Line element-less method (LEM) for beam torsion solution (truly no-mesh method)

In this paper a new numerical method for finding approximate solutions of the torsion problem is proposed. The method takes full advantage of the theory of analytic complex function. A new potential function directly in terms of shear stresses is proposed and expanded in the double-ended Laurent series involving harmonic polynomials. A novel element-free weak form procedure, labelled Line Element-Less Method (LEM), has been developed imposing that the square of the net flux across the border is minimum with respect to coefficients expansion. Numerical implementation of the LEM results in systems of linear algebraic equations involving symmetric and positive-definite matrices without resorti…

Iterative closure method for non-linear systems driven by polynomials of Gaussian filtered processes

This paper concerns the statistical characterization of the non-Gaussian response of non-linear systems excited by polynomial forms of filtered Gaussian processes. The non-Gaussianity requires the computation of moments of any order. The problem is solved profiting from both the stochastic equivalent linearization (EL), and the moment equation approach of Ito's stochastic differential calculus through a procedure divided into two parts. The first step requires the linearization of the system, while retaining the non-linear excitation; the response statistical moments are calculated exactly, and constitute a first estimate of the moments of the actual non-linear system. In the second step, t…

Higher order statistics of the response of MDOF linear systems under polynomials of filtered normal white noises

This paper exploits the work presented in the companion paper in order to evaluate the higher order statistics of the response of linear systems excited by polynomials of filtered normal processes. In fact, by means of a variable transformation, the original system is replaced by a linear one excited by external and linearly parametric white noise excitations. The transition matrix of the new enlarged system is obtained simply once the transition matrices of the original system and of the filter are evaluated. The method is then applied in order to evaluate the higher order statistics of the approximate response of nonlinear systems to which the pseudo-force method is applied.

A generalized model of elastic foundation based on long-range interactions: Integral and fractional model

The common models of elastic foundations are provided by supposing that they are composed by elastic columns with some interactions between them, such as contact forces that yield a differential equation involving gradients of the displacement field. In this paper, a new model of elastic foundation is proposed introducing into the constitutive equation of the foundation body forces depending on the relative vertical displacements and on a distance-decaying function ruling the amount of interactions. Different choices of the distance-decaying function correspond to different kind of interactions and foundation behavior. The use of an exponential distance-decaying function yields an integro-d…

Digital generation of multivariate wind field processes

Abstract A very efficient procedure for the generation of multivariate wind velocity stochastic processes by wave superposition as well as autoregressive time series is proposed in this paper. The procedure starts by decomposing the wind velocity field into a summation of fully coherent independent vector processes using the frequency dependent eigenvectors of the Power Spectral Density matrix. It is shown that the application of the method allows to show some very interesting physical properties that allow to reduce drastically the computational effort. Moreover, using a standard finite element procedure for approximating the frequency dependent eigenvectors, the generation procedure requi…

Solutions for the Design and Increasing of Efficiency of Viscous Dampers

In last decades many strategies for seismic vulnerability mitigation of structures have been studied and experimented; in particular energy dissipation by external devices assumes a great importance for the relative simplicity and efficacy. Among all possible approaches the use of fluid viscous dampers are very interesting, because of their velocity-dependent behaviour and relatively low costs. Application on buildings requires a specific study under seismic excitation and a particular attention on structural details. Nevertheless seismic codes give only general information and in most case the design of a such protection systems results difficult; this problem is relevant also in Italy whe…

Filter equation by fractional calculus

Aim of this paper is to represent a causal filter equation for any kind of linear system in the general form L=f(t), where f(t) is the forcing function, x(t) is the output and L is a summation of fractional operators. The exact form of the operator L is obtained by using Mellin transform in complex plane.

Path integral solution handled by Fast Gauss Transform

Abstract The path integral solution method is an effective tool for evaluating the response of non-linear systems under Normal White Noise, in terms of probability density function (PDF). In this paper it has been observed that, using short-time Gaussian approximation, the PDF at a given time instant is the Gauss Transform of the PDF at an earlier close time instant. Taking full advantage of the so-called Fast Gauss Transform a new integration method is proposed. In order to overcome some unsatisfactory trends of the classical Fast Gauss Transform, a new version termed as Symmetric Fast Gauss Transform is also proposed. Moreover, extensions to the two Fast Gauss Transform to MDOF systems ar…

Fractional visco-elastic Euler–Bernoulli beam

Abstract Aim of this paper is the response evaluation of fractional visco-elastic Euler–Bernoulli beam under quasi-static and dynamic loads. Starting from the local fractional visco-elastic relationship between axial stress and axial strain, it is shown that bending moment, curvature, shear, and the gradient of curvature involve fractional operators. Solution of particular example problems are studied in detail providing a correct position of mechanical boundary conditions. Moreover, it is shown that, for homogeneous beam both correspondence principles also hold in the case of Euler–Bernoulli beam with fractional constitutive law. Virtual work principle is also derived and applied to some c…

