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

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## 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

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**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 $\left(\right)$ 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.

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**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