Authors: Cajić, Milan 
Karličić, Danilo 
Lazarević, Mihailo
Title: Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters
Journal: Meccanica
Volume: 52
Issue: 1-2
First page: 363
Last page: 382
Issue Date: 1-Jan-2017
Rank: M22
ISSN: 0025-6455
DOI: 10.1007/s11012-016-0417-z
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.
Keywords: Fractional derivatives | Nanobeams | Nanocomposites | Nonlocal elasticity | Viscoelasticity
Publisher: Springer Link
Project: Dynamics of hybrid systems with complex structures. Mechanics of materials. 
Dynamic stability and instability of mechanical systems subjected to stochastic excitations 
Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design 

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