Stochastic seismic analysis of multidegree of freedom systems

Abstract A unconditionally stable step-by-step procedure is proposed to evaluate the mean square response of a linear system with several degrees of freedom, subjected to earthquake ground motion. A non-stationary modulated random process, obtained as the product of a deterministic time envelope function and a stationary noise, is used to simulate earthquake acceleration. The accuracy of the procedure and its extension to nonlinear systems are discussed. Numerical examples are given for a hysteretic system, a duffing oscillator and a linear system with several degrees of freedom.

Fractional viscoelastic beam under torsion

Abstract This paper introduces a study on twisted viscoelastic beams, having considered fractional calculus to capture the viscoelastic behaviour. Further another novelty of this paper is extending a recent numerical approach, labelled line elementless method (LEM), to viscoelastic beams. The latter does not require any discretization neither in the domain nor in the boundary. Some numerical applications have been reported to demonstrate the efficiency and accuracy of the method.

Structural performances of pultruded GFRP emergency structures – Part 1: Experimental characterization of materials and substructure

Abstract This paper presents an experimental study in the field of structures made of pultruded fiber reinforced polymers (FRP) elements to be used for emergency purposes. A preliminary design of a 3D pultruded glass fiber reinforced polymer structure is presented with the mechanical characterization of the constituting elements. The axial and flexural properties of laminate and I-shaped GRFP profiles are discussed considering the short term creep. In a companion paper, the benefits, the limits and the reliability of the structure analyzed for emergency applications are discussed. In details, the numerical structural analysis of the full-scale 3D model is described followed by the experimen…

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

SUMMARY 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 interac…

Statistic moments of the total energy of potential systems and application to equivalent non-linearization

In this paper some properties of the total energy moments of potential systems, subjected to external white noise processes, are shown. Potential systems with a polynomial form of energy-dependent damping have been considered. It is shown that the analytical relations between the statistical moments of the energy associated with such systems can be obtained with the aid of the standard Ito calculus. Furthermore, it is shown that, for the stationary case, these analytical relations are very useful for the application of the equivalent non-linearization technique.

Fractional Tajimi–Kanai model for simulating earthquake ground motion

The ground acceleration is usually modeled as a filtered Gaussian process. The most common model is a Tajimi–Kanai (TK) filter that is a viscoelastic Kelvin–Voigt unit (a spring in parallel with a dashpot) carrying a mass excited by a white noise (acceleration at the bedrock). Based upon the observation that every real material exhibits a power law trend in the creep test, in this paper it is proposed the substitution of the purely viscous element in the Kelvin Voigt element with the so called springpot that is an element having an intermediate behavior between purely elastic (spring) and purely viscous (dashpot) behavior ruled by fractional operator. With this choice two main goals are rea…

Linear Systems Excited by Polynomials of Filtered Poission Pulses

The stochastic differential equations for quasi-linear systems excited by parametric non-normal Poisson white noise are derived. Then it is shown that the class of memoryless transformation of filtered non-normal delta correlated process can be reduced, by means of some transformation, to quasi-linear systems. The latter, being excited by parametric excitations, are frst converted into ltoˆ stochastic differential equations, by adding the hierarchy of corrective terms which account for the nonnormality of the input, then by applying the Itoˆ differential rule, the moment equations have been derived. It is shown that the moment equations constitute a linear finite set of differential equatio…

A method for the probabilistic analysis of nonlinear systems

Abstract The probabilistic description of the response of a nonlinear system driven by stochastic processes is usually treated by means of evaluation of statistical moments and cumulants of the response. A different kind of approach, by means of new quantities here called Taylor moments, is proposed. The latter are the coefficients of the Taylor expansion of the probability density function and the moments of the characteristic function too. Dual quantities with respect to the statistical cumulants, here called Taylor cumulants, are also introduced. Along with the basic scheme of the method some illustrative examples are analysed in detail. The examples show that the proposed method is an a…

Performance and optimal design of Tuned Mass Damper Inerter for base isolated systems

In this paper, the use of a Tuned Mass Damper Inerter to mitigate the seismic response of a base-isolated structure is studied and compared to other passive control devices. The control performance of four different types of hybrid passive control strategies aiming to reduce base displacements of isolated buildings is investigated. Specifically, the Tuned Mass Damper (TMD), the Tuned Liquid Column Damper (TLCD), and finally, the Tuned Mass Damper Inerter, each one associated to a Base Isolated structure (BI system) have been considered. This study has been carried out using as optimal design parameters of the TMDI those estimated by a proposed direct optimization procedure, considering a wh…

Non-Stationary Probabilistic Response of Linear Systems Under Non-Gaussian Input

The probabilistic characterization of the response of linear systems subjected to non-normal input requires the evaluation of higher order moments than two. In order to obtain the equations governing these moments, in this paper the extension of the Ito’s differential rule for linear systems excited by non-normal delta correlated processes is presented. As an application the case of the delta correlated compound Poisson input process is treated.

