# Study on the Dynamic Characteristics of Gears Considering Surface Topography in a Mixed Lubrication State

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## Abstract

**:**

## 1. Introduction

## 2. Gear Stiffness and Damping under Mixed Lubrication Consideration

#### 2.1. Gear Parameters and Lubrication Analysis

**σ**) and tribology analysis (F,

**v**, η, ρ, and r

_{e}) at each contact point during gear engagement are established, the contact stiffness and damping in the process of gear meshing can be computed using the developed model [24]. The main variables that change over time during gear engagement are the meshing force, velocity, and radius of curvature. These parameters should be initially calculated and discussed. The load-sharing coefficients for these engaged tooth pairs can be determined through research conducted by Sánchez et al. [45], while velocity and radius of curvature can be obtained through gear meshing motion parameter analysis. The results have been listed in Appendix A.

#### 2.2. Calculation of Stiffness and Damping

_{c}represents the average film thickness. Considering the impact of asperity contact and the fact that lubricant only exists on part of the entire surface, the damping calculation expression is adjusted accordingly as follows:

_{b}, shear stiffness K

_{t}, and compressive stiffness K

_{a}:

_{x}, A

_{x}, and G denote moments of inertia, area moments, and shear module of the gear, which are calculated as follows:

_{f}caused by fillets and base deformation is:

_{m}denotes the pressure angle, u

_{f}represents the distance from the point of engagement to the root radius, and s

_{f}refers to the radius of the root fillet. L*, M*, P*, and Q* are coefficients determined by gear geometric parameters, the values of which are detailed in the literature [48] and thus not repeated here. The total mesh stiffness, considering the effects of root fillets and base deformation, is expressed as:

_{h}represents the contact stiffness. Taking into account surface deformations due to Hertzian contact, it can be said that:

_{m}would also become a time-varying quantity.

_{1}and m

_{2}represent the masses of the two engaged gears, respectively, and $\overline{K}$

_{sys}represents the average value of the system engagement stiffness K

_{sys}. It should be noted that the stiffness K

_{sys}represents the stiffness of the gear engagement system without considering the contact stiffness:

## 3. The Dynamics Model of Gears and Its Energy Dissipation

_{m}and damping c

_{m}along the meshing line. It should be noted that these values represent periodic changes under mixed lubrication conditions, taking into account factors such as dynamic transmission errors e resulting from the processing and installation of the gear pair as well as side clearance bn during the meshing process. The gear base is supported by a pair of spring-damping elements, with k

_{x}and c

_{x}representing stiffness and damping in the x-direction and k

_{y}and c

_{y}representing stiffness and damping in the y-direction. These elements represent the supporting effect of the shaft and bearing on the gear. Since torsion needs to be considered, the torsional stiffness k

_{θ}and torsional damping c

_{θ}of the support shaft must also be included in the overall dynamics system. Assuming a load torque T

_{g}is applied, the following set of equations can be obtained based on Figure 3:

_{bp}and r

_{bg}represent the base circle radius, x and y denote the displacement along the coordinate axis, i = one, two identifies the meshing gear pair, F

_{mi}represents the dynamic meshing force of the gear, and F

_{fi}denotes the frictional force on the gear surface. L

_{pi}and L

_{gi}represent the arm length of friction torques of the driving gear and driven gear, respectively.

_{i}denotes the dynamic transmission error during gear engagement. As a dynamic variable, its physical meaning is the difference between the relative vibration displacement in the meshing gear pair along the meshing line direction and the static transmission error, so its expression can be written as:

_{u}(x,t) represents the dynamic friction coefficient, F

_{m}(x,t) represents the dynamic meshing force, v

_{s}(x,t) represents the relative sliding speed, and b is the tooth width. In practical calculations, the values of each parameter along the tooth width direction can be replaced by a discrete value, so the above formula can be rewritten as a calculation formula directly related to time:

_{u}(t) is the gear engagement friction coefficient calculated by the mixed lubrication model, while F

_{m}(t) is the meshing force obtained through the aforementioned dynamic model. Based on previous analysis, the relative sliding speed v

_{s}(t) is easily determined. After solving the dynamic model, the energy loss can then be calculated according to Equation (21).

## 4. Simulation Results and Analysis

#### 4.1. Gear Meshing Stiffness and Damping Analysis

_{1}= 0.35644 μm, σ

_{2}= 0.63991 μm, and σ

_{3}= 0.92883 μm, the non-dimensional oil film thickness and pressure distribution from tooth engagement to disengagement were solved. Rough surfaces with micro-topography were generated through numerical simulation (as shown in Figure 2, and their roughness could be altered by adjusting control parameters. The three different roughness values presented in the text were obtained by calculating the statistical parameters (standard deviation) of the surface roughness.

