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

Cottone, Giulio; Paola, Mario Di

doi:10.1016/j.probengmech.2010.04.003

Cited by 20 publications

(14 citation statements)

References 24 publications

(40 reference statements)

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“…Although the SMs return important information on the nature of a stochastic process, they do not allow fully describing the PSD or the correlation of the process itself [21]. Cottone and Di Paola [22] proposed an extension of the traditional integer order SMs to complex order moments, by defining the so-called Fractional Spectral Moments (FSMs) as the Mellin transform of the one-sided PSD of a stochastic process. The advantage of these complex quantities is that they are able to reconstruct both the PSD and the correlation functions, and, therefore, can be seen as an alternative representation of the process itself [22,23].…”

Section: Exact Closed-form Fractional Spectral Moments For Linear Framentioning

confidence: 99%

“…Cottone and Di Paola [22] proposed an extension of the traditional integer order SMs to complex order moments, by defining the so-called Fractional Spectral Moments (FSMs) as the Mellin transform of the one-sided PSD of a stochastic process. The advantage of these complex quantities is that they are able to reconstruct both the PSD and the correlation functions, and, therefore, can be seen as an alternative representation of the process itself [22,23]. Additionally, Cottone and Di Paola [24] have shown that FSMs can be used as coefficients of a time series defined in terms of the Riesz fractional derivatives of a white noise for the digital simulation of realizations of a stochastic process with assigned PSD.…”

Section: Exact Closed-form Fractional Spectral Moments For Linear Framentioning

confidence: 99%

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Exact Closed-Form Fractional Spectral Moments for Linear Fractional Oscillators Excited by a White Noise

Artale

Navarra

Ricciardello

et al. 2017

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering

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In the last decades the research community has shown an increasing interest in the engineering applications of fractional calculus, which allows to accurately characterize the static and dynamic behaviour of many complex mechanical systems, e.g. the non-local or non-viscous constitutive law. In particular, fractional calculus has gained considerable importance in the random vibration analysis of engineering structures provided with viscoelastic damping. In this case, the evaluation of the dynamic response in the frequency domain presents significant advantages, once a probabilistic characterization of the input is provided. On the other hand, closed-form expressions for the response statistics of dynamical fractional systems are not available even for the simplest cases. Taking advantage of the Residue Theorem, in this paper the exact expressions of the spectral moments of integer and complex orders (i.e. fractional spectral moments) of linear fractional oscillators driven by acceleration time histories obtained as samples of stationary Gaussian white noise processes are determined. IntroductionThere is an increasing amount of research on the use of fractional operators to describe viscoelastic properties of materials in structural dynamics [1]. In fact, it has been shown that fractional integrals and derivatives are suited to mathematically model constitutive equations of viscoelastic materials, returning creep and relaxation functions whose general shapes well fit experimental data [2][3][4][5][6][7][8][9]. In civil and mechanical engineering, when fractional operators are used to model dissipative forces in dynamic systems, the latter are indicated with the term fractional oscillators. Several numerical methods to integrate the equations of motion of fractional systems have been proposed. A comprehensive review of papers dealing with the dynamic behaviour of fractional linear and nonlinear systems, single and multi-degrees-of-freedom, including vibration of rods, beams, plates and shells, among others, can be found in [1] and [10]. Among the various studies on fractional oscillators in literature, Rüdinger [11] proposes their use as viscoelastic tuned mass dampers to reduce vibrations of systems excited by an external white noise. The fractional oscillator is constituted by a mass linked to the main structure through a linear spring placed in parallel to a viscoelastic damping element exerting a force

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Section: Exact Closed-form Fractional Spectral Moments For Linear Framentioning

confidence: 99%

Section: Exact Closed-form Fractional Spectral Moments For Linear Framentioning

confidence: 99%

Exact Closed-Form Fractional Spectral Moments for Linear Fractional Oscillators Excited by a White Noise

Artale

Navarra

Ricciardello

et al. 2017

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering

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show abstract

“…The problem is solved on basis of a recently developed method which allows to represent PSD and correlation function (AC) in closed form by means of a generalized Taylor expansion using fractional spectral moments (FSMs) [11]. The concept is used in [15] to derive a linear fractional differential equation, whose output is a stationary colored Gaussian process with target PSD, e.g.…”

Section: Motivations and Aim Of The Papermentioning

confidence: 99%

“…As shown in [11] both the AC and the PSD function can be reconstructed by fractional spectral moments (FSMs) defined as (3) with γ ∈ C chosen such that the integral converge, that is with the real part γ 0 < Reγ < γ 1 .…”

Section: Fractional Representation Of Stationary Gaussian Processesmentioning

confidence: 99%

Treatment of arbitrarily autocorrelated load functions in the scope of parameter identification

Runtemund

Cottone

Müller

2013

Computers & Structures

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Ambient vibration tests are output-only tests based on the structural response recorded under natural excitation, such as wind or traffic loads. In contrast to forced vibration test, they do not require service interruption and expensive devices to excite the structure. To this aim, we combine the recently proposed H-fractional spectral moments representation of stationary processes with a modification of the Kalman filter to the scope of structural parameter identification. This paper shows that the method is particularly suited to dealing with longcorrelated loads, i.e. stochastic processes with inverse power-law correlation, where many existing methods are not applicable or insufficiently accurate.

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“…Physical meaning of SMs quantities has been given in Di Paola (1985). In Cottone and Di Paola, 2010a, the authors introduced a generalized class of SM, called the Fractional Spectral Moments (FSMs) of the one-sided PSD, that is…”

Section: Fractional Moments For Stochastic Processesmentioning

confidence: 99%

Fractional spectral moments for digital simulation of multivariate wind velocity fields

Cottone

Paola

2011

Journal of Wind Engineering and Industrial Aerodynamics

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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. * Publication info: Cottone G., Di Paola M., Fractional spectral moments for digital simulation of multivariate wind velocity fields,

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

Cited by 20 publications

References 24 publications

Exact Closed-Form Fractional Spectral Moments for Linear Fractional Oscillators Excited by a White Noise

Exact Closed-Form Fractional Spectral Moments for Linear Fractional Oscillators Excited by a White Noise

Treatment of arbitrarily autocorrelated load functions in the scope of parameter identification

Fractional spectral moments for digital simulation of multivariate wind velocity fields

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