1. Introduction
Gear mechanisms are widely used in machinery due to their high transmission efficiency, long service life, and stable transmission ratio. However, gear failures, such as wear and pitting, can lead to reduced gear life, increased clearance, decreased efficiency, and noise generation. Contact stress during gear meshing has a significant impact on gear performance. Therefore, studying the contact stress of gears is crucial for precise gear design and improving the quality of gear transmission mechanisms.
This article focuses on involute spur cylindrical modified gears. It aims to calculate the maximum contact stress, analyze its influencing factors, and provide a basis for accurate gear design.
2. Significance of Gear Contact Stress Research
Gear failures can cause serious problems in various mechanical systems. For example, in wind turbines, gear failures account for 59% of turbine failures, and in helicopters, 19.1% of transmission failures are due to gear problems. These failures not only affect the normal operation of the equipment but also increase maintenance costs and downtime.
By studying gear contact stress, researchers can understand the stress distribution and variation laws during gear meshing, which helps to optimize gear design, improve gear strength and durability, and reduce the occurrence of failures.
3. Current Research Status
Many researchers have conducted studies on gear contact problems and achieved fruitful results. For instance, PUNEETH and MALLESH analyzed the contact process of a pair of spur gears and used the finite element software Abaqus to calculate the contact stress, applying the results to gear design. Hu Aiping et al. simplified the calculation method of the maximum contact stress for standard gear transmission by expressing it as the product of the contact stress at the node and a stress ratio. Tang Jin-yuan et al. analyzed the impact problem caused by the change of transient contact speed during gear meshing based on contact dynamics theory. Yang Shenghua used the finite element method to calculate the deformation and contact stress of gears. ZHAO et al. established a fractal surface contact model of rough tooth surfaces and analyzed the influence of tooth surface roughness on gear transmission performance. LIU et al. compared the bearing capacities of different types of hypoid gears by studying their contact and bending stresses.
However, the existing research still has some limitations. For example, the calculation method of the maximum contact stress for modified gear transmission in national standards does not fully consider the influence of gear modification, and there is a lack of a unified and accurate calculation method for the maximum contact stress of modified gears.
4. Hertz Stress Calculation during Gear Meshing
4.1 Basic Assumptions and Parameters
In the analysis of gear meshing, assume a pair of involute spur cylindrical modified gears with parameters such as transmission torque \(T_{1}\), rotational speed \(n_{1}\), load correction factor K, normal force of single – pair tooth meshing \(F_{n}\), tooth width b, tangential force \(F_{t1}\), module m, transmission ratio u, number of teeth \(Z_{1}\) and \(Z_{2}\), base circle radii \(r_{b1}\) and \(r_{b2}\), radii of any meshing point on the meshing tooth profiles \(r_{1i}\) and \(r_{2i}\), pressure angles \(\alpha_{1i}\) and \(\alpha_{2i}\), pitch circle radii \(r_{1}’\) and \(r_{2}’\), and meshing angle \(\alpha’\). Also, assume that the influence of the contact ratio on the contact line length b is ignored.
4.2 Hertz Formula and Its Derivation
According to the Hertz formula, the contact stress at any point on the tooth profile is \(\sigma_{Hi}=\sqrt{\frac{KF_{n}}{\pi b}\frac{1/\rho_{1i} + 1/\rho_{2i}}{\frac{1-\nu_{1}^{2}}{E_{1}}+\frac{1-\nu_{2}^{2}}{E_{2}}}}\)
The curvature radii \(\rho_{1i}\) and \(\rho_{2i}\) can be expressed as \(\rho_{1i}=r_{b1}\tan\alpha_{1i}=\frac{1}{2}mZ_{1}\cos\alpha\tan\alpha_{1i}\) and \(\rho_{2i}=r_{b2}\tan\alpha_{2i}=\frac{1}{2}mZ_{2}\cos\alpha\tan\alpha_{2i}\)
The comprehensive curvature \(\frac{1}{\rho}=\frac{1}{\rho_{1i}}+\frac{1}{\rho_{2i}}=\frac{2}{mZ_{1}\cos\alpha\tan\alpha_{1i}}+\frac{2}{mZ_{2}\cos\alpha\tan\alpha_{2i}}=\frac{L}{x(L – x)}\)
where \(L=(r_{b1}+r_{b2})\tan\alpha’\) represents the length of the limit meshing line, and \(x=\sqrt{r_{1i}^{2}-r_{b1}^{2}}=\frac{1}{2}mZ_{1}\cos\alpha\tan\alpha_{1i}\) represents the distance from any meshing point of the driving gear to the meshing limit point.
