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

[+] Author and Article Information
Valeria Artale

Engineering and Architecture Faculty,
University of Enna “Kore,”
via delle Olimpiadi,
Enna 94100, Italy
e-mail: valeria.artale@unikore.it

Giacomo Navarra

Engineering and Architecture Faculty,
University of Enna “Kore,”
via delle Olimpiadi,
Enna 94100, Italy
e-mail: giacomo.navarra@unikore.it

Angela Ricciardello

Engineering and Architecture Faculty,
University of Enna “Kore,”
via delle Olimpiadi,
Enna 94100, Italy
e-mail: angela.riccardello@unikore.it

Giorgio Barone

School of Civil and Building Engineering,
Loughborough University,
Loughborough, Leicestershire LE11 3TU, UK
e-mail: g.barone@lboro.ac.uk

1Corresponding author.

Manuscript received June 17, 2016; final manuscript received December 10, 2016; published online June 12, 2017. Assoc. Editor: Mario Di Paola.

ASME J. Risk Uncertainty Part B 3(3), 030901 (Jun 12, 2017) (6 pages) Paper No: RISK-16-1091; doi: 10.1115/1.4036700 History: Received June 17, 2016; Revised December 10, 2016

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 behavior of many complex mechanical systems, e.g., the nonlocal or nonviscous 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.

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Rossikhin, Y. A. , and Shitikova, M. V. , 2010, “ Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results,” ASME Appl. Mech. Rev., 63(1), p. 010801. [CrossRef]
Shimizu, N. , and Zhang, W. , 1999, “ Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials,” JSME Int. J., Ser. C, 42(4), pp. 825–837. [CrossRef]
Bagley, R. L. , and Torvik, P. J. , 1983, “ A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27(3), pp. 201–210. [CrossRef]
Koeller, R. C. , 1984, “ Applications of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51(2), pp. 299–307. [CrossRef]
Koh, C. G. , and Kelly, J. M. , 1990, “ Application of Fractional Derivatives to Seismic Analysis of Base-Isolated Models,” Earthquake Eng. Struct. Dyn., 19(2), pp. 229–241. [CrossRef]
Makris, N. , and Constantinou, M. C. , 1991, “ Fractional Derivative Maxwell Model for Viscous Dampers,” J. Struct. Eng., 117(9), pp. 2708–2724. [CrossRef]
Lewandowski, R. , and Chorazyczewski, B. , 2010, “ Identification of the Parameters of the Kelvin–Voigt and the Maxwell Fractional Models, Used to Modelling of Viscoelastic Dampers,” Comput. Struct., 88, pp. 1–17. [CrossRef]
Di Paola, M. , and Zingales, M. , 2012, “ Exact Mechanical Models of Fractional Hereditary Materials,” J. Rheol., 56(5), pp. 983–1004. [CrossRef]
Di Paola, M. , Pirrotta, A. , and Valenza, A. , 2011, “ Viscoelastic Behaviour Through Fractional Calculus: An Easier Method for Best Fitting Experimental Results,” Mech. Mater., 43(12), pp. 799–806. [CrossRef]
Rossikhin, Y. A. , and Shitikova, M. V. , 1997, “ Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids,” ASME Appl. Mech. Rev., 50(1), pp. 15–67. [CrossRef]
Rüdinger, F. , 2006, “ Tuned Mass Damper With Fractional Derivative Damping,” Eng. Struct., 28(13), pp. 1774–1779. [CrossRef]
Barone, G. , Lo Iacono, F. , and Navarra, G. , 2014, “ Passive Control of Fractional Viscoelastic Structures by Fractional Tuned Mass Dampers,” Computational Stochastic Mechanics Conference (CSM7), Santorini, Greece, June 15–18, pp. 97–106.
Barone, G. , Lo Iacono, F. , and Navarra, G. , 2014, “ Dynamic Characterization of Fractional Oscillators for Fractional Tuned Mass Dampers Tuning,” International Conference on Fractional Differentiation and Its Applications (ICFDA), Catania, Italy, June 23–25, pp. 1–5.
Barone, G. , Di Paola, M. , Lo Iacono, F. , and Navarra, G. , 2015, “ Viscoelastic Bearings With Fractional Constitutive Law for Fractional Tuned Mass Dampers,” J. Sound Vib., 344, pp. 18–27. [CrossRef]
Barone, G. , Palmeri, A. , and Lombardo, M. , 2015, “ Stochastic Analysis of Fractional Oscillators by Equivalent System Definition,” First Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP), Crete Island, Greece, May 25–27, pp. 319–328.
Vanmarcke, E. H. , 1972, “ Properties of Spectral Moments With Applications to Random Vibration,” ASCE J. Eng. Mech. Div., 98(2), pp. 425–446.
Michaelov, G. , Sarkani, S. , and Luted, L. D. , 1999, “ Spectral Characteristic of Nonstationary Random Processes—A Critical Review,” Struct. Saf., 21(3), pp. 223–244. [CrossRef]
Di Paola, M. , and Petrucci, G. , 1990, “ Spectral Moments and Pre-Envelope Covariances of Non Separable Processes,” ASME J. Appl. Mech., 57(1), pp. 218–224. [CrossRef]
Muscolino, G. , 1991, “ Nonstationary Pre-Envelope Covariances of Nonclassically Damped Systems,” J. Sound Vib., 149(1), pp. 107–123. [CrossRef]
Spanos, P. D. , and Miller, S. M. , 1994, “ Hilbert Transform Generalization of a Classical Random Vibration Integral,” ASME J. Appl. Mech., 61(3), pp. 575–581. [CrossRef]
Di Paola, M. , 2015, “ Complex Fractional Moments and Their Use in Earthquake Engineering,” Encyclopedia of Earthquake Engineering, Springer, Berlin, pp. 446–461.
Cottone, G. , and Di Paola, M. , 2010, “ A New Representation of Power Spectral Density and Correlation Function by Means of Fractional Spectral Moments,” Probab. Eng. Mech., 25(3), pp. 348–353. [CrossRef]
Cottone, G. , Di Paola, M. , and Santoro, R. , 2010, “ A Novel Exact Representation of Stationary Colored Gaussian Processes (Fractional Differential Approach),” J. Phys. A: Math. Theor., 43(8), p. 085002. [CrossRef]
Cottone, G. , and Di Paola, M. , 2011, “ Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields,” J. Wind Eng. Ind. Aerodyn., 99(6), pp. 741–747. [CrossRef]
Mitrinović, D. S. , and Keckić, J. D. , 1984, The Cauchy Method of Residues: Theory and Applications, Springer, Dordrecht, The Netherlands.
Ablowitx, M. J. , and Fokas, A. S. , 2003, Complex Variables: Introduction and Applications, Cambridge University Press, Cambridge, UK.


Grahic Jump Location
Fig. 2

Fractional oscillator

Grahic Jump Location
Fig. 3

FSMs for classic linear oscillators with varying damping ratios

Grahic Jump Location
Fig. 4

Existence dominion for FSMs of fractional oscillators

Grahic Jump Location
Fig. 5

Real-order FSMs for fractional oscillators

Grahic Jump Location
Fig. 6

SMs of order 0, 1, and 2 for fractional oscillators




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