Stability of an elastic rod on a fractional derivative type of foundation

2004 ◽  
Vol 277 (1-2) ◽  
pp. 149-161 ◽  
Author(s):  
T.M. Atanackovic ◽  
B. Stankovic
Keyword(s):  
Fractional Derivative ◽  
Elastic Rod ◽  
Derivative Type

Dynamic Interaction of Viscoelastic Soil and Fractional Derivative Type Lining with a Circular Tunnel

2012 ◽  
Vol 170-173 ◽  
pp. 1542-1545
Author(s):  
Min Jie Wen ◽  
Zi Ping Su ◽  
Hui Tuan He
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Dynamic Interaction ◽  
Constitutive Behavior ◽  
Steady State Response ◽  
Circular Tunnel ◽  
Resonance Effects ◽  
Material Parameters ◽  
State Response ◽  
Derivative Type

Coupled harmonic vibration of viscoelastic soil and fractional derivative type lining system with a deeply buried circular tunnel is investigated in the frequency domain. Based on theory of elastic and fractional derivative, steady state response of the viscoelastic soil and lining system is studied. Regarding the lining as a medium with fractional derivative constitutive behavior, and the analytical expressins of the displacement and stress of the soil and lining are respectively obtained by the continuity conditions on the inner boundary of lining and the interface between the soil and the lining. The order of fractional derivative model has a greater influence on system dynamic response, and it dependent on the material parameters of lining. With the frequency increasing, the resonance effects of system decrease.

Vertical Vibration Amplification of a Saturated Fractional Derivative Type Viscoelastic Soil Layer

2012 ◽  
Vol 170-173 ◽  
pp. 1142-1146
Author(s):  
Min Jie Wen ◽  
Hui Tuan He
Keyword(s):  
Fractional Derivative ◽  
Layer Thickness ◽  
Water Pressure ◽  
Soil Layer ◽  
Vertical Vibration ◽  
Saturated Soil ◽  
Amplification Coefficient ◽  
Geometrical Parameters ◽  
Derivative Type ◽  
Soil Skeleton

Regarding the soil skeleton as viscoelastic medium with fractional derivative constitutive behavior, the influences of the soil skeleton viscosity and the soil layer thickness of saturated fractional derivative viscoelastic soil layer on the vertical vibration amplification coefficient is studied in the frequency domain by using the theory of Biot and one dimensional wave. The analytical expressions of the displacement, stress and pore water pressure of saturated soil layer are obtained by decoupling dynamic control equations and bounding the soil layer boundary conditions. The influences of physical and geometrical parameters of the saturated soil on vertical vibration amplification are investigated, and it is revealed that the vertical vibration amplification of the saturated classic elastic, fractional derivative type viscoelastic saturated soil and saturated classic viscoelastic soil are different when the soil layer thickness are changed; the material parameters of the fractional derivative model have great influences on the vertical vibration amplification coefficient.

Dynamics of a Fractional Derivative Type of a Viscoelastic Rod With Random Excitation

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Teodor Atanacković ◽  
Marko Nedeljkov ◽  
Stevan Pilipović ◽  
Danijela Rajter-Ćiri
Keyword(s):  
Fractional Derivative ◽  
Weak Solutions ◽  
Constitutive Equations ◽  
Fractional Derivatives ◽  
Random Excitation ◽  
Bounded Noise ◽  
White Noise Process ◽  
Derivative Type ◽  
Axial Vibrations ◽  
Viscoelastic Rod

AbstractThe axial vibrations of a viscoelastic rod with a body attached to its end are investigated. The problem is modelled by the constitutive equations with fractional derivatives as well as with the perturbations involving a bounded noise and a white noise process. The weak solutions for the equations given below in two cases of constitutive equations as well as their stochastic moments are determined.

scholarly journals A generalization of truncated M-fractional derivative and applications to fractional differential equations

2020 ◽  
Vol 5 (1) ◽  
pp. 171-188 ◽  
Author(s):  
Esin İlhan ◽  
İ. Onur Kıymaz
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integer Order ◽  
Derivative Type ◽  
Fractional Differential

AbstractIn this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

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