The comprehensive elastic influence coefficient \(Z_{E}=\sqrt{\frac{1}{\pi(\frac{1-\nu_{1}^{2}}{E_{1}}+\frac{1-\nu_{2}^{2}}{E_{2}})}}\)
After substituting relevant formulas, the contact stress can be expressed as \(\sigma_{H}=Z_{E}\sqrt{\frac{KT_{1}}{b r_{1}’\cos\alpha’}\frac{L}{x(L – x)}}\)
Let \(B = Z_{E}\sqrt{\frac{KT_{1}}{b r_{1}’\cos\alpha’}}=Z_{E}\sqrt{\frac{KT_{1}}{b r_{1}\cos\alpha}}\), then \(\sigma_{H}=B\sqrt{\frac{L}{x(L – x)}}\)
4.3 Variation Law of Contact Stress
By studying the function \(\frac{1}{\rho}=\sqrt{\frac{L}{Lx – x^{2}}}\), it can be found that the comprehensive curvature is larger near the limit meshing point of the meshing line, and the mid – point of the limit meshing line has a minimum value of the comprehensive curvature. In general, the closer to the end – point of the actual meshing line, the larger the contact stress. However, the area near the end – point of the actual meshing line is the double – tooth meshing area, and the contact stress is not the largest. The contact stress in the single – tooth meshing area is relatively large. Therefore, the maximum contact stress usually appears near the boundary point between the single – tooth meshing area and the double – tooth meshing area, close to the limit meshing point of the small gear.
| Parameter | Influence on Contact Stress |
|---|---|
| Comprehensive Curvature | Larger comprehensive curvature leads to larger contact stress |
| Meshing Position | Contact stress is larger in the single – tooth meshing area near the limit meshing point of the small gear |
5. Calculation of Maximum Contact Stress during Single – Pair Tooth Meshing
5.1 Determination of the Meshing Position of the Maximum Contact Stress
As shown in the figure of the limit meshing line, when the tooth profile contacts at point C (the boundary point between the single – tooth meshing area and the double – tooth meshing area near the limit meshing point of the small gear), \(N_{1}C=N_{1}B-\pi m\cos\alpha=r_{b1}\tan\alpha_{a1}-\pi m\cos\alpha\), that is, \(x = r_{b1}\tan\alpha_{a1}-\pi m\cos\alpha\)
5.2 Calculation Formula of the Maximum Contact Stress
Substitute \(x = r_{b1}\tan\alpha_{a1}-\pi m\cos\alpha\) into the contact stress formula \(\sigma_{H}=B\sqrt{\frac{L}{x(L – x)}}\), and the maximum contact stress \(\sigma_{HC}=B\sqrt{\frac{m(Z_{1}+Z_{2})\cos\alpha\tan\alpha’}{(r_{b1}\tan\alpha_{a1}-\pi m\cos\alpha)[m(Z_{1}+Z_{2})\cos\alpha\tan\alpha’-2r_{hf}\tan\alpha_{nl}+2\pi m\cos\alpha]}}\)
6. Contact Stress at the Node Meshing Point
6.1 Node Meshing Position and Parameters
For a pair of modified gears with modification coefficients \(x_{1}\) and \(x_{2}\), when the pitch circle and the 分度圆 do not coincide (assuming no errors in manufacturing and installation), the distance from the node P to the meshing limit point of the gear is \(x=\sqrt{(r_{1}’)^{2}-(r_{b1})^{2}}=\frac{1}{2}mZ_{1}\cos\alpha\tan\alpha’\)
