1. Introduction
Gear transmission is a fundamental component in a wide range of mechanical systems, from automotive engines to industrial machinery. Its efficiency and durability are significantly influenced by the lubrication mechanism. In this article, we will delve into the world of elastohydrodynamic lubrication (EHL) design for involute cylindrical spur gears, exploring how various factors impact gear lubrication and offering practical insights for optimizing gear performance.
1.1 The Significance of Gear Lubrication
Gear lubrication serves multiple crucial purposes. It reduces friction between meshing teeth, minimizing wear and tear, and thus extending the lifespan of the gear system. In addition, it helps dissipate heat generated during operation, preventing overheating and potential damage. In harsh operating conditions such as low – speed, heavy – load, variable – load, and shock – prone environments, proper lubrication becomes even more critical as gears are more likely to experience dry friction or boundary lubrication, which can lead to premature failure.
1.2 Research Background and Objectives
Over the years, researchers have made significant progress in understanding gear lubrication. However, there is still a need for a comprehensive and accessible guide that integrates the latest research findings and practical design considerations. The objective of this article is to provide engineers and designers with a clear understanding of EHL design for involute cylindrical spur gears, enabling them to make informed decisions and improve gear performance.
2. Gear Lubrication Mechanism
2.1 How Lubricants Work in Gear Systems
When gears are in operation, lubricants are introduced between the meshing teeth. They form a continuous dynamic pressure oil film that separates the tooth surfaces, preventing direct contact and reducing friction. This oil film also distributes the load evenly across the teeth, enhancing the gear’s load – carrying capacity. The lubrication process involves a combination of rolling and sliding motions. Rolling friction is beneficial as it aids in the movement of the oil film, while sliding friction can have both positive and negative effects. On one hand, it can drive the flow of the oil film, reducing the risk of pitting; on the other hand, excessive sliding friction under heavy loads can rupture the oil film, leading to surface wear, scuffing, or even seizure.
2.2 Relative Sliding Speed Analysis
The relative sliding speed between meshing teeth plays a crucial role in gear lubrication. As shown in Figure 1 (Gear Pair Meshing Schematic), except at the meshing node where pure rolling occurs, there is relative sliding at all other meshing points.
| Parameter | Symbol | Explanation |
|---|---|---|
| Angular velocity of gear 1 | \(\omega_{1}\) | Rotational speed of gear 1 |
| Angular velocity of gear 2 | \(\omega_{2}\) | Rotational speed of gear 2 |
| Pitch circle pressure angle | \(\alpha\) | Angle at the pitch circle |
| Pitch circle radius of gear 1 | \(r_{p1}\) | Radius of gear 1’s pitch circle |
| Pitch circle radius of gear 2 | \(r_{p2}\) | Radius of gear 2’s pitch circle |
| Distance from the node P to the meshing point E | S | Represents the position of the meshing point E on the actual meshing line |
The relative sliding speed \(v_{H}\) at point E is the difference between the sliding speeds of the two tooth surfaces along the common tangent direction. It can be calculated using the formula \(v_{H}=v_{E2}-v_{E1}\), where \(v_{E1}=\omega_{1}(r_{p1}\sin\alpha – S)\) and \(v_{E2}=\omega_{2}(r_{p2}\sin\alpha+S)\).
Figure 2 shows the variation of the relative sliding speed \(v_{H}\) along the meshing line for a specific gear pair. It can be observed that \(v_{H}\), \(v_{E1}\), and \(v_{E2}\) change linearly along the meshing line. The speed \(v_{E1}\) of the large gear \(z_{1}\) increases from the meshing – in point to the meshing – out point, while the speed \(v_{E2}\) of the small gear \(z_{2}\) decreases. At the node, the relative sliding speed \(v_{H}\) is zero. The absolute value of \(v_{H}\) forms a V – shape, with the maximum values occurring at the meshing – in and meshing – out points.
| Gear Parameter | Value |
|---|---|
| Addendum coefficient \(h_{a}^*\) | 0.95 |
| Module m | 2 |
| Pressure angle \(\alpha\) (°) | 25 |
| Clearance coefficient \(c^*\) | 0.2 |
| Number of teeth of gear 1 \(z_{1}\) | 20 |
| Number of teeth of gear 2 \(z_{2}\) | 17 |
| Modification coefficient of gear 1 \(x_{1}\) | 0.23 |
| Modification coefficient of gear 2 \(x_{2}\) | 0.299 |
| Contact ratio \(\varepsilon_{a}\) | 1.224 |
| Meshing angle \(\alpha’\) (°) | 28.06 |
| Input speed of the driving gear (r/min) | 300 |
| Meshing center distance \(a’\) (mm) | 38 |
| Input power of the driving gear (W) | 1000 |
| Face width b (mm) | 24 |
3. Elastohydrodynamic Lubrication Analysis of Gear Pairs
3.1 Minimum Oil Film Thickness Formula
The minimum oil film thickness in gear transmission is calculated using the Dowson – Higginson formula: \(h_{min}=2.65\frac{(\alpha^{*})^{0.54}(\eta_{0}u)^{0.7}R^{0.43}}{E^{\prime0.03}W^{\prime0.13}}\), where R is the equivalent curvature radius at the meshing point, \(\alpha\) is the pressure – viscosity coefficient of the lubricant, \(E’\) is the equivalent elastic modulus of the gear material, \(\eta_{0}\) is the dynamic viscosity of the lubricant, u is the entrainment speed at the meshing point, and W is the meshing load per unit face width of the gear.
