×

Bifurcation analysis of a vibro-impact viscoelastic oscillator with fractional derivative element. (English) Zbl 1414.34042

Summary: To the best of authors’ knowledge, little work has been focused on the noisy vibro-impact systems with fractional derivative element. In this paper, stochastic bifurcation of a vibro-impact oscillator with fractional derivative element and a viscoelastic term under Gaussian white noise excitation is investigated. First, the viscoelastic force is approximately replaced by damping force and stiffness force. Thus the original oscillator is converted to an equivalent oscillator without a viscoelastic term. Second, the nonsmooth transformation is introduced to remove the discontinuity of the vibro-impact oscillator. Third, the stochastic averaging method is utilized to obtain analytical solutions of which the effectiveness can be verified by numerical solutions. We also find that the viscoelastic parameters, fractional coefficient and fractional derivative order can induce stochastic bifurcation.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A08 Fractional ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
34F10 Bifurcation of solutions to ordinary differential equations involving randomness
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen, Y., Petras, I. & Xue, D. [2009] “ Fractional order control — A tutorial,” American Control Conf. 2009. ACC’09, pp. 1397-1411.
[2] Colinas-Armijo, N., Di Paola, M. & Di Matteo, A. [2018] “ Fractional viscoelastic behaviour under stochastic temperature process,” Probab. Engin. Mech.54, 37-43. · Zbl 1510.65177
[3] Deng, M. & Zhu, W. [2016] “ Response of MDOF strongly nonlinear systems to fractional Gaussian noises,” Chaos26, 084313. · Zbl 1378.37090
[4] Di Bernardo, M., Kowalczyk, P. & Ordmark, A. [2003] “ Sliding bifurcations: A novel mechanism for the sudden onset of chaos in dry friction oscillators,” Int. J. Bifurcation and Chaos13, 2935-2948. · Zbl 1099.70504
[5] Dimentberg, M. & Iourtchenko, D. [2004] “ Random vibrations with impacts: A review,” Nonlin. Dyn.36, 229-254. · Zbl 1125.70019
[6] Feng, J., Xu, W. & Wang, R. [2008] “ Stochastic responses of vibro-impact Duffing oscillator excited by additive Gaussian noise,” J. Sound Vibr.309, 730-738.
[7] Feng, J., Xu, W., Rong, H. & Wang, R. [2009] “ Stochastic responses of Duffing-van der Pol vibro-impact system under additive and multiplicative random excitations,” Int. J. Non-Lin. Mech.44, 51-57. · Zbl 1203.74061
[8] Huang, Z. & Jin, X. [2009] “ Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative,” J. Sound Vibr.319, 1121-1135.
[9] Iourtchenko, D. V. & Song, L. L. [2006] “ Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts,” Int. J. Non-Lin. Mech.41, 447-455. · Zbl 1160.70361
[10] Kougioumtzoglou, I. A., dos Santos, K. R. & Comerford, L. [2017] “ Incomplete data based parameter identification of nonlinear and time-variant oscillators with fractional derivative elements,” Mech. Syst. Sign. Process.94, 279-296.
[11] Kumar, P., Narayanan, S. & Gupta, S. [2016a] “ Investigations on the bifurcation of a noisy Duffing-van der Pol oscillator,” Probab. Engin. Mech.45, 70-86.
[12] Kumar, P., Narayanan, S. & Gupta, S. [2016b] “ Stochastic bifurcations in a vibro-impact Duffing-van der Pol oscillator,” Nonlin. Dyn.85, 439-452.
[13] Kumar, P., Narayanan, S. & Gupta, S. [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.
