In the field of gear transmission systems, internal gear pairs play a critical role in various industrial applications due to their compact design and high power density. As an internal gear manufacturer, we constantly seek to optimize tooth profiles to enhance performance, durability, and efficiency. One promising approach involves replacing the conjugate external gear tooth profile with an optimized arc profile when meshing with an internal gear featuring a rectilinear tooth profile. This modification addresses challenges in hard tooth surface machining for internal gears, which is a common concern for internal gear manufacturers. In this study, we analyze the meshing efficiency of such rectilinear-arc tooth profile internal gear pairs, focusing on geometric and meshing parameters. We derive formulas for重合度 (coincidence degree) and meshing efficiency, supported by computational examples and comparisons with traditional involute profiles. Our findings indicate that while the重合度 of rectilinear-arc internal gears is lower than that of involute internal gears with the same modulus and tooth numbers, the meshing efficiency is significantly higher under similar conditions. This makes rectilinear-arc profiles an attractive option for internal gear manufacturers aiming to improve transmission systems.
The development of internal gear pairs has evolved significantly, with rectilinear tooth profiles for internal gears gaining attention for their ease of grinding and hard surface processing. However, pairing these with conventional involute external gears can lead to transmission errors, necessitating profile modifications that complicate manufacturing. By optimizing the external gear tooth profile into an arc shape, we achieve minimal single-tooth transmission errors—less than 5 μm at the pitch circle—making it a viable solution for internal gear manufacturers. This paper delves into the meshing characteristics, specifically重合度 and efficiency, which are crucial for ensuring smooth and continuous operation in applications such as gear pumps and precision machinery. We employ mathematical modeling and simulation to provide insights that can guide internal gear manufacturers in design and production.
To begin, we examine the重合度 of rectilinear-arc internal gear pairs, which is a key indicator of transmission smoothness and continuity.重合度, defined as the ratio of the端面作用角 (face action angle) to the tooth pitch angle, must generally exceed 1 for reliable operation. For internal gears, this involves analyzing the meshing process from initial contact to disengagement. Let O1 and O2 represent the centers of the external and internal gears, respectively. The meshing process includes three typical positions: start of meshing at point P1, midpoint at C, and end at P2. The rectilinear tooth profile of the internal gear always touches a circle centered at O2 with radius Rb, termed the internal gear tooth profile tangent circle. The angles α1 and α2 correspond to the rotations of the internal gear during the meshing phases from P1 to C and C to P2, respectively.
The derivation of α1 and α2 involves solving nonlinear equations based on geometric relationships. For α1, the vertical coordinate of