# An Equivalent Damping Numerical Prediction Method for the Ring Damper Used in Gears under Axial Vibration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vibration Equation of the GEAR-Ring Damper System

**u**is the vibration displacement; (·) is the differentiation of time;

**M**,

**C**, and

**K**are the mass matrix, damping matrix, and stiffness matrix of the gear system, respectively; ${\mathit{F}}_{\mathrm{ext}}$ is the external exciting force with time; ${\mathit{F}}_{\mathrm{nl}}$ is the nonlinear friction force on the surface of the groove, which can be expressed as the equivalent damping and equivalent stiffness form:

**Φ**is the gear system modal matrix;

**q**is the modal displacement vector.

**Φ**

^{T}:

**I**is the unit matrix, and

**Z**,

**Λ**,

**Z**

_{eq},

**Λ**

_{eq}are diagonal matrices. When the system vibrates at the jth natural frequency, the jth mode is dominant, and the contribution of other modes is negligible. Therefore, Equation (1) can be expressed as:

_{d}represents the mass of the ring damper and m

_{g}represents the mass of the gear.

_{re}is:

_{re}in Equation (7) can be obtained by forced response analysis. For a gear system with a damper ring, the response amplitude at the jth order resonant frequency q

_{re}is:

_{re}is linear with the vibration loads σ

_{re}. Therefore, the relationship among the vibration loads σ, the excitation amplitude of the jth order mode of the gear, and the damping ratio can be written as:

_{j}represents the proportional coefficient under the jth mode, and the proportional coefficient is hardly affected by the damper.

_{a}, allowable vibration amplitude q

_{a}, reference modal stress σ

_{ref}, and reference modal displacement q

_{ref}can be obtained:

## 3. Theoretical Analysis of Friction Energy Dissipation

#### 3.1. Motion State of the Slider

_{1}, the normal force is F

_{N}, and the displacement equation of the plate is:

_{0}is the circular vibration frequency of the plate; the motion state of the plate is not affected by the slider. The displacement of the slider is ${x}_{1}(t)$, and the relative motion (trend) between the plate and the slider creates friction. This is called a viscous state when there is no relative displacement between the slider and the plate; it is called a sliding state when there is relative displacement between the slider and the plate.

#### 3.1.1. Equation of Motion of the Slider in the Viscous-Sliding State

_{1}indicates the end of the viscous phase and begins to enter the sliding phase.

_{1}exists only when $\left|k\right|<1$. Equation (14) also gives the upper limit of the frictional force in the presence of relative displacement, i.e., ${F}_{\mathrm{f}}=A{\omega}_{0}^{2}{m}_{1}$. When $\left|k\right|\ge 1$, the slider and the plate are completely viscous, and displacement, velocity, and acceleration are the same.

_{2}can be obtained by Equations (12), (14) and (15).

_{1}, t

_{2}gives the sliding interval in half cycle, and likewise t

_{3}, t

_{4}gives the sliding interval in the corresponding other half cycle. Equation (18) can be obtained by Equations (15) and (17):

#### 3.1.2. The Motion State of the Slider in the Fully Sliding State

#### 3.2. Friction Energy Dissipation Model

- (1)
- There is optimum friction to maximize friction energy dissipation during a vibration cycle;
- (2)
- The mass of the slider has a positive effect on energy dissipation, i.e., increasing the mass of the damper can increase the friction energy dissipation.

_{0}, the equivalent damping ratio due to friction can be expressed as:

_{1}. The maximum damping ratio due to friction is:

## 4. Ring Damper Friction Energy Dissipation Model

_{0}represents the vibration circle frequency (rad/s).

_{1}is a function of the circumferential position angle θ. Since the normalized frictional force of different circumferential positions of the ring damper is variable, the minimum normalized frictional force occurs at the maximum amplitude, defined as:

_{0}= 1, thereby obtaining the critical amplitude under given conditions.

_{r}is the ring damper mass. The physical meaning of the normalized friction energy dissipation is the ratio of the friction energy dissipation to the maximum kinetic energy of the ring damper. Damping ratio can be expressed as:

_{eq}is the equivalent mass of the gear in a given mode, relating to the vibration form and gear.

