Abstract
To reveal the influence of asymmetric pitch deviation on herringbone gear transmission, a load-bearing contact analysis method for large periods of herringbone gears with asymmetric pitch deviation is proposed. The comprehensive mesh stiffness, axial displacement, comprehensive mesh error, and mesh-in impact force for a large period of the herringbone gear pair are obtained. A dynamic model of the herringbone gear with asymmetric pitch deviation is established. The dynamic responses of the herringbone gear pair under different loads and speeds are compared, and the influence of asymmetric mesh-in impacts on the dynamic responses of the left and right helical gear pairs is analyzed. The results show that in the frequency spectrum of the 3D vibration displacement of herringbone gears with asymmetric pitch deviation, the shaft frequency component is the most prominent, while the mesh frequency and its harmonics are relatively smaller. In the frequency spectrum of the end-face vibration acceleration, the mesh frequency and its harmonics are more evident, and a series of sidebands are distributed on both sides of the mesh frequency and its harmonics.
1. Introduction
Herringbone gears are widely used in heavy machinery, aviation, ships, and other important mechanical transmission devices due to their advantages of high contact ratio, high load capacity, low axial force, and compact structure. The left and right helical gears of a herringbone gear are processed separately, and inevitable manufacturing and installation deviations cause asymmetric deviations in the herringbone gear pair. These asymmetric deviations result in different meshing states for the left and right helical gear pairs, significantly affecting the transmission performance of the herringbone gear pair.
The following table summarizes previous research on the vibration characteristics of straight and helical gear transmissions affected by manufacturing and installation deviations:
| Researcher | Method | Focus | Key Findings |
|---|---|---|---|
| Zhang Tao et al. | Contact finite element analysis | Influence of tooth profile and pitch errors on dynamic transmission error and angular acceleration characteristics of straight gear pairs | The errors affect the dynamic transmission error and angular acceleration characteristics |
| Wang Qibin et al. | Finite element model of straight gear rotor system | Influence of pitch deviation on vibration response of straight gear systems | Pitch deviation impacts vibration response |
| Guo et al. | Dynamics model of helical gear-rotor-bearing system | Influence of different accuracy levels of pitch cumulative deviation on dynamic meshing force of gear systems | Accuracy level affects dynamic meshing force |
| Liu et al. | Nonlinear dynamics model with pitch deviation | Influence of pitch deviation on dynamic meshing force and meshing state of straight gear pairs | Pitch deviation influences dynamic meshing force and state |
| Dai Rihui et al. | Analysis of spiral angle error, time-varying mesh stiffness, and tooth profile error | Influence on axial vibration of herringbone gear transmission | Different excitations affect axial vibration |
| Kang et al. | Overall dynamics model of herringbone gear transmission system | Variation law of 3D vibration characteristics of herringbone gear transmission system under different tooth left-right misalignment angles | Misalignment angles affect 3D vibration characteristics |
| Yuan Bing et al. | Dynamics model of herringbone gear system | Influence of pitch cumulative deviation on system dynamic characteristics | Pitch cumulative deviation impacts dynamic characteristics |
2. Mathematical Description and Periodicity Analysis of Asymmetric Pitch Deviation
Pitch cumulative deviation is measured in the tangential direction of the gear’s end-face pitch circle. Converting the relative pitch deviation of the left and right helical gear pairs to the normal direction of the tooth surface, the relative pitch deviation of the herringbone gear pair along the normal meshing line direction can be expressed as:
λ = (Fpt1 – Fpt2)cosβbcosαt
Where Fpt1 and Fpt2 are the pitch cumulative deviations of the pinion and gear, βb is the base helix angle, and αt is the transverse pressure angle.
The fluctuation period of short-period deviation is one meshing period, while the fluctuation period of long-period deviation is determined by the tooth numbers of the pinion and gear and the revolution periods of their axes. Pitch cumulative deviation belongs to long-period deviation, so the fluctuation period of asymmetric pitch deviation is the large period.
