Abstract
In this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.
Issue Section:
Research Papers
References
1.
Makris
, N.
, and Constantinou
, M. C.
, 1991
, “Fractional-Derivative Maxwell Model for Viscous Dampers
,” J. Struct. Eng.
, 117
(9
), pp. 2708
–2724
. 2.
Zhu
, S.
, Cai
, C.
, and Spanos
, P. D.
, 2015
, “A Nonlinear and Fractional Derivative Viscoelastic Model for Rail Pads in the Dynamic Analysis of Coupled Vehicle–Slab Track Systems
,” J. Sound Vib.
, 335
, pp. 304
–320
. 3.
Chang
, T. S.
, and Singh
, M. P.
, 2002
, “Seismic Analysis of Structures With a Fractional Derivative Model of Viscoelastic Dampers
,” Earthq. Eng. Eng. Vib.
, 1
(2
), pp. 251
–260
. 4.
Zhang
, H.
, Zhe Zhang
, Q.
, Ruan
, L.
, Duan
, J.
, Wan
, M.
, and Insana
, M. F.
, 2018
, “Modeling Ramp-Hold Indentation Measurements Based on Kelvin–Voigt Fractional Derivative Model
,” Meas. Sci. Technol.
, 29
(3
), p. 035701
. 5.
Sherief
, H. H.
, El-Sayed
, A. M.
, Behiry
, S. H.
, and Raslan
, W. E.
, 2012
, “Using Fractional Derivatives to Generalize the Hodgkin–Huxley Model,” Fractional Dynamics and Control
, D.
Baleanu
, J.A.T.
Machado
, and A.C.J.
Luo
, eds., Springer
, New York
, pp. 275
–282
.6.
Djordjević
, V. D.
, Jarić
, J.
, Fabry
, B.
, Fredberg
, J. J.
, and Stamenović
, D.
, 2003
, “Fractional Derivatives Embody Essential Features of Cell Rheological Behavior
,” Ann. Biomed. Eng.
, 31
(6
), pp. 692
–699
. /10.1114/1.15740267.
Chen
, Q.
, Suki
, B.
, and An
, K. N.
, 2004
, “Dynamic Mechanical Properties of Agarose Gels Modeled by a Fractional Derivative Model
,” ASME J. Biomech. Eng.
, 126
(5
), pp. 666
–671
. 8.
Chang
, T. K.
, Rossikhin
, Y. A.
, Shitikova
, M. V.
, and Chao
, C. K.
, 2011
, “Application of Fractional-Derivative Standard Linear Solid Model to Impact Response of Human Frontal Bone
,” Theor. Appl. Fract. Mec.
, 56
(3
), pp. 148
–153
. 9.
Petromichelakis
, I.
, Psaros
, A. F.
, and Kougioumtzoglou
, I. A.
, 2021
, “Stochastic Response Analysis and Reliability-Based Design Optimization of Nonlinear Electromechanical Energy Harvesters With Fractional Derivative Elements
,” ASCE-ASME J. Risk Uncert. Eng. Syst. Part B: Mech. Eng.
, 7
(1
), p. 010901
. 10.
Agrawal
, O. P.
, 1999
, “An Analytical Scheme for Stochastic Dynamic Systems Containing Fractional Derivatives
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Vol. 19777
, American Society of Mechanical Engineers
, Sept. 12
, pp. 243
–249
.11.
Agrawal
, O. P.
, 2001
, “Stochastic Analysis of Dynamic Systems Containing Fractional Derivatives
,” J. Sound Vib.
, 5
(247
), pp. 927
–938
. 12.
Cao
, Q.
, Hu
, S. L.
, and Li
, H.
, 2021
, “Nonstationary Response Statistics of Fractional Oscillators to Evolutionary Stochastic Excitation
,” Commun. Nonlinear Sci. Numer. Simul.
, 103
, p. 105962
. 13.
Su
, C.
, and Xian
, J.
, 2022
, “Nonstationary Random Vibration Analysis of Fractionally-Damped Systems by Numerical Explicit Time-Domain Method
,” Probabilistic Eng. Mech.
