Search results for "Fractional Calculus"
showing 10 items of 128 documents
Deterministic and Random Vibration of Linear Systems with Singular Parameter Matrices and Fractional Derivative Terms
2021
Both time- and frequency-domain solution techniques are developed for determining the response of linear multi-degree-of-freedom systems exhibiting singular parameter matrices and endowed with derivative terms of noninteger orders modeled as rational numbers. This is done based on the Moore-Penrose matrix inverse theory, in conjunction with a state variable formulation and with a complex modal analysis treatment. It is worth noting that, for the class of systems considered herein, this treatment also yields decoupled governing equations, thus facilitating further their numerical solution. Next, a generalization of the standard frequency-domain input-output (excitation-response) relationship…
Innovative modeling of tuned liquid column damper controlled structures
2016
In this paper a different formulation for the response of structural systems controlled by Tuned Liquid Column Damper (TLCD) devices is developed, based on the mathematical tool of fractional calculus. Although the increasing use of these devices for structural vibration control, it has been demonstrated that existing model may lead to inaccurate prediction of liquid motion, thus reflecting in a possible imprecise description of the structural response. For this reason the recently proposed fractional formulation introduced to model liquid displacements in TLCD devices, is here extended to deal with TLCD controlled structures under base excitations. As demonstrated through an extensive expe…
Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral
2014
A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the…
Fractional generalized cumulative entropy and its dynamic version
2021
Following the theory of information measures based on the cumulative distribution function, we propose the fractional generalized cumulative entropy, and its dynamic version. These entropies are particularly suitable to deal with distributions satisfying the proportional reversed hazard model. We study the connection with fractional integrals, and some bounds and comparisons based on stochastic orderings, that allow to show that the proposed measure is actually a variability measure. The investigation also involves various notions of reliability theory, since the considered dynamic measure is a suitable extension of the mean inactivity time. We also introduce the empirical generalized fract…
Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields
2012
In this paper, a method for the digital simulation of wind velocity fields by Fractional Spectral Moment function is proposed. It is shown that by constructing a digital filter whose coefficients are the fractional spectral moments, it is possible to simulate samples of the target process as superposition of Riesz fractional derivatives of a Gaussian white noise processes. The key of this simulation technique is the generalized Taylor expansion proposed by the authors. The method is extended to multivariate processes and practical issues on the implementation of the method are reported.
Dynamic Finite Element analysis of fractionally damped structural systems in the time domain
2015
Visco-elastic material models with fractional characteristics have been used for several decades. This paper provides a simple methodology for Finite-Element-based dynamic analysis of structural systems with viscosity characterized by fractional derivatives of the strains. In particular, a re-formulation of the well-known Newmark method taking into account fractional derivatives discretized via the Grunwald–Letnikov summation allows the analysis of structural systems using standard Finite Element technology.
Finite element method on fractional visco-elastic frames
2016
Viscoelastic behavior is defined by fractional operators.Quasi static FEM analysis of frames with fractional constitutive law is performed.FEM solution is decoupled into a set of fractional Kelvin Voigt elements.Proposed approach could be easily integrated in existing FEM codes. In this study the Finite Element Method (FEM) on viscoelastic frames is presented. It is assumed that the Creep function of the constituent material is of power law type, as a consequence the local constitutive law is ruled by fractional operators. The Euler Bernoulli beam and the FEM for the frames are introduced. It is shown that the whole system is ruled by a set of coupled fractional differential equations. In q…
A non-local model of thermal energy transport: The fractional temperature equation
2013
Abstract Non-local models of thermal energy transport have been used in recent physics and engineering applications to describe several “small-scale” and/or high frequency thermodynamic processes as shown in several engineering and physics applications. The aim of this study is to extend a recently proposed fractional-order thermodynamics ( [5] ), where the thermal energy transfer is due to two phenomena: A short-range heat flux ruled by a local transport equation; a long-range thermal energy transfer that represents a ballistic effects among thermal energy propagators. Long-range thermal energy transfer accounts for small-scale effects that are assumed proportional to the product of the in…
The fractal model of non-local elasticity with long-range interactions
2010
The mechanically-based model of non-local elasticity with long-range interactions is framed, in this study, in a fractal mechanics context. Non-local interactions are modelled introducing long-range central body forces between non-adjacent volume elements of the elastic continuum. Such long-range interactions are modelled as proportional to the product of interacting volumes, to the relative displacements of the centroids and to a distance-decaying function that is monotonically-decreasing with the distance. The choice of the decaying function is a key aspect of the model and it has been proved that any continuous function, strictly positive, is thermodynamically consistent and it leads to …