The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10 years 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. DOI: 10.1115/1.4000563

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

Review: On the fractional signals and systems

The Legendre wavelet method for solving fractional differential equations

Application of generalized differential transform method to multi-order fractional differential equations

A brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of the fractional derivatives. We will show that the computation of the impulse and step responses is very similar to the classic. The main difference lies in the substitution of the exponential by the Mittag-Leffler function. We will present also the main formulae defining the fractional derivatives.

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An introduction to the fractional continuous-time linear systems: the 21st century systems

The integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator. The solution is obtained in terms of Mittag–Leffler functions using Laplace transforms. The expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained. Numerical application and the phase plane representation of the dynamics are discussed.

Dynamics of the fractional oscillator

The integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus. The Green's function solution for the fractional oscillator is obtained in terms of Mittag–Leffler functions using Laplace transforms. The response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function.

Response characteristics of a fractional oscillator

A study of sinusoidally forced oscillations of a fractional oscillator shows that the system exhibits a rich variety of damping characteristics. While some aspects of the damping mimic the characteristic features of a damped harmonic oscillator, there are others, which do not find any parallel in the damped harmonic oscillator system. It is clearly demonstrated that the “free” and “forced” oscillations of a fractional oscillator are characterized by different damping parameters. While both depend on the fractional index α , the “free” oscillation damping depends on the “natural frequency”, ω 0 , of the oscillator, the “forced” oscillation damping depends in addition, on the “driving frequency”, ω . Furthermore, there is a different power-law tail associated with each of these cases.

Damping characteristics of a fractional oscillator

Mittag–Leffler functions and transmission lines

The well-known equations pertaining to an infinitely long capacitor–resistor transmission line are recast following the methods of fractional calculus and the transmission characteristics expressed in terms of Mittag–Leffler functions. The response to various input driving functions will be shown.

Ted Clarke

Papers

Dynamics of the fractional oscillator

Response characteristics of a fractional oscillator

Damping characteristics of a fractional oscillator

Mittag–Leffler functions and transmission lines

Mittag-Leffler Functions and Transmission Lines