# The Basin Stability of Bi-Stable Friction-Excited Oscillators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Concept of Basin Stability

`bSTAB`[36] available at https://github.com/TUHH-DYN/bSTAB/tree/v1.0.

## 3. Bi-Stable Oscillator with Falling Friction Slope

## 4. Bi-Stable Oscillator with Mode-Coupling

## 5. Bi-Stable Oscillator with Isolated Periodic Solution

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

**Open Access Publishing**of Hamburg University of Technology (TUHH).

## Conflicts of Interest

## Abbreviations

DOF | degree-of-freedom |

FIV | friction-induced vibrations |

FP | fixed point |

LC | limit cycle |

## Appendix A. Single-DOF Oscillator

#### Appendix A.1. Equations of Motion

#### Appendix A.2. Convergence of Basin Stability Values

**Figure A1.**Effect of increasing the number of samples for estimating the basin stability values at ${\tilde{v}}_{\mathrm{d}}=1.5$. For each value n, the calculation has been repeated ten times. Mean values $\delta $ and the standard deviation $\sigma $ are reported along with the analytical values.

## Appendix B. Mode-Coupling Instability Oscillator

## Appendix C. Mode-Coupling Instability Oscillator with Isolated Solutions

## References

- Ibrahim, R.A. Friction-Induced Vibration, Chatter, Squeal, and Chaos—Part I: Mechanics of Contact and Friction. Appl. Mech. Rev.
**1994**, 47, 209. [Google Scholar] [CrossRef] - Kinkaid, N.M.; O’Reilly, O.M.; Papadopoulos, P. Automotive disc brake squeal. J. Sound Vib.
**2003**, 267, 105–166. [Google Scholar] [CrossRef] - von Wagner, U.; Hochlenert, D.; Hagedorn, P. Minimal models for disk brake squeal. J. Sound Vib.
**2007**, 302, 527–539. [Google Scholar] [CrossRef] - Hoffmann, N.; Gaul, L. Friction Induced Vibrations of Brakes: Research Fields and Activities. In Friction Induced Vibrations of Brakes: Research Fields and Activities; SAE Technical Paper Series; SAE International: Warrendale, PA, USA, 2008. [Google Scholar] [CrossRef]
- Awrejcewicz, J.; Grzelcyk, D. Modeling and Analysis of Thermal Processes in Mechanical Friction Clutch—Numerical and Experimental Investigations. Int. J. Struct. Stab. Dyn.
**2013**, 13, 1340004. [Google Scholar] [CrossRef] - Ritto, T.G.; Escalante, M.R.; Sampaio, R.; Rosales, M.B. Drill-string horizontal dynamics with uncertainty on the frictional force. J. Sound Vib.
**2013**, 332, 145–153. [Google Scholar] [CrossRef] - Weiss, C.; Gdaniec, P.; Hoffmann, N.P.; Hothan, A.; Huber, G.; Morlock, M.M. Squeak in hip endoprosthesis systems: An experimental study and a numerical technique to analyze design variants. Med. Eng. Phys.
**2010**, 32, 604–609. [Google Scholar] [CrossRef] - Hoffmann, N.; Fischer, M.; Allgaier, R.; Gaul, L. A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations. Mech. Res. Commun.
**2002**, 29, 197–205. [Google Scholar] [CrossRef] - Hoffmann, N. Transient Growth and Stick-Slip in Sliding Friction. J. Appl. Mech.
**2006**, 73, 642. [Google Scholar] [CrossRef] - Sinou, J.J.; Jézéquel, L. Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping. Eur. J. Mech. A/Solids
**2007**, 26, 106–122. [Google Scholar] [CrossRef][Green Version] - Kruse, S.; Tiedemann, M.; Zeumer, B.; Reuss, P.; Hetzler, H.; Hoffmann, N. The influence of joints on friction induced vibration in brake squeal. J. Sound Vib.
**2015**, 340, 239–252. [Google Scholar] [CrossRef][Green Version] - Papangelo, A.; Ciavarella, M.; Hoffmann, N. Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: Analytical results and comparison with experiments. Nonlinear Dyn.
**2017**, 90, 2037–2046. [Google Scholar] [CrossRef][Green Version] - Jahn, M.; Stender, M.; Tatzko, S.; Hoffmann, N.; Grolet, A.; Wallaschek, J. The extended periodic motion concept for fast limit cycle detection of self-excited systems. Comput. Struct.
**2019**, 106–139. [Google Scholar] [CrossRef] - Stender, M.; Jahn, M.; Hoffmann, N.; Wallaschek, J. Hyperchaos co-existing with periodic orbits in a frictional oscillator. J. Sound Vib.
**2020**, 472, 115–203. [Google Scholar] [CrossRef] - Habib, G.; Cirillo, G.I.; Kerschen, G. Uncovering detached resonance curves in single-degree-of-freedom systems. Procedia Eng.
**2017**, 199, 649–656. [Google Scholar] [CrossRef] - Gräbner, N.; Tiedemann, M.; von Wagner, U.; Hoffmann, N. Nonlinearities in Friction Brake NVH-Experimental and Numerical Studies; SAE: Warrendale, PA, USA, 2014. [Google Scholar]
- Papangelo, A.; Hoffmann, N.; Grolet, A.; Stender, M.; Ciavarella, M. Multiple spatially localized dynamical states in friction-excited oscillator chains. J. Sound Vib.
