Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion

Pinnola, Francesco Paolo

doi:10.1016/j.cnsns.2016.03.013

Cited by 27 publications

(8 citation statements)

References 58 publications

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“…In order to make the analysis extensible to systems with several degrees of freedom or subjected to any type of excitation for which analytical solutions are not available, the process to solve the differential equation (1) numerically is presented. The analytical response of a 1 DOF system to the cited excitations-initial conditions or impulse and step forces-can be found, for instance, in [2,20,29,31].…”

Section: Dynamic Responsementioning

confidence: 99%

An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

et al. 2019

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Fractional derivative models are widely used to easily characterise more complex damping behaviour than the viscous one, although the underlying properties are not trivial. Several studies about the mathematical properties can be found, but are usually far from the most daily applications. Thus, this paper studies the properties of structural systems whose damping is represented by a fractional model from the point of view of a mechanical engineer. First, a single-degree-of-freedom system with fractional damping is analysed. Specifically, the distribution of the poles and the dynamic response to several excitations is studied for different model parameter values highlighting dissimilarities from systems with conventional viscous damping. In fact, thanks to fractional models, particular behaviours are observed that cannot be reproduced by classical ones. Finally, the dynamics of a machine shaft supported by two bearings presenting fractional damping is analysed. The study is carried out by the Finite Element method, deriving in a system with symmetric matrices. Eigenvalues and eigenvectors are obtained by means of an iterative method, and the effect of damping is visualised on the mode shapes. In addition, the response to a perturbation is computed, revealing the influence of the model parameters on the resulting vibration.

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Section: Dynamic Responsementioning

confidence: 99%

An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

et al. 2019

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“…That is, the PSD, and the CF. It has been shown that for the fractional differential equations forced by Gaussian white noise the stationary PSD can be evaluated in closed form but the CF can be obtained just in approximated way by the discrete inverse Fourier transform or by using the complex spectral moments [51].…”

Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

“…where R U j U k (τ) represents the CCF of the processes U j (t) and U k (t). Observe that from Eq.s (31) is not possible to obtain the CCFs in analytical form because the second integral can be solved just in numerical way when fractional powers of order α of iω appear in the PSDs and in the CPSDs [51]. By Eqs.…”

Section: Stationary Response Characterizationmentioning

confidence: 99%

Stochastic Analysis of a Nonlocal Fractional Viscoelastic Bar Forced by Gaussian White Noise

Alotta

Failla

Pinnola

2017

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

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Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.

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“…Roughly, four distinct strategies in the literature have been pursued to compute the response of a fractional oscillator. As a common way to obtain the transient response of a linear system to any loading is through the Duhamel integral, which employs the impulse response function (IRF) of the system (Suarez and Shokooh 1995), a popular strategy has been attempting to obtain the fractional oscillator's IRF first (Gaul et al 1989;Suarez and Shokooh 1995;Agrawal 1999;Agrawal 2001;Achar et al 2002;Ye et al 2003;Achar et al 2004;Huang et al 2010;Naber 2010;Pinnola 2016). While considering an oscillator with fractional order q = 1/2, Gaul et al (Gaul et al 1989) computed the system's IRF by taking the inverse Fourier transform analytically from the system's frequency response function (FRF).…”

Section: Introductionmentioning

confidence: 99%

Frequency/Laplace domain methods for computing transient responses of fractional oscillators

Cao

2022

Nonlinear Dyn

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Although computing the transient response of fractional oscillators, characterized by secondorder differential equations with fractional derivatives for the damping term, to external loadings has been studied, most existing methodologies have dealt with cases with either restricted fractional orders or simple external loadings. In this paper, considering complicated irregular loadings acting on oscillators with any fractional order between 0 and 1, efficient frequency/Laplace domain methods for getting transient responses are developed. The proposed methods are based on pole-residue operations. In the frequency domain approach, "artificial" poles located along the imaginary axis of complex plane are designated. In the Laplace domain approach, the "true" poles are extracted through two phases: (1) a discrete impulse response function (IRF) is produced by taking the inverse Fourier transform of the corresponding frequency response function (FRF) that is readily obtained from the exact TF, and (2) a complex exponential signal decomposition method, i.e., the Prony-SS method, is invoked to extract the poles and residues. Once the poles/residues of the system are known, those of the response can be determined by simple pole-residue operations.Sequentially, the response time history is readily obtained. Two fractional oscillators with rational and irrational derivatives, respectively, subjected to sinusoidal and complicated earthquake loading are presented to illustrate the procedure and verify the correctness of the proposed method. The verification is conducted by comparing the results from both the Laplace and the frequency domain

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Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion

Cited by 27 publications

References 58 publications

An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

An Analysis of the Dynamical Behaviour of Systems with Fractional Damping for Mechanical Engineering Applications

Stochastic Analysis of a Nonlocal Fractional Viscoelastic Bar Forced by Gaussian White Noise

Frequency/Laplace domain methods for computing transient responses of fractional oscillators

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