3.2 Key Parameters Affecting Oil Film Thickness
3.2.1 Equivalent Curvature Radius
The equivalent curvature radius R at the meshing point of the gear pair is calculated based on the curvature radii of the two contacting cylinders. Let the curvature radii of the tooth surfaces of the large and small gears at the meshing point E be \(R_{1}\) and \(R_{2}\) respectively. For external meshing, \(R_{1}=r_{p1}\sin\alpha – S\) and \(R_{2}=r_{p2}\sin\alpha+S\), and \(R = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\).
The variation of the equivalent curvature radius R along the meshing line is shown in Figure 3. It is a parabolic distribution. The maximum value of R occurs at a certain point on the meshing line. The position and value of R are closely related to the transmission ratio and modification coefficient. When the transmission ratio \(i > 1\) (deceleration transmission), the R – parabola is skewed, with the minimum value at the \(B_{1}\) point. When \(i < 1\) (acceleration transmission), the minimum value is at the \(B_{2}\) point. When \(i = 1\) (constant – speed transmission), the R – curve is a symmetric parabola.
3.2.2 Entrainment Speed
The entrainment speed u is the average speed of the two tooth surfaces at the meshing point. It can be calculated as \(u=\frac{u_{1} + u_{2}}{2}\), where \(u_{1}=v_{E1}\) and \(u_{2}=v_{E2}\). After substitution and simplification, \(u=\frac{\pi n_{1}}{30}(\frac{L}{1 + i}+\frac{1 – i}{2i}S)\), where \(n_{1}\) is the input speed of the gear, \(L=a’\sin\alpha\) is the length of the meshing line, and i is the transmission ratio.
Figure 4 shows the variation of the entrainment speed u along the meshing line. The entrainment speed changes linearly with the position of the meshing point S. Its magnitude is affected by factors such as the transmission ratio i and the input speed \(n_{1}\).
3.2.3 Load per Unit Contact Length
For general involute cylindrical spur gear transmissions with a contact ratio in the range of \(1<\varepsilon<2\), there are single – tooth – pair and double – tooth – pair meshing regions in a gear meshing cycle. In the single – tooth – pair meshing region, the normal tooth surface load \(F_{N}\) is borne entirely by one pair of teeth. In the double – tooth – pair meshing region, the normal load \(F_{N}\) is distributed between the two pairs of meshing teeth according to a certain rule.
The load per unit contact length \(W’\) at the meshing point of the gear pair is calculated as \(W’=\frac{W_{1}}{b}\), where \(W_{1}\) is the normal load borne by one of the meshing teeth and b is the face width of the gear.
4. Influence of Gear Pair Geometric Parameters on Lubricating Oil Film
4.1 Influence of Transmission Ratio
When studying the influence of the transmission ratio on the minimum oil film thickness, we conduct calculations for different transmission ratios while keeping other parameters constant. Figure 6 shows the results for transmission ratios \(i = 0.75\), \(i = 1\), and \(i = 1.45\).
As the transmission ratio i increases, the entrainment speed curve rotates clockwise around the entrainment speed point \(u_{0}\) at the meshing line coordinate 0 (node P). The equivalent curvature radius R also changes, with its curve changing from downward – sloping to upward – sloping. Consequently, the minimum oil film thickness curve changes from downward – sloping to upward – sloping, and the average minimum oil film thickness increases.
| Transmission Ratio i | Entrainment Speed Curve Trend | Equivalent Curvature Radius Curve Trend | Minimum Oil Film Thickness Curve Trend | Average Minimum Oil Film Thickness Change |
|---|---|---|---|---|
| 0.75 | Downward – sloping | Downward – sloping | Downward – sloping | Smaller |
| 1 | – | – | – | – |
| 1.45 | Upward – sloping | Upward – sloping | Upward – sloping | Larger |
4.2 Influence of Modification Coefficient
4.2.1 High – Modification Transmission
For high – modification transmission with a constant center distance and meshing angle, we consider different modification coefficient combinations. Figure 7 shows the changes in the entrainment speed, equivalent curvature radius, and minimum oil film thickness.
