Search results for "fractional calculu"
showing 10 items of 145 documents
Fractional calculus approach to the statistical characterization of random variables and vectors
2009
Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. A…
Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements
2019
In this paper, the steady-state dynamic response of hysteretic oscillators comprising fractional derivative elements and subjected to harmonic excitation is examined. Notably, this problem may arise in several circumstances, as for instance, when structures which inherently exhibit hysteretic behavior are supplemented with dampers or isolators often modeled by employing fractional terms. The amplitude of the steady-state response is determined analytically by using an equivalent linearization approach. The procedure yields an equivalent linear system with stiffness and damping coefficients which are related to the amplitude of the response, but also, to the order of the fractional derivativ…
Einstein-Smoluchowsky equation handled by complex fractional moments
2014
In this paper the response of a non linear half oscillator driven by α-stable white noise in terms of probability density function (PDF) is investigated. The evolution of the PDF of such a system is ruled by the so called Einstein-Smoluchowsky equation involving, in the diffusive term, the Riesz fractional derivative. The solution is obtained by the use of complex fractional moments of the PDF, calculated with the aid of Mellin transform operator. It is shown that solution can be found for various values of stability index α and for any nonlinear function of the drift term in the stochastic differential equation.
Fractional viscoelastic behaviour under stochastic temperature process
2018
Abstract This paper deals with the mechanical behaviour of a linear viscoelastic material modelled by a fractional Maxwell model and subject to a Gaussian stochastic temperature process. Two methods are introduced to evaluate the response in terms of strain of a material under a deterministic stress and subjected to a varying temperature. In the first approach the response is determined making the material parameters change at each time step, due to the temperature variation. The second method, takes advantage of the Time–Temperature Superposition Principle to lighten the calculations. In this regard, a stochastic characterisation for the Time–Temperature Superposition Principle method is p…
Fractional visco-elastic Timoshenko beam from elastic Euler-Bernoulli beam
2014
The Euler–Bernoulli beam theory is well established in such a way that engineers are very confident with the determination of the stress field or deflections of the elastic beam based on this theory. In contrast, Timoshenko theory is not so much used by engineers. However, in some cases, Euler–Bernoulli theory, which neglects the effect of transversal shear deformation, yields unacceptable results. For instance, when dealing with visco-elastic behavior, shear deformations play a fundamental role. Recent studies on the response evaluation of a visco-elastic Euler–Bernoulli beam under quasi-static and dynamic loads have been stressed that for better capturing of the visco-elastic behavior, a …
Fractional visco-elastic Timoshenko beam deflection via single equation
2015
SUMMARY This paper deals with the response determination of a visco-elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco-elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace-transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald–Letni…
Finite-Element Formulation of a Nonlocal Hereditary Fractional-Order Timoshenko Beam
2017
AbstractA mechanically-based nonlocal Timoshenko beam model, recently proposed by the authors, hinges on the assumption that nonlocal effects can be modeled as elastic long-range volume forces and moments mutually exerted by nonadjacent beam segments, which contribute to the equilibrium of any beam segment along with the classical local stress resultants. Long-range volume forces/moments linearly depend on the product of the volumes of the interacting beam segments, and on pure deformation modes of the beam, through attenuation functions governing the space decay of nonlocal effects. This paper investigates the response of this nonlocal beam model when viscoelastic long-range interactions a…
Timoshenko vs Euler-Bernoulli beam: fractional visco-elastic behaviour
2013
The Euler-Bernoulli beam theory is well established in such a way that engineers are very confident with the determination of stress field or def lections of the elastic beam based on this theory. Conversely Timoshenko theory is not so much used by engineers. However in such cases Euler-Bernoulli theory that n eglects the effect of transversal shear deformation leads to unacceptable results. For inst ance when dealing with the visco-elastic behaviour the shear deformations play a fundamental role. Recent studies [1]-[2] on the response evaluation of visco-elastic Euler-Bernoulli beam under quasi-static and dynamic loads, have been stressed that for better capturing the visco-elastic behavio…
Fractional viscoelastic beam under torsion
2017
Abstract This paper introduces a study on twisted viscoelastic beams, having considered fractional calculus to capture the viscoelastic behaviour. Further another novelty of this paper is extending a recent numerical approach, labelled line elementless method (LEM), to viscoelastic beams. The latter does not require any discretization neither in the domain nor in the boundary. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method.
Fractional Modeling of the AC Large-Signal Frequency Response in Magnetoresistive Current Sensors
2013
Fractional calculus is considered when derivatives and integrals of non-integer order are applied over a specific function. In the electrical and electronic domain, the transfer function dependence of a fractional filter not only by the filter order n, but additionally, of the fractional order α is an example of a great number of systems where its input-output behavior could be more exactly modeled by a fractional behavior. Following this aim, the present work shows the experimental ac large-signal frequency response of a family of electrical current sensors based in different spintronic conduction mechanisms. Using an ac characterization set-up the sensor transimpedance function is obtain…