Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme
- 49 Downloads
In this paper, a modified short-time Gaussian approximation (STGA) scheme is incorporated into the generalized cell mapping (GCM) method to study the stochastic response and bifurcations of a nonlinear dry friction oscillator with both periodic and Gaussian white noise excitations. Firstly, the Fokker–Planck–Kolmogorov equation and the moment equations with Gaussian closure are derived. Because of the periodic force and the singularity of moment equations in the initial condition being a Dirac delta function, a set of novel STGA solutions of the conditional probability density functions (PDFs) over each small fraction of one period is then constructed in order to form the solution over the whole period. At last, the transient and steady-state response PDFs are computed by the GCM method. The accuracy of the results is demonstrated by the direct Monte Carlo simulations. The transient PDFs are presented when evolving from a Gaussian initial distribution to a non-Gaussian steady-state one. The effects of periodic excitation and dry friction damping on the steady-state stochastic response are particularly discussed. When the amplitude of periodic excitation is changed, both stochastic P-bifurcation and D-bifurcation are detected. It is also found that the increase in the dry friction damping coefficient makes the chaotic response disappear gradually, inducing stochastic D-bifurcation but no stochastic P-bifurcation.
KeywordsStochastic bifurcations Dry friction damping Periodic excitation Short-time Gaussian approximation Generalized cell mapping
This work was funded by the National Natural Science Foundation of China (Grant Nos. 11672230, 11532011, 11702214) and the Fundamental Research Funds for the Central Universities (Grant Nos. 2662017QD024, 2662018JC007).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
- 6.Feeny, B., Guran, A., Hinrichs, N., Popp, K.: A historical review on dry friction and stick-slip phenomena. Appl. Mech. Rev. 51(5), 321–341 (1998)Google Scholar
- 11.Rong, H.W., Wang, X.D., Xu, W., Fang, T.: Resonance response of a single-degree-of-freedom nonlinear dry system to a randomly disordered periodic excitation. Acta Phys. Sin. 58(11), 7558–7564 (2009)Google Scholar
- 12.Chen, Y., Just, W.: First-passage time of Brownian motion with dry friction. Phys. Rev. E 89(2), 022103 (2014)Google Scholar
- 13.Tian, Y.P., Wang, Y., Jin, X.L., Huang, Z.L.: Optimal load resistance of a randomly excited nonlinear electromagnetic energy harvester with Coulomb friction. Smart Mater. Struct. 23(9), 095001 (2014)Google Scholar
- 14.Sun, J.J., Xu, W., Lin, Z.F., Zhou, Y.: Random responses analysis of friction systems with viscoelastic forces under Gaussian colored noise excitation. Appl. Math. Mech. 38(1), 109–117 (2017)Google Scholar
- 16.Kumar, P., Narayanan, S., Gupta, S.: Stochastic bifurcation analysis of a Duffing oscillator with Coulomb friction excited by Poisson white noise. Proc. Eng. 144, 998–1006 (2016)Google Scholar
- 17.Sun, J.-Q.: Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method. J. Sound Vib. 180(5), 785–795 (1995)Google Scholar
- 20.Gan, C.B., Lei, H.: Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations. J. Sound Vib. 330(10), 2174–2184 (2011)Google Scholar
- 25.Kashani, H.: Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic, impulse and random excitations. Nonlinear Dyn. 11, 1–13 (2017)Google Scholar
- 26.Liu, W.Y., Zhu, W.Q., Chen, L.C.: Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations. Probab. Eng. Mech. 53, 109–115 (2018)Google Scholar
- 31.Yue, X.L., Xu, W., Wang, L., Zhou, B.C.: Transient and steady-state responses in a self-sustained oscillator with harmonic and bounded noise excitations. Probab. Eng. Mech. 30, 70–76 (2012)Google Scholar
- 37.Wehner, M.F., Wolfer, W.G.: Numerical evaluation of path-integral solutions to Fokker–Planck equations. Phys. Rev. A 27(5), 2663–2670 (1983)Google Scholar
- 39.Sun, J.-Q.: Stochastic Dynamics and Control. Elsevier, Amsterdam (2006)Google Scholar