scholarly journals Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations

2018 ◽  
Vol 2 (3) ◽  
pp. 21 ◽  
Author(s):  
Guy Eyebe ◽  
Gambo Betchewe ◽  
Alidou Mohamadou ◽  
Timoleon Kofane
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Nonlinear Vibration ◽  
Fractional Order ◽  
Nonlocal Elasticity ◽  
Multiple Scales ◽  
Approximate Analytical Solution ◽  
Nonlocal Elasticity Theory ◽  
Pasternak Foundation ◽  
D’Alembert Principle

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

scholarly journals Nonlinear vibration of a nonlocal nanobeam resting on fractional order viscoelastic pasternak foundations

2018 ◽  
Author(s):  
Guy joseph Eyebe ◽  
Gambo Betchewe ◽  
Mohamadou Alidou ◽  
Timoleon crepin Kofane
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Nonlinear Vibration ◽  
Fractional Order ◽  
Parametric Study ◽  
Nonlocal Elasticity ◽  
Multiple Scales ◽  
Approximate Analytical Solution ◽  
Nonlocal Elasticity Theory ◽  
D’Alembert Principle

In the present study, nonlinear vibration of a nanobeam resting on fractional order viscoelastic Winkler-Pasternak foundaion is studied using nonlocal elasticity theory. D'Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. Detailled parametric study is conducted, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations

Nonlinear Vibration Analysis of Viscoelastic Smart Sandwich Plates Through the use of Fractional Derivative Zener Model

2021 ◽  
pp. 2150061
Author(s):  
Hamid Reza Talebi Amanieh ◽  
Seyed Alireza Seyed Roknizadeh ◽  
Arash Reza
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Nonlinear Vibration ◽  
Fractional Order ◽  
Sandwich Plate ◽  
Functionally Graded ◽  
Viscoelastic Layer ◽  
Sandwich Plates ◽  
Multiple Time ◽  
Nonlinear Frequency

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

scholarly journals Nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT

2021 ◽  
Author(s):  
Reza Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Frequency Ratio ◽  
Vibration Analysis ◽  
Multiple Scales ◽  
Numerical Technique ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Nonlinear Frequency

Abstract In this paper, the nonlinear vibration analysis of the nanobeams subjected to magneto-electro-thermo loading based on a novel HSDT is studied. Nonlocal elasticity theory is applied to consider the small scale effect. The nonlinear equations of motion are derived using Hamilton’s principle. First, a Galerkin-based numerical technique is applied to reduce the nonlinear governing equation into a set of Duffing-type time-dependent differential equations. Afterward, the analytical solutions are derived based on the method of multiple scales (MMS) and perturbation technique. All of the mechanical properties of the beam are temperature dependent. The impacts of the several variables are investigated on the nonlinear frequency ratio of the nanobeams. The results illustrate that when maximum deflection is smaller/ greater than 0.2, its impact on the nonlinear frequency ratio will decrease/increase.

scholarly journals Analytical solution for non-linear local fractional Bratu-type equation in a fractal space

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3941-3947
Author(s):  
Shao-Wen Yao ◽  
Wen-Jie Li ◽  
Kang-Le Wang
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Variational Formulation ◽  
Decomposition Method ◽  
Approximate Analytical Solution ◽  
Type Equation ◽  
Transform Method ◽  
Inverse Transform ◽  
Non Linear ◽  
Fractal Space

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method.

scholarly journals On the Solution of Burgers’ Equation with the New Fractional Derivative

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Ali Kurt ◽  
Yücel Çenesiz ◽  
Orkun Tasbozan
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Burgers Equation ◽  
Approximate Analytical Solution ◽  
Initial Conditions ◽  
Conformable Fractional Derivative ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Homotopy Analysis

AbstractFirstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

Nonlinear vibration of magnetoelectroelastic nanoscale shells embedded in elastic media in thermoelectromagnetic fields

2019 ◽  
Vol 30 (15) ◽  
pp. 2331-2347 ◽  
Author(s):  
Yan Qing Wang ◽  
Yun Fei Liu ◽  
Jean W Zu
Keyword(s):  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Nonlinear Ordinary Differential Equation ◽  
Elastic Media ◽  
Nonlinear Frequency ◽  
Governing Equations ◽  
Nonlinear Frequency Response ◽  
The Galerkin Method

This study investigates the nonlinear vibration of magnetoelectroelastic composite cylindrical nanoshells embedded in elastic media for the first time. The small-size effect and thermoelectromagnetic loadings are considered. Based on the nonlocal elasticity theory and Donnell’s nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived using Hamilton’s principle. Then, the Galerkin method is utilized to transform the governing equations into a nonlinear ordinary differential equation and subsequently the method of multiple scales is employed to obtain an approximate analytical solution to nonlinear frequency response. The present results are verified by the comparison with the published ones in the literature. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external magnetic potential, the external electric potential, the temperature change, and the elastic media on the nonlinear vibration characteristics of magnetoelectroelastic composite nanoshells.

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