scholarly journals Analysis of damped vibrations of thin bodies embedded into a fractional derivative viscoelastic medium

2013 ◽  
Vol 21 (5-6) ◽  
pp. 155-159
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Viscoelastic Medium ◽  
Integral Transform ◽  
Viscous Friction ◽  
Middle Surface ◽  
Thin Body ◽  
Governing Equations ◽  
Transform Method ◽  
External Forces ◽  
The Given

AbstractDamped vibrations of elastic thin bodies, such as plates and circular cylindrical shells, embedded into a viscoelastic medium, the rheological features of which are described by fractional derivatives, are considered in the present article. Besides the forces of viscous friction, a thin body is subjected to the action of external forces dependent on the coordinates of the middle surface and time. The boundary conditions are proposed in such a way that the governing equations allow the Navier-type solution. The Laplace integral transform method and the method of expansion of all functions entering into the set of governing equations in terms of the eigenfunctions of the given problem are used as the methods of solution. It is shown that as a result of such a procedure, the systems of equations in the generalized coordinates could be reduced to infinite sets of uncoupled equations, each of which describes damped vibrations of a mechanical oscillator based on the fractional derivative Kelvin-Voigt model.

Transient electro-osmotic flow of generalized Maxwell fluids in a straight pipe of circular cross section

Open Physics ◽  
2014 ◽  
Vol 12 (6) ◽  
Author(s):  
Shaowei Wang ◽  
Moli Zhao ◽  
Xicheng Li
Keyword(s):  
Fractional Derivative ◽  
Integral Transform ◽  
Capillary Tube ◽  
Circular Cross Section ◽  
Maxwell Fluid ◽  
Navier Stokes Equation ◽  
Transform Method ◽  
Osmotic Flow ◽  
Poisson Boltzmann Equation ◽  
Electro Osmotic Flow

AbstractThe transient electro-osmotic flow of a generalized Maxwell fluid with fractional derivative in a narrow capillary tube is examined. With the help of an integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of relaxation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically and analyzed numerically. The velocity overshoot and oscillation are observed and discussed.

scholarly journals An Investigation of Fractional Bagley–Torvik Equation

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 28 ◽  
Author(s):  
Azhar Ali Zafar ◽  
Grzegorz Kudra ◽  
Jan Awrejcewicz
Keyword(s):  
Fractional Derivative ◽  
Mechanical System ◽  
Caputo Fractional Derivative ◽  
Integral Transform ◽  
Rolling Bearing ◽  
Degree Of Freedom ◽  
Free Oscillations ◽  
Derivative Operator ◽  
Transform Method ◽  
Integral Transform Method

In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

scholarly journals Generalization of Thermal and Mass Fluxes for the Flow of Differential Type Fluid with Caputo–Fabrizio Approach of Fractional Derivative

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Asima Razzaque ◽  
Anam Rani ◽  
Mudassar Nazar
Keyword(s):  
Fractional Derivative ◽  
Flow Model ◽  
Integral Transform ◽  
Research Work ◽  
Vertical Surface ◽  
Physical Parameters ◽  
Governing Equations ◽  
Mass Fluxes ◽  
Generalized Momentum ◽  
Differential Type

In this research work, generalized thermal and mass transports for the unsteady flow model of an incompressible differential type fluid are considered. The Caputo–Fabrizio fractional derivative is used for the respective generalization of Fourier’s and Fick’s laws. A MHD fluid flow is considered near a flat vertical surface subject to unsteady mechanical, thermal, and mass conditions at boundary. The governing equations of flow model are solved by integral transform, and closed form results for generalized momentum, thermal, and concentration fields are obtained. Generalized thermal and mass fluxes at boundary are quantified in terms of Nusselt and Sherwood numbers, respectively, and presented in tabular form. The significance of the physical parameters over the momentum, thermal, and concentration profiles is characterized by sketching the graphs.

