Stochastic P-bifurcation analysis of a novel type of unilateral vibro-impact vibration system. (English) Zbl 1485.34150
Summary: This work investigated the stochastic P-bifurcation (SPB) of a new class of unilateral impact vibration (VI) system under random parametric excitations, in which the elastoplastic stage of impact process is described by the restitution coefficient dependent with velocity model. By dint of the transformation of non-smooth, a new system without impact could be transformed from the original VI system. Subsequently, the stationary probability density functions (PDFs) of new system could be calculated by the associated equivalent nonlinear system. And then the critical parameters condition of SPB could be derived by means of the catastrophe theory. In the course of the study, it is verified that the new parameters induce the bifurcation of the system, which can be used to accurately measure the bifurcation behavior of the system quantitatively. And the system will tend to be stable under the precondition of enhancing the noise intensity. In addition, the validation of analytical results is examined by relative figures.
MSC:
| 34F10 | Bifurcation of solutions to ordinary differential equations involving randomness |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
vibro-impact system; velocity-dependent restitution coefficient; equivalent nonlinear system; stochastic bifurcationReferences:
| [1] | Ma, J.; Xu, Y.; Li, Y.; Tian, R.; Ma, S.; Kurths, J., Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions, Appl Math Mech, 42, 65-84 (2021) · Zbl 1485.37049 |
| [2] | Zhang, XY; Xu, Y.; Liu, Q.; Kurths, J., Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise, Sci China Technol Sci, 63, 2315-2327 (2020) |
| [3] | Liu, Q.; Xu, Y.; Kurths, J., Bistability and stochastic jumps in an airfoil system with viscoelastic material property and random fluctuations, Commun Nonlinear Sci Numer Simul, 84, Article 105184 pp. (2020) · Zbl 1450.74014 |
| [4] | Horsthemke, W., (Vidal, C.; Pacault, A., Noise Induced Transitions BT - Non-Equilibrium Dynamics in Chemical Systems (1984), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 150-160, editors · Zbl 0567.58001 |
| [5] | Liu, X.; Chen, D., The researches on the stability and bifurcation of nonlinear stochastic dynamical systems, Adv Mech, 26, 437-452 (1996) · Zbl 1077.37516 |
| [6] | Xu, Y.; Gu, R.; Zhang, H.; Xu, W.; Duan, J., Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise, Phys Rev E Stat Nonlin Soft Matter Phys, 83, 56215 (2011) |
| [7] | Xu, Y.; Li, Y.; Li, J.; Jing, F.; Zhang, H., The Phase Transition in a Bistable Duffing System Driven by Lévy Noise, J Stat Phys, 158, 120-131 (2014) · Zbl 1317.82036 |
| [8] | Li, B.; Liang, H.; He, Q., Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model, Chaos, Solitons & Fractals, 146, Article 110856 pp. (2021) · Zbl 1498.39019 |
| [9] | Bilazeroğlu, Ş.; Merdan, H., Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications, Chaos, Solitons & Fractals, 142, Article 110391 pp. (2021) · Zbl 1496.37047 |
| [10] | Arnold, L.; Sri Namachchivaya, N.; Schenk-Hoppe, KR, Toward an understanding of stochastic Hopf bifurcation, Int J Bifurc Chaos, 06, 1947-1975 (1996) · Zbl 1298.37038 |
| [11] | Han, P.; Wang, L.; Xu, W.; Zhang, H.; Ren, Z., The stochastic P-bifurcation analysis of the impact system via the most probable response, Chaos, Solitons & Fractals, 144, Article 110631 pp. (2021) · Zbl 1498.34158 |
| [12] | Yang, Y-G; Xu, W.; Sun, Y-H; Xiao, Y., Stochastic response of nonlinear vibroimpact system with fractional derivative excited by Gaussian white noise, Commun Nonlinear Sci Numer Simul, 42, 62-72 (2016) · Zbl 1473.70053 |
| [13] | Kumar, P.; Narayanan, S.; Gupta, S., Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers, Int J Mech Ences, 127, 103-117 (2016) |
| [14] | 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, 1-10 (2019) · Zbl 1477.34085 |
| [15] | Liu, L.; Xu, W.; Yue, X.; Huang, D., Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact System with Coulomb Friction under Real Noise, Symmetry (Basel), 11, 4 (2018) · Zbl 1423.70053 |
| [16] | Liu, R.; Niu, J.; Shen, Y.; Yang, S., Stability and bifurcation analysis of two-degrees-of-freedom vibro-impact system with fractional-order derivative, Int J Non Linear Mech, 126, Article 103570 pp. (2020) |
| [17] | Xu, W.; Yang, G.; Yue, X., P-bifurcations of a Duffing-Rayleigh vibroimpact system under stochastic parametric excitation, Acta Phys Sin, 65, Article 210501 pp. (2016) |
| [18] | Dimentberg, MF; Gaidai, O.; Naess, A., Random vibrations with strongly inelastic impacts: response PDF by the path integration method, Int J Non Linear Mech, 44, 791-796 (2009) · Zbl 1203.74145 |
| [19] | Zhu, HT., Stochastic response of a vibro-impact Duffing system under external Poisson impulses, Nonlinear Dyn, 82, 1001-1013 (2015) · Zbl 1348.74165 |
| [20] | Wang, D.; Xu, W.; Gu, X.; Pei, H., Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations, Int J Non Linear Mech, 86, 55-65 (2016) |
| [21] | Chen, L.; Qian, J.; Zhu, H.; Sun, JQ., The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators, J Sound Vib, 439, 260-270 (2018) |
| [22] | Johnson, KL., Contact Mechanics (1985), UKCambridge Univ Press: UKCambridge Univ Press Cambridge · Zbl 0599.73108 |
| [23] | Thornton, C., Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres, J Appl Mech Asme, 64, 383-386 (1997) · Zbl 0883.73072 |
| [24] | Stronge, WJ., Impact Mechanics (2004), Cambridge Univ Press: Cambridge Univ Press Cambridge, UK |
| [25] | Ma, D.; Liu, C., Contact law and coefficient of restitution in elastoplastic spheres, J Appl Mech, 82, 121006-121015 (2015) |
| [26] | Qian, J.; Chen, L., Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations, Mech Syst Signal Process, 147, Article 107082 pp. (2021) |
| [27] | Zhu, W.; Soong, T.; Lei, Y., Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems, J Appl Mech Asme, 61, 618-623 (1994) · Zbl 0825.70095 |
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