Response Correlations of Linear Systems with White Noise Linearly Parametric Inputs

Relationships between moments and correlations of the response of linear systems subjected to linearly parametric normal white noise inputs are here reported. They are obtained by extensively using the properties of the stochastic integral calculus.

Non-stationary spectral moments of base excited MDOF systems

The paper deals with the evaluation of non-stationary spectral moments of multi-degree-of-freedom (MDOF) line systems subjected to seismic excitations. The spectral moments of the response are evaluated in incremental form solution by means of an unconditionally stable step-by-step procedure. As an application, the statistics of the largest peak of the response are also evaluated.

Digital simulation of multivariate earthquake ground motions

In this paper a new generation procedure of multivariate earthquake ground motion is presented. The technique takes full advantage of the decomposition of the power spectral density matrix by means of its eigenvectors. The application of the method to multivariate ground accelerations shows some very interesting physical properties which allows one to obtain significant reduction of the computational effort in the generation of sample functions relative to multivariate earthquake ground motion processes. Copyright © 2000 John Wiley & Sons, Ltd.

Stochastic Response on Non-Linear Systems under Parametric Non-Gaussian Agencies

The probabilistic response characterization of non-linear systems subjected to non-normal delta correlated parametric excitation is obtained. In order to do this an extension of both Ito’s differential rule and the Fokker-Planck equation is presented, enabling one to account for the effect of the non-normal input. The validity of the approach reported here is confirmed by results obtained by means of a Monte Carlo simulation.

Higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses

The higher order statistics of the response of linear systems excited by polynomials of filtered Poisson pulses are evaluated by means of knowledge of the first order statistics and without any further integration. This is made possible by a coordinate transformation which replaces the original system by a quasi-linear one with parametric Poisson delta-correlated input; and, for these systems, a simple relationship between first order and higher order statistics is found in which the transition matrix of the dynamical new system, incremented by the correction terms necessary to apply the Ito calculus, appears.

Stochastic seismic analysis of structures with nonlinear viscous dampers

Fluid damper devices inserted in buildings or bridges are commonly used as energy sinks for seismic protection. In the response analysis of structures with filled damper devices the main problem exists in the strong nonlinear behavior of such equipment, as a consequence the differential equation of motion remains nonlinear and the response spectrum analysis still cannot be applied. In this note, by using the concept of power spectral density function coherent with the elastic response spectrum and by using the statistical linearization technique, expressions for finding the equivalent linear damping have been found. Comparisons with results obtained by Monte Carlo simulations confirm that f…

Approximate analytical mean-square response of an impacting stochastic system oscillator with fractional damping

The paper deals with the stochastic dynamics of a vibroimpact single-degree-of-freedom system under a Gaussian white noise. The system is assumed to have a hard type impact against a one-sided motionless barrier, located at the system's equilibrium. The system is endowed with a fractional derivative element. An analytical expression for the system's mean squared response amplitude is presented and compared with the results of numerical simulations.

Fractional mechanical model for the dynamics of non-local continuum

In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both…

A non-local model of fractional heat conduction in rigid bodies

In recent years several applications of fractional differential calculus have been proposed in physics, chemistry as well as in engineering fields. Fractional order integrals and derivatives extend the well-known definitions of integer-order primitives and derivatives of the ordinary differential calculus to real-order operators. Engineering applications of fractional operators spread from viscoelastic models, stochastic dynamics as well as with thermoelasticity. In this latter field one of the main actractives of fractional operators is their capability to interpolate between the heat flux and its time-rate of change, that is related to the well-known second sound effect. In other recent s…

Extended Entropy Functional for Nonlinear Systems in Stochastic Dynamics

Direct evaluation of jumps for nonlinear systems under external and multiplicative impulses

In this paper the problem of the response evaluation of nonlinear systems under multiplicative impulsive input is treated. Such systems exhibit a jump at each impulse occurrence, whose value cannot be predicted through the classical differential calculus. In this context here the correct jump evaluation of nonlinear systems is obtained in closed form for two general classes of nonlinear multiplicative functions. Analysis has been performed to show the different typical behaviors of the response, which in some cases could diverge or converge to zero instantaneously, depending on the amplitude of the Dirac's delta.