_{sys}without considering mixed lubrication. The values of speed, load, lubricant viscosity, and surface hardness were given as 500 rpm, 50 N·m, 0.096 Pa·s, and 2.38 GPa, respectively. Figure 5 shows the engagement stiffness and damping under different surface roughnesses. It can be seen from the figure that the greater the surface roughness, the smaller the damping and stiffness. As shown in Figure 5a, the overall amplitude of damping changes decreases with increasing surface roughness. When the engagement gear pair undergoes a sudden change in damping from double-tooth engagement to single-tooth engagement, the relative change amplitude increases with increasing surface roughness. Not only is the amplitude of damping changes greatly affected by the load, but the overall pattern of change in damping in the double-tooth engagement area is also different. According to Figure 5b, the higher the roughness, the smaller the stiffness, and there is almost no sudden change when the gear transitions from double-tooth engagement to single-tooth engagement. Additionally, it should be noted that when surface roughness and lubrication are not considered, i.e., the contact stiffness and damping are not included in the overall stiffness and damping of the gear, its damping is Csys, which is a constant value. Moreover, its stiffness, Ksys, is smaller than the stiffness of the rough tooth surface.

#### 4.2. Dynamic Characteristics and Energy Loss Analysis

_{1}= 0.35644 μm, σ

_{2}= 0.63991 μm, and σ

_{3}= 0.92883 μm.

_{1}and σ

_{2}but more comparable to that of gears with roughness σ

_{3}. Furthermore, after the onset of steady, periodic variation in vibration displacement (post 0.008 s in the graph), the amplitude of the control group exceeds that of gears with roughness σ

_{3}for about half of the duration.

_{3}display the highest amplitudes. Except for frequency 3f, at all other harmonics, gears with smooth surfaces exhibit the greatest amplitudes. Moreover, apart from frequency 3f, the amplitude tends to increase with increasing roughness level.

_{1}and σ

_{2}, but closer to that of gears with roughness σ

_{3}. However, there are slight differences in the temporal patterns of vibration displacement change, corresponding to different levels of roughness. Gears with smaller roughness exhibit local extrema at earlier times, with only the vibration displacement change pattern of gears with roughness σ

_{3}being more consistent with the smooth surface gear. This indicates that roughness not only affects the amplitude of vibration displacement change but also influences its change pattern. Therefore, roughness has a greater impact on the y-directional vibration.

_{1}and σ

_{2}. Although the smooth surface gear has a similar angular vibration amplitude to the gear with roughness σ

_{3}, there are time intervals where the amplitude is significantly greater than that of the gear with roughness σ

_{3}. In contrast to the y-directional vibration displacement, the smaller the roughness, the later the local extrema appear in the rotational angular vibration displacement.

_{3}have a significantly larger amplitude value compared to those with other roughness levels. Based on the analysis of rotational vibration displacement and its frequency spectrum, combined with the previous analysis of x- and y-directional vibrations, it can be concluded that roughness has the greatest impact on rotational vibration displacement.

_{3}have significantly greater energy losses than others, and gears with roughness level σ

_{1}correspond to minimum energy loss.

## 5. Conclusions

- (1).
- Surface roughness has a direct impact on gear meshing stiffness and damping. As the roughness increases, both the contact stiffness and damping decrease;
- (2).
- Different surface roughness results in distinct vibration characteristics, with the overall trend being that the greater the roughness, the larger the amplitude of vibration. Although gears with different roughness have dissimilar frequency components, they all exhibit a clear harmonic characteristic. When the roughness is small, gear vibrations under rough surface conditions are less than those of smooth surface gear without considering roughness;
- (3).
- A higher surface roughness leads to increased mesh force, where the mesh force of the rough surface gear surpasses that of the smoother surface gear. The energy loss is similar to the vibration displacement under varying levels of roughness, i.e., increasing roughness means more significant energy loss. Yet interestingly, the energy loss in the smooth surface gear control group, which does not factor in roughness, outweighs that in cases of lower surface roughness;
- (4).
- While the gear tribo-dynamics model established in this study has connected surface micro-topography with gear dynamic characteristics, providing a theoretical basis for improving gear service performance, there are still some shortcomings. Firstly, the coupling relationship between tribology and dynamics is not tight enough, and parameter exchange has not been realized through real-time iteration. Secondly, the evolution of surface topography under actual conditions has not been sufficiently considered. Lastly, the analyzed working conditions and surface topographies are insufficient. In our subsequent research, we will mainly focus on addressing these three shortcomings.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Gear Parameters

_{1}and O

_{2}, respectively, represent the centers of the pinion and wheel, while V

_{1}and V

_{2}denote the speeds of the small and large gears at point O. The common tangent of the base circle is denoted by K

_{1}K

_{2}, with r

_{b}

_{1}and r

_{b}

_{2}representing the radii of the base circles and α and β indicating the rolling angles. According to the principle of involute gears, the radius of curvature at contact point O can be expressed as follows:

_{e}= r

_{1}r

_{2}/(r

_{1}+ r

_{2}). Given the rotational speeds n

_{1}and n

_{2}, the tangential velocity at the engagement point can be expressed according to the geometric relationship shown in Figure A1:

_{s}denotes the relative sliding speed.