, 68
, p. 103228
. 14.
Di Paola
, M.
, Failla
, G.
, and Pirrotta
, A.
, 2012
, “Stationary and Non-Stationary Stochastic Response of Linear Fractional Viscoelastic Systems
,” Probabilistic Eng. Mech.
, 28
, pp. 85
–90
. 15.
Failla
, G.
, and Pirrotta
, A.
, 2012
, “On the Stochastic Response of a Fractionally-Damped Duffing Oscillator
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(12
), pp. 5131
–5142
. 16.
Spanos
, P. D.
, Di Matteo
, A.
, Cheng
, Y.
, Pirrotta
, A.
, and Li
, J.
, 2016
, “Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements
,” ASME J. Appl. Mech.
, 83
(12
), p. 121003
. 17.
Di Matteo
, A.
, Spanos
, P. D.
, and Pirrotta
, A.
, 2018
, “Approximate Survival Probability Determination of Hysteretic Systems With Fractional Derivative Elements
,” Probabilistic Eng. Mech.
, 54
, pp. 138
–146
. 18.
Spanos
, P. D.
, Kougioumtzoglou
, I. A.
, dos Santos
, K. R.
, and Beck
, A. T.
, 2018
, “Stochastic Averaging of Nonlinear Oscillators: Hilbert Transform Perspective
,” J. Eng. Mech.
, 144
(2
), p. 04017173
. 19.
dos Santos
, K. R.
, Kougioumtzoglou
, I. A.
, and Spanos
, P. D.
, 2019
, “Hilbert Transform–Based Stochastic Averaging Technique for Determining the Survival Probability of Nonlinear Oscillators
,” J. Eng. Mech.
, 145
(10
), p. 04019079
. 0.1061/(ASCE)EM.1943-7889.000165120.
Di Matteo
, A.
, Kougioumtzoglou
, I. A.
, Pirrotta
, A.
, Spanos
, P. D.
, and Di Paola
, M.
, 2014
, “Stochastic Response Determination of Nonlinear Oscillators With Fractional Derivatives Elements Via the Wiener Path Integral
,” Probabilistic Eng. Mech.
, 38
, pp. 127
–135
. 21.
Zhang
, Y.
, Li
, S.
, and Kong
, F.
, 2021
, “Survival Probability of Nonlinear Oscillators Endowed With Fractional Derivative Element and Subjected to Evolutionary Excitation: A Stochastic Averaging Treatment With Path Integral Concepts
,” Probabilistic Eng. Mech.
, 66
, p. 103156
. 22.
Spanos
, P. D.
, and Malara
, G.
, 2014
, “Nonlinear Random Vibrations of Beams With Fractional Derivative Elements
,” J. Eng. Mech.
, 140
(9
), p. 04014069
. 23.
Spanos
, P. D.
, and Malara
, G.
, 2017
, “Random Vibrations of Nonlinear Continua Endowed With Fractional Derivative Elements
,” Procedia Eng.
, 199
, pp. 18
–27
. 24.
Malara
, G.
, and Spanos
, P. D.
, 2018
, “Nonlinear Random Vibrations of Plates Endowed With Fractional Derivative Elements
,” Probabilistic Eng. Mech.
, 54
, pp. 2
–8
. 25.
Spanos
, P. D.
, and Malara
, G.
, 2020
, “Nonlinear Vibrations of Beams and Plates With Fractional Derivative Elements Subject to Combined Harmonic and Random Excitations
,” Probabilistic Eng. Mech.
, 59
, p. 103043
. 26.
Malara
, G.
, Pomaro
, B.
, and Spanos
, P. D.
, 2021
, “Nonlinear Stochastic Vibration of a Variable Cross-Section Rod With a Fractional Derivative Element
,” Int. J. Non-Linear Mech.
, 135
, p. 103770
. 27.
Malara
, G.
, Spanos
, P. D.
, and Jiao
, Y.
, 2020
, “Efficient Calculation of the Response Statistics of Two-Dimensional Fractional Diffusive Systems
,” Probabilistic Eng. Mech.
, 59
, p. 103036
. 28.