As the modification coefficient of gear \(z_{1}\) changes, the pressure angle at the meshing start point \(B_{1}\) changes, and the point \(B_{1}\) moves. The overall entrainment speed decreases, the equivalent curvature radius curve shifts, and the minimum oil film thickness curve also shifts, with the lowest point of the minimum oil film thickness at the \(B_{1}\) point.
| Modification Coefficient Combination | Pressure Angle Change at \(B_{1}\) | Movement of \(B_{1}\) | Entrainment Speed Change | Equivalent Curvature Radius Curve Shift | Minimum Oil Film Thickness Curve Shift | Lowest Point of Minimum Oil Film Thickness |
|---|---|---|---|---|---|---|
| \(x_{1}=+0.23,x_{2}=-0.23\) | Decreases | Moves towards \(N_{1}\) | Decreases | Shifts towards \(N_{1}\) | Shifts towards \(N_{1}\) | \(B_{1}\) |
| \(x_{1}=0,x_{2}=0\) | – | – | – | – | – | – |
| \(x_{1}=-0.23,x_{2}=+0.23\) | Increases | Moves away from \(N_{1}\) | – | – | – | – |
4.2.2 Angular – Modification Transmission
For angular – modification transmission, which causes changes in the meshing center distance and meshing angle, we calculate the parameter changes for different modification coefficient combinations. As shown in Table 2 and Figure 8, as the sum of the modification coefficients \(\sum x\) decreases, the meshing angle \(\alpha’\), the meshing center distance \(a’\), and the meshing length L all decrease. The entrainment speed u and the equivalent curvature radius R decrease and shift towards the \(N_{1}\) end, and the minimum oil film thickness curve decreases and shifts with the meshing start point \(B_{1}\) towards the \(N_{1}\) end.
| Modification Situation | Center Distance \(a’\) (mm) | Meshing Angle \(\alpha’\) (°) | Meshing Line Length L (mm) | Entrainment Speed Change | Equivalent Curvature Radius Change | Minimum Oil Film Thickness Curve Change |
|---|---|---|---|---|---|---|
| Situation 1 | 38 | 28.06 | 17.88 | Decreases and shifts towards \(N_{1}\) | Decreases and shifts towards \(N_{1}\) | Decreases and shifts towards \(N_{1}\) |
| Situation 2 | 37 | 25 | 15.64 | – | – | – |
| Situation 3 | 36.5 | 23.27 | 14.42 | – | – | – |
4.3 Influence of Pressure Angle
We calculate the minimum oil film thickness for different pressure angles. Figure 9 shows that the entrainment speed u, the equivalent curvature radius R, and the minimum oil film thickness increase with the increase of the pressure angle \(\alpha\). Larger pressure angles are beneficial for forming a flowing oil film. In aviation gear transmission, a pressure angle of \(25^{\circ}\) is often used to meet the requirements of good lubrication, high load – carrying capacity, and long service life.
| Pressure Angle \(\alpha\) (°) | Entrainment Speed u | Equivalent Curvature Radius R | Minimum Oil Film Thickness \(h_{min}\) |
|---|---|---|---|
| 20 | Smaller | Smaller | Smaller |
| 22.5 | – | – | – |
| 25 | Larger | Larger | Larger |
4.4 Influence of Center Distance
For standard gear transmissions with a transmission ratio \(i = 1\), we calculate the minimum oil film thickness for different center distances. Figure 10 shows that as the center distance increases, the entrainment speed and the equivalent curvature radius increase, and the oil film thickness at each meshing point also increases. The minimum oil film thickness in the double – tooth – pair meshing region is greater than that in the single – tooth – pair meshing region.
| Center Distance a (mm) | Entrainment Speed | Equivalent Curvature Radius | Oil Film Thickness at Meshing Points | Minimum Oil Film Thickness Comparison |
|---|---|---|---|---|
| 60 | Smaller | Smaller | Smaller | – |
| 63 | – | – | – | Double – tooth – pair > Single – tooth – pair |
| 69 | Larger | Larger | Larger | – |
5. Tooth Surface Lubrication State Analysis
5.1 Lubrication State Criteria
The lubrication state of gears is described by the oil film thickness ratio \(\lambda\), which is the ratio of the minimum oil film thickness \(h_{min}\) to the combined roughness of the two tooth surfaces, \(\lambda=\frac{h_{min}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\), where \(\sigma_{1}\) and \(\sigma_{2}\) are the arithmetic mean deviations of the surface roughness of the two meshing gears.
| Lubrication State | Oil Film Thickness Ratio \(\lambda\) | Characteristics | Lubricant Requirement | Gear Condition |
|---|---|---|---|---|
| Full – Film Lubrication | \(\lambda>4\) | The minimum oil film thickness is much larger than the tooth surface roughness. The meshing surfaces are completely separated by the dynamic pressure oil film, and the load is borne entirely by the oil film. The friction coefficient is very small. | No anti – wear agent is needed. | No |
