# Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Model

#### 2.1. Primary System

#### 2.2. Mechanical Model of the Host System with the Absorber

#### 2.3. Friction Force

## 3. Linear Stability Analysis

#### 3.1. Linear Stability of the Host System without the DVA

#### 3.2. Linear Stability of the Host System with DVA

#### 3.2.1. Analytical Optimal Solution

#### 3.2.2. Numerical Validation

#### 3.3. Evaluation of the Absorber’s Performance

## 4. Bifurcation Analysis of the Host System without the DVA

## 5. Bifurcation Analysis of the Host System with the DVA

`MatCont`[28], a

`MATLAB`-based toolbox for numerical continuation) of the system smoothed assuming that ${v}_{\mathrm{rel}}$ is always positive. This assumption makes the considered system unable to exhibit stick–slip oscillations, but keeps it equivalent to the original system for $v>{z}_{3}$. In contrast, the stable branches were obtained from direct numerical simulations of the full system. Therefore, inaccuracies of the smoothed system in the proximity of the onset of stick–slip motions are possible.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The weakening friction law with ${\mu}_{\mathrm{s}}=1$, ${\mu}_{\mathrm{d}}=0.5$, ${v}_{0}=0.5$.

**Figure 4.**Stability diagrams for different values of ${\zeta}_{2}$ and $\epsilon =0.05$: (

**a**) ${\zeta}_{2}=0.07$; (

**b**) ${\zeta}_{2}=1/2\sqrt{\epsilon /(1+\epsilon )}=0.109109$; and (

**c**) ${\zeta}_{2}=0.13$.

**Figure 5.**Vanishing loop of the ${\Delta}_{3}=0$ curve. The blue line was obtained utilizing the optimal damping as defined by the analytical procedure ${\zeta}_{2}={\zeta}_{2\mathrm{opt}}$, the green line corresponds to the optimal solution obtained by the numerical procedure and the yellow and red lines are obtained for ${\zeta}_{2}$ values slightly larger and smaller, respectively, than ${\zeta}_{2\mathrm{opt}}$.

**Figure 6.**Comparison of host system with and without DVA with varying ${\zeta}_{1}$, other parameters as in Table 3: (

**a**) critical velocities; and (

**b**) improvement curve.

**Figure 7.**Comparison of host system with and without DVA with varying $\epsilon $: (

**a**) critical velocities; and (

**b**) improvement curve.

**Figure 8.**(

**a**) Bifurcation diagram for the host system without DVA; the thin red line marks analytical solutions, black lines numerical ones and dashed lines the numerical unstable solutions. (

**b**) Steady state solutions of the system for $v=1.3$; solid line is the stable solution and dashed line is the unstable solution. (

**c**) Time series of the system leading to the steady state solutions represented in (

**b**) with initial conditions ${\mathit{z}}_{h}={(1.367,0)}^{\mathrm{T}}$ (blue line) and ${\mathit{z}}_{h}={(1.36,0)}^{\mathrm{T}}$ (red line). Other parameter values are as in Table 3.

**Figure 9.**Bifurcation diagrams for the host system with the DVA for parameter values as in Table 3, $\gamma ={\gamma}_{\mathrm{opt}}$ and ${\zeta}_{2}={\zeta}_{2\mathrm{opt}}$: (

**a**) ${k}_{\mathrm{nl}2}=0$, (

**b**) ${k}_{\mathrm{nl}2}=-0.01$; and (

**c**) ${k}_{\mathrm{nl}2}=0.01$. Solid lines are stable solutions, dashed lines are unstable solutions and thin red lines are analytical solutions.

Parameter | min | max | Step |
---|---|---|---|

${\zeta}_{2}$ | 0.1 | 0.11 | ${10}^{-5}$ |

$\gamma $ | 0.97589 | 0.97591 | ${10}^{-6}$ |

$\psi $ | 0.1124 | 0.1126 | ${10}^{-5}$ |

Parameter | Numerical | Analytical | Relative Error [%] |
---|---|---|---|

${\zeta}_{2}$ | 0.10977 | 0.109109 | 0.605867 [%] |

$\gamma $ | 0.975899 | 0.975900 | 0.000109945 [%] |

$\psi $ | 0.11247 | 0.111803 | 0.596226 [%] |

${v}_{\mathrm{cr}}$ | 1.36678 | 1.36883 | 0.364409 [%] |

${\mathit{\mu}}_{\mathbf{s}}$ | ${\mathit{\mu}}_{\mathbf{d}}$ | ${\mathit{v}}_{0}$ | $\mathit{\epsilon}$ | ${\mathit{\zeta}}_{1}$ |
---|---|---|---|---|

1 | 0.5 | 0.5 | 0.05 | 0.05 |

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**MDPI and ACS Style**

Hu, J.L.; Habib, G.
Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy. *Lubricants* **2020**, *8*, 100.
https://doi.org/10.3390/lubricants8110100

**AMA Style**

Hu JL, Habib G.
Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy. *Lubricants*. 2020; 8(11):100.
https://doi.org/10.3390/lubricants8110100

**Chicago/Turabian Style**

Hu, Jia Lin, and Giuseppe Habib.
2020. "Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy" *Lubricants* 8, no. 11: 100.
https://doi.org/10.3390/lubricants8110100