# Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact System with Coulomb Friction under Real Noise

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Survey

#### 1.3. Formulation of the Problem

#### 1.4. Contribution of this Study

#### 1.5. Organization

## 2. System Description and Non-Smooth Transformation

#### 2.1. System Description

#### 2.2. Non-Smooth Transformation

## 3. Stochastic Averaging Procedure

## 4. Response Probability Density Functions

## 5. Example

#### 5.1. The Effect of System Parameters on Response

#### 5.2. Stochastic Bifurcations

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ibrahim, R.A. Modeling Mapping and Application. In Vibro-impact Dynamics; Springer: Berlin, Germany, 2009. [Google Scholar]
- Di Bernardo, M.; Nordmark, A.; Olivar, G. Discontinity-induced bifurcations of quilibria in piecewise smooth and impacting dynamical systems. Phys. D Nonlinear Phenom.
**2008**, 237, 119–136. [Google Scholar] [CrossRef] - Luo, G.W.; Chu, Y.D.; Zhang, Y.L.; Zhang, J.G. Double Neimark-Sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops. J. Sound Vib.
**2006**, 298, 154–179. [Google Scholar] [CrossRef] - Wagg, D.J.; Bishop, S.R. Chatter sticking and chaotic impacting motion in a two degree of freedom impact oscillator. Int. J. Bifurc. Chaos
**2001**, 11, 57–71. [Google Scholar] [CrossRef] - Namachchivaya, N.S.; Park, J.H. Stochastic dynamics of impact oscillators. J. Appl. Mech.
**2005**, 72, 862–870. [Google Scholar] [CrossRef] - Park, J.H.; Namachchivaya, N.S. Noisy impact oscillators. In Proceedings of the ASME 2004 International Mechanical Engineering Congress and Exposition, Anaheim, CA, USA, 13–19 November 2004. [Google Scholar]
- Huang, Z.; Liu, Z.; Zhu, W. Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitation. J. Sound Vib.
**2004**, 275, 223–240. [Google Scholar] [CrossRef] - Xu, M.; Wang, Y.; Jin, X.; Huang, Z.; Yu, T. Random response of vibro-impact systems with inelastic contact. Int. J. Non-Linear Mech.
**2013**, 52, 26–31. [Google Scholar] [CrossRef] - Rong, H.; Wang, X.; Xu, W.; Fang, T. Subharmonic response of a single-degree-of freedom nonlinear 8vibro-impact system to a randomly disordered periodic excitation. J. Sound Vib.
**2009**, 327, 173–182. [Google Scholar] [CrossRef] - Yang, G.; Xu, W.; Jia, W.; He, M. Random vibrations of Rayleigh vibroimpact oscillator under parametric poisson white noise. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 33, 19–29. [Google Scholar] [CrossRef] - Zhu, H. Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. J. Sound Vib.
**2013**, 333, 945–961. [Google Scholar] [CrossRef] - Zhu, H. Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier. Phys. A Stat. Mech. Appl.
**2014**, 40, 335–344. [Google Scholar] [CrossRef] - Green, P.L.; Worden, K.; Sims, N.D. On the identification and modeling of friction in a randomly excited energy harvester. J. Sound Vib.
**2013**, 332, 4696–4708. [Google Scholar] [CrossRef] - 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.
**1995**, 180, 785–795. [Google Scholar] [CrossRef] - Kumar, P.; Narayanan, S.; Gupta, S. Stochastic bifurcation analysis of a Duffing oscillator with Coulomb friction excited by Poisson White noise. Procedia Eng.
**2016**, 144, 998–1006. [Google Scholar] [CrossRef] - Sun, J.J.; Xu, W.; Lin, Z.F. Research on the reliability of friction system under combined additive and multiplicative random excitations. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 54, 1–12. [Google Scholar] [CrossRef] - Rigatos, G.G.; Siano, P. Sensorless control of electric motors with Kalman Filters: Applications to robotic and industrial system. Int. J. Adv. Robot. Syst.
**2011**, 8, 71. [Google Scholar] [CrossRef] - Rigatos, G.; Siano, P. Sensorless nonlinear control of induction motors using Unscented Kalman Filtering. In Proceedings of the IECON 2012-38th Annual Conference on IEEE Industrial Electronics Society, Montreal, QC, Canada, 25–28 October 2012; pp. 4654–4659. [Google Scholar]
- Bryson, A.E.; Ho, Y.-C. Applied Optimal Control: Optimization, Estimation and Control. Routledge: London, UK, 1935. [Google Scholar]
- Pappalardo, C.M.; Guida, D. Use of the Adjoint Method for Controlling the Mechanical Vibrations of Nonlinear Systems. Machines
**2018**, 6, 19. [Google Scholar] [CrossRef] - Pappalardo, C.M.; Guida, D. System algorithm for computing the Modal Parameters of linear mechanical Systems. Machines
**2018**, 6. [Google Scholar] [CrossRef] - Ibrahim, R.A. Vibro-Impact Dynamics Modeling, Mapping and Applications; Springer: Berlin, Germany, 2009. [Google Scholar]
- Dimentberg, M.F.; Iourtchenko, D.V. Random vibrations with impacts: A review. Nonlinear Dyn.
**2004**, 36, 229–254. [Google Scholar] [CrossRef] - Zhu, W.Q.; Huang, Z.L.; Suzuki, Y. Response and stability of strongly non-linear oscillators under wide-band random excitation. Int. J. Non-Linear Mech.
**2011**, 36, 1235–1250. [Google Scholar] [CrossRef] - Zhu, W.; Cai, G. Random vibration of viscoelastic system under broad-band excitations. Int. J. Non-Linear Mech.
**2011**, 46, 720–726. [Google Scholar] [CrossRef] - Ling, Q.; Jin, X.; Huang, Z. Response and stability of SDOF viscoelastic system under wideband noise excitations. J. Franklin Inst.
**2008**, 345, 499–507. [Google Scholar] [CrossRef] - Zhuravlev, V.F. A method for analyzing vibration-impact systems by means of special function. Mech. Solids
**1976**, 11, 23–27. [Google Scholar] - Stratonovich, R.L. Topics in the Theory of Random Noise; Gordon Breach: New York, NY, USA, 1963. [Google Scholar]
- Khasminskii, R.Z. A limit theorem for the solution of differential equations with random right-band sides. Theory Probab. Appl.
**1966**, 11, 390–405. [Google Scholar] [CrossRef] - Xu, W.; He, Q.; Rong, H.; Fang, T. Global analysis of stochastic bifurcation in Ueda system. In Proceedings of the Fifth International Conference on stochastic Structural Dynamics-SSD03, Hangzhou, China, 26–28 May 2003; pp. 509–515. [Google Scholar]

