Analysis of multi degree of freedom systems with fractional derivative elements of rational order

In this paper a novel method based on complex eigenanalysis in the state variables domain is proposed to uncouple the set of rational order fractional differential equations governing the dynamics of multi-degree-of-freedom system. The traditional complex eigenanalysis is appropriately modified to be applicable to the coupled fractional differential equations. This is done by expanding the dimension of the problem and solving the system in the state variable domain. Examples of applications are given pertaining to multi-degree-of-freedom systems under both deterministic and stochastic loads.

The Multiscale Stochastic Model of Fractional Hereditary Materials (FHM)

Abstract In a recent paper the authors proposed a mechanical model corresponding, exactly, to fractional hereditary materials (FHM). Fractional derivation index 13 E [0,1/2] corresponds to a mechanical model composed by a column of massless newtonian fluid resting on a bed of independent linear springs. Fractional derivation index 13 E [1/2, 1], corresponds, instead, to a mechanical model constituted by massless, shear-type elastic column resting on a bed of linear independent dashpots. The real-order of derivation is related to the exponent of the power-law decay of mechanical characteristics. In this paper the authors aim to introduce a multiscale fractance description of FHM in presence …

Direct Derivation of Corrective Terms in SDE Through Nonlinear Transformation on Fokker–Planck Equation

This paper examines the problem of probabilistic characterization of nonlinear systems driven by normal and Poissonian white noise. By means of classical nonlinear transformation the stochastic differential equation driven by external input is transformed into a parametric-type stochastic differential equation. Such equations are commonly handled with Ito-type stochastic differential equations and Ito's rule is used to find the response statistics. Here a different approach is proposed, which mainly consists in transforming the Fokker–Planck equation for the original system driven by external input, in the transformed probability density function of the new state variable. It will be shown …

A physically based connection between fractional calculus and fractal geometry

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the m…

Fractional differential equations solved by using Mellin transform

In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.

Innovative modeling of tuned liquid column damper controlled structures

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…

Aspetti di dinamica dei ponti

Inverse Mellin Transform to characterize the nonlinear system PDF response to Poisson white noise

Ito and Stratonovich integrals for delta-correlated processes

Abstract In this paper the generalization of the Itd and Stratonovich integrals for the case of non-linear systems excited by parametric delta-correlated processes is presented. This generalization gives a new light on the corrective coefficients in the stochastic differential equations driven by parametric delta-correlated processes. The full significance of these corrective terms is evidenced by means of some examples.

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

Abstract In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is anal…

STOCHASTIC ANALYSIS OF ONE-DIMENSIONAL HETEROGENEOUS SOLIDS WITH LONG-RANGE INTERACTIONS

Random mass distribution in one-dimensional (1D) elastic solids in the presence of long-range interactions is studied in this paper. Besides the local Cauchy contact forces among adjacent elements, long-range forces depending on the product of interacting masses, as well as on their relative displacements, are considered. In this context, the random fluctuations of the mass distribution involve a stochastic model of the nonlocal interactions, and the random displacement field of the body is provided as the solution of a stochastic integro-differential equation. The presence of the random field of mass distribution is reflected in the random kernel of the solving integro-differential equatio…

Poisson white noise parametric input and response by using complex fractional moments

Abstract In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.

Damage detection based on the analytical signal representation

Earthquake ground motion artificial simulations through Fractional Tajimi-Kanai Model

Non-linear systems under parametric alpha-stable LÉVY WHITE NOISES

In this study stochastic analysis of nonlinear dynamical systems under a-stable, multiplicative white noise has been performed. Analysis has been conducted by means of the Ito rule extended to the case of α-stable noises. In this context the order of increments of Levy process has been evaluated and differential equations ruling the evolutions of statistical moments of either parametrically and external dynamical systems have been obtained. The extended Ito rule has also been used to yield the differential equation ruling the evolution of the characteristic function for parametrically excited dynamical systems. The Fourier transform of the characteristic function, namely the probability den…

Rollio delle navi in presenza di onde modellate come processi gaussiani e poissoniani agenti simultaneamente.

Obiettivo del presente lavoro è l’estensione del metodo della path integral solution (PIS) per lo studio della dinamica del rollio delle navi in presenza di onde modellate come processi gaussiani e poissionani agenti simultaneamente. Si è proceduto dapprima a mostrare come la PIS consenta di valutare l’evoluzione temporale della funzione densità di probabilità (PDF) del processo di risposta, applicando il metodo ad equazioni differenziali stocastiche soggette a forzanti esterne gaussiane e poissoniane. Successivamente si è trattato il caso di un sistema non lineare soggetto ad entrambi i rumori gaussiano e poissoniano agenti contestualmente. Si è infine affrontato sia analiticamente che num…

Un metodo per il progetto di massima di dissipatori isteretici e viscosi e applicazione a casi concreti

Stability analysis of a non-local column with non-central interactions

Stochastic dynamics of nonlinear systems driven by non-normal delta-correlated processes

In this paper, nonlinear systems subjected to external and parametric non-normal delta-correlated stochastic excitations are treated. A new interpretation of the stochastic differential calculus allows first a full explanation of the presence of the Wong-Zakai or Stratonovich correction terms in the Itoˆ’s differential rule. Then this rule is extended to take into account the non-normality of the input. The validity of this formulation is confirmed by experimental results obtained by Monte Carlo simulations.

Impulsive Tests on Historical Structures: The Dome of Teatro Massimo in Palermo

Cultural heritage is the set of things, that having particular historical cultural and aesthetic are of public interest and constitute the wealth and civilization of a place and its people. Sharpen up methodologies aimed at safeguarding of monuments is crucial because the future may have in mind the historical past. Italy is a country that has invested heavily on its historical memory returned in large part by the historical building or the monuments. Furthermore, culture represents a fundamental indicator of the growth of the culture of a country. Consider a monitoring project of one of the most Impressive theater in the world, like “Teatro Massimo” in Palermo (Italy), means to add value t…

Fractional visco-elastic systems under normal white noise

In this paper an original method is presented to compute the stochastic response of singledegree- of-freedom structural systems with viscoelastic fractional damping. The key-idea stems from observing that, based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion can be reverted to a coupled linear system involving additional degrees of freedom, the number of which depends on the discretization adopted for the fractional derivative operator. The method applies for fractional damping of arbitrary order a (0 < α < 1). For most common input correlation functions, including a Gaussian white noise, …

Electrical analogous in viscoelasticity

In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these viscoelastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. Moreover, a comparison with numerical expe…

Waves propagation in a fractional viscoelastic continuum

In this paper the analysis of waves scattering in a fractional-type viscoelastic material is analyzed. Such a material involves, in the constitutive equation, the presence of noninteger order derivatives of the strain filed yielding a memory-type behavior of the material model. The presence of such a term has been also justified experimentally reporting the relaxation modulus of polymeric materials, obtained from experimental test, that are well-fitted by a powerlaw of fractional order. Some numerical applications reporting the standing-waves condition of an 1D solid varying the fractional differentiation order has also been reported in the paper.