Parameters | Pinion | Wheel |
---|---|---|

Number of teeth | 30 | 36 |

Module (mm) | 4 | 4 |

Pressure angle (°) | 20 | 20 |

Radius of the base circle (mm) | 56.382 | 67.658 |

Rotation speed (rpm) | 500 | 416.667 |

**Figure A3.**The meshing force sharing coefficient of the gear pair. SAP: start of the active profile; Tip: top point of gear; LPSTC and HPSTC: lowest point and highest point of single-tooth contact.

#### Appendix A.2. Equation of Lubrication and Friction Analysis

_{1}(x, y, t) and S

_{2}(x, y, t) have been determined, while the original gap h

_{0}(t) and contact geometry f(x, y, t) can be derived based on the geometric relationship of the contacting pair. The elastic deformation V(x, y, t) requires particular consideration in calculation as it is associated with pressure. Its calculation formula can be expressed as follows:

**τ**

_{0}and

**τ**

_{1}, respectively, denote the shear stress under standard conditions and the ultimate shear stress, while

**η**indicates the viscosity related to pressure, which can be derived through the following equation:

_{L}and G

_{x}are the limiting shear stress and limiting elastic modulus, respectively. Given τ

_{L}and G

_{x}, the shear stress can be obtained by solving Equation (A12), and then integrating the shear stress gives the friction coefficient. In the asperity contact area, the friction coefficient can be taken as a constant depending on the interface material, typically ranging from 0.08 to 0.15.

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**Figure 1.**Force analysis diagram of a single-tooth. F: meshing force; a: pressure angle; d: distance from acting point to root circle; and h

_{x}: tooth thickness.

**Figure 4.**The distribution of pressure and film thickness at the pitches of different tooth surfaces.

**Figure 5.**Damping (

**a**) and stiffness (

**b**) of different rough surfaces: σ

_{1}= 0.35644 μm, σ

_{2}= 0.63991 μm, and σ

_{3}= 0.92883 μm. Csys and Ksys: damping and stiffness without considering roughness and lubrication.

**Figure 6.**Vibration displacement in the x-direction (

**a**) and its frequency spectrum (

**b**) on different rough surfaces.

**Figure 7.**Vibration displacement in the y-direction (

**a**) and its frequency spectrum (

**b**) on different rough surfaces.

**Figure 8.**Vibration displacement in rotation-direction (

**a**) and its frequency spectrum (

**b**) for different rough surfaces.

Parameters | Pinion | Wheel |
---|---|---|

Number of teeth | 30 | 36 |

Module (mm) | 4 | 4 |

Pressure angle (°) | 20 | |

Load (N·m) | 50 | |

Rotation speed (rpm) | 500 | 416.667 |

Mass (kg) | 1.582 | 2.3566 |

Moment of inertia (kg·m^{2}) | 0.001753 | 0.003536 |

Bearing stiffness (N/m)/damping (N·s/m) | 6.9127 × 10^{8}/1804 | |

Torsional stiffness (N/m)/damping (N·s/m) | 7.6712 × 10^{8}/2209 |

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## Share and Cite

**MDPI and ACS Style**

Cheng, G.; Ma, J.; Li, J.; Sun, K.; Wang, K.; Wang, Y.
Study on the Dynamic Characteristics of Gears Considering Surface Topography in a Mixed Lubrication State. *Lubricants* **2024**, *12*, 7.
https://doi.org/10.3390/lubricants12010007

**AMA Style**

Cheng G, Ma J, Li J, Sun K, Wang K, Wang Y.
Study on the Dynamic Characteristics of Gears Considering Surface Topography in a Mixed Lubrication State. *Lubricants*. 2024; 12(1):7.
https://doi.org/10.3390/lubricants12010007

**Chicago/Turabian Style**

Cheng, Gong, Jianzuo Ma, Junyang Li, Kang Sun, Kang Wang, and Yun Wang.
2024. "Study on the Dynamic Characteristics of Gears Considering Surface Topography in a Mixed Lubrication State" *Lubricants* 12, no. 1: 7.
https://doi.org/10.3390/lubricants12010007