Dynamical stability of viscoelastic column with fractional derivative constitutive relation

2001 ◽  
Vol 22 (3) ◽  
pp. 294-303 ◽  
Author(s):  
Li Gen-guo ◽  
Zhu Zheng-you ◽  
Cheng Chang-jun
Keyword(s):  
Fractional Derivative ◽  
Constitutive Relation ◽  
Dynamical Stability ◽  
Fractional Derivative Constitutive Relation ◽  
Viscoelastic Column

scholarly journals The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Katica (Stevanovic) Hedrih
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Constitutive Relation ◽  
Continuous Functions ◽  
Beam Dynamic ◽  
Fractional Differential ◽  
Elastic Form ◽  
Partial Fractional Differential Equation ◽  
Derivative Order

We considered the problem on transversal oscillations of two-layer straight bar, which is under the action of the lengthwise random forces. It is assumed that the layers of the bar were made of nonhomogenous continuously creeping material and the corresponding modulus of elasticity and creeping fractional order derivative of constitutive relation of each layer are continuous functions of the length coordinate and thickness coordinates. Partial fractional differential equation and particular solutions for the case of natural vibrations of the beam of creeping material of a fractional derivative order constitutive relation in the case of the influence of rotation inertia are derived. For the case of natural creeping vibrations, eigenfunction and time function, for different examples of boundary conditions, are determined. By using the derived partial fractional differential equation of the beam vibrations, the almost sure stochastic stability of the beam dynamic shapes, corresponding to thenth shape of the beam elastic form, forced by a bounded axially noise excitation, is investigated. By the use of S. T. Ariaratnam's idea, as well as of the averaging method, the top Lyapunov exponent is evaluated asymptotically when the intensity of excitation process is small.

Research on Fractional Derivative Viscoelastic Constitutive Relation of Asphalt Mixture

2012 ◽  
Vol 446-449 ◽  
pp. 2560-2566
Author(s):  
Hua Yin ◽  
Yi Li ◽  
Nai Zhou Wang
Keyword(s):  
Mechanical Properties ◽  
Fractional Derivative ◽  
Phase Angle ◽  
Constitutive Relation ◽  
Viscoelastic Model ◽  
Asphalt Mixture ◽  
Maxwell Model ◽  
Dynamic Mechanical Properties ◽  
Dynamic Mechanical ◽  
Definition Of

Based on the definition of fractional derivative, the paper proposed a unique new idea to describe the viscoelastic property of asphalt mixture with fractional calculus. According to the SPT (Simple Performance Tests) test results, the dynamic modulus and phase angle of asphalt mixture were determined. The result of the test was fitted with the classical Kelvin model, the Maxwell model, the solid model with three elements, respectively. It showed that the classical viscoelastic model did not simulate the dynamic mechanical properties of asphalt mixture properly. Since the existing constitutive relation cannot describe well the dynamic viscoelastic properties of asphalt mixture, the fractional derivative viscoelastic model with three elements was adopted and its fitting effect analyzed. The result shown a good fitting for the fractional derivative viscoelastic model with three elements, and a few test parameters were required to build the mode. In addition, these simulating parameters were significant in physics. The order  of the fractional derivative has good correlation with the phase angle, incarnating the viscoelastic proportion of asphalt mixture. So the fractional derivative viscoelastic model with three elements can accurately describe the dynamic mechanical properties of asphalt mixture.

MIXED FRACTIONAL DIFFERENTIATION OPERATORS IN HÖLDER SPACES DEFINED BY USUAL HÖLDER CONDITION

2019 ◽  
Author(s):  
T. Mamatov ◽  
R. Sabirova ◽  
D. Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Differentiation ◽  
Hölder Condition ◽  
Hölder Spaces ◽  
Main Interest ◽  
Hölder Class ◽  
Holder Class ◽  
Function Of Two Variables ◽  
Holder Spaces

We study mixed fractional derivative in Marchaud form of function of two variables in Hölder spaces of different orders in each variables. The main interest being in the evaluation of the latter for the mixed fractional derivative in the cases Hölder class defined by usual Hölder condition

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