Probabilistic analysis of non-local random media

Computational stochastic methods have been devoted over the last years to analysis and quantification of the mechanical response of engineering systems involving random media. Specifically analysis of random, heterogeneous media is getting more and more important with the emergence of new complex materials requiring reliable methods to provide accurate probabilistic response. Advanced materials, often used at nano or meso-levels possess strong non-local characters showing that long-range forces between non-adjacent volume elements play an important role in mechanical response. Moreover long and short-range molecular interactions may have random nature due to unpredictable fabrication proces…

On the dynamics of non-local fractional viscoelastic beams under stochastic agencies

Abstract Non-local viscoelasticity is a subject of great interest in the context of non-local theories. In a recent study, the authors have proposed a non-local fractional beam model where non-local effects are represented as viscoelastic long-range volume forces and moments, exchanged by non-adjacent beam segments depending on their relative motion, while local effects are modelled by elastic classical stress resultants. Long-range interactions have been given a fractional constitutive law, involving the Caputo's fractional derivative. This paper introduces a comprehensive numerical approach to calculate the stochastic response of the non-local fractional beam model under Gaussian white no…

Finite element method on fractional visco-elastic frames

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…

Exact mechanical models of fractional hereditary materials

Fractional Viscoelasticity is referred to materials, whose constitutive law involves fractional derivatives of order β R such that 0 β 1. In this paper, two mechanical models with stress-strain relation exactly restituting fractional operators, respectively, in ranges 0 β 1 / 2 and 1 / 2 β 1 are presented. It is shown that, in the former case, the mechanical model is described by an ideal indefinite massless viscous fluid resting on a bed of independent springs (Winkler model), while, in the latter case it is a shear-type indefinite cantilever resting on a bed of independent viscous dashpots. The law of variation of all mechanical characteristics is of power-law type, strictly related to th…

Fractional multiphase hereditary materials: Mellin Transforms and Multi-Scale Fractances

The rheological features of several complex organic natural tissues such as bones, muscles as well as of complex artificial polymers are well described by power-laws. Indeed, it is well-established that the time-dependence of the stress and the strain in relaxation/creep test may be well captured by power-laws with exponent β ∈ [0, 1]. In this context a generalization of linear springs and linear dashpots has been introduced in scientific literature in terms of a mechanical device dubbed spring-pot. Recently the authors introduced a mechanical analogue to spring-pot built upon a proper arrangements of springs and dashpots that results in Elasto-Viscous (EV) materials, as β ∈ [0, 1/2] and Vi…

A mechanical approach to fractional non-local thermoelasticity

In recent years fractional di erential calculus applications have been developed in physics, chemistry as well as in engineering elds. Fractional order integrals and derivatives ex- tend the well-known de nitions of integer-order primitives and derivatives of the ordinary di erential calculus to real-order operators. Engineering applications of these concepts dealt with viscoelastic models, stochastic dy- namics as well as with the, recently developed, fractional-order thermoelasticity [3]. In these elds the main use of fractional operators has been concerned with the interpolation between the heat ux and its time-rate of change, that is related to the well-known second sound e ect. In othe…

Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises

In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based …

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF…

On the behavior of a three-dimensional fractional viscoelastic constitutive model

In this paper a three-dimensional isotropic fractional viscoelastic model is examined. It is shown that if different time scales for the volumetric and deviatoric components are assumed, the Poisson ratio is time varying function; in particular viscoelastic Poisson ratio may be obtained both increasing and decreasing with time. Moreover, it is shown that, from a theoretical point of view, one-dimensional fractional constitutive laws for normal stress and strain components are not correct to fit uniaxial experimental test, unless the time scale of deviatoric and volumetric are equal. Finally, the model is proved to satisfy correspondence principles also for the viscoelastic Poisson’s ratio a…

Prestress and experimental tests on fractional viscoelastic materials

Creep and/or Relaxation tests on viscoelastic materials show a power-law trend. Based upon Boltzmann superposition principle the constitutive law with a power-law kernel is ruled by the Caputo’s fractional derivative. Fractional constitutive law posses a long memory and then the parameters obtained by best fitting procedures on experimental data are strongly influenced by the prestress on the specimen. As in fact during the relaxation test the imposed history of deformation is not instantaneously applied, since a unit step function may not be realized by the test machine. Aim of this paper, it is shown that, the experimental procedure, and in particular the initial ramp to reach the constan…

First-passage problem for nonlinear systems under Lévy white noise through path integral method

In this paper, the first-passage problem for nonlinear systems driven by $$\alpha $$ -stable Levy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of $$\alpha $$ -stable random variables and processes, PIS is extended to deal with Levy white noises with any value of the stability index $$\alpha $$ . Application to linear and nonlinear systems considering different values of $$\alpha $$ is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Non-local continuum: Fractional Calculus Approach

Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

Abstract A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis becomes very straightforward and simple since, based on stand…

Rocking of rigid block on nonlinear flexible foundation

Abstract The two prime models used currently to describe rocking of rigid bodies, the Housner’s model and the Winkler foundation model, can capture some of the salient features of the physics of this important problem. These two models involve either null or linear interaction between the block and the foundation. Hopefully, some additional aspects of the problem can be captured by an enhanced nonlinear model for the base-foundation interaction. In this regard, what it is adopted in this paper is the Hunt and Crossley’s nonlinear impact force model in which the impact/contact force is represented by springs in parallel with nonlinear dampers. In this regard, a proper mathematical formulatio…

Linear and non-linear systems under α-stable sub-Gaussian white-noise

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

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…

Viscoelasticity: an electrical point of view

Time dependent hereditary properties of complex materials are well described by power-laws with real order exponent. This experimental observation and analogous electrical experiments, yield a description of these properties by using fractional-order operators. In this paper, elasto-viscous and visco-elastic behaviors of fractional order hereditary materials are firstly described by using fractional mathematical operators, based on recent work of some of the authors. Then, electrical analogous models are introduced. Viscoelastic models have elastic and viscous components which can be obtained by combining springs and dashpots: these models can be equivalently viewed as electrical circuits, …

The fractal model of non-local elasticity with long-range interactions

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 …

Einstein-Smoluchowsky equation handled by complex fractional moments

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.

Path Integral Solution Handled by Fractional Calculus

THE MECHANICAL MODEL OF FRACTIONAL VISCOELASTICITY

Viscoelastic materials have been more and more used nowadays for their low-cost productions as well as for their dissipative capabilities that may be coupled with others, more performing materials, to form complex-type engineering elements. The main feature of viscoelastic behavior is the relaxation of the stress state and the creep of the strain field that may be experienced, respectively, in hard or soft test devices. Such phenomenological consideration has been extensively analyzed yet at the beginning of the twentieth century and simple rheological models representing linear, constitutive, stress-velocity relations of the studied material have been proposed. Moreover the rheological rel…

Non-local finite element method for the analysis of elastic continuum with long-range central interactions.

In this paper the Finite Element Method (FEM) for the mechanically-based non-local elastic continuum model is proposed. In such a model non-adjacent elements are considered mutually interacting by means of central body forces that are monotonically decreasing with their interdistance and proportional to the product of the interacting volume elements. The resulting governing equation is an integro-differential one and for such a model both kinematical and mechanical boundary conditions are exactly coincident with the classical boundary conditions of the continuum mechanics. The solution of the integro-differential problem is framed in the paper by the finite element method. Finally, the solu…

Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields

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.