Kougioumtzoglou
, I. A.
, and Spanos
, P. D.
, 2016
, “Harmonic Wavelets Based Response Evolutionary Power Spectrum Determination of Linear and Non-Linear Oscillators With Fractional Derivative Elements
,” Int. J. Non-Linear Mech.
, 80
, pp. 66
–75
. 29.
Kong
, F.
, Zhang
, Y.
, and Zhang
, Y.
, 2022
, “Non-Stationary Response Power Spectrum Determination of Linear/Non-Linear Systems Endowed With Fractional Derivative Elements Via Harmonic Wavelet
,” Mech. Syst. Signal Process
, 162
, p. 108024
. 30.
Kong
, F.
, Zhang
, H.
, Zhang
, Y.
, Chao
, P.
, and He
, W.
, 2022
, “Stationary Response Determination of MDOF Fractional Nonlinear Systems Subjected to Combined Colored Noise and Periodic Excitation
,” Commun. Nonlinear Sci. Numer. Simul.
, 110
, p. 106392
. 31.
Kong
, F.
, Han
, R.
, and Zhang
, Y.
, 2022
, “Approximate Stochastic Response of Hysteretic System With Fractional Element and Subjected to Combined Stochastic and Periodic Excitation
,” Nonlinear Dyn.
, 107
(1
), pp. 375
–390
. 32.
Kong
, F.
, and Spanos
, P. D.
, 2020
, “Response Spectral Density Determination for Nonlinear Systems Endowed With Fractional Derivatives and Subject to Colored Noise
,” Probabilistic Eng. Mech.
, 59
, p. 103023
. 33.
Spanos
, P. D.
, and Evangelatos
, G. I.
, 2010
, “Response of a Non-Linear System With Restoring Forces Governed by Fractional Derivatives—Time Domain Simulation and Statistical Linearization Solution
,” Soil Dyn. Earthq. Eng.
, 30
(9
), pp. 811
–821
. 34.
Katsikadelis
, J. T.
, 2009
, “Numerical Solution of Multi-term Fractional Differential Equations
,” ZAMM Z. fur Angew. Math. Mech.
, 89
(7
), pp. 593
–608
. 35.
Katsikadelis
, J. T.
, 2014
, “Numerical Solution of Distributed Order Fractional Differential Equations
,” J. Comput. Phys.
, 259
, pp. 11
–22
. 36.
Katsikadelis
, J. T.
, 2011
, “The BEM for Numerical Solution of Partial Fractional Differential Equations
,” Comput. Math. Appl.
, 62
(3
), pp. 891
–901
. 37.
Katsikadelis
, J. T.
, 2018
, “Numerical Solution of Variable Order Fractional Differential Equations
,” arXiv preprint
. https://arxiv.org/abs/1802.0051938.
Spanos
, P. D.
, and Zhang
, W.
, 2022
, “Nonstationary Stochastic Response Determination of Nonlinear Oscillators Endowed With Fractional Derivatives
,” Int. J. Non-Linear Mech.
, 146
, p. 104170
. 39.
Schmidt
, A.
, and Gaul
, L.
, 2006
, “On a Critique of a Numerical Scheme for the Calculation of Fractionally Damped Dynamical Systems
,” Mech. Res. Commun.
, 33
(1
), pp. 99
–107
. 40.
Spanos
, P. D.
, Di Matteo
, A.
, and Pirrotta
, A.
, 2019
, “Steady-State Dynamic Response of Various Hysteretic Systems Endowed With Fractional Derivative Elements
,” Nonlinear Dyn.
, 98
(4
), pp. 3113
–3124
. 41.
Hassani
, V.
, Tjahjowidodo
, T.
, and Do
, T. N.
, 2014
, “A Survey on Hysteresis Modeling, Identification and Control
,” Mech. Syst. Signal Process
, 49
(1–2
), pp. 209
–233
. 42.
Spanos
, P. T.
, 1979
, “Hysteretic Structural Vibrations Under Random Load
,” J. Acoust. Soc. Am.
, 65
(2
), pp. 404
–410
. 43.
Bouc
, R.