The meshing period Tm, revolution periods T1 and T2 of the pinion and gear, and the large period Tl can be expressed as:
Tm = 60/n1Z1, T1 = 60/n1, T2 = 60Z2/n1Z1, Tl = Tm * lcm(Z1, Z2)
Table 1: Definitions and Formulas Related to Periodic Analysis
| Symbol | Definition | Formula |
|---|---|---|
| λ | Relative pitch deviation | λ = (Fpt1 – Fpt2)cosβbcosαt |
| Tm | Meshing period | Tm = 60/n1Z1 |
| T1 | Revolution period of pinion | T1 = 60/n1 |
| T2 | Revolution period of gear | T2 = 60Z2/n1Z1 |
| Tl | Large period | Tl = Tm * lcm(Z1, Z2) |
3. Load-Bearing Contact Analysis Model for Large Periods of Herringbone Gears with Asymmetric Pitch Deviation
When considering the asymmetric pitch deviation of a herringbone gear pair, the tooth surface clearance for multiple pairs of teeth simultaneously meshing in one meshing period includes the following three parts:
- Initial tooth surface clearance calculated by geometric contact analysis of the herringbone gear.
- Continuously varying meshing clearance of the left and right helical gear pairs produced by asymmetric pitch deviation.
- Normal clearance increment of the left and right helical gear pairs caused by axial displacement of the pinion.
Assuming there are m and q discrete points on the meshing tooth surfaces of the left and right helical gear pairs, respectively, at a certain meshing position and instant, with a total of N (m+q=N) discrete points on both meshing tooth surfaces. Then, after considering the asymmetric pitch deviation of the herringbone gear pair, the new displacement coordination conditions for a single meshing period of the herringbone gear with asymmetric pitch deviation can be represented as [details omitted for brevity].
4. Calculation of Mesh-In Impact Force for Large Periods of Herringbone Gears with Asymmetric Pitch Deviation
Accurately solving the position of the initial mesh-in point is crucial for calculating the mesh-in impact force. Based on the position of the initial mesh-in point, the mesh stiffness and relative velocity at the mesh-in point of the two tooth surfaces can be further calculated.
After considering asymmetric pitch deviation, what affects the actual mesh-in point position of the left and right helical gear pairs is the equivalent clearance of the current meshing tooth pair and the elastic deformation of the other tooth pairs. The elastic deformation of the other tooth pairs can be obtained based on the load-bearing contact analysis model of the herringbone gear with asymmetric pitch deviation for a single meshing period, while the equivalent clearance of the current meshing tooth pair is the difference between the relative pitch deviation of the current meshing tooth pair and the relative pitch deviation of the previous tooth pair at the instant of meshing.
The maximum impact force fs of the left and right helical gear pairs of the herringbone gear can be calculated by:
fs = |c + 12I1I2(I1r2b2 + I2r2b1)v2sk1/cs|c/(c+1)
Where I1 and I2 are the rotational inertias of the left and right pinions and gears of the herringbone gear, ks is the mesh stiffness at the mesh-in point, c is the deformation coefficient in the static state, vs is the speed difference between the pinion and gear at the mesh-in point, and rb1 and rb2 are the base circle radii of the pinion and gear.
5. Dynamic Model of Herringbone Gear with Asymmetric Pitch Deviation
The relative vibration displacement of the left and right helical gear pairs of the herringbone gear along the normal meshing line direction can be represented as [details omitted for brevity].
Conclusion
This paper proposes a load-bearing contact analysis method for large periods of herringbone gears with asymmetric pitch deviation. By establishing a dynamic model, the dynamic responses of herringbone gear pairs under different loads and speeds are analyzed, and the influence of asymmetric mesh-in impacts on the dynamic responses of left and right helical gear pairs is investigated. The results provide insights into the vibration characteristics of herringbone gears with asymmetric pitch deviation and contribute to improving the design and performance of such gears in practical applications.