Long-range cohesive interactions of non-local continuum faced by fractional calculus

Abstract A non-local continuum model including long-range forces between non-adjacent volume elements has been studied in this paper. The proposed continuum model has been obtained as limit case of two fully equivalent mechanical models: (i) A volume element model including contact forces between adjacent volumes as well as long-range interactions, distance decaying, between non-adjacent elements. (ii) A discrete point-spring model with local springs between adjacent points and non-local springs with distance-decaying stiffness connecting non-adjacent points. Under the assumption of fractional distance-decaying interactions between non-adjacent elements a fractional differential equation in…

Non-stationary response of fractionally-damped viscoelastic systems

Non-local stiffness and damping models for shear-deformable beams

This paper presents the dynamics of a non-local Timoshenko beam. The key assumption involves modeling non-local effects as long-range volume forces and moments mutually exerted by non-adjacent beam segments, that contribute to the equilibrium of any beam segment along with the classical local stress resultants. Elastic and viscous long-range volume forces/moments are endowed in the model. They are built as linearly depending on the product of the volumes of the interacting beam segments and on generalized measures of their relative motion, based on the pure deformation modes of the beam. Attenuation functions governing the space decay of the non-local effects are introduced. Numerical resul…

Misura delle vibrazioni sul simulacro argenteo dell'Immacolata processione

Non-Local Thermoelasticity: The Fractional Heat conduction

Non-linear systems under parametric a-stable Levý white noises

Fractional model of concrete hereditary viscoelastic behaviour

The evaluation of creep effects in concrete structures is addressed in the literature using different predictive models, supplied by specific codes, and applying the concepts of linear viscoelastic theory with ageing. The expressions used in the literature are mainly based on exponential laws, which are introduced in the integral expression of the Boltzmann principle; this approach leads to the need of finding approximated numerical solutions of the viscoelastic response. In this study, the hereditary fractional viscoelastic model is applied to concrete elements, underlining the convenience of using creep or relaxation functions expressed by power laws. The full reciprocal character of cree…

A Galerkin approach for power spectrum determination of nonlinear oscillators

A numerical method to estimate spectral properties of nonlinear oscillators with random input is presented. The stationary system response is expanded into a trigonometric Fourier series. A set of nonlinear algebraic equations, solved by Newton's method, leads to the determination of the unknown Fourier series coefficients of single samples of the response process. For cubic polynomial nonlinearities, closed-form expressions are used to find the nonlinear terms at each step of the solution scheme. Further, a simple procedure yields an approximation of an arbitrary nonlinearity by a cubic polynomial. Power spectral density estimates for the response process are constructed by averaging the s…

Elastic waves propagation in 1D fractional non-local continuum

Aim of this paper is the study of waves propagation in a fractional, non-local 1D elastic continuum. The non-local effects are modeled introducing long-range central body interactions applied to the centroids of the infinitesimal volume elements of the continuum. These non-local interactions are proportional to a proper attenuation function and to the relative displacements between non-adjacent elements. It is shown that, assuming a power-law attenuation function, the governing equation of the elastic waves in the unbounded domain, is ruled by a Marchaud-type fractional differential equation. Wave propagation in bounded domain instead involves only the integral part of the Marchaud fraction…

Frequency domain identification of the fractional Kelvin-Voigt’s parameters for viscoelastic materials

Abstract In this work, a new innovative method is used to identify the parameters of fractional Kelvin-Voigt constitutive equation. These parameters are: the order of fractional derivation operator, 0 ≤ α ≤ 1, the coefficient of fractional derivation operator, CV, and the stiffness of the model, KV. A particular dynamic test setup is developed to capture the experimental data. Its outputs are time histories of the excitation and excited accelerations. The investigated specimen is a polymeric cubic silicone gel material known as α-gel. Two kinds of experimental excitations are used as random frequencies and constant frequency harmonic excitations. In this study, experimental frequency respon…

Fokker Planck equation solved in terms of complex fractional moments

Abstract In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones, the solution of the FPK equation in terms of a finite number of complex moments may be easily found. It is shown that the probability density function (PDF) solution of the FPK equation is restored in the whole domain, including the trend at infinity with the exception of the value of the PDF in zero.

Corrigendum to “Fractional differential equations solved by using Mellin transform” [Commun Nonlinear Sci Numer Simul 19(7) (2014) 2220–2227]

Special Section on Fractional Operators in the Analysis of Mechanical Systems Under Stochastic Agencies

Passive Control of Linear Structures Equipped with non-Linear Viscous Dampers and Amplification Mechanism

Fluid damper devices in civil structures such as buildings or bridges are commonly used as energy absorbers for seismic protection. The problem in the response analysis of structures with filled dampers mainly consists in the fact that, due to the strongly nonlinear behavior of such equipments, the response spectrum technique fails. Moreover, in order to enhance the damping effect various toggle brace configurations have been recently proposed. In this paper by using the concept of power spectral density function compatible with the elastic response spectrum and the stochastic linearization technique, the equivalent damping ratio is obtained. It is shown that once the system is linearized r…

CFD approach for the induced effects of free wake past rivulets on cables of stayed bridges

Large-amplitude oscillations of stayed bridge cables appear under the combined effects of crosswind and rain. "Wave-like" oscillations along the whole cable length are born with large amplitudes even under low-speed wind and only the presence of the rain rivulet motion on the cable cross-section which guarantees an "all-or-none" amplification of the dynamic response of the cable. Even though this peculiar behavior has been studied, in a recent past, through dynamic and structural approaches, the phenomenon is not yet quite well understood, i.e., the quasi-cyclic oscillations seem to be activated to events (rain rivulets motion along the cables) compromising the circumferential geometry shap…

Non-linear transformation on Kolmogorov-Feller equation

Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach

Fractional power-law nonlinear drift arises in many applications of engineering interest, as in structures with nonlinear fluid viscous–elastic dampers. The probabilistic characterization of such structures under external Gaussian white noise excitation is still an open problem. This paper addresses the solution of such a nonlinear system providing the equation governing the evolution of the characteristic function, which involves the Riesz fractional operator. An efficient numerical procedure to handle the problem is also proposed.

Ito calculus extended to non-linear systems under alpha-stable Lévy white noise

Amplification of Interstory Drift and Velocity for the Passive Control of Structural Vibrations

Mitigation of structural damage due to earthquake ground motion may be performed by inserting dampers in the structure. In order to enhance the damping effect various toggle brace configurations have been recently proposed. In this paper these equipments are analyzed in detail and compared with a new one here proposed. The analysis is performed by taking into account the inherent nonlinearities of the damper by means of stochastic analysis and validated by using time histories of recorder accelerograms and by the stochastic analysis using spectrum consistent power spectral density.

