Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters
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 f...requencies, 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:
Viscoelasticity / Nonlocal elasticity / Nanocomposites / Nanobeams / Fractional derivativesSource:
Meccanica, 2017, 52, 1-2, 363-382Publisher:
- Springer, Dordrecht
Funding / projects:
- Dynamics of hybrid systems with complex structures. Mechanics of materials. (RS-MESTD-Basic Research (BR or ON)-174001)
- Dynamic stability and instability of mechanical systems subjected to stochastic excitations (RS-MESTD-Basic Research (BR or ON)-174011)
- Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design (RS-MESTD-Technological Development (TD or TR)-35006)
DOI: 10.1007/s11012-016-0417-z
ISSN: 0025-6455
WoS: 000389603900024
Scopus: 2-s2.0-84962154864
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Cajić, Milan AU - Karličić, Danilo AU - Lazarević, Mihailo PY - 2017 UR - https://machinery.mas.bg.ac.rs/handle/123456789/2527 AB - 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. PB - Springer, Dordrecht T2 - Meccanica T1 - Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters EP - 382 IS - 1-2 SP - 363 VL - 52 DO - 10.1007/s11012-016-0417-z ER -
@article{ author = "Cajić, Milan and Karličić, Danilo and Lazarević, Mihailo", year = "2017", 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.", publisher = "Springer, Dordrecht", journal = "Meccanica", title = "Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters", pages = "382-363", number = "1-2", volume = "52", doi = "10.1007/s11012-016-0417-z" }
Cajić, M., Karličić, D.,& Lazarević, M.. (2017). Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters. in Meccanica Springer, Dordrecht., 52(1-2), 363-382. https://doi.org/10.1007/s11012-016-0417-z
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. in Meccanica. 2017;52(1-2):363-382. doi:10.1007/s11012-016-0417-z .
Cajić, Milan, Karličić, Danilo, Lazarević, Mihailo, "Damped vibration of a nonlocal nanobeam resting on viscoelastic foundation: fractional derivative model with two retardation times and fractional parameters" in Meccanica, 52, no. 1-2 (2017):363-382, https://doi.org/10.1007/s11012-016-0417-z . .