Stochastic differential calculus for wind-exposed structures with autoregressive continuous (ARC) filters

In this paper, an alternative method to represent Gaussian stationary processes describing wind velocity fluctuations is introduced. The technique may be considered the extension to a time continuous description of the well-known discrete-time autoregressive model to generate Gaussian processes. Digital simulation of Gaussian random processes with assigned auto-correlation function is provided by means of a stochastic differential equation with time delayed terms forced by Gaussian white noise. Solution of the differential equation is a specific sample of the target Gaussian wind process, and in this paper it describes a digitally obtained record of the wind turbolence. The representation o…

On the numerical implementation of a 3D fractional viscoelastic constitutive model

The aim of this paper is the implementation of a 3D fraction al viscoelastic constitutive law in a user material subroutine (UMAT) in the finite element software Abaqus. Essential to the implementation of the model is access to the strain history at each Gauss point of each element in a constructive manner. Details of the UMAT and comparison with some analytical results are presented in order to show that the fractional viscoelastic constitutive law has been successfully implemented.

Nonlinear System Response for Impulsive Parametric Input

In engineering applications when the intensity of external forces depends on the response of the system, the input is called parametric. In this paper dynamical systems subjected to a parametric deterministic impulse are dealt with. Particular attention has been devoted to the evaluation of the discontinuity of the response when the parametric impulse occurs. The usual forward difference and trapezoidal integration schemes have been shown to provide only approximated solutions of the jump of the response; hence, the exact solution has been pursued and presented under the form of a numerical series. The impulse is represented throughout the paper by means of a classical Dirac’s delta functio…

Stochastic Differential Calculus

In many cases of engineering interest it has become quite common to use stochastic processes to model loadings resulting from earthquake, turbulent winds or ocean waves. In these circumstances the structural response needs to be adequately described in a probabilistic sense, by evaluating the cumulants or the moments of any order of the response (see e.g. [1, 2]). In particular, for linear systems excited by normal input, the response process is normal too and the moments or the cumulants up to the second order fully characterize the probability density function of both input and output processes. Many practical problems involve processes which are approximately normal and the effect of the…

On the convergent parts of high order spectral moments of stationary structural responses

The paper deals with the evaluation of the convergent parts of the high spectral moments of linear systems subjected to stationary random input. An adequate physical meaning of these quantities in both the time and frequency domains is presented. Recurrence formulas to obtain the high convergent cross spectral moments of any order are given in the case of white noise input.

Stochastic dynamics of linear elastic trusses in presence of structural uncertainties (virtual distortion approach)

Structures involving uncertainties in material and/or in geometrical parameters are referred to as uncertain structures. Reliability analysis of such structures strongly depends on variation of parameters and probabilistic approach is often used to characterize structural uncertainties. In this paper dynamic analysis of linearly elastic system in presence of random parameter variations will be performed. In detail parameter fluctuations have been considered as inelastic, stress and parameter dependent superimposed strains. Analysis is then carried out via superposition principle accounting for response to external agencies and parameter dependent strains. Proposed method yields asymptotic s…

Path integral solution for nonlinear systems under parametric Poissonian white noise input

Abstract In this paper the problem of the response evaluation in terms of probability density function of nonlinear systems under parametric Poisson white noise is addressed. Specifically, extension of the Path Integral method to this kind of systems is introduced. Such systems exhibit a jump at each impulse occurrence, whose value is obtained in closed form considering two general classes of nonlinear multiplicative functions. Relying on the obtained closed form relation liking the impulses amplitude distribution and the corresponding jump response of the system, the Path Integral method is extended to deal with systems driven by parametric Poissonian white noise. Several numerical applica…

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

In this paper, the probabilistic characterization of a nonlinear system enforced by Poissonian white noise in terms of complex fractional moments (CFMs) is presented. The main advantage in using such quantities, instead of the integer moments, relies on the fact that, through the CFMs the probability density function (PDF) is restituted in the whole domain. In fact, the inverse Mellin transform returns the PDF by performing integration along the imaginary axis of the Mellin transform, while the real part remains fixed. This ensures that the PDF is restituted in the whole range with exception of the value in zero, in which singularities appear. It is shown that using Mellin transform theorem…

Dynamics of non-local systems handled by fractional calculus

Mechanical vibrations of non-local systems with long-range, cohesive, interactions between material particles have been studied in this paper by means of fractional calculus. Long-range cohesive forces between material particles have been included in equilibrium equations assuming interaction distance decay with order α . This approach yields as limiting case a partial fractional differential equation of order α involving space-time variables. It has been shown that the proposed model may be obtained by a discrete, mass-spring model that includes non-local interactions by non-adjacent particles and the mechanical vibrations of the particles have been obtained by an approximation fractional …

Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments

Abstract The probability density function of the response of a nonlinear system under external α -stable Levy white noise is ruled by the so called Fractional Fokker–Planck equation. In such equation the diffusive term is the Riesz fractional derivative of the probability density function of the response. The paper deals with the solution of such equation by using the complex fractional moments. The analysis is performed in terms of probability density for a linear and a non-linear half oscillator forced by Levy white noise with different stability indexes α . Numerical results are reported for a wide range of non-linearity of the mechanical system and stability index of the Levy white nois…

Path integral solution for non-linear system enforced by Poisson White Noise

Abstract In this paper the response in terms of probability density function of non-linear systems under Poisson White Noise is considered. The problem is handled via path integral (PI) solution that may be considered as a step-by-step solution technique in terms of probability density function. First the extension of the PI to the case of Poisson White Noise is derived, then it is shown that at the limit when the time step becomes an infinitesimal quantity the Kolmogorov–Feller (K–F) equation is fully restored enforcing the validity of the approximations made in obtaining the conditional probability appearing in the Chapman Kolmogorov equation (starting point of the PI). Spectral counterpa…

The Hu-Washizu variational principle for the identification of imperfections in beams

This paper presents a procedure for the identification of imperfections of structural parameters based on displacement measurements by static tests. The proposed procedure is based on the well-known Hu–Washizu variational principle, suitably modified to account for the response measurements, which is able to provide closed-form solutions to some inverse problems for the identification of structural parameter imperfections in beams. Copyright © 2008 John Wiley & Sons, Ltd.

A correction method for dynamic analysis of linear continuous systems

A method to improve the dynamic response analysis of continuous classically damped linear system is proposed. As in fact usually, following a classical approach, a reduced number of eigenfunctions are accounted for and the response is evaluated by integrating the uncoupled differential equations of motion in modal space, neglecting the contribution of high frequency modes (truncation procedure). Here, starting from the given system, it is proposed to set up two differential equations governing the motion of two new continuous systems: the first one contains only the first m non-zero eigenvalues of the given system and the second one contains the remainder non-zero infinity - m eigenvalues. …

Random analysis of geometrically non-linear FE modelled structures under seismic actions

Abstract In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Ito's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal simi…

A Wiener Path Integral Technique for Non-Stationary Response Determination of Nonlinear Oscillators with Fractional Derivative Elements

In this paper a novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators with fractional derivative 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. In this manner, the analytical Wi…

Stochastic ship roll motion via path integral method

ABSTRACTThe response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple …

Approximate solution of the Fokker-Planck-Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker-Planck-Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions i…

Ideal Elastic-Plastic Oscillators Subjected to Stochastic Input

Abstract The paper deals with the evaluation of the probabilistic response of an ideal elastic-plastic single degree of freedom oscillator subjected to a normal white noise. The analysis has been conducted on the hypothesis that accumulated plastic displacements are a compound homogeneous Poisson process independent of the external excitation. In this case plastic displacements can be treated as an additional external noise, to be identified, acting on a linear system. In the paper a time domain approach to obtain the two variable non stationary correlation function is proposed. Hence the evolutionary power spectral density function is also obtained. A numerical example is presented in orde…

Stationary and Nontationary Response Probability Density Function of a Beam under Poisson White Noise

In this paper an approximate explicit probability density function for the analysis of external oscillations of a linear and geometric nonlinear simply supported beam driven by random pulses is proposed. The adopted impulsive loading model is the Poisson White Noise , that is a process having Dirac’s delta occurrences with random intensity distributed in time according to Poisson’s law. The response probability density function can be obtained solving the related Kolmogorov-Feller (KF) integro-differential equation. An approximated solution, using path integral method, is derived transforming the KF equation to a first order partial differential equation. The method of characteristic is the…

Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique

This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation o…

Viscoelastic bearings with fractional constitutive law for fractional tuned mass dampers

Abstract The paper aims at studying the effects of the inherent fractional constitutive law of viscoelastic bearings used as devices for tuned mass dampers. First, the proper constitutive law of the viscoelastic supports is determined by the local constitutive law. Then, the characteristic force–displacement relationship at the top of the bearing is found. Taking advantage of the whole bearing constitutive laws, the tuning of the mass damper is proposed by defining the damped fractional frequency, which is analogous to the classical damped frequency. The effectiveness of the optimal tuning procedure is validated by a numerical application on a system subjected to a Gaussian white noise.