6.2 Calculation Formula of the Contact Stress at the Node
Substitute \(x=\frac{1}{2}mZ_{1}\cos\alpha\tan\alpha’\) and \(L=(r_{b1}+r_{b2})\tan\alpha’\) into \(\frac{L}{x(L – x)}\), and then substitute the result into the contact stress formula \(\sigma_{H}=B\sqrt{\frac{L}{x(L – x)}}\), the contact stress at the node P is \(\sigma_{HP}=B\sqrt{\frac{2(1 + u)}{umZ_{1}\cos\alpha\tan\alpha’}}\)
7. Ratio of Maximum Contact Stress to Contact Stress at the Node (Stress Ratio)
7.1 Calculation Formula of the Stress Ratio
The ratio of the maximum contact stress \(\sigma_{Hmax}\) (formula (11)) to the contact stress at the node \(\sigma_{HP}\) (formula (14)) is \(\lambda=\frac{\sigma_{Hmax}}{\sigma_{HP}}=\sqrt{\frac{[(1 + u)\tan\alpha’-\tan\alpha_{a1}+\frac{2\pi}{Z_{1}}](\tan\alpha_{a1}-\frac{2\pi}{Z_{1}})}{}}\)
where \(\tan\alpha_{a1}=\frac{\sqrt{r_{a1}^{2}-r_{b1}^{2}}}{r_{b1}}=\sqrt{(\frac{Z_{1}+2 + 2x_{1}}{Z_{1}\cos\alpha})^{2}-1}\), and \(\cos\alpha’=\frac{Z_{1}(1 + u)\cos\alpha}{Z_{1}(1 + u)+2(x_{1}+x_{2})}\)
7.2 Expression of the Maximum Contact Stress Based on the Stress Ratio
The maximum contact stress of modified gear transmission can be expressed as \(\sigma_{max}=\lambda\sigma_{HP}=B\lambda\sqrt{\frac{2(1 + u)}{umZ_{1}\cos\alpha\tan\alpha’}}\)
The calculation formula for accurately designing gears according to the maximum contact stress is \(d_{1}\geq\sqrt[3]{\frac{KT_{1}}{\phi_{d}}\frac{4}{\cos^{2}\alpha\times\tan\alpha’}\frac{1 + u}{u}(\frac{\lambda Z_{E}}{[\sigma_{H}]})}\)
8. Stress Ratio Analysis
8.1 Isometric Modified Gear Transmission
For isometric modified gear transmission, assume the number of teeth of the driving gear \(Z_{1}\) ranges from 10 – 17, and the transmission ratio ranges from 1.7 – 6.
- When \(Z_{1}<17\), as \(Z_{1}\) increases, the modification coefficient of the small gear gradually decreases, the stress ratio increases, and the maximum contact stress increases.
- As the transmission ratio of isometric modified gear transmission increases, the stress ratio increases, and the maximum contact stress also increases.
- When the transmission ratio is greater than or equal to 3 and the number of teeth of the small gear is greater than or equal to 13 (less than 17), the maximum contact stress is 8% or more than the contact stress at the node.
| Parameter Change | Influence on Stress Ratio in Isometric Modified Gear Transmission |
|---|---|
| Increase of \(Z_{1}\) (\(Z_{1}<17\)) | Stress ratio increases |
| Increase of Transmission Ratio | Stress ratio increases |
8.2 Unisometric Modified Gear Transmission
Unisometric modified gear transmission can be divided into positive transmission and negative transmission (\(\vert x_{1}+x_{2}\vert\geq0\)).
- In positive transmission (\(Z_{1}<17\)), as the number of teeth of the small gear \(Z_{1}\) increases, the stress ratio increases; as the modification coefficient \(x_{1}\) increases, the stress ratio decreases.