**2018**, 417, 56–64. [Google Scholar] [CrossRef][Green Version] - Awrejcewicz, J. Chaos in simple mechanical systems with friction. J. Sound Vib.
**1986**, 109, 178–180. [Google Scholar] [CrossRef] - Coudeyras, N.; Sinou, J.J.; Nacivet, S. A new treatment for predicting the self-excited vibrations of nonlinear systems with frictional interfaces: The Constrained Harmonic Balance Method, with application to disc brake squeal. J. Sound Vib.
**2009**, 319, 1175–1199. [Google Scholar] [CrossRef][Green Version] - Gdaniec, P.; Weiß, C.; Hoffmann, N. On chaotic friction induced vibration due to rate dependent friction. Mech. Res. Commun.
**2010**, 37, 92–95. [Google Scholar] [CrossRef] - de Witte, V.; Govaerts, W.; Kuznetsov, Y.; Meijer, H. Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms. Phys. D Nonlinear Phenom.
**2014**, 269, 126–141. [Google Scholar] [CrossRef][Green Version] - Hetzler, H.; Schwarzer, D.; Seemann, W. Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise. Commun. Nonlinear Sci. Numer. Simul.
**2007**, 12, 83. [Google Scholar] [CrossRef][Green Version] - Hetzler, H. Bifurcations in autonomous mechanical systems under the influence of joint damping. J. Sound Vib.
**2014**, 333, 5953–5969. [Google Scholar] [CrossRef] - Grolet, A.; Thouverez, F. Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases. Mech. Syst. Signal Process.
**2015**, 52–53, 529–547. [Google Scholar] [CrossRef][Green Version] - Hoffmann, N. Linear stability of steady sliding in point contacts with velocity dependent and LuGre type friction. J. Sound Vib.
**2007**, 301, 1023–1034. [Google Scholar] [CrossRef] - Hetzler, H.; Schwarzer, D.; Seemann, W. Steady-state stability and bifurcations of friction oscillators due to velocity-dependent friction characteristics. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
**2007**, 221, 401–412. [Google Scholar] [CrossRef][Green Version] - Nakano, K.; Maegawa, S. Safety-design criteria of sliding systems for preventing friction-induced vibration. J. Sound Vib.
**2009**, 324, 539–555. [Google Scholar] [CrossRef] - Stender, M.; Tiedemann, M.; Hoffmann, L.; Hoffmann, N. Determining growth rates of instabilities from time-series vibration data: Methods and applications for brake squeal. Mech. Syst. Signal Process.
**2019**, 129, 250–264. [Google Scholar] [CrossRef] - Stender, M.; Di Bartolomeo, M.; Massi, F.; Hoffmann, N. Revealing transitions in friction-excited vibrations by nonlinear time-series analysis. Nonlinear Dyn.
**2019**, 47, 209. [Google Scholar] [CrossRef] - Milnor, J. On the concept of attractor. Commun. Math. Phys.
**1985**, 99, 177–195. [Google Scholar] [CrossRef] - Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed.; Studies in Nonlinearity; Perseus Books: Cambridge, MA, USA, 2001. [Google Scholar]
- Menck, P.J.; Heitzig, J.; Marwan, N.; Kurths, J. How basin stability complements the linear-stability paradigm. Nat. Phys.
**2013**, 9, 89–92. [Google Scholar] [CrossRef][Green Version] - Schultz, P.; Menck, P.J.; Heitzig, J.; Kurths, J. Potentials and limits to basin stability estimation. New J. Phys.
**2017**, 19, 023005. [Google Scholar] [CrossRef] - Mitra, C.; Choudhary, A.; Sinha, S.; Kurths, J.; Donner, R.V. Multiple-node basin stability in complex dynamical networks. Phys. Rev. E
**2017**, 95, 032317. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rakshit, S.; Bera, B.K.; Majhi, S.; Hens, C.; Ghosh, D. Basin stability measure of different steady states in coupled oscillators. Sci. Rep.
**2017**, 7, 45909. [Google Scholar] [CrossRef] [PubMed][Green Version] - Stender, M.; Hoffmann, N. bSTAB (V1). 2020. Available online: https://www.preprints.org/manuscript/202011.0234/v1 (accessed on 6 November 2020). [CrossRef]

