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Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters

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Abstract

In this paper, we investigate the free damped vibration of a nanobeam resting on viscoelastic foundation. Nanobeam and viscoelastic foundation are modeled using nonlocal elasticity and fractional order viscoelasticity theories. Motion equation is derived using D’Alambert’s principle and involves two retardation times and fractional order derivative parameters regarding to a nanobeam and viscoelastic foundation. The analytical solution is obtained using the Laplace transform method and it is given as a sum of two terms. First term denoting the drift of the system’s equilibrium position is given as an improper integral taken along two sides of the cut of complex plane. Two complex conjugate roots located in the left half-plane of the complex plane determine the second term describing the damped vibration around equilibrium position. Results for complex roots of characteristic equation obtained for a single nanobeam without viscoelastic foundation, where imaginary parts represent damped frequencies, are validated with the results found in the literature for natural frequencies of  a single-walled carbon nanotube obtained from molecular dynamics simulations. In order to examine the effects of nonlocal parameter, fractional order parameters and retardation times on the behavior of characteristic equation roots in the complex plane and the time-response of nanobeam, several numerical examples are given.

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Acknowledgments

This research was sponsored by the research Grants of the Serbian Ministry of Education, Science and Technological Development under the Numbers OI 174001, OI 174011 and TR 35006.

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Authors and Affiliations

  1. Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001, Belgrade, Serbia

    Milan Cajić

  2. Faculty of Mechanical Engineering, University of Niš, A. Medvedeva 14, 18000, Niš, Serbia

    Danilo Karličić

  3. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000, Belgrade, Serbia

    Mihailo Lazarević

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  1. Milan Cajić
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Cajić, M., Karličić, D. & Lazarević, M. Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters. Meccanica 52, 363–382 (2017). https://doi.org/10.1007/s11012-016-0417-z

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