, 1967
, “Forced Vibrations of Mechanical Systems With a Hysteresis
,” Proceedings of the Fourth Conference on Nonlinear Oscillations
, Prague, Czech Republic
.44.
Wen
, Y. K.
, 1980
, “Equivalent Linearization for Hysteretic Systems Under Random Excitation
,” ASME J. Appl. Mech.
, 47
(1
), pp. 150
–154
. 45.
Abe
, M.
, Yoshida
, J.
, and Fujino
, Y.
, 2004
, “Multiaxial Behaviors of Laminated Rubber Bearings and Their Modeling. I: Experimental Study
,” J. Struct. Eng.
, 130
(8
), pp. 1119
–1132
. 46.
Sireteanu
, T.
, Giuclea
, M.
, and Mitu
, A. M.
, 2010
, “Identification of an Extended Bouc–Wen Model With Application to Seismic Protection Through Hysteretic Devices
,” Comput. Mech.
, 45
(5
), pp. 431
–441
. 47.
Shi
, Y.
, Wang
, M.
, and Wang
, Y.
, 2011
, “Experimental and Constitutive Model Study of Structural Steel Under Cyclic Loading
,” J. Constr. Steel Res.
, 67
(8
), pp. 1185
–1197
. /10.1016/j.jcsr.2011.02.01148.
Suzuki
, Y.
, and Minai
, R.
, 1988
, “Application of Stochastic Differential Equations to Seismic Reliability Analysis of Hysteretic Structures
,” Probabilistic Eng. Mech.
, 3
(1
), pp. 43
–52
. 49.
Kashani
, H.
, 2017
, “Analytical Parametric Study of Bi-Linear Hysteretic Model of Dry Friction Under Harmonic, Impulse and Random Excitations
,” Nonlinear Dyn.
, 89
(1
), pp. 267
–279
. 50.
Balasubramanian
, P.
, Franchini
, G.
, Ferrari
, G.
, Painter
, B.
, Karazis
, K.
, and Amabili
, M.
, 2021
, “Nonlinear Vibrations of Beams With Bilinear Hysteresis at Supports: Interpretation of Experimental Results
,” J. Sound Vib.
, 499
, p. 115998
. 51.
Katsaras
, C. P.
, Panagiotakos
, T. B.
, and Kolias
, B.
, 2008
, “Restoring Capability of Bilinear Hysteretic Seismic Isolation Systems
,” Earthq. Eng. Struct. Dyn.
, 37
(4
), pp. 557
–575
. 52.
Hughes
, D.
, and Wen
, J. T.
, 1997
, “Preisach Modeling of Piezoceramic and Shape Memory Alloy Hysteresis
,” Smart Mater. Struct.
, 6
(3
), pp. 287
–300
. 53.
Iwan
, W. D.
, 1966
, “A Distributed-Element Model for Hysteresis and Its Steady-State Dynamic Response
,” ASME J. Appl. Mech.
, 33
(4
), pp. 893
–900
. 54.
Korman
, C. E.
, and Mayergoyz
, I. D.
, 1997
, “Review of Preisach Type Models Driven by Stochastic Inputs as a Model for After-Effect
,” Phys. B
, 233
(4
), pp. 381
–389
. 55.
Spanos
, P. D.
, Cacciola
, P.
, and Muscolino
, G.
, 2004
, “Stochastic Averaging of Preisach Hysteretic Systems
,” J. Eng. Mech.
, 130
(11
), pp. 1257
–1267
. 56.
Doong
, T.
, and Mayergoyz
, I.
, 1985
, “On Numerical Implementation of Hysteresis Models
,” IEEE Trans. Magn.
, 21
(5
), pp. 1853
–1855
. 57.
Ubale
, P. V.
, 2012
, “Numerical Solution of Boole’s Rule in Numerical Integration by Using General Quadrature Formula
,” Bull. Soc. Math. Serv. Stand. (B SO MA SS)
, 1
(2
), pp. 1
–5
. 58.
Roberts
, J. B.
, and Spanos
, P. D.
, 2003
, Random Vibration and Statistical Linearization
, Dover Publications
, Mineola, NY
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