On fallacies in the decision between the Caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator

2012 ◽  
Vol 45 ◽  
pp. 22-27 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Dynamic Response ◽  
Fractional Derivatives ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Oscillator

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

2005 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu ◽  
Hiroshi Nasuno
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Dynamical Behavior ◽  
Variable Coefficients ◽  
Rate Of Change ◽  
Order Derivative ◽  
Nonlinear Viscoelastic ◽  
Higher Order Derivative ◽  
The Difference ◽  
Different Order

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.

scholarly journals Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

scholarly journals Numerical calculating linear vibrations of third order systems involving fractional operators

2012 ◽  
Vol 34 (2) ◽  
pp. 91-99
Author(s):  
Nguyen Van Khang ◽  
Tran Dinh Son ◽  
Bui Thi Thuy
Keyword(s):  
Dynamic Response ◽  
Numerical Method ◽  
Numerical Algorithm ◽  
Fractional Derivatives ◽  
Integration Method ◽  
Fractional Operators ◽  
Dynamic Calculation ◽  
Third Order ◽  
Linear Vibrations ◽  
Definition Of

This paper presents a numerical method for dynamic calculation of third order systems involving fractional operators. Using the Liouville-Rieman's definition of fractional derivatives, a numerical algorithm is developed on base of the well-known Newmark integration method to calculate dynamic response of third order systems. Then, we apply this method to calculate linear vibrations of viscoelastic systems containing fractional derivatives.

Initial Condition Problems of Fractional Viscoelastic Equations

2003 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Initial Condition ◽  
Initial Values ◽  
Viscoelastic Equation ◽  
Key Issues ◽  
Fractional Differential ◽  
Viscoelastic Equations ◽  
Viscoelastic Oscillator ◽  
Numerical And Analytical Methods

The 2nd order differential equation with fractional derivatives describing dynamic behavior of a single-degree-of-freedom viscoelastic oscillator, referred to as fractional viscoelastic equation (FVE), is considered. Some types of viscoelastic damped mechanical systems may be described by FVE. The differential equation with fractional derivatives is often called the fractional differential equation (FDE). FDE can be solved for zero initial values, but it can not generally be solved for non-zero initial values. How to solve the problem is one of the key issues in this field. This is called “Initial condition (value) problems” of FDE. In this paper, initial condition problems of FVE are solved by making use of the prehistory functions of unknowns which are specified before the initial instance (referred to as the initial functions) starts. Introduction of initial functions into FDE reflects the physical state in giving the initial values. In this paper, several types of initial function are used to solve unique solutions for a type of FVE (referred to as FVE-I). The solutions of FVE-I are obtained by means of both numerical and analytical methods. Implication of the solutions to viscoelastic material will also be discussed.

scholarly journals Vibration Control of Fractionally-Damped Beam Subjected to a Moving Vehicle and Attached to Fractionally-Damped Multiabsorbers

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hashem S. Alkhaldi ◽  
Ibrahim M. Abu-Alshaikh ◽  
Anas N. Al-Rabadi
Keyword(s):  
Dynamic Response ◽  
Fractional Derivatives ◽  
Single Degree Of Freedom ◽  
Moving Vehicle ◽  
Single Degree ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Damped Systems ◽  
Special Case

This paper presents the dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam. The beam is attached to multi single-degree-of-freedom (SDOF) fractionally-damped systems, and it is subjected to a vehicle moving with a constant velocity. The damping characteristics of the beam and SDOF systems are described in terms of fractional derivatives. Three coupled second-order fractional differential equations are produced and then they are solved by combining the Laplace transform with the decomposition method. The obtained numerical results show that the dynamic response decreases as (a) the number of absorbers attached to the beam increases and (b) the damping-ratios of used absorbers and beam increase. However, there are some critical values of fractional derivatives which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivatives model. Furthermore, the obtained results show very good agreements with special case studies that were published in the literature.

Evaluation of Nonlinear Damping Properties of Magnetorheological Gels Under Shear Loading

2013 ◽  
Author(s):  
Yotsugi Shibuya ◽  
Hiroshi Nasuno ◽  
Hirohisa Sakurai ◽  
Katsuaki Sunakoda
Keyword(s):  
Magnetic Field ◽  
Dynamic Response ◽  
Hysteresis Loop ◽  
Viscoelastic Model ◽  
Shear Loading ◽  
Material System ◽  
Damping Properties ◽  
Stress Strain ◽  
Nonlinear Viscoelastic ◽  
Nonlinear Viscoelastic Model

Rheological properties of magnetorheological gels can be changed reversibly by applied magnetic fields. Magnetorheological gels with different material system are characterized the dynamic response of the material by shearing test in magnetic field. Nonlinear behavior is observed in the dynamic response of the material. To understand mechanism of the behavior, dynamic properties of magnetorheological gels are evaluated by experiment and nonlinear viscoelastic model. Magnetorheological gels used in this study consist of three types of paramagnetic particles and a cyclic-poly-siloxane gel matrix. Three material systems of magnetic particles are chosen: Fe-Si-Ni, Si-Fe and Fe-Si-B-Cr types. Shear testing is conducted in magnetic field 0mT, 105mT and 211mT. The stress-strain response under shear deformation is characterized by non-ellipsoidal hysteresis loop due to nonlinearity of the response. To identify the nonlinear properties, analysis in frequency domain is applied to identify the dynamic response of the material. Nonlinear viscoelastic model with high order components is made and phenomenon of the non-ellipsoidal hysteresis loop in the stress-strain relation and damping properties are illustrated.

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