# Friction-Induced Vibration in a Bi-Stable Compliant Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Model

## 3. Linearized Stability

## 4. Mass-Belt Detachment

## 5. Numerical Results

## 6. Spiral Source–Saddle–Spiral Source

## 7. Spiral Source–Saddle–Spiral Sink

## 8. Spiral Sink–Saddle–Spiral Sink

## 9. Case of One Equilibrium Point

## 10. Discussion

- Spiral source–saddle–spiral source at a relatively low belt speed.
- Spiral source–saddle–spiral sink at an intermediate belt speed.
- Spiral sink–saddle–spiral sink at a relatively high belt speed.

## 11. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The mechanical model consists of an oscillator with mass “$m$” whose tangential displacement is measured by $x$. Spring and dashpot can rotate freely around the joints.

**Figure 2.**Bifurcation diagrams: (

**a**) versus $V$ for $n=1$ and $\alpha =3$; (

**b**) versus $n$ for $V=0.5$ and $\alpha =3$; (

**c**) versus $\alpha \text{}$ for $n=1$ and $V=0.5$. Dashed lines show saddle points and solid lines stand for spirals.

**Figure 3.**Real part of eigenvalues corresponding to two equilibrium points, left spiral ($*$) and right spiral ($o$). Red, blue and black colors stand for $\zeta =0$, $0.01$ and $0.05$, respectively. $\alpha =3$, $n=1$.

**Figure 4.**Free diagram of the oscillator. ${F}_{d}$ and ${F}_{k}$ are viscous and elastic forces, respectively.

**Figure 5.**Phase portrait of the oscillator with one saddle ($\times $) located between two spirals (+). Dash lines represent separatrices of the saddle. $V=0.1$, $\zeta =0.01$, $\alpha =3$, $n=1$.

**Figure 6.**Global analysis of the system response. Dots shows initial conditions. (

**a**) Phase portrait of the oscillator with one saddle ($\times $) located between two spirals (+). The left spiral was a source and the right one was a spiral sink. Dashed lines represent separatrices of the saddle; (

**b**) a limit cycle around the spiral sink was shown by a thick solid line and a transient response inside the stick-slip orbit was illustrated by a dashed line. $V=0.33$, $\zeta =0.01$, $\alpha =3$, $n=1$.

**Figure 7.**Global analysis of the system response. Dots shows initial conditions. (

**a**) Phase portrait of the oscillator with one saddle ($\times $) located between two spirals (+). Both spirals were stable. Dashed lines represent separatrices of the saddle; (

**b**) a limit cycle around the left spiral sink is shown by a thick solid line. Initial conditions (dot) outside and inside the limit cycle are visible. The former attracted towards the stick-slip orbit while the latter was asymptotically stable; (

**c**) a limit cycle around the right spiral sink is shown by a thick solid line. Initial conditions (dot) outside and inside the limit cycle are visible. The former attracted towards the stick-slip orbit while the latter was asymptotically stable. $V=0.5$, $\zeta =0.01$, $\alpha =3$, $n=1$.

**Figure 8.**Phase portrait of the oscillator with one equilibrium point (+): (

**a**) a spiral source for $V=0.1$; (

**b**) a spiral sink for $V=0.5$. $\alpha =3$, $n=3$.

**Figure 9.**Net force on the slider (${F}_{net}$) stuck to the belt (${u}_{2}=V$) for various spring pre-compression values. ${F}_{net}=0$ indicates that friction force reached its maximum. $n=1$, $\zeta =0.01$ and $V=5$.

**Figure 10.**Evolution of the stable limit cycle around the (

**a**) left spiral; (

**b**) right spiral. $\zeta =0.01$, $\alpha =3$, $n=1$.

© 2019 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/).

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**MDPI and ACS Style**

Niknam, A.; Farhang, K.
Friction-Induced Vibration in a Bi-Stable Compliant Mechanism. *Vibration* **2019**, *2*, 285-299.
https://doi.org/10.3390/vibration2040018

**AMA Style**

Niknam A, Farhang K.
Friction-Induced Vibration in a Bi-Stable Compliant Mechanism. *Vibration*. 2019; 2(4):285-299.
https://doi.org/10.3390/vibration2040018

**Chicago/Turabian Style**

Niknam, Alborz, and Kambiz Farhang.
2019. "Friction-Induced Vibration in a Bi-Stable Compliant Mechanism" *Vibration* 2, no. 4: 285-299.
https://doi.org/10.3390/vibration2040018