**M**represents the physical mass matrix of the gear and

**Φ**represents the mode matrix.

## 5. Effect of Ring Damper Parameters

^{3}, Poisson’s ratio of 0.3 and an elastic modulus of 210 Gpa (room temperature), as shown in Figure 2a. According to the modal calculation, the gear has a 4 nodal diameter resonance at a working speed of 10,590 r/min; a vibration mode diagram is presented in Figure 11. Taking the 4 nodal diameter vibration of the gear as an example, the effect of mass on equivalent damping is shown in Figure 12, and the effect of friction coefficient on equivalent damping is shown in Figure 13.

## 6. Conclusions

- (1)
- Under the axial component of the nodal diameter vibration, friction energy dissipation is caused by the relative motion between the gear and the ring damper.
- (2)
- In the flat plate-slider model, there is a critical amplitude that causes a relative motion between the slider and the plate. When the amplitude is greater than the critical amplitude, the friction energy dissipation increases with the amplitude. An optimum amplitude maximizes the equivalent damping provided by the slider, and the optimum amplitude is $\mathsf{\pi}/\sqrt{2}$ times of the critical amplitude.
- (3)
- For any given model, there is a critical vibration load. When the vibration load does not reach the critical value, the ring damper gets stuck in the gear, and there is no damping effect. When the vibration load is greater than the critical vibration load, the contact surface of the gear and the ring damper slide relative to each other, and the vibration energy is dissipated by the frictional force, thus enabling the ring damper to work. The optimal amplitude is 2.74 times that of the critical amplitude.
- (4)
- When the ring damper mass is much smaller than the gear mass, the equivalent damping provided by the ring damper is proportional to its mass.
- (5)
- When other parameters are constant, the friction coefficient is linear with the critical vibration load, but does not affect the peak damping.

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

u | Vibration displacement |

C | Damping matrix |

${\mathit{F}}_{\mathrm{ext}}$ | External exciting force |

${\mathit{C}}_{\mathrm{eq}}$ | Equivalent damping |

Φ | Gear system modal matrix |

$\zeta $ | Damping ratio of the gear |

$\beta $ | Damping ring mass rate |

${m}_{\mathrm{g}}$ | Mass of the gear |

α | Proportional coefficient |

F_{N} | Normal force |

ω_{0} | Circular vibration frequency of plate |

k | Normalized frictional parameter |

η | Normalized friction energy dissipation |

θ | Circumferential position |

$\Omega $ | Angular velocity of gear |

$r$ | Radius of ring damper |

M | Mass matrix |

K | Stiffness matrix of the gear system |

${\mathit{F}}_{\mathrm{nl}}$ | Nonlinear friction force |

${\mathit{K}}_{\mathrm{eq}}$ | Equivalent stiffness |

q | Modal displacement vector |

${\zeta}_{\mathrm{eq}}$ | Equivalent damping ratio |

${m}_{\mathrm{d}}$ | Mass of the ring damper |

σ | Vibration loads |

m_{1} | Mass of the slider |

A | Amplitude of structure |

$x(t)$ | Displacement of slider |

${F}_{\mathrm{f}}$ | Maximum static friction force |

$\Delta W$ | Dissipated energy of any period in the steady state |

$z(\theta ,t)$ | Axial displacement of ring damper |

$\mu $ | Friction coefficient |

$\Delta s(\theta )$ | Relative displacement of the ring damper to the gear |

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**Figure 5.**Acceleration, velocity, and displacement of slider and plate in critical state (k = 0.537).

**Figure 9.**The relationship between the minimum normalized frictional force and the normalized energy dissipation of the slider (black line) and the ring damper (red line).

**Figure 10.**The relationship between the normalized amplitude and the normalized energy dissipation of the slider (black line) and the ring damper (red line).

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**MDPI and ACS Style**

Wang, S.; Wang, X.; Wang, Y.; Ye, H.
An Equivalent Damping Numerical Prediction Method for the Ring Damper Used in Gears under Axial Vibration. *Symmetry* **2019**, *11*, 1469.
https://doi.org/10.3390/sym11121469

**AMA Style**

Wang S, Wang X, Wang Y, Ye H.
An Equivalent Damping Numerical Prediction Method for the Ring Damper Used in Gears under Axial Vibration. *Symmetry*. 2019; 11(12):1469.
https://doi.org/10.3390/sym11121469

**Chicago/Turabian Style**

Wang, Shuai, Xiaolei Wang, Yanrong Wang, and Hang Ye.
2019. "An Equivalent Damping Numerical Prediction Method for the Ring Damper Used in Gears under Axial Vibration" *Symmetry* 11, no. 12: 1469.
https://doi.org/10.3390/sym11121469