- In negative transmission (\(Z_{1}<17\)), as the number of teeth of the small gear \(Z_{1}\) increases, the stress ratio increases; as the modification coefficient \(x_{1}\) increases, the stress ratio decreases.
| Transmission Type | Parameter Change | Influence on Stress Ratio in Unisometric Modified Gear Transmission |
|---|---|---|
| Positive Transmission | Increase of \(Z_{1}\) | Stress ratio increases |
| Positive Transmission | Increase of \(x_{1}\) | Stress ratio decreases |
| Negative Transmission | Increase of \(Z_{1}\) | Stress ratio increases |
| Negative Transmission | Increase of \(x_{1}\) | Stress ratio decreases |
8.3 Maximum Contact Stress Analysis of Standard Gear Transmission
For standard gear transmission, take \(x_{1}=x_{2}=0\) and \(\alpha’=\alpha\) in the stress ratio calculation formula.
- The stress ratio of standard gear transmission (\(Z_{1}\geq17\)) decreases as the number of teeth of the small gear increases.
- The stress ratio increases as the transmission ratio increases.
- The maximum contact stress of standard gear transmission is equal to the product of the stress at the node and the stress ratio, \(\sigma_{Hmax}=\lambda\sigma_{H}=\lambda B\sqrt{\frac{2}{mZ_{1}\sin\alpha}\frac{1 + u}{u}}\)
| Parameter Change | Influence on Stress Ratio in Standard Gear Transmission |
|---|---|
| Increase of \(Z_{1}\) (\(Z_{1}\geq17\)) | Stress ratio decreases |
| Increase of Transmission Ratio | Stress ratio increases |
8.4 Conditions Requiring Precise Contact Fatigue Strength Design Calculation
When the ratio of the maximum contact stress to the contact stress at the node is greater than or equal to 8%, precise contact fatigue strength design calculation is required. When \(Z_{1}\geq23\) and \(Z_{1}\leq11\), precise contact fatigue strength design calculation according to the maximum contact stress is usually not required. The specific conditions are shown in the following table:
9. Example Analysis
9.1 Gear Design Calculation Based on Contact Strength
Given an external meshing involute spur cylindrical modified gear transmission with input power \(P_{1}=10kW\), transmission ratio \(u = 3\), small gear rotational speed \(n_{1}=960r/min\), and other parameters such as material properties and allowable stress.
- First, calculate the torque \(T_{1}=9.55\times10^{6}P_{1}/n_{1}=9.9479\times10^{4}N\cdot mm\)
- Select the number of teeth of the small gear \(Z_{1}=15\), then the number of teeth of the large gear \(Z_{2}=uZ_{1}=45\), and the modification coefficients \(x_{1}=0.15\), \(x_{2}=-0.11\)
- Calculate the stress ratio \(\lambda = 1.085\), and then calculate the gear pitch circle diameters according to the node meshing and the maximum stress design methods. The results show that the pitch circle diameter designed by the maximum stress method is about 5.59% larger than that designed by the node meshing method.
- Calculate the module according to the pitch circle diameter and the number of teeth, and round up to the standard module. The modules designed by the two methods are \(m = 4.5mm\) and \(m = 5mm\) respectively.
9.2 Verification of the Stress Ratio Calculation Formula by Finite Element Analysis
Use Abaqus finite – element software to simulate and calculate the contact stress of the gear transmission in the above example.
- Determine the geometric dimensions and parameters of the gear, such as the tooth tip circle diameter, base circle diameter, meshing angle, and pitch circle radius.
- Calculate the normal force during gear meshing \(F_{n}=\frac{2KT_{1}}{Z_{1}m\cos\alpha}=4940.3N\)
- Simulate the contact of two semi – cylinders with the curvature radii of the tooth profiles at the meshing point, and calculate the ratio of the maximum contact stresses at the boundary point of the single – tooth meshing area and the double – tooth meshing area and the node meshing point.