Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

Abstract

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance

“…The results obtained in the field critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice in the state-of-the-art article by Rossikhin and Shitikova [8].…”

“…This set of the differential equations in amplitudes and phases approximates the previous system of equations (8)(9)(10) in the case of the additive combinational resonance…”

“…Thus, utilizing the fractional derivative Kelvin-Voigt model and the procedure proposed in [8] for decoupling the linear parts of differential equations of motion of the cylindrical shell (1)-(3), the system of equations of three predominating modes of vibrations, which could be coupled by some conditions of internal resonance, have been derived in [6,7,9] and written in the following dimensionless form:…”

“…Rossikhin and Shitikova [8] suggested a new idea to model nonlinear damped vibrations of plates and shells in a viscoelastic medium representing viscous resistance forces via fractional order time derivatives, as distinct to the traditional way [14] when damping forces are assumed to be proportional to first order time-derivatives of displacements. It has been shown in [15] that viscoelastic fractional derivative models used for dynamic response of suspension bridges produce better fit of theoretical investigation with experimental results.…”

“…Recently Rossikhin and Shitikova [8] proposed a new approach for studying nonlinear dynamic response of thin fractionally damped cylindrical shells embedded into a fractional derivative viscoelastic medium. The dynamic behavior of the shell is described by a set of three coupled nonlinear differential equations The displacement functions are determined in terms of eigenfunctions of linear vibrations.…”