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Rayleigh’s quotients and eigenvalue bounds for linear dynamical systems

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Abstract

The primary objective of this article is to demonstrate that Rayleigh’s quotient and its variants retain the usual properties of boundedness and stationarity even when the linear vibratory system is non-classically damped, extending previously accepted results that these quotients could attain stationarity when damping was proportional or the modal damping matrix was diagonally dominant. This conclusion is reached by allowing the quotients to be defined in complex space and using complex differentiation. A secondary objective is to show how these quotients and their associated eigenvalue problems can be combined to generate bounds on the system’s eigenvalues, an immediate consequence that follows from establishing boundedness and stationarity in complex space. The reported bounds are simple to compute and appear to be tighter than previous bounds reported in the literature.

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Acknowledgements

The authors want to acknowledge Matheus Basílio Rodrigues Fernandes, doctorate candidate at the Aeronautics Institute of Technology, for his comments on an earlier draft of this manuscript.

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The authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Federal University of Rio Grande do Norte, Natal, Rio Grande do Norte, 59078-970, Brazil

    Rubens G. Salsa Jr.

  2. Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN, 47803, USA

    Daniel T. Kawano

  3. Department of Mechanical Engineering, University of California, Berkeley, CA, 94720, USA

    Fai Ma

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  1. Rubens G. Salsa Jr.
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  2. Daniel T. Kawano
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  3. Fai Ma
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Correspondence to Rubens G. Salsa Jr..

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Salsa, R.G., Kawano, D.T. & Ma, F. Rayleigh’s quotients and eigenvalue bounds for linear dynamical systems. Arch Appl Mech 92, 679–689 (2022). https://doi.org/10.1007/s00419-022-02105-5

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