# On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Traveling Wave Vibration of the Rotating Gear

#### 2.2. Mechanical Model of the Gear Self-Excited Force

_{z}(t).

_{0}, take the pitch diameter d

_{0}instead of the mesh point diameter d

_{1}, assuming that the height of the gear tooth is small enough compared to the gear diameter. Accordingly, the mesh force F

_{t}(t) for the engaged gear can be expressed as:

_{t}(t) is not only time-dependent when observed from the whole gear, but is also periodic. Thus, two different mesh points should be considered.

_{0}= 2T

_{0}/d

_{0}, and after the coordinate transformation t′ = 2πnzt/60, the expressions of the mesh force at mesh point 1 and mesh point 2 are shown in Figure 3, where n is the rotational speed, z is the number of gear teeth, and ε is the gear contact ratio.

#### 2.3. The Work of the Self-Excited Force on the Vibration

_{b}with w

_{f}in Equation (9) and the other related equations. Moreover, if the object of the analysis is the driving gear, the term F

_{t}(t) must be replaced with −F

_{t}(t). That is, the relationship of the excitation work between the driving gear and the driven gear is:

#### 2.4. The Work of the Damping Force on the Vibration

_{max}is the initial system energy, which is approximately equal to the maximum system kinetic energy when the damping is relatively low, which is U

_{max}= T

_{max}.

_{0}·R(r). According to Equations (18) and (19), and the relationship η = 2ζ for the situation of low damping [32], the damping work during one vibration cycle can be determined as:

_{i}and R

_{(i)}are the values of the radius and normalized deformation at point i, respectively.

#### 2.5. Theoretical Prediction of the Self-Excited Vibration

_{0}when the self-excited vibration occurs can then be determined by solving the equation:

_{0}is obtained, the critical transmission power, which also represents the minimum transmission power that can cause self-excited vibration, or the maximum transmission power for safe operation, can then be predicted according to the definition of the transmission power.

_{0}, then self-excited vibration will occur and the system will be unstable.

## 3. Results and Discussion

_{1}= 95 mm, radius of the hole r

_{2}= 25 mm, thickness h = 4 mm, and the number of teeth z = 50. The finite element model is shown in Figure 7. The gear material properties are: Young modulus E = 212 GPa, density ρ = 7.85 × 10

^{3}kg/m

^{3}, and Poisson’s ratio μ = 0.284. Several different parameters were also chosen for the numerical simulation in order to demonstrate the effects of the nodal diameter and damping ratio on the self-excited vibration stability. The maximum simulation speed was 10,000 rpm, and the results for each speed were calculated.

#### 3.1. Convergence Analyses

#### 3.2. Stability Boundary Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

A(r) | vibration deformation of gear web | U_{max} | initial system energy |

A_{0} | vibration amplitude at the gear rim | w | vibration displacement |

BTW | backward traveling wave | w_{f} | displacement of FTW |

d_{0} | pitch diameter | w_{b} | displacement of BTW |

d_{1} | diameter of mesh point | W_{1cbF} | excitation work of BTW at mesh point 1 |

F_{t}(t) | mesh force | W_{2cbF} | excitation work of BTW at mesh point 2 |

F_{0} | amplitude of the mesh force | W_{cbF} | excitation work of BTW for driven gear |

F_{z}(t) | self-excited force of driven gear | W_{zbF} | excitation work of BTW for driving gear |

FTW | forward traveling wave | W_{F} | excitation work |

h | thickness of the gear web | W_{D} | damping work |

k | excitation order | β | axial angle |

m | number of nodal diameters | β′ | central angle of one gear tooth |

n | rotational speed (r/min) | ΔU | energy dissipated in one damping cycle |

P_{0} | transmission power | ε | contact ratio |

r | radius of the gear | $\varphi $ | phase angle between adjacent teeth |

R(r) | normalized deformation function | φ | circumferential angle of the mesh point |

r_{1} | radius of the inner hole | η | loss factor |

r_{2} | radius of the gear rim | Ω | angular speed of the gear |

T_{max} | maximum kinetic of the gear | θ | circumferential coordinates |

T_{0} | torque | ζ | damping ratio |

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**Figure 3.**Expressions of the mesh force: (

**a**) Mesh point 1, and (

**b**) mesh point 2. The abscissa indicates time, and the ordinate indicates the meshing force.

**Figure 9.**Excitation work of the vibration with two nodal diameters: (

**a**) Backward traveling wave (BTW) and (

**b**) forward traveling wave (FTW).

**Figure 10.**(

**a**) Critical mesh force and (

**b**) critical transmission power for the vibration with three nodal diameters.

**Figure 11.**Influences of the nodal diameters: (

**a**) Critical mesh force and (

**b**) critical transmission power.

**Figure 12.**Influences of the damping ratios, m = 3, ε = 1.75 for (

**a**) critical mesh force and (

**b**) critical transmission power.

Parameter | Value |
---|---|

r_{1} | 25 mm |

r_{2} | 95 mm |

h | 4 mm |

z | 50 |

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**MDPI and ACS Style**

Wang, Y.; Ye, H.; Yang, L.; Tian, A.
On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method. *Symmetry* **2018**, *10*, 664.
https://doi.org/10.3390/sym10120664

**AMA Style**

Wang Y, Ye H, Yang L, Tian A.
On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method. *Symmetry*. 2018; 10(12):664.
https://doi.org/10.3390/sym10120664

**Chicago/Turabian Style**

Wang, Yanrong, Hang Ye, Long Yang, and Aimei Tian.
2018. "On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method" *Symmetry* 10, no. 12: 664.
https://doi.org/10.3390/sym10120664