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Review Articles

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

[+] Author and Article Information
Yuriy A. Rossikhin

Department of Theoretical Mechanics, Voronezh State University of Architecture and Civil Engineering, Voronezh 394006, Russia

Marina V. Shitikova1

Department of Theoretical Mechanics, Voronezh State University of Architecture and Civil Engineering, Voronezh 394006, Russiashitikova@vmail.ru

1

Corresponding author.

Appl. Mech. Rev 63(1), 010801 (Dec 02, 2009) (52 pages) doi:10.1115/1.4000563 History: Received December 04, 2008; Revised October 11, 2009; Published December 02, 2009; Online December 02, 2009

## 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.

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## Figures

Figure 4

The behavior of the complex conjugate roots for a fractional oscillator for Maxwell model (curves located inside the unit circle) and the Kelvin–Voigt model (curves located outside the unit circle)

Figure 5

The behavior of the complex conjugate roots for a fractional oscillator based on the fractional operator model

Figure 6

Vector diagram e″(e′) for the fractional Kelvin–Voigt model 35

Figure 7

Vector diagram e″(e′) for the fractional operator model 36

Figure 11

The behavior of the complex conjugate roots of Eq. 71 in the case of equal fractional parameters

Figure 12

The locus of the complex conjugate roots of Eq. 71 at ξ=0 and α1=α2

Figure 13

The stream-lines of the nonlinear fractional oscillator 92 in the phase plane

Figure 14

Scheme of a linear 2DOF system

Figure 15

Scheme of a nonlinear 2DOF system

Figure 16

Behavior of the roots of the characteristic equation for a 2DOF system based on the generalized Maxwell+Maxwell model (bounded curves) and on the generalized Voigt+Voigt model (unbounded curves) for (a) γ1=0.4,γ2=0.8, (b) γ1=0.8,γ2=0.4, and (c) γ1=γ2=0.8; 1 and 2 indicate the first and second roots, respectively

Figure 17

Dynamic response of the 2DOF system based on the generalized Maxwell+Maxwell model at γ1=0.4,γ2=0.8: (a) the time-dependence of the drift of the equilibrium position and (b) the time-dependence of the displacements of the first mass --- and the second mass—

Figure 18

Dynamic response of the 2DOF system based on the generalized “Voigt+Voigt” model at γ1=0.4, γ2=0.8: (a) the time-dependence of the drift of the equilibrium position and (b) the time-dependence of the displacements of the first mass --- and the second mass—

Figure 19

Phase portrait of a 2DOF system in the case of the one-to-one internal resonance ω0=Ω0=0.3583

Figure 20

Scheme of a cantilever rod

Figure 21

Scheme for determining the roots of the characteristic equation 152 using the basic curve

Figure 22

Root locus plot for the characteristic equation at γ=0.28 and γ=0.25

Figure 23

Scheme of a multibeam system

Figure 24

Scheme of a multiplate system

Figure 25

Scheme of the shock interaction of a rigid body with a viscoelastic Uflyand–Mindlin plate: (a) before interaction, (b) during interaction, and (c) a plan view

Figure 26

Root locus plot of Eq. 201

Figure 27

Suspension bridge scheme

Figure 3

The locus of the complex conjugate roots for a Kelvin–Voigt fractional oscillator

Figure 8

Behavior of the characteristic equation roots for a fractional oscillator based on the generalized fractional derivative standard linear solid model 34 at ω0=1 for ξβ=1∕50 and β=0.8. The figures near the curves denote the magnitudes of the fractional parameter α.

Figure 9

The behavior of the characteristic equation roots for the fractional oscillator: (a) model 42 and (b) model 42

Figure 10

The behavior of the complex conjugate roots of Eq. 71 for α1≠α2

Figure 1

Contour of Integration

Figure 2

Behavior of the complex conjugate roots p1,2=−α±iω for a fractional oscillator based on (a) the fractional derivative Maxwell model and (b) fractional derivative standard linear solid model

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