Fractional characteristic times and dissipated energy in fractional linear viscoelasticity

Abstract In fractional viscoelasticity the stress–strain relation is a differential equation with non-integer operators (derivative or integral). Such constitutive law is able to describe the mechanical behavior of several materials, but when fractional operators appear, the elastic and the viscous contribution are inseparable and the characteristic times (relaxation and retardation time) cannot be defined. This paper aims to provide an approach to separate the elastic and the viscous phase in the fractional stress–strain relation with the aid of an equivalent classical model (Kelvin–Voigt or Maxwell). For such equivalent model the parameters are selected by an optimization procedure. Once …

Probabilistic analysis of truss structures with uncertain parameters (virtual distortion method approach)

A new approach for probabilistic characterization of linear elastic redundant trusses with uncertainty on the various members subjected to deterministic loads acting on the nodes of the structure is presented. The method is based on the simple observation that variations of structural parameters are equivalent to superimposed strains on a reference structure depending on the axial forces on the elastic modulus of the original structure as well as on the uncertainty (virtual distortion method approach). Superposition principle may be applied to separate contribution to mechanical response due to external loads and parameter variations. Statically determinate trusses dealt with the proposed m…

On the use of fractional calculus for the probabilistic characterization of random variables

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are o…

On the derivation of the Fokker-Plank equation by using of Fractional calculus

In this paper, fractional calculus has been used to find the spectral counterpart of the Fokker- Planck equations for non-linear systems driven by Lévy white noise processes. In particular it is shown that one can obtain the equation ruling the characteristic function of the response to a non-linear system, without using the Itô formula. Indeed, it is possible to reproduce the well-known results, already known in literature, by means of the characteristic function representation in terms of complex moments, recently proposed by the first two authors. The case of a-stable Lévy driven stochastic differential equation is also treated introducing an associated process constructed from the stabl…

Le strutture portanti di copertura - The roof support structures

A Further Insight on the Intrinsic Mode Function through Stochastic Analysis

Path integral method for first-passage probability determination of nonlinear systems under levy white noise

In this paper the problem of the first-passage probabilities determination of nonlinear systems under alpha-stable Lévy white noises is addressed. Based on the properties of alpha-stable random variables and processes, the Path Integral method is extended to deal with nonlinear systems driven by Lévy white noises with a generic value of the stability index alpha. Furthermore, the determination of reliability functions and first-passage time probability density functions is handled step-by-step through a modification of the Path Integral technique. Comparison with pertinent Monte Carlo simulation reveals the excellent accuracy of the proposed method.

Mellin transform approach for the solution of coupled systems of fractional differential equations

In this paper, the solution of a multi-order, multi-degree-of-freedom fractional differential equation is addressed by using the Mellin integral transform. By taking advantage of a technique that relates the transformed function, in points of the complex plane differing in the value of their real part, the solution is found in the Mellin domain by solving a linear set of algebraic equations. The approximate solution of the differential (or integral) equation is restored, in the time domain, by using the inverse Mellin transform in its discretized form.

Step-by-step integration for fractional operators

Abstract In this paper, an approach based on the definition of the Riemann–Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grunwald–Letnikov operators. The proposed Riemann–Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function f(t). It has been shown that, as f(t) is discretised as stepwise or piecewise function, the Riemann–Liouville fractional integral and derivative are governing by operators very similar to the Grunwald–Letnikov operators. In order to show the accuracy and capabilities of the proposed Riemann–Liouville discretisation technique and th…

Sistemi non lineari eccitati da processi di Lévy risolti mediante momenti frazionari

Fractional differential calculus for 3D mechanically based non-local elasticity

This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium proble…

Non-linear viscoelastic behavior of polymer melts interpreted by fractional viscoelastic model

Very recently, researchers dealing with constitutive law pertinent viscoelastic materials put forward the successful idea to introduce viscoelastic laws embedded with fractional calculus, relating the stress function to a real order derivative of the strain function. The latter consideration leads to represent both, relaxation and creep functions, through a power law function. In literature there are many papers in which the best fitting of the peculiar viscoelastic functions using a fractional model is performed. However there are not present studies about best fitting of relaxation function and/or creep function of materials that exhibit a non-linear viscoelastic behavior, as polymer melt…

Nonlinear rocking of rigid blocks on flexible foundation: Analysis and experiments

Abstract Primarily, two models are commonly used to describe rocking of rigid bodies; the Housner model, and the Winkler foundation model. The first deals with the motion of a rigid block rocking about its base corners on a rigid foundation. The second deals with the motion of a rigid block rocking and bouncing on a flexible foundation of distributed linear springs and dashpots (Winkler foundation). These models are two-dimensional and can capture some of the features of the physics of the problem. Clearly, there are additional aspects of the problem which may be captured by an enhanced nonlinear model for the base-foundation interaction. In this regard, what it is adopted in this paper is …

An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus

If the attenuation function of strain is expressed as a power law, the formalism of fractional calculus may be used to handle Eringen non-local elastic model. Aim of the present paper is to provide a mechanical interpretation to this non-local fractional elastic model by showing that it is equivalent to a discrete, point-spring model. A one-dimensional geometry is considered; static, kinematic and constitutive equations as well as the proper boundary conditions are derived and discussed.

The finite element method for the mechanically-based model of non-local continuum

In this paper the finite element method (FEM) for the mechanically based non-local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter-distance and proportional to the product of the interacting volume elements. The constitutive relations of the long-range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance-decaying function, which accounts for the decrement of the long-range interactions as…

Spectral Moments and Pre-Envelope Covariances of Nonseparable Processes

A critical review of the definition of the spectral moments of a stochastic process in the nonstationary case is presented. An adequate time-domain representation of the spectral moments in the stationary case is first established, showing that the spectral moments are related to the variances of the stationary analytical pre-envelope processes. The extension to the nonstationary case is made in the time domain evaluating the covariances of the nonstationary pre-envelope showing the differences between the proposed definition and the classical one made introducing the evolutionary power.

A Mellin transform approach to wavelet analysis

The paper proposes a fractional calculus approach to continuous wavelet analysis. Upon introducing a Mellin transform expression of the mother wavelet, it is shown that the wavelet transform of an arbitrary function f(t) can be given a fractional representation involving a suitable number of Riesz integrals of f(t), and corresponding fractional moments of the mother wavelet. This result serves as a basis for an original approach to wavelet analysis of linear systems under arbitrary excitations. In particular, using the proposed fractional representation for the wavelet transform of the excitation, it is found that the wavelet transform of the response can readily be computed by a Mellin tra…

On the moving load problem in Euler–Bernoulli uniform beams with viscoelastic supports and joints

This paper concerns the vibration response under moving loads of Euler–Bernoulli uniform beams with translational supports and rotational joints, featuring Kelvin–Voigt viscoelastic behaviour. Using the theory of generalized functions to handle the discontinuities of the response variables at the support/joint locations, exact beam modes are obtained from a characteristic equation built as determinant of a (Formula presented.) matrix, for any number of supports/joints. Based on pertinent orthogonality conditions for the deflection modes, the response under moving loads is built in the time domain by modal superposition. Remarkably, all response variables are built in a closed analytical for…

Funzione densità di probabilità della risposta di strutture esposte al vento (PIS)

Vibration mitigation of the Silver Madonna during the procession in Palermo: preliminary study

A correction method for dynamic analysis of linear systems

Abstract This paper proposes an analytical method to improve the accuracy of the dynamic response of classically damped linear systems, as given by a standard truncated modal analysis. Upon computing the first m undamped modes of a n-degree-of-freedom system, two sets of equations in the Rn nodal space are built, which are uncoupled and govern the contribution to the response of the m computed modes and the remaining (n−m) unknown modes, respectively. The first set is solved in the Rm modal space by using the m available modes; the second set is solved in a reduced R(n−m) nodal space, without computing additional modes. Specifically, it is shown that the particular solution of the second se…