**Figure 1.**Schematic of the basin stability calculation. In the two-dimensional state space, two stable attractors $\mathrm{EP}$ (equilibrium position) and $\mathrm{LC}$ (limit cycle) co-exist. The respective basins of attraction ${\mathcal{B}}_{\mathrm{EP}}$ and ${\mathcal{B}}_{\mathrm{LC}}$ are separated by an unstable periodic orbit (indicated by the dashed line). The steady-state behaviors of $n=100$ randomly sampled states are used to estimate the volume shares of the basins of attraction in the subset $\mathcal{Q}$. The resulting basin stability estimates are ${\mathcal{S}}_{\mathcal{B}}\left(\mathrm{EP}\right)=0.37$, ${\mathcal{S}}_{\mathcal{B}}\left(\mathrm{LC}\right)=0.63$ for this example.

**Figure 2.**(

**a**) single-degree-of-freedom frictional oscillator, (

**b**) bifurcation diagram for the non-dimensional belt velocity ${\tilde{v}}_{\mathrm{d}}$, and (

**c**) phase plane for ${\tilde{v}}_{\mathrm{d}}=1.5$. Stable (unstable) solutions are indicated by solid (dashed) lines. The stable steady sliding state (blue spiral trajectory) co-exists with the unstable periodic orbit (black dashed line) and the stable stick-slip limit cycle (red trajectory). The non-dimensional system $\left(\tilde{\xb7}\right)$ is evaluated for ${\mu}_{\mathrm{d}}=0.5$, ${\mu}_{\mathrm{st}}=1.0$, $\xi =0.005$, $N=1.0$ and ${\tilde{v}}_{0}=0.5$.

**Figure 3.**Bifurcation diagram (

**top**), real eigenvalue (

**middle**) and basin stability (

**bottom**) of the single-DOF friction oscillator along the relative sliding velocity.

**Figure 4.**(

**a**) Frictional oscillator with nonlinear joint and mode-coupling instability [11]. (

**b**) Trajectories obtained in the reference configuration (see Appendix B) for two different initial conditions of the horizontal displacement x (all other states were kept at 0).

**Figure 5.**(

**a**) bifurcation diagram for the horizontal stiffness parameter, (

**b**) eigenvalues’ real parts and (

**c**) basin stability of the fixed point and limit cycle solution. $\widehat{x}$ denotes the maximum amplitude of $x\left(t\right)$ along one vibration period. Solid and dashed lines indicate stable and unstable solutions, respectively.

**Figure 6.**Basin stability values in the bi-stability range for the reference sets of initial conditions ${\mathcal{Q}}_{1}$ (

**a**), ${\mathcal{Q}}_{2}$ (

**b**), and ${\mathcal{Q}}_{3}$ (

**c**) defined in Equation (6).

**Figure 7.**Bifurcation diagram for the weakly damped friction oscillator exhibiting an isolated solution branch (

**a**) and basin stability values (

**b**) for all three stable solutions along the horizontal stiffness ${k}_{\mathrm{x}}$. Initial conditions for each solution are given in Appendix C.

**Figure 8.**State space of all DOFs (horizontal direction in (

**a**), vertical direction in (

**b**) and diagonal direction in (

**c**)) at ${k}_{\mathrm{x}}=27.0$ for the weakly damped oscillator.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stender, M.; Hoffmann, N.; Papangelo, A. The Basin Stability of Bi-Stable Friction-Excited Oscillators. *Lubricants* **2020**, *8*, 105.
https://doi.org/10.3390/lubricants8120105

**AMA Style**

Stender M, Hoffmann N, Papangelo A. The Basin Stability of Bi-Stable Friction-Excited Oscillators. *Lubricants*. 2020; 8(12):105.
https://doi.org/10.3390/lubricants8120105

**Chicago/Turabian Style**

Stender, Merten, Norbert Hoffmann, and Antonio Papangelo. 2020. "The Basin Stability of Bi-Stable Friction-Excited Oscillators" *Lubricants* 8, no. 12: 105.
https://doi.org/10.3390/lubricants8120105