Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems

Rossikhin, Yury A.; Shitikova, Marina V.

doi:10.1007/bf01174319

Cited by 175 publications

(90 citation statements)

References 11 publications

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“…The solution to the governing equations is based on the method presented by Beyer and Kempfle (1995) or by Rossikhin and Shitikova (1997). The solution is achieved with the aid of a convolution integral of fractional Green's function and the forcing function.…”

Section: Discussionmentioning

confidence: 99%

“…The function g(p) has only two poles lying in the left complex plane (Beyer and Kempfle, 1995;Rossikhin and Shitikova, 1997). Moreover, the function g(p) has a branch point at the origin, and the branch cut follows the negative real axis, then the negative real half-axis have to be encircled by the integration contour.…”

Section: A Appendixmentioning

confidence: 99%

“…(2.13) can be found by the inverse Laplace transform of the expression below (Bagley and Torvik, 1983b;Beyer and Kempfle, 1995;Rossikhin and Shitikova, 1997;Podlubny, 1999) …”

Section: A Appendixmentioning

confidence: 99%

“…In some cases of vibration analysis, K 2n could be neglected in comparison with K 1n (Kempfle et al, 2002;Rossikhin and Shitikova, 1997). …”

Section: Problem Formulationmentioning

confidence: 99%

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Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load

Freundlich

2016

jtam

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In the paper, the dynamic response of a simply supported viscoelastic beam of the fractional derivative type to a moving force load is studied. The Bernoulli-Euler beam with the fractional derivative viscoelastic Kelvin-Voigt material model is considered. The RiemannLiouville fractional derivative of the order 0 < α 1 is used. The forced-vibration solution of the beam is determined using the mode superposition method. A convolution integral of fractional Green's function and forcing function is used to achieve the beam response. Green's function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second term describes the drift of the equilibrium position. The solution is obtained analytically whereas dynamic responses are calculated numerically. A comparison between the results obtained using the fractional and integer viscoelastic material models is performed. Next, the effects of the order of the fractional derivative and velocity of the moving force on the dynamic response of the beam are studied. In the analysed system, the effect of the term describing the drift of the equilibrium position on the beam deflection is negligible in comparison with the first term and therefore can be omitted. The calculated responses of the beam with the fractional material model are similar to those presented in works of other authors.

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Section: Discussionmentioning

confidence: 99%

Section: A Appendixmentioning

confidence: 99%

“…(2.13) can be found by the inverse Laplace transform of the expression below (Bagley and Torvik, 1983b;Beyer and Kempfle, 1995;Rossikhin and Shitikova, 1997;Podlubny, 1999) …”

Section: A Appendixmentioning

confidence: 99%

“…In some cases of vibration analysis, K 2n could be neglected in comparison with K 1n (Kempfle et al, 2002;Rossikhin and Shitikova, 1997). …”

Section: Problem Formulationmentioning

confidence: 99%

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Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load

Freundlich

2016

jtam

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“…It has been found that the differential equations involving fractional derivatives in time are more realistic when it comes to describing many phenomena in practical cases than those of integer order in time. For instance, the fractional calculus concepts have been used in the modeling of neurons [23] and viscoelastic materials [27]. Other examples from fractional order dynamics can be found in [1,9,12,20,25,26] and the references therein.…”

Section: Introductionmentioning

confidence: 99%