Fractional differential equations and related exact mechanical models

Creep and relaxation tests, performed on various materials like polymers, rubbers and so on are well-fitted by power-laws with exponent β ∈ [0, 1] (Nutting (1921), Di Paola et al. (2011)). The consequence of this observation is that the stress-strain relation of hereditary materials is ruled by fractional operators (Scott Blair (1947), Slonimsky (1961)). A large amount of researches have been performed in the second part of the last century with the aim to connect constitutive fractional relations with some mechanical models by means of fractance trees and ladders (see Podlubny (1999)). Recently, Di Paola and Zingales (2012) proposed a mechanical model that corresponds to fractional stress-…

Fractional viscoelastic characterization of laminated glass beams under time-varying loading

Abstract Laminated glass is a composite made of elastic glass layers sandwiching thin viscoelastic polymeric interlayers. There are several types of polymers, traditionally modelled as linear viscoelastic materials using a Prony’s series of units in the Maxwell-Wiechert arrangement. We show that one single element with fractional viscoelastic properties (two constitutive parameters that depend on environmental temperature), is sufficient to provide an accurate description of the polymer response under arbitrary time-varying actions. This is a great advantage over the classical viscoelastic characterization, which requires at least 10–15 terms in the Prony’s series, each one characterized by…

FRACTIONAL MOMENTS AND PATH INTEGRAL SOLUTION FOR NON LINEAR SYSTEMS DRIVEN BY NORMAL WHITE NOISE

Digital simulation of wind field velocity

Abstract In this paper some computational aspects on the generation procedure of n -variate wind velocity vectors are discussed in detail. Decompositions of the power spectral density matrix are also discussed showing the physical significance of eigenquantities of this matrix.

A discrete mechanical model of fractional hereditary materials

Fractional hereditary materials are characterized for the presence, in the stress-strain relations, of fractional-order operators with order beta a[0,1]. In Di Paola and Zingales (J. Rheol. 56(5):983-1004, 2012) exact mechanical models of such materials have been extensively discussed obtaining two intervals for beta: (i) Elasto-Viscous (EV) materials for 0a parts per thousand currency sign beta a parts per thousand currency sign1/2; (ii) Visco-Elastic (VE) materials for 1/2a parts per thousand currency sign beta a parts per thousand currency sign1. These two ranges correspond to different continuous mechanical models. In this paper a discretization scheme based upon the continuous models p…

Random vibration mitigation of beams via tuned mass dampers with spring inertia effects

The dynamics of beams equipped with tuned mass dampers is of considerable interest in engineering applications. Here, the purpose is to introduce a comprehensive framework to address the stochastic response of the system under stationary and non-stationary loads, considering inertia effects along the spring of every tuned mass damper applied to the beam. For this, the key step is to show that a tuned mass damper with spring inertia effects can be reverted to an equivalent external support, whose reaction force on the beam depends only on the deflection of the attachment point. On this basis, a generalized function approach provides closed analytical expressions for frequency and impulse res…

On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials

Creep and/or Relaxation tests on viscoelastic materials show a power-law trend. Based upon Boltzmann superposition principle the constitutive law with a power-law kernel is ruled by the Caputo's fractional derivative. Fractional constitutive law posses a long memory and then the parameters obtained by best fitting procedures on experimental data are strongly influenced by the prestress on the specimen. As in fact during the relaxation test the imposed history of deformation is not instantaneously applied, since a unit step function may not be realized by the test machine. Usually an initial ramp is present in the deformation history and the time at which the deformation attains the maximum …

Time delay induced effects on control of non-linear systems under random excitation

On the vibrations of a mechanically based non-local beam model

The vibration problem of a Timoshenko non-local beam is addressed. The beam model involves assuming that the equilibrium of each volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are taken as depending on the product of the interacting volume elements and on their relative displacement through a material-dependent distance-decaying function. To derive the motion equations and the related mechanical boundary conditions, the Hamilton's principle is applied The vibration problem of a Timoshenko non-local beam …

Cross-correlation and cross-power spectral density representation by complex spectral moments

Abstract A new approach to provide a complete characterization of normal multivariate stochastic vector processes is presented in this paper. Such proposed method is based on the evaluation of the complex spectral moments of the processes. These quantities are strictly related to the Mellin transform and they are the generalization of the integer-order spectral moments introduced by Vanmarcke. The knowledge of the complex spectral moments permits to obtain the power spectral densities and their cross counterpart by a complex series expansions. Moreover, with just the aid of some mathematical properties the complex fractional moments permit to obtain also the correlation and cross-correlatio…

Ideal and physical barrier problems for non-linear systems driven by normal and Poissonian white noise via path integral method

Abstract In this paper, the probability density evolution of Markov processes is analyzed for a class of barrier problems specified in terms of certain boundary conditions. The standard case of computing the probability density of the response is associated with natural boundary conditions, and the first passage problem is associated with absorbing boundaries. In contrast, herein we consider the more general case of partially reflecting boundaries and the effect of these boundaries on the probability density of the response. In fact, both standard cases can be considered special cases of the general problem. We provide solutions by means of the path integral method for half- and single-degr…

Stochastic response of offshore structures by a new approach to statistical cubicization

This study presents a new statistical cubicization approach for predicting the stochastic response of offshore platforms subjected to a Morison-type nonlinear drag loading. Statistics of the original system are obtained from an equivalent nonlinear system, which is constructed by replacing the Morison drag force by a cubic polynomial function of the relative fluid-structure velocity, up to cubic order. A Volterra series expansion with a finite Fourier series representation is used to approximate the response of the equivalent system. Exact solutions are developed to express the Fourier coefficients of the second and third-order response as functions of the Fourier coefficients of the first-…

A new representation of power spectral density and correlation function by means of fractional spectral moments

In this paper, a new perspective for the representation of both the power spectral density and the correlation function by a unique class of function is introduced. We define the moments of order gamma (gamma being a complex number) of the one sided power spectral density and we call them Fractional Spectral Moments (FSM). These complex quantities remain finite also in the case in which the ordinary spectral moments diverge, and are able to represent the whole Power Spectral Density and the corresponding correlation function.

Stochastic seismic analysis of MDOF structures with viscous damper devices

Fluid damper devices in civil structures such as buildings or bridges are commonly used as energy absorbers for seismic protection. The problem in the response analysis of structures with filled dampers mainly consists in the fact that, due to the strongly nonlinear behavior of such equipments, the response spectrum technique fails. In this paper by using the concept of power spectral density function compatible with the elastic response spectrum and the stochastic linearization technique, the equivalent damping ratio is obtained. It is shown that once the system is linearized results obtained by Monte Carlo simulation and those obtained by stochastic analysis are in good agreement. Respons…

Constructing transient response probability density of non-linear system through complex fractional moments

Abstract The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Ito stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments an…

The finite element implementation of 3D fractional viscoelastic constitutive models

Abstract The aim of this paper is to present the implementation of 3D fractional viscoelastic constitutive theory presented in Alotta et al., 2016 [1]. Fractional viscoelastic models exactly reproduce the time dependent behaviour of real viscoelastic materials which exhibit a long “fading memory”. From an implementation point of view, this feature implies storing the stress/strain history throughout the simulations which may require a large amount of memory. We propose here a number of strategies to effectively limit the memory required. The form of the constitutive equations are summarized and the finite element implementation in a Newton-Raphson integration scheme is described in detail. …

Path integral solution by fractional calculus

In this paper, the Path Integral solution is developed in terms of complex moments. The method is applied to nonlinear systems excited by normal white noise. Crucial point of the proposed procedure is the representation of the probability density of a random variable in terms of complex moments, recently proposed by the first two authors. Advantage of this procedure is that complex moments do not exhibit hierarchy. Extension of the proposed method to the study of multi degree of freedom systems is also discussed.

