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A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation

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Abstract

This paper aims to apply a transformation method that replaces the elastic forces of the original equation of motion with a power-form elastic term. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the finite-amplitude damped, forced vibration of a vertically suspended load body supported by incompressible, homogeneous, and isotropic viscohyperelastic elastomer materials. Numerical integrations of the original equations of two oscillators described by neo-Hookean and Mooney–Rivlin viscohyperelastic elastomer material models, and their equivalent equations of motion, are compared to the frequency–amplitude steady-state solutions obtained from the harmonic balance and the averaging methods. It is shown from numerical integrations and approximate steady-state solutions that the equivalent equations predict well the original system dynamic response despite having higher system nonlinearities.

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Acknowledgements

The authors would like to thank financial support from Tecnológico de Monterrey-Campus Monterrey, through the Research Group in Nanotechnology and Devices Design. Also, the authors are thankful to the reviewers for their valuable suggestions that help us to improve the quality of our paper.

Funding

This research was funded by Tecnológico de Monterrey through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, 296176, and National Lab in Additive Manufacturing, 3D Digitizing and Computed Tomography (MADiT) LN299129.

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Authors and Affiliations

  1. Mechanical Engineering and Advanced Materials Department, School of Engineering and Science, Tecnologico de Monterrey, Ave. Eugenio Garza Sada 2501, 64849, Monterrey, Mexico

    Alex Elías-Zúñiga, Santiago Puma-Araujo, Oscar Martínez-Romero & Daniel Olvera Trejo

  2. Tecnológico Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca 42080, Hidalgo, Mexico

    Luis Manuel Palacios-Pineda

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  1. Alex Elías-Zúñiga
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  2. Luis Manuel Palacios-Pineda
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  3. Santiago Puma-Araujo
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  4. Oscar Martínez-Romero
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Correspondence to Alex Elías-Zúñiga.

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Elías-Zúñiga, A., Palacios-Pineda, L.M., Puma-Araujo, S. et al. A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation. Nonlinear Dyn 104, 3065–3075 (2021). https://doi.org/10.1007/s11071-021-06461-9

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