Vibration of a Beam-Oscillator System Subjected to a Moving Vehicle: Fractional Derivative Approach

2012 ◽  
Author(s):  
Hashem S. Alkhaldi ◽  
Ibrahim Abu-Alshaikh ◽  
Anas N. Al-Rabadi
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Moving Load ◽  
Moving Vehicle ◽  
Beam Dynamic ◽  
Damping Characteristics ◽  
Born Series ◽  
The Laplace Transform ◽  
Derivatives Of

The dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam is investigated. The beam is appended at its mid-span by a single-degree-of-freedom (SDOF) fractionally-damped oscillator. The beam is further subjected to a vehicle modeled as a spring-dashpot system moves with a constant velocity over the beam. Hence, the damping characteristics of the beam and SDOF attached-oscillator are formally described in terms of fractional derivatives of arbitrary orders. In the analysis, the beam, SDOF oscillator, and the vehicle are assumed to be initially at rest. A system of three coupled differential equations is produced. These equations are handled by combining the Laplace transform with the Born series. Thereafter, curves are plotted to show the effect of the moving vehicle and the fractional derivatives behavior on the dynamic response of the beam. The numerical results show that the dynamic response decreases as the damping-ratios of the used absorber and beam increase. However, there are some optimal values of fractional derivative orders which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivative model. A comparison between the moving load and moving vehicle shows a significant reduction in the beam dynamic response in the case when vehicle is compared with the running load.

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 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.

Dynamic Response of a Beam With Absorber Exposed to a Running Force: Fractional Calculus Approach

2012 ◽  
Author(s):  
Ibrahim Abu-Alshaikh ◽  
Anas N. Al-Rabadi ◽  
Hashem S. Alkhaldi
Keyword(s):  
Fractional Calculus ◽  
Dynamic Response ◽  
Fractional Derivative ◽  
Initial Conditions ◽  
Damping Ratio ◽  
Experimental Models ◽  
Damping Characteristics ◽  
Damping Behavior ◽  
The Laplace Transform ◽  
Vibration Parameters

This paper analyzes the transverse vibration of Bernoulli-Euler homogeneous isotropic simply-supported beam. The beam is assumed to be fractionally-damped and attached to a single-degree-of-freedom (SDOF) absorber with fractionally-damping behavior at the mid-span of the beam. The beam is also exposed to a running force with constant velocity. The fractional calculus is introduced to model the damping characteristics of both the beam and absorber. The Laplace transform accompanied by the used decomposition method is applied to solve the handled problem with homogenous initial conditions. Subsequently, curves are depicted to measure the dynamic response of the utilized beam under different set of vibration parameters and different values of fractional derivative orders for both of the beam and absorber. The results obtained show that the dynamic response decreases as both the damping-ratio of the absorber and beam increase. The results reveal that there are critical values of fractional derivative orders which are different from unity. At these optimal values, the beam behaves with less dynamic response than that obtained for the full-order derivatives model of unity order. Therefore, the fractional derivative approach provides better damping models for fractionally-damped structures and materials which may allow researchers to choose suitable mathematical models that precisely fit the corresponding experimental models for many engineering applications.

scholarly journals Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 475
Author(s):  
Ewa Piotrowska ◽  
Krzysztof Rogowski
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Electric Circuit ◽  
Theoretical Models ◽  
Time Domain Analysis ◽  
Electrical Circuit ◽  
State Variable ◽  
State Space System ◽  
Derivatives Of ◽  
Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

Fractional derivatives of multidimensional Colombeau generalized stochastic processes

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Danijela Rajter-Ćirić ◽  
Mirjana Stojanović
Keyword(s):  
Stochastic Process ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Compact Support ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
One Dimensional ◽  
Compactly Supported ◽  
Derivatives Of ◽  
Do So

AbstractWe consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝn. We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support.

Two compartmental fractional derivative model with fractional derivatives of different order

2013 ◽  
Vol 18 (9) ◽  
pp. 2507-2514 ◽  
Author(s):  
Jovan K. Popović ◽  
Stevan Pilipović ◽  
Teodor M. Atanacković
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Derivative Model ◽  
Derivatives Of ◽  
Different Order

scholarly journals A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Advantages And Disadvantages ◽  
Derivatives Of

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

Random Base Excitation of Timoshenko Beam Traversed by Moving Load

2011 ◽  
Author(s):  
Fahim Javid ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Base Excitation ◽  
Lateral Direction ◽  
Beam Dynamic

The study of dynamic response of Timoshenko beam traversed by moving load subjected to random base excitation is carried out. By applying the theory of dynamic response of Timoshenko beam as well as finite element theory, beam finite element governing equations of motion are developed and they are solved using Galerkin method. To validate the model, some results of the model are compared with those available in literatures and very close agreement is achieved. The beam is subjected to travelling load and random base excitation in lateral direction simultaneously. Three types of boundary conditions, namely, hinged-hinged, hinged-clamped, and the clamped-clamped ends, are considered and beam dynamic behavior; such as deflection, velocity, and bending moment of beam midpoint, with all so-called boundary conditions are studied. To get better understanding of base excitation effects on the beam dynamic performance, all the results are presented with and without base excitation, in which considerably difference is observed. Moreover, the effect of base excitation on beam with different span-length is monitored.

scholarly journals New approach to the fractional derivatives

2003 ◽  
Vol 2003 (5) ◽  
pp. 315-325 ◽  
Author(s):  
Kostadin Trenčevski
Keyword(s):  
Positive Integer ◽  
Fractional Derivative ◽  
Taylor Expansion ◽  
Fractional Derivatives ◽  
Analytical Continuation ◽  
New Approach ◽  
Analytical Functions ◽  
Summation Of Divergent Series ◽  
Summation Of Series ◽  
Derivatives Of

We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives off, it is not sufficient to know the Taylor expansion off, but we should also know the constants of all consecutive integrations off. For example, any fractional derivative ofexisexonly if we assume that thenth consecutive integral ofexisexfor each positive integern. The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions.

scholarly journals On applications of Caputo k-fractional derivatives

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Naveed Latif ◽  
Matloob Anwar ◽  
Ali Imran ◽  
Muhammad Ozair ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Chebyshev Inequality ◽  
Absolutely Continuous ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Type Variable

Abstract This research explores Caputo k-fractional integral inequalities for functions whose nth order derivatives are absolutely continuous and possess Grüss type variable bounds. Using Chebyshev inequality (Waheed et al. in IEEE Access 7:32137–32145, 2019) for Caputo k-fractional derivatives, several integral inequalities are derived. Further, Laplace transform of Caputo k-fractional derivative is presented and Caputo k-fractional derivative and Riemann–Liouville k-fractional integral of an extended generalized Mittag-Leffler function are calculated. Moreover, using the extended generalized Mittag-Leffler function, Caputo k-fractional differential equations are presented and their solutions are proposed by applying the Laplace transform technique.

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