Modellazione e risposta di strutture continue a parametri incerti mediante processi Poissoniani filtrati

An integral equation for damage identification of Euler-Bernulli beams under static loads

Power-law hereditariness of hierarchical fractal bones

In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ≤ β ≤1. The rheological behavior of the material has therefore been obtained, using the Boltzmann-Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related to the e…

Free energy and states of fractional-order hereditariness

AbstractComplex materials, often encountered in recent engineering and material sciences applications, show no complete separations between solid and fluid phases. This aspect is reflected in the continuous relaxation time spectra recorded in cyclic load tests. As a consequence the material free energy cannot be defined in a unique manner yielding a significative lack of knowledge of the maximum recoverable work that can extracted from the material. The non-uniqueness of the free energy function is removed in the paper for power-laws relaxation/creep function by using a recently proposed mechanical analogue to fractional-order hereditariness.

Probabilistic Analysis of Non-Local 1-D Continuum under Random Load

Multivariate stochastic wave generation

Abstract In this paper, for the case of the fluid particle velocity, a procedure that substantially reduces the computational effort to generate a multivariate stochastic process is proposed. It is shown that, for a fully coherent wave field, it is possible to decompose the Power Spectral Density (PSD) matrix into the eigenvectors of the matrix itself. This leads to generate each field's process as independent, and the time generation increases linearly with the processes' number in the field. A numerical example to evaluate the statistical properties, in terms of correlation and cross-correlation functions, of the processes is also presented.

A mechanically based approach to non-local beam theories

A mechanically based non-local beam theory is proposed. The key idea is that the equilibrium of each beam volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are modeled as depending on the product of the interacting volume elements, their relative displacement and a material-dependent distance-decaying function. To derive the beam equilibrium equations and the pertinent mechanical boundary conditions, the total elastic potential energy functional is used based on the Timoshenko beam theory. In this manner, t…

The mechanically-based approach to 3D non-local linear elasticity theory: Long-range central interactions

Abstract This paper presents the generalization to a three-dimensional (3D) case of a mechanically-based approach to non-local elasticity theory, recently proposed by the authors in a one-dimensional (1D) case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range forces exerted by non-adjacent elements. Specifically, the long-range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements, measured along the line connecting the centroids. Further, the long-range forces are assumed to be proportional to a proper, material-dependent, dis…

Modello frazionario per il comportamento elasto-viscoso delle strutture in calcestruzzo

A novel exact representation of stationary colored Gaussian processes (fractional differential approach)

A novel representation of functions, called generalized Taylor form, is applied to the filtering of white noise processes. It is shown that every Gaussian colored noise can be expressed as the output of a set of linear fractional stochastic differential equations whose solution is a weighted sum of fractional Brownian motions. The exact form of the weighting coefficients is given and it is shown that it is related to the fractional moments of the target spectral density of the colored noise.

Influence of protecting devices on the optimal design of elastic plastic structures

The paper concerns the minimum volume design of structures constituted by elastic perfectly plastic material. The relevant optimal design problem is formulated on the grounds of a statical approach and two different resistance limits are considered: in particular, it is required that the optimal structure satisfies the elastic shakedown limit and the instantaneous collapse limit, imposing for each different condition a suitably chosen safety factor. For sake of generality, the structure is thought as discretized into compatible finite elements and subjected to loads quasi-statically acting as well as to dynamic (seismic) loads. The effects of the dynamic actions are studied on the grounds o…

Power-law hereditariness of hierarchical fractal bones

SUMMARY In this paper, the authors introduce a hierarchic fractal model to describe bone hereditariness. Indeed, experimental data of stress relaxation or creep functions obtained by compressive/tensile tests have been proved to be fit by power law with real exponent 0 ⩽ β ⩽1. The rheological behavior of the material has therefore been obtained, using the Boltzmann–Volterra superposition principle, in terms of real order integrals and derivatives (fractional-order calculus). It is shown that the power laws describing creep/relaxation of bone tissue may be obtained by introducing a fractal description of bone cross-section, and the Hausdorff dimension of the fractal geometry is then related …

Modeling of the viscoelastic behavior of paving bitumen using fractional derivatives

The paving grade bitumen used in the production of asphalt mixtures for road construction is significantly able to affect the in-service performances of flexible road pavements. It has been proved that, when subjected to loading conditions comparable with most pavement operating conditions, bituminous binders behave as linear visco-elastic materials. The aim of this paper is to propose a model based on fractional differential equations which is able to describe the behavior of bituminous binders in the linear viscoelastic range. Shear creep testing and creep recovery testing were carried out over a range of temperatures and by applying a stress level which makes it possible to maintain the …

Stochastic response of fractional visco-elastic beams

Stochastic response of a fractional vibroimpact system

Abstract The paper proposes a method to investigate the stochastic dynamics of a vibroimpact single-degree-of-freedom fractional system under a Gaussian white noise input. It is assumed that the system has a hard type impact against a one-sided motionless barrier, which is located at the system’s equilibrium position; furthermore, the system under study is endowed with an element modeled with fractional derivative. The proposed method is based on stochastic averaging technique and overcome the particular difficulty due to the presence of fractional derivative of an absolute value function; particularly an analytical expression for the system’s mean squared response amplitude is presented an…

The mechanically based non-local elasticity: an overview of main results and future challenges

The mechanically based non-local elasticity has been used, recently, in wider and wider engineering applications involving small-size devices and/or materials with marked microstructures. The key feature of the model involves the presence of non-local effects as additional body forces acting on material masses and depending on their relative displacements. An overview of the main results of the theory is reported in this paper.

Fractional models for capturing both relaxation and creep phase

Non-linear systems under Levy White Noise Handled by path integration method

Aim of this paper is an investigation on the consistency of the Path Integration (PI) method already proposed by Naess & Johansen, 1991,1993 for non-linear systems driven by α-stable white noise. It is shown that in the limit, as τ→0, the Einstein-Smoluchowsky (ES) equation is fully restored. Once the consistency of the PI is demonstrated for the half oscillator, then the extension of the ES equation for MDOF system is found starting from the PI method.