[14] Li, C. & Zeng, F. [2015] Numerical Methods for Fractional Calculus (CRC Press). · Zbl 1326.65033
[15] Li, Z., Liu, L., Dehghan, S., Chen, Y. & Xue, D. [2017] “ A review and evaluation of numerical tools for fractional calculus and fractional order controls,” Int. J. Contr.90, 1165-1181. · Zbl 1367.93205
[16] Ling, Q., Jin, X. & Huang, Z. [2011] “ Response and stability of SDOF viscoelastic system under wideband noise excitations,” J. Franklin Instit.348, 2026-2043. · Zbl 1358.74011
[17] Luo, G., Chu, Y., Zhang, Y. & Zhang, J. [2006] “ Double Neimark-Sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops,” J. Sound Vibr.298, 154-179. · Zbl 1243.70028
[18] Ma, L. & Li, C. [2016] “ Center manifold of fractional dynamical system,” J. Comput. Nonlin. Dyn.11, 021010.
[19] Ma, L. & Li, C. [2017] “ On Hadamard fractional calculus,” Fractals25, 1750033. · Zbl 1371.26009
[20] Ma, L. & Li, C. [2018] “ On finite part integrals and Hadamard-type fractional derivatives,” J. Comput. Nonlin. Dyn.13, 090905.
[21] Machado, J. T., Kiryakova, V. & Mainardi, F. [2011] “ Recent history of fractional calculus,” Commun. Nonlin. Sci. Numer. Simul.16, 1140-1153. · Zbl 1221.26002
[22] Malara, G. & Spanos, P. D. [2018] “ Nonlinear random vibrations of plates endowed with fractional derivative elements,” Probab. Engin. Mech.54, 2-8.
[23] Namachchivaya, N. S. & Park, J. H. [2005] “ Stochastic dynamics of impact oscillators,” J. Appl. Mech.72, 862-870. · Zbl 1111.74572
[24] Nguyen, V.-D., Duong, T.-H., Chu, N.-H. & Ngo, Q.-H. [2017] “ The effect of inertial mass and excitation frequency on a Duffing vibro-impact drifting system,” Int. J. Mech. Sci.124, 9-21.
[25] Rossikhin, Y. A. & Shitikova, M. V. [1997] “ Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Appl. Mech. Rev.50, 15-67.
[26] Rossikhin, Y. A. & Shitikova, M. V. [2010] “ Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results,” Appl. Mech. Rev.63, 010801.
[27] Rossikhin, Z. H. [2015] “ Stochastic response of a vibro-impact Duffing system under external Poisson impulses,” Nonlin. Dyn.82, 1001-1013. · Zbl 1348.74165
[28] Xu, M., Wang, Y., Jin, X. & Huang, Z. [2014] “ Random vibration with inelastic impact: Equivalent nonlinearization technique,” J. Sound Vibr.333, 189-199.
[29] Yang, Y., Xu, W., Jia, W. & Han, Q. [2015] “ Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation,” Nonlin. Dyn.79, 139-146.
[30] Yang, Y., Xu, W., Sun, Y. & Xiao, Y. [2017] “ Stochastic bifurcations in the nonlinear vibro-impact system with fractional derivative under random excitation,” Commun. Nonlin. Sci. Numer. Simul.42, 62-72. · Zbl 1473.70053
[31] Yang, Y., Xu, W. & Yang, G. [2018] “ Bifurcation analysis of a noisy vibro-impact oscillator with two kinds of fractional derivative elements,” Chaos28, 043106. · Zbl 1390.34186
[32] Yurchenko, D., Burlon, A., Di Paola, M., Failla, G. & Pirrotta, A. [2017] “ Approximate analytical mean-square response of an impacting stochastic system oscillator with fractional damping,” ASCE-ASME J. Risk Uncertainty Engin. Syst., Part B: Mech. Engin.3, 030903.
[33] Zhu, W. & Cai, G. [2011] “ Random vibration of viscoelastic system under broad-band excitations,” Int. J. Non-Lin. Mech.46, 720-726.
[34] Zhuravlev, V. [1976] “ A method for analyzing vibration-impact systems by means of special functions,” Mech. Solids11, 23-27.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.