459 (1991)
Neild, S., Wagg, D.: Applying the method of normal forms to second-order nonlinear vibration problems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 467, 1141–1163 (2015)
Peeters, M., Kerschen, G., Golinval, J.-C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330(3), 486–509 (2011)
Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dyn. 86(3), 1493–1534 (2016)
Cabré, X., Fontich, E., de la Llave, R.: The parameterization method for invariant manifolds i: manifolds associated to non-resonant subspaces. Univ. Indiana Math. J. 52, 283–328 (2003)
Haro, A., de la Llave, R.: A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differ. Equ. 228(2), 530–579 (2006)
Szalai, R., Ehrhardt, D., Haller, G.: Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations. Proc. R. Soc. A. 473, (2017)
Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. J. Sound Vib. 298(4), 958–981 (2006)
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© 2019 The Society for Experimental Mechanics, Inc.
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Breunung, T., Haller, G. (2019). Time-varying Spectral Submanifolds: Analytic Calculation of Backbone Curves and Forced Response. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-74280-9_12
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DOI: https://doi.org/10.1007/978-3-319-74280-9_12
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