Spectral Approach to Equivalent Statistical Quadratization and Cubicization Methods for Nonlinear Oscillators

Random vibrations of nonlinear systems subjected to Gaussian input are investigated by a technique based on statistical quadratization, and cubicization. In this context, and depending on the nature of the given nonlinearity, statistics of the stationary response are obtained via an equivalent system with a polynomial nonlinearity of either quadratic or cubic order, which can be solved by the Volterra series method. The Volterra series response is expanded in a trigonometric Fourier series over an adequately long interval T, and exact expressions are derived for the Fourier coefficients of the second- and third-order response in terms of the Fourier coefficients of the first-order, Gaussian…

Fractional calculus for the solution of non-linear stochastic oscillators with viscous dampers devices

Fluid viscoelastic dampers are of great interest in different fields of engineering. Examples of their applications can be found in seismic mitigation design of structures or in vibration absorption in airplane suspension. Such devices introduce a non-linear dissipative term in the equation of motion and therefore, the solution of even a single degree of freedom system excited by a white noise process, becomes prohibitive. The solution is usually obtained by approximated methods, like the stochastic linearization technique. In this paper it is shown that, by means of fractional operators, it is possible to find the solution of oscillators provided with fluid viscoelastic devices, approachin…

Materials Science Forum

This paper deals with the generalization to three-dimensional elasticity of the physically-based approach to non-local mechanics, recently proposed by the authors in one-dimensional case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range central forces exerted by non-adjacent elements. Specifically, the long-range forces are modeled as central body forces depending on the relative displacements between the centroids of the volume elements, measured along the line connecting the centroids. Furthermore, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decay…

Mechanically Based Nonlocal Euler-Bernoulli Beam Model

AbstractThis paper presents a nonlocal Euler-Bernoulli beam model. It is assumed that the equilibrium of a beam segment is attained because of the classical local stress resultants, along with long-range volume forces and moments exchanged by the beam segment with all the nonadjacent beam segments. Elastic long-range volume forces/moments are considered, built as linearly depending on the product of the volumes of the interacting beam segments and on generalized measures of their relative motion, based on the pure deformation modes of the beam. Attenuation functions governing the space decay of the nonlocal effects are introduced. The motion equations are derived in an integro-differential …

Seismic behavior of structures equipped with variable friction dissipative (VFD) systems

Usually, to mitigate the stresses in framed structures, different strategies are used. Among them, base isolation, viscous/friction/metallic yielding dampers and tuned mass dumpers have been widely investigated. Fluid Viscous Dampers (FVD) probably result the most diffused for the simplicity in the applications. However, these type of dampers request limited interstorey drifts to avoid dangerous effects. Further, they have an elevate cost. On the contrary, friction dampers are not so expensive but request high interstorey drifts to give a significant contribute in the dissipation of energy during an earthquake. In this paper an approach for the energy dissipation by friction, modified with …

CROSS-POWER SPECTRAL DENSITY AND CROSS-CORRELATION REPRESENTATION BY USING FRACTIONAL SPECTRAL MOMENTS

Truly no-mesh method for beam torsion solution

Theoretical and experimental analysis of viscoelastic behavior of biomaterials

Probabilistic characterization of nonlinear systems under parametric Poisson white noise via complex fractional moments

Evaluation of the temperature effect on the fractional linear viscoelastic model for an epoxy resin

The paper deals with the evolution of the parameters of a fractional model for different values of temperature. An experimental campaign has been performed on epoxy resin at different levels of temperature. It is shown that epoxy resin is very sensitive to the temperature.

Efficient solution of the first passage problem by Path Integration for normal and Poissonian white noise

Abstract In this paper the first passage problem is examined for linear and nonlinear systems driven by Poissonian and normal white noise input. The problem is handled step-by-step accounting for the Markov properties of the response process and then by Chapman–Kolmogorov equation. The final formulation consists just of a sequence of matrix–vector multiplications giving the reliability density function at any time instant. Comparison with Monte Carlo simulation reveals the excellent accuracy of the proposed method.

Evaluation of the shakedown limit load multiplier for stochastic seismic actions

A new approach for the evaluation of the shakedown limit load multiplier for structures subjected to a combination of quasi-statically variable loads and seismic actions is presented. The common case of frame structures constituted by elastic perfectly plastic material is considered. The acting load history during the lifetime of the structure will be defined as a suitable combination of never ending quasi-statical loads, variable within an appropriate given domain, and stochastic seismic actions occurring for limited time interval. The proposed approach utilizes the Monte Carlo method in order to generate a suitable large number of seismic acceleration histories and the corresponding shake…

A physical approach to the connection between fractal geometry and fractional calculus

Our goal is to prove the existence of a connection between fractal geometries and fractional calculus. We show that such a connection exists and has to be sought in the physical origins of the power laws ruling the evolution of most of the natural phenomena, and that are the characteristic feature of fractional differential operators. We show, with the aid of a relevant example, that a power law comes up every time we deal with physical phenomena occurring on a underlying fractal geometry. The order of the power law depends on the anomalous dimension of the geometry, and on the mathematical model used to describe the physics. In the assumption of linear regime, by taking advantage of the Bo…

Self-similarity and response of fractional differential equations under white noise input

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann-Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t -&gt; at (a &gt; 0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coeffi…

Fractional moments of non-linear systems under Lévy white noise processes

Linear and nonlinear fractional hereditary constitutive laws of asphalt mixtures

The aim of this paper is to propose a fractional viscoelastic and viscoplastic model of asphalt mixtures using experimental data of several tests such as creep and creep recovery performed at different temperatures and at different stress levels. From a best fitting procedure it is shown that both the creep one and recovery curve follow a power law model. It is shown that the suitable model for asphalt mixtures is a dashpot and a fractional element arranged in series. The proposed model is also available outside of the linear domain but in this case the parameters of the model depend on the stress level.

Vibrations of elastic structures with external nonlinear visco-elastic damping devices

Fractional Viscoelasticity Under Combined Stress and Temperature Variations

Nowadays polymeric materials or composites with polymeric matrices are widely used in a very wide range of applications such as aerospace, automotive, biomedical and also civil engineering. From a mechanical point of view, polymers are characterized by high viscoelastic properties and high sensitiveness of mechanical parameters from temperature. Analytical predictions in real-life conditions of mechanical behaviour of such a kind of materials is not trivial for the intrinsic hereditariness that imply the knowledge of all the history of the material at hand in order to predict the response to applied external loads. If temperature variations are also present in the materials, a reliable eval…

Stochastic analysis of linear and nonlinear systems under α-stable Lèvy white noise

Fractional viscoelastic behaviour under stochastic temperature process

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…

Representation of Stationary Multivariate Gaussian Processes Fractional Differential Approach

In this paper, the fractional spectral moments method (H-FSM) is used to generate stationary Gaussian multivariate processes with assigned power spectral density matrix. To this aim, firstly the N-variate process is expressed as sum of N fully coherent normal random vectors, and then, the representation in terms of HFSM is used.

