# A Prediction Method for the Damping Effect of Ring Dampers Applied to Thin-Walled Gears Based on Energy Method

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## Abstract

**:**

## 1. Introduction

## 2. Vibration Analysis of The Gear-Ring Damper System

#### 2.1. The Equations of Motion

_{eq}and the equivalent stiffness matrix K

_{eq}depend on the motion of the gear.

^{T}:

^{T}MΦ, Z = Φ

^{T}CΦ, Λ = Φ

^{T}KΦ, Z

_{eq}= Φ

^{T}C

_{eq}Φ, Λ

_{eq}= Φ

^{T}Λ

_{eq}, Q(t) = Φ

^{T}F(t)

_{eq}, and Λ

_{eq}are all diagonal. In the vicinity of the jth natural frequency, Equation (5) can be rewritten as

_{j}and ζ

_{j,}

_{eq}are the modal damping ratio and the equivalent damping ratio caused by the ring damper for the jth mode, respectively; k

_{j}and k

_{j,}

_{eq}are the modal stiffness and equivalent stiffness for the jth mode, respectively; and ${k}_{j}={\mathsf{\omega}}_{j}^{2}$; ω

_{j}is the jth natural frequency of the undamped system.

_{j,}

_{eq}is much smaller than k

_{j}. Generally, for the ring damper, k

_{j,}

_{eq}is two orders of magnitude lower than k

_{j}. In other words, the ring damper does not affect the shape of the vibration mode; rather, it affects only the vibration amplitude. Moreover, the influence of the damper on the resonance frequency of the primary structure can be neglected. However, the equivalent damping matrix is of the same order of magnitude or even larger with respect to the damping matrix because the structural damping is usually small (For steel, the damping ratio is 1~5×10

^{−4}). The results of other scholars [2,3,20,25,26,27] also showed that the influence of the ring damper on the frequency is negligible. With or without ring dampers, the frequency variation is less than 1%. Thus, the damper ring reduces the resonant amplitude of the gear, primarily by providing damping, rather than changing the stiffness of the gear system.

#### 2.2. Modal Analysis

- The modal amplitude has an integer number of harmonic distributions along the circumferential direction.
- The nodal line passes through the center of rotation, and the vibration amplitude of the nodal line is zero.
- For thin-walled gears, the gear rim vibrates mainly in the radial direction.

## 3. Theoretical Model of Equivalent Damping Ratio of The Ring Damper

#### 3.1. Energy Dissipated by Frictional Force

_{f}is defined as the frictional force per unit length, where F

_{f}is a function of circumferential angle θ.

_{f}can be written as

_{f max}will appears at θ = π/2N. And ${F}_{\mathrm{f}\text{}\mathrm{max}}=\frac{B{A}_{\mathrm{d}}E}{{R}_{\mathrm{d}}}(\frac{{c}_{\mathrm{d}}}{{R}_{\mathrm{d}}^{2}}+\frac{{c}_{\mathrm{g}}}{{R}_{\mathrm{g}}^{2}})N(1-{N}^{2})$. When tangential force is greater than the maximum static friction, slipping occurs at θ < π/2N, and over the zone θ

_{0}< θ < π/2N, F

_{f max}= μP. Where μ is friction coefficient, and P is normal pressure on the contact surface.

_{0}represents the angle where slippage starts, which is called the critical slip angle.

_{c}.

_{d}and A

_{d}are respectively radius and the cross-sectional area of the ring damper.

_{f}and the relative displacement s(θ) in the slip region.

_{0}. According to Equation (21), θ

_{0}is a nonlinear function of B. Therefore, ΔW is a function of B.

#### 3.2. Equivalent Damping Ratio

_{j,}

_{eq}in Equation (7) can be rewritten as

## 4. Application and Discussion

^{3}kg/m

^{3}). The mass of the gear is 425 g. Figure 4 shows the mode shape of the model with 3 ND. The corresponding natural frequency is 3758 Hz. For reasons of confidentiality, some of results are given in a normalized form.

#### 4.1. Method Validation

_{c}, sliding appears in θ

_{0}= π/2N. When B increases, the critical slip angle decreases and the slip area increases. When the vibration amplitude is large enough, the critical slip angle approaches 0, and the ring damper is approximately full-slip. In this case, the energy dissipation caused by the ring damper is approximately linear with the vibration amplitude, as shown in Figure 8.