Stochastic dynamic analysis of fractional viscoelastic systems

A method is presented to compute the non-stationary response of single-degree-of-freedom structural systems with fractional damping. Based on 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 half oscillators, the number of which depends on the discretization of the fractional derivative operator. In this context, it is shown that such a set of oscillators can be given a proper fractal representation, with a Mandelbrot dimension depending on the fractional derivative order a. It is then seen that the response second-order statistics of the derived set of c…

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself…

A novel approach to nonlinear variable-order fractional viscoelasticity.

This paper addresses nonlinear viscoelastic behaviour of fractional systems with variable time-dependent fractional order. In this case, the main challenge is that the Boltzmann linear superposition principle, i.e. the theoretical basis on which linear viscoelastic fractional operators are formulated, does not apply in standard form because the fractional order is not constant with time. Moving from this consideration, the paper proposes a novel approach where the system response is derived by a consistent application of the Boltzmann principle to an equivalent system, built at every time instant based on the fractional order at that instant and the response at all the previous ones. The ap…

La Cupola del Teatro Massimo: Il contemporaneo nel 1876

Non Linear Systems Under Complex α-Stable Le´vy White Noise

The problem of predicting the response of linear and nonlinear systems under Levy white noises is examined. A method of analysis is proposed based on the observation that these processes have impulsive character, so that the methods already used for Poisson white noise or normal white noise may be also recast for Levy white noises. Since both the input and output processes have no moments of order two and higher, the response is here evaluated in terms of characteristic function.Copyright © 2003 by ASME

Stochastic integro-differential and differential equations of non-linear systems excited by parametric Poisson pulses

Abstract The connection between stochastic integro-differential equation and stochastic differential equation of non-linear systems driven by parametric Poisson delta correlated processes is presented. It is shown that the two different formulations are fully equivalent in the case of external excitation. In the case of parametric type excitation the two formulation are equivalent if the non-linear argument in the integral representation is related by means of a series to the corresponding non-linear parametric term in the stochastic differential equation. Differential rules for the two representations to find moment equations of every order of the response are also compared.

Mechanically-based approach to non-local elasticity: Variational principles

Abstract The mechanically-based approach to non-local elastic continuum, will be captured through variational calculus, based on the assumptions that non-adjacent elements of the solid may exchange central body forces, monotonically decreasing with their interdistance, depending on the relative displacement, and on the volume products. Such a mechanical model is investigated introducing primarily the dual state variables by means of the virtual work principle. The constitutive relations between dual variables are introduced defining a proper, convex, potential energy. It is proved that the solution of the elastic problem corresponds to a global minimum of the potential energy functional. Mo…

Fractional-order nonlinear hereditariness of tendons and ligaments of the human knee

In this paper the authors introduce a nonlinear model of fractional-order hereditariness used to capture experimental data obtained on human tendons of the knee. Creep and relaxation data on fibrous tissues have been obtained and fitted with logarithmic relations that correspond to power-laws with nonlinear dependence of the coefficients. The use of a proper nonlinear transform allows one to use Boltzmann superposition in the transformed variables yielding a fractional-order model for the nonlinear material hereditariness. The fundamental relations among the nonlinear creep and relaxation functions have been established, and the results from the equivalence relations have been contrasted wi…

Monte Carlo simulation for the response analysis of long-span suspended cables under wind loads

This paper presents a time-domain approach for analyzing nonlinear random vibrations of long-span suspended cables under transversal wind. A consistent continuous model of the cable, fully accounting for geometrical nonlinearities inherent in cable behavior, is adopted. The effects of spatial correlation are properly included by modeling wind velocity fluctuation as a random function of time and of a single spatial variable ranging over cable span, namely as a one-variate bi-dimensional (1V-2D) random field. Within the context of a Galerkin`s discretization of the equations governing cable motion, a very efficient Monte Carlo-based technique for second-order analysis of the response is prop…

Analytic evaluation of spectral moments

In this paper an analytic procedure that drastically reduces the computational effort in evaluating the spectral moments of the response of multi-degree-of-freedom systems is presented. It is shown that the cross-spectral moments of any order of two oscillators subjected to a filtered stochastic process can be obtained in a recursive manner as a linear combination of the spectral moment of each oscillator up to the third order separately taken. A numerical procedure is also presented in order to evaluate such first few spectral moments.