The fractional tajimi-kanai model of earthquake gound motion

Damage Identification of Beams Using Static Test Data

A damage identification procedure for beams under static loads is presented. Damage is modelled through a damage distribution function which determines a variation of the beam stiffness with respect to a reference condition. Using the concept of the equivalent superimposed deformation, the equations governing the static problem are recast in a Fredholm’s integral equation of the second kind in terms of bending moments. The solution of this equation is obtained through an iterative procedure as well as in closed form. The latter is explicitly dependent from the damage parameters, thus, it can be conveniently used to set-up a damage identification procedure. Some numerical results are present…

Variational Aspects of the Physically-Based Approach to 3D Non-Local Continuum Mechanics

This paper deals with the generalization to three-dimensional elasticity of the physically-based approach to non-local mechanics, recently proposed by the authors in one-dimensional case. The proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long-range central forces exerted by non-adjacent elements. Specifically, the long-range forces are modeled as central body forces depending on the relative displacements between the centroids of the volume elements, measured along the line connecting the centroids. Furthermore, the long-range forces are assumed to be proportional to a proper, material-dependent, distance-decay…

Stochastic response of linear and non-linear systems to α-stable Lévy white noises

Abstract The stochastic response of linear and non-linear systems to external α -stable Levy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed. Approximate CF solutions to the …

Laminated Glass Members in Compression: Experiments and Modeling

It is well known that structural glass members are made by assembling thin laminated panels, which can be connected with different bonding techniques; for instance, with steel devices or with structural adhesives. The latter are very commonly used because they do not reduce the transparency of the member and make it possible to avoid stress concentrations because of the presence of holes. This technique is used to make up columns in glazing structures and different applications of the technique can be found in contemporary architecture. As evidenced by the literature, one of the most important problems in such members is caused by buckling phenomena, resulting from the slenderness of the pa…

De Saint-Venant flexure-torsion problem handled by Line Element-less Method (LEM)

In this paper, the De Saint-Venant flexure-torsion problem is developed via a technique by means of a novel complex potential function analytic in all the domain whose real and imaginary parts are related to the shear stresses. The latter feature makes the complex analysis enforceable for the shear problem. Taking full advantage of the double-ended Laurent series involving harmonic polynomials, a novel element-free weak form procedure, labelled Line Element-less Method (LEM), is introduced, imposing that the square of the net flux across the border is minimized with respect to expansion coefficients. Numerical implementation of the LEM results in systems of linear algebraic equations involv…

Il modello autoregressivo continuo (ARC) per l'analisi dinamica di strutture esposte al vento

Le strutture portanti verticali - The vertical support structures

Stochastic Response Of Fractionally Damped Beams

Abstract This paper aims at introducing the governing equation of motion of a continuous fractionally damped system under generic input loads, no matter the order of the fractional derivative. Moreover, particularizing the excitation as a random noise, the evaluation of the power spectral density performed in frequency domain highlights relevant features of such a system. Numerical results have been carried out considering a cantilever beam under stochastic loads. The influence of the fractional derivative order on the power spectral density response has been investigated, underscoring the damping effect in reducing the power spectral density amplitude for higher values of the fractional de…

Fractional calculus application to visco-elastic solid

It is widely known that fractional derivative is the best mathematical tool to describe visco-elastic constitutive law. In this paper it is shown that as soon as we assume the creep compliance function as power law type, as in the linearized version of the Nutting equation, then the fractional constitutive law appears in a natural way. Moreover, using Nutting equation for the creep function, the relaxation modulus is also of power law type whose coefficients (intensity and exponent) are strictly related to those of the creep compliance. It follows that by a simple creep test (or relaxation test) by means of a best fitting procedure we may easily evaluate the parameters of Nutting equation a…

Stochastic analysis of external and parametric dynamical systems under sub-Gaussian Levy white-noise

In this study stochastic analysis of non-linear dynamical systems under α-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of α-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with α/2- stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function …

Fractional calculus in solid mechanics: local versus non-local approach

Several enriched continuum mechanics theories have been proposed by the scientific community in order to develop models capable of describing microstructural effects. The aim of the present paper is to revisit and compare two of these models, whose common denominator is the use of fractional calculus operators. The former was proposed to investigate damage in materials exhibiting a fractal-like microstructure. It makes use of the local fractional derivative, which turns out to be a powerful tool to describe irregular patterns such as strain localization in heterogeneous materials. On the other hand, the latter is a non-local approach that models long-range interactions between particles by …

Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables

Abstract The aim of this paper is the probabilistic representation of the probability density function (PDF) or the characteristic function (CF) in terms of fractional moments of complex order. It is shown that such complex moments are related to Riesz and complementary Riesz integrals at the origin. By invoking the inverse Mellin transform theorem, the PDF or the CF is exactly evaluated in integral form in terms of complex fractional moments. Discretization leads to the conclusion that with few fractional moments the whole PDF or CF may be restored. Application to the pathological case of an α -stable random variable is discussed in detail, showing the impressive capability to characterize…

Fractional calculus approach to the statistical characterization of random variables and vectors

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…

Optimal shakedown design of base isolated structures

The optimal design problem of finite element discretized elastic perfectly plastic structures is studied. In particular, a minimum volume formulation is developed, based on a statical approach, and two different resistance limits are considered: the elastic shakedown limit and the instantaneous collapse limit, imposing for each a suitably chosen safety factor. The structure is thought as subjected to quasi-static loads as well as to dynamic actions. The dynamic features of the relevant structure are determined taking also into account an appropriate base isolation system in order to reduce the seismic effects. According with the Italian code related to the structural analysis and design, th…

Fractional calculus approach for the representation of power spectral densities and correlation functions in wind engineering

The paper deals with the digital simulation of wind velocity samples by Fractional Spectral Moment function. It is shown that such a function represents a third useful way to characterize a stationary Gaussian stochastic process, alongside the power spectral density and the correlation function. The method is applied to wind velocity fields whose power spectra is given by the Kaimal’s, the Davenport’s and the Solari’s representation. 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 Gaussian white noise processes.

Interventi di miglioramento e adeguamento sismico di edifici in c.a. dell’Azienda Ospedaliera Universitaria Policlinico “Paolo Giaccone” di Palermo

The paper shows the results of an experimental investigation on the behaviour of compressed concrete columns subjected to the overcoring technique. Extraction of cylindrical concrete cores in perpendicular direction to the column surface, was achieved by rotary drilling using a diamond tipped hollow barrel. Core is gradually isolated from the stress field in the surrounding concrete and at every advancing step the re-equilibrium deformation response is measured by means of four strain-gauges, placed on its external surface and near the hole. Finally the original stress state is computed in the elastic phase on the basis of recorded strains, knowing the elastic properties of concrete and by …

Health monitoring of civil and aerospace structural components by guided ultrasonic waves

Moment stability of parametrically perturbed systems via path integral solution

Non-linear oscillators under parametric and external poisson pulses

The extended Ito calculus for non-normal excitations is applied in order to study the response behaviour of some non-linear oscillators subjected to Poisson pulses. The results obtained show that the non-normality of the input can strongly affect the response, so that, in general, it can not be neglected.

Il Filtro Integrale Auto-Regressivo Continuo (I-ARC) per l’Analisi di Strutture Esposte al Vento

In questo studio viene proposto un metodo per la rappresentazione di processi aleatori Gaussiani e stazionari, utile a modellare la turbolenza della velocità del vento, introducendo la versione integrale del modello auto-regressivo discreto già proposto in precedenza. La rappresentazione di un processo aleatorio di assegnata funzione di correlazione viene condotta integrando un’equazione integro-differenziale in cui viene coinvolto un nucleo, che rappresenta la memoria del processo, in presenza di un rumore bianco Gaussiano. La soluzione dell’equazione rappresenta un campione del processo aleatorio della turbolenza della velocità del vento. E’ stato mostrato che il modello I-ARC fornisce, n…