_{c}, the contact state is stick. Thus, no relative motion occurs on the contact surface. Frictional force is a function of θ, and the maximum frictional force appears at the position of the nodal line. When B = B

_{c}, slip appears at the position of the nodal line, as shown in Figure 9b. When B > B

_{c}, the slip region expands to both sides as B increases, as shown in Figure 9c. When B ≫ B

_{c}, the slip region increases slowly as the vibration amplitude increases. In this case, the contact status is approximately full-slip, as shown in Figure 9d. This observation is highly consistent with other studies [9,20], according to those studies, when the excitation frequency is far from the natural frequency, the response amplitude is small and the contact status is stick. When the excitation frequency gradually approaches the natural frequency, relative slip appears on the contact surface and the slip region gradually increases.

#### 4.2. Effect of Ring Damper Parameters

#### 4.2.1. Effect of Rotating Speed or Normal Pressure

#### 4.2.2. Effect of Temperature

#### 4.2.3. Effect of the Ring Damper Density

#### 4.2.4. Effect of the Friction Coefficient

_{f max}= μP, the maximum frictional force on the contact surface F

_{f max}decreased with a decrease in μ. For a given vibration stress, there is an optimum density that maximizes frictional damping. When the density is greater than 3.7 times the optimal density, the ring damper will cease to be effective again, as shown in Figure 13b.

#### 4.2.5. Effect of the Cross-Sectional Area of the Ring Damper

## 5. Conclusions

_{c}, the ring damper is ineffective. When B is greater than B

_{c}, the ring damper can provide friction damping. By increasing B, slip first appears at position of the nodal line, and the slip region expands to both sides as B increases. At approximately 3.7 times the critical vibration amplitude, the efficiency of the damper is theoretically maximized.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

B | vibration amplitude | Subscript g | gear |

B_{c} | critical vibration amplitude | Subscript d | ring damper |

C | damping matrices of the gear | Subscript eq | equivalent |

c | half-width of the gear rim or the ring damper | W | total energy of the system |

E | Young’s modulus | w | radial displacement of the groove |

F(t) | external periodic force | X | displacement vector |

${\mathrm{F}}_{nl}(\mathrm{X},\dot{\mathrm{X}},t)$ | nonlinear frictional force | z | number of teeth of the gear |

F_{f} | frictional force per unit length | ε | strain |

I | sectional moment of inertia | η | loss coefficient |

K | stiffness matrices of the gear | κ | curvature |

M | mass matrices of the gear | μ | friction coefficient |

M | bending moment | θ | circumferential angle |

N | number of nodal diameters | θ_{0} | critical slip angle |

P | normal pressure | ρ | density of the ring damper |

P’ | normalized normal pressure | ζ | damping ratio |

R | radius | ζ_{eq} | equivalent damping ratio provided by the ring damper |

s | relative displacement | ΔW | energy dissipated per cycle by the ring damper |

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**Figure 3.**Typical gear resonance failure [1].

**Figure 6.**Validation of the proposed method by [34] results: Resonance amplitude by normalized normal pressure.

**Figure 8.**Energy dissipated per cycle by the ring damper and maximum kinetic energy of the system versus normalized amplitude.

**Figure 9.**Normalized frictional force and contact state: (

**a**) B < B

_{c}; (

**b**) B = B

_{c}; (

**c**) B > B

_{c}; (

**d**) B ≫ B

_{c}.

**Figure 10.**Effect of the rotating speed: (

**a**) Friction damping at various rotating speed; (

**b**) friction damping for normalized rotating speed (for a given vibration stress).

**Figure 13.**Effect of the friction coefficient:(

**a**) Friction damping at various friction coefficient; (

**b**) friction damping for normalized friction coefficient (for a given vibration stress).

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**MDPI and ACS Style**

Wang, Y.; Ye, H.; Jiang, X.; Tian, A.
A Prediction Method for the Damping Effect of Ring Dampers Applied to Thin-Walled Gears Based on Energy Method. *Symmetry* **2018**, *10*, 677.
https://doi.org/10.3390/sym10120677

**AMA Style**

Wang Y, Ye H, Jiang X, Tian A.
A Prediction Method for the Damping Effect of Ring Dampers Applied to Thin-Walled Gears Based on Energy Method. *Symmetry*. 2018; 10(12):677.
https://doi.org/10.3390/sym10120677

**Chicago/Turabian Style**

Wang, Yanrong, Hang Ye, Xianghua Jiang, and Aimei Tian.
2018. "A Prediction Method for the Damping Effect of Ring Dampers Applied to Thin-Walled Gears Based on Energy Method" *Symmetry* 10, no. 12: 677.
https://doi.org/10.3390/sym10120677