Some properties of multi-degree-of-freedom potential systems and application to statistical equivalent non-linearization

This paper presents some properties of two restricted classes of multi-degree-of-freedom potential systems subjected to Gaussian white-noise excitations. Specifically, potential systems which exhibit damping terms with energy-dependent polynomial form are referred to. In this context, first systems with coupled stiffness terms and damping terms depending on the total energy are investigated. Then, systems with uncoupled stiffness terms and damping terms depending on the total energy in each degree-of-freedom are considered. For these two classes, it is found that algebraic relations among the stationary statistical moments of the energy functions can be derived by applying standard tools of…

Hysteretic Systems Subjected to Delta Correlated Input

The paper deals with the evaluation of the probabilistic response of a single degree of freedom elastic-perfectly plastic system subjected to a delta correlated input process. The probabilistic characterisation of the response is here obtained by considering the accumulated plastic deformations as a compound homogeneous Poisson process independent of the external input. In this case the former can be considered as an external noise acting on the linear system. A closed form solution is also obtained and the analytic expression is compared with the customary Monte-Carlo method.

Ship Roll Motion under Stochastic Agencies Using Path Integral Method

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamica…

Effect of epicentral direction on seismic response of asymmetric buildings

The paper deals with the influence of the epicentral direction on the displacement and stress response of multistorey asymmetric buildings to earthquake horizontal ground motion. A method is given for computing for each plane frame of the complex structure a particular direction of the bidirectional stationary random input for which the horizontal floor displacement of the given frame is maximized. It is shown that this direction can be considered conservative for the corresponding non-stationary process.

Stochastic seismic analysis of hydrodynamic pressure in dam reservoir systems

Hydrodynamic seismic-induced pressure requires careful consideration in the aseismic design of dams. Effects induced by earthquake excitation may cause many-fold increments of hydrostatic pressure. In this study earthquake excitation has been modelled by means of random process theory obtaining the response statistics of a dam-reservoir dynamical system. The analysis has been conducted assuming a rigid retaining wall of the reservoir and dissipative fluid. Copyright © 2002 John Wiley & Sons, Ltd.

Mechanical Behavior Of Fractional Visco-Elastic Beams

Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations

The aim of this paper is the evaluation of higher order statistics of the response of linear systems subjected to external excitations and to linearly parametric white noise. The external excitations considered are deterministic or filtered white noise processes. The procedure implies the knowledge of the transition matrix connected to the linear system; this, however, has already been evaluated for obtaining the statistics at single times. The method, which avoids making further integrations for the evaluation of the higher order statistics, is very advantageous from a computational point of view.

Nonstationary response envelope probability densities of nonlinear oscillators

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh dis…

Stochastic seismic analysis of MDOF structures with nonlinear viscous dampers

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 (RS) technique fails. In this paper by using the concept of power spectral density function compatible with the elastic RS and the statistical 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. Copyright © 200…

Itô calculus extended to systems driven by -stable Lévy white noises (a novel clip on the tails of Lévy motion)

Abstract The paper deals with probabilistic characterization of the response of non-linear systems under α -stable Levy white noise input. It is shown that, by properly selecting a clip in the probability density function of the input, the moments of the increments of Levy motion process remain all of the same order ( d t ) , like the increments of the Compound Poisson process. It follows that the Ito calculus extended to Poissonian input, may also be used for α -stable Levy white noise input processes. It is also shown that, when the clip on the tails of the probability of the increments of the Levy motion approaches to infinity, the Einstein–Smoluchowsky equation is restored. Once these c…

A representation of wind velocity by means of fractional spectral moments

This paper deals with the definition of a new function that is a link between Power Spectral Density (PSD) and correlation function, called the Fractional Spectral Moments function. This is defined as the moment of complex order g of the one-sided PSD. It is shown that by means of this complex function both the correlation function and PSD can be represented with great accuracy.

Stochastic response of MDOF wind-excited structures by means of Volterra series approach

Abstract The role played by the quadratic term of the forcing function in the response statistics of multi-degree-of-freedom (MDOF) wind-excited linear-elastic structures is investigated. This is accomplished by modeling the structural response as a Volterra series up to the second order and neglecting the wind-structure interaction. In order to reduce the computational effort due to the calculation of a large number of multiple integrals, required by the used approach, a recent model of the wind stochastic field is adopted.

What is Differential Stochastic Calculus?

Some well known concepts of stochastic differential calculus of non linear systems corrupted by parametric normal white noise are here outlined. Ito and Stratonovich integrals concepts as well as Ito differential rule are discussed. Applications to the statistics of the response of some linear and non linear systems is also presented.

