Study exponential and polynomial stability of Timoshenko beam with boundary dissipative conditions of fractional derivative type

2022 ◽  
Author(s):  
C. Messikh ◽  
S. Labidi
Keyword(s):  
Fractional Derivative ◽  
Timoshenko Beam ◽  
Polynomial Stability ◽  
Derivative Type

scholarly journals Energy decay for a degeneratewave equation under fractional derivative controls

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6045-6072
Author(s):  
Abbes Benaissa ◽  
Chahira Aichi
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Boundary Control ◽  
Semigroup Theory ◽  
Energy Decay ◽  
Linear Operators ◽  
Polynomial Stability ◽  
One Dimensional ◽  
Spectrum Method ◽  
Derivative Type

In this article, we consider a one-dimensional degenerate wave equation with a boundary control condition of fractional derivative type. We show that the problem is not uniformly stable by a spectrum method and we study the polynomial stability using the semigroup theory of linear operators.

Free Vibration of Elastic Timoshenko Beam on Fractional Derivative Winkler Viscoelastic Foundation

2011 ◽  
Vol 368-373 ◽  
pp. 1034-1037 ◽  
Author(s):  
Qi Fang Yan ◽  
Zi Ping Su
Keyword(s):  
Free Vibration ◽  
Fractional Derivative ◽  
Shear Deformation ◽  
Timoshenko Beam ◽  
Shape Factor ◽  
Viscoelastic Foundation ◽  
Deformation Function ◽  
Numerical Example ◽  
Foundation Model ◽  
Derivative Order

The fractional derivative Winkler viscoelastic foundation model is established by introducing the concept of fractional derivative. The control equations of free vibration of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are also built by considering the shear deformation and rotary inertia, and the control equations of elastic Timoshenko beam are decoupled by using the deformation function and considering the properties of fractional derivative, and the expressions of deflection and section corner of elastic Timoshenko beam on fractional derivative Winkler viscoelastic foundation are obtained. The influences of fractional derivative order and shear shape factor on the free vibration of elastic Timoshenko beam are discussed by numerical example.

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.

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