Abstract
Impact together with friction can be widely found in the mechanical engineering. Although some scholars have investigated the stochastic systems with impact and friction, they only involve the integer-order systems and do not consider the fractional-order cases. In fact, for the system with viscoelastic material, the damping term depends not only on the current time and position but also on the previous states. The memory property of viscoelastic material is characterized by a power-law kernel function which is associated with the fractional derivative. Based on this viewpoint, in this article, we focus on the friction-damped system with fractional derivative damping under Gaussian white noise excitation. We propose an approximate approach to investigate the stochastic response and bifurcation of a fractional-order friction-damped system with the help of variable transformations and stochastic averaging method. One example is employed to verify the effectiveness of the proposed approach. We also explore the stochastic bifurcation phenomenon induced by the fractional order, fractional coefficient and other system parameters through the critical conditions. At last, the difference of bifurcation regions for the fractional-order model and the integer-order model are presented.
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References
W Xu L Wang J Feng Y Qiao P Han 2018 Some new advance on the research of stochastic non-smooth systems Chin. Phys. B. 27 110503
JQ Sun 1995 Random vibration analysis of a non-linear system with dry friction damping by the short-time gaussian cell mapping method J. Sound Vib. 180 785 795
Q Han X Yue H Chi S Chen 2019 Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme Nonlinear Dyn. 96 2001 2011
X Jin Y Wang Z Huang 2018 Approximately analytical technique for random response of LuGre friction system Int. J. Non-Linear Mech. 104 1 7
Y Wang XL Luan XL Jin ZL Huang 2016 Random response evaluation of mono-stable and bi-stable Duffing systems with Dahl friction Arch. Appl. Mech. 86 1827 1840
M Dimentberg D Iourtchenko 2004 Random vibrations with impacts: a review Nonlinear Dyn. 36 229 254
X Yue W Xu L Wang 2013 Global analysis of boundary and interior crises in an elastic impact oscillator Commun. Nonlinear Sci. Numer. Simul. 18 3567 3574
L Wang S Ma C Sun W Jia W Xu 2018 The stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method J. Appl. Mech. 85 054502
V Zhuravlev 1976 A method for analyzing vibration-impact systems by means of special functions Mech. Solids. 11 23 27
A Ivanov 1994 Impact oscillations: linear theory of stability and bifurcations J. Sound Vib. 178 361 378
J Qian L Chen 2021 Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations Mech. Syst. Signal Process. 147 107082
J Feng W Xu R Wang 2008 Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise J. Sound Vib. 309 730 738
D Liu M Li J Li 2018 Probabilistic response and analysis for a vibro-impact system driven by real noise Nonlinear Dyn. 91 1261 1273
P Kumar S Narayanan S Gupta 2017 Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers Int. J. Mech. Sci. 127 103 117
Yang Y-G, Sun Y-H, Xu W. Stochastic bifurcations of a fractional-order vibro-impact system driven by additive and multiplicative gaussian white noises. Complexity. 2019 (2019).
P Kumar S Narayanan S Gupta 2016 Stochastic bifurcations in a vibro-impact Duffing-Van der Pol oscillator Nonlinear Dyn. 85 439 452
MF Dimentberg O Gaidai A Naess 2009 Random vibrations with strongly inelastic impacts: response PDF by the path integration method Int. J. Non-Linear Mech. 44 791 796
Z Ren W Xu D Wang 2019 Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method Nonlinear Dyn. 97 391 402
HT Zhu 2015 Stochastic response of a vibro-impact Duffing system under external Poisson impulses Nonlinear Dyn. 82 1001 1013
I Podlubny 1998 Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications Elsevier Amsterdam
C Li M Cai 2019 Theory and numerical approximations of fractional integrals and derivatives SIAM Delhi
W Zhu M Lu Q Wu 1993 Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation J. Sound Vib. 165 285 304
W Li M-T Zhang J-F Zhao 2017 Stochastic bifurcations of generalized Duffing–van der Pol system with fractional derivative under colored noise Chin. Phys. B. 26 090501
A Zakharova T Vadivasova V Anishchenko A Koseska J Kurths 2010 Stochastic bifurcations and coherencelike resonance in a self-sustained bistable noisy oscillator Phys. Rev. E. 81 011106
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This work was supported by the National Natural Science Foundation of China (No. 11902081, 12002089 and 11872305).
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Yang, YG., Sun, YH. & Xu, W. Stochastic bifurcation analysis of a friction-damped system with impact and fractional derivative damping. Nonlinear Dyn 105, 3131–3138 (2021). https://doi.org/10.1007/s11071-021-06806-4
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DOI: https://doi.org/10.1007/s11071-021-06806-4