**Figure 2.**Numerical and analytical stationary probability density function (PDF) for ${\alpha}_{1}=2.0,$ $\mu =0.001$. (

**a**) PDF of amplitude $A$. (

**b**) PDF of displacement ${x}_{1}$. (

**c**) PDF of velocity ${x}_{2}$.

**Figure 3.**Numerical and analytical stationary PDF for $D=0.01,$ $\mu =0.001$. (

**a**) PDF of amplitude $A$. (

**b**) PDF of displacement ${x}_{1}$. (

**c**) PDF of velocity ${x}_{2}$.

**Figure 4.**Numerical and analytical stationary PDF for ${\alpha}_{1}=2.0,$ $D=0.01$. (

**a**) PDF of amplitude $A$. (

**b**) PDF of displacement ${x}_{1}$. (

**c**) PDF of velocity ${x}_{2}$.

**Figure 5.**Joint PDFs of the displacement ${x}_{1}(t)$ and the velocity ${x}_{2}(t)$ for $r=0.975$. (

**a**) $\mu =0.08$. (

**b**) $\mu =0.0035$. (

**c**) $\mu =0.001$. (

**d**) Section graphs of joint PDFs on the surface ${x}_{1}=0$ for different values $\mu $.

**Figure 6.**Joint PDFs of the displacement ${x}_{1}(t)$ and the velocity ${x}_{2}(t)$ for $\mu =0.001$. (

**a**) $r=0.96$. (

**b**) $r=0.975$. (

**c**) Section graphs of joint PDFs on the surface ${x}_{1}=0$ for different values $r$.

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**MDPI and ACS Style**

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* **2019**, *11*, 4.
https://doi.org/10.3390/sym11010004

**AMA Style**

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*. 2019; 11(1):4.
https://doi.org/10.3390/sym11010004

**Chicago/Turabian Style**

Liu, Li, Wei Xu, Xiaole Yue, and Dongmei Huang.
2019. "Stochastic Bifurcation of a Strongly Non-Linear Vibro-Impact System with Coulomb Friction under Real Noise" *Symmetry* 11, no. 1: 4.
https://doi.org/10.3390/sym11010004