Markovian approximation of linear systems with fractional viscoelastic term

It is well known that the response of a linear system enforced by a Gaussian white noise is Markovian. The order of Markovianity is n-1 being n the maximum order of the derivative of the equation ruling the evolution of the system. However when a fractional operator appears, the order of Markovianity of the system becomes infinite. Then the main aim developed in the proposed paper, consists of rewriting the system with fractional term of order r with an "equivalent" one, in which the fractional operator is substituted by two classical differential terms with integer order of derivative int(r) and int(r + 1) (for a real r). In this way the fractional differential equation reverts into a clas…

Structural performances of pultruded GFRP emergency structures – Part 2: Full-scale experimental testing

Abstract This paper presents an experimental testing of a pultruded glass fiber reinforced polymer (FRP) structure used for emergency applications continuing the discussion presented in a previous paper (part 1) where the study of the characteristics of material and elements are presented. First, the design of the composite structure and components and the evaluation of the structural behavior by means of numerical and analytical approach according to current regulatory codes are described. In this frame, the global and local response was observed according to load paths deriving from the design loads at limit states. Then, the experimental test on a full-scale 2D model is presented at diff…

Representation of Strongly Stationary Stochastic Processes

A generalization of the orthogonality conditions for a stochastic process to represent strongly stationary processes up to a fixed order is presented. The particular case of non-normal delta correlated processes, and the probabilistic characterization of linear systems subjected to strongly stationary stochastic processes are also discussed.

Non-linear systems under impulsive parametric input

In this paper the problem of the response of non-linear systems excited by an impulsive parametric input is treated. For such systems the response exhibits a jump depending on the amplitude of the impulse as well as on the value of the state variables immediately before the impulse occurrence. Recently, the jump prediction has been obtained in a series form. Here the incremental rule for any scalar real valued function is obtained in an analytical form involving the jump of the state variables. It is also shown that the formulation for the jump evaluation is also able to give a new step-by-step integration technique.

Modal analysis for random response of MDOF systems

The usefulness of the mode-superposition method of multidegrees of freedom systems excited by stochastic vector processes is here presented. The differential equations of moments of every order are written in compact form by means of the Kronecker algebra; then the method for integration of these equations is presented for both classically and non-classically damped systems, showing that the fundamental operator available for evaluating the response in the deterministic analysis is also useful for evaluating the response in the stochastic analysis.

A correction method for the analysis of continuous linear one-dimensional systems under moving loads

A new correction procedure for dynamic analysis of linear, proportionally damped, continuous systems under traveling concentrated loads is proposed; both cases of non-parametric (moving forces) and parametric (moving mass) loads are considered. Improvement in the evaluation of the dynamic response is obtained by separating the contribution of the low-frequency (LF) modes from that of the high-frequency (HF) modes. The former is calculated, as usual, by classical modal analysis, while the latter is taken into account using a new series expansion of the corresponding particular solution. The advantage of the suggested method is immediately shown in the calculation of the stress distribution s…

Non-linear Systems Under Poisson White Noise Handled by Path Integral Solution

An extension of the path integral to non-linear systems driven by a Poissonian white noise process is presented. It is shown that at the limit when the time increment becomes infinitesimal the Kolmogorov— Feller equation is fully restored. Applications to linear and non-linear systems with different distribution of the Dirac's deltas occurrences are performed and results are compared with analytical solutions (when available) and Monte Carlo simulation.

On the Characterization of Dynamic Properties of Random Processes by Spectral Parameters

This paper deals with the general problem of directly relating the distribution of ranges of wide band random processes to the power spectral density (PSD) by means of closed-form expressions. Various attempts to relate the statistical distribution of ranges to the PSD by means of the irregularity factor or similar parameters have been done by several authors but, unfortunately, they have not been fully successful. In the present study, introducing the so-called analytic processes, the reasons for which these parameters are insufficient to an unambiguous determination of the range distribution and the fact that parameters regarding the time-derivative processes are needed have been explaine…

L’IDENTIFICAZIONE DINAMICA DELLE CARATTERISTICHE MODALI E MECCANICHE DELLA STRUTTURA DELLA CUPOLA DEL TEATRO MASSIMO

Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results

In capturing visco-elastic behavior, experimental tests play a fundamental rule, since they allow to build up theoretical constitutive laws very useful for simulating their own behavior. The main challenge is representing the visco-elastic materials through simple models, in order to spread their use. However, the wide used models for capturing both relaxation and creep tests are combinations of simple models as Maxwell and/or Kelvin, that depend on several parameters for fitting both creep and relaxation tests. This paper, following Nutting and Gemant idea of fitting experimental data through a power law function, aims at stressing the validity of fractional model. In fact, as soon as rela…

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

Recursive formulas in terms of statistics of the response of linear systems with time delay under normal white noise input are developed. Two alternative methods are presented, in order to capture the time delay effects. The first is given in an approximate solution obtained by expanding the control force in a Taylor series. The second, available for the stationary solution (if it exists) gets the variance of the controlled system, with time delay in an analytical form. The efficacy loss in terms of statistics of the response is discussed in detail.