scholarly journals The Analysis of Fractional-Order Kersten–Krasil Shchik Coupled KdV System, via a New Integral Transform

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1592
Author(s):  
Nehad Ali Shah ◽  
Asiful H. Seikh ◽  
Jae Dong Chung
Keyword(s):  
Linear System ◽  
Water Waves ◽  
Integral Transform ◽  
Shallow Water Waves ◽  
Transform Method ◽  
Electric Circuits ◽  
Homotopy Perturbation ◽  
Non Linear ◽  
Non Linear System ◽  
The Given

In this article, we use the homotopy perturbation transform method to find the fractional Kersten–Krasil’shchik coupled Korteweg–de Vries (KdV) non-linear system. This coupled non-linear system is typically used to describe electric circuits, traffic flow, shallow water waves, elastic media, electrodynamics, etc. The homotopy perturbation method is modified with the help of the ρ-Laplace transformation to investigate the solution of the given examples to show the accuracy of the current technique. The solution of the given technique and the actual results are shown and analyzed with figures.

scholarly journals Study of Fuzzy Fractional Nonlinear Equal Width Equation in the Sense of Novel Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Muhammad Naeem ◽  
Ahmed A. Khammash ◽  
Ibrahim Mahariq ◽  
Ghaylen Laouini ◽  
Jeevan Kafle
Keyword(s):  
Fractional Derivative ◽  
Fuzzy Theory ◽  
Approximate Solutions ◽  
Transform Method ◽  
Numerical Examples ◽  
Suggested Technique ◽  
Wide Range ◽  
The Laplace Transform ◽  
Uncertainty Parameters ◽  
The Given

In this paper, we designed an algorithm by applying the Laplace transform to calculate some approximate solutions for fuzzy fractional-order nonlinear equal width equations in the sense of Atangana-Baleanu-Caputo derivatives. By analyzing the fuzzy theory, the suggested technique helps the solution of the fuzzy nonlinear equal width equations be investigated as a series of expressions in which the components can be effectively recognised and produce a pair of numerical results with the uncertainty parameters. Several numerical examples are analyzed to validate convergence outcomes for the given problem to show the proposed method’s utility and capability. The simulation outcomes reveal that the fuzzy iterative transform method is an effective method for accurately and precisely studying the behavior of suggested problems. We test the developed algorithm by three different problems. The analytical analysis provided that the results of the models converge to their actual solutions at the integer-order. Furthermore, we find that the fractional derivative produces a wide range of fuzzy results.

Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

1953 ◽  
Vol 20 (4) ◽  
pp. 469-474
Author(s):  
W. A. Nash
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Middle Surface ◽  
Elastic Buckling ◽  
Elastic Instability ◽  
Total Potential ◽  
External Forces ◽  
Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

Reflection of an Acoustic Step Wave From an Elastic Cylinder

1958 ◽  
Vol 25 (1) ◽  
pp. 103-108
Author(s):  
Richard Skalak ◽  
M. B. Friedman
Keyword(s):  
Integral Transform ◽  
Formal Solution ◽  
Elastic Cylinder ◽  
Transform Method ◽  
Pressure Fields ◽  
Plane Step ◽  
Single Integral ◽  
Short Time ◽  
Acoustic Fluid

Abstract An elastic cylinder, circular in section and infinite in length, is considered in an infinite acoustic fluid. The object of this paper is the determination of the reflected and diffracted pressure fields at large distances resulting from a plane step wave of pressure impinging on the cylinder and moving in a direction normal to the axis of the cylinder. A formal solution is obtained for the general case of an elastic cylinder. Numerical results are computed for rigid, fixed cylinders, and for rigid, floating cylinders. Two different methods are used to achieve results in the different ranges of time which are of interest. A short time approximation is developed by the use of a double integral-transform method. A mode approach and a single integral transform are used for later times. The results show that the reflected pulse decays quickly, within a time on the order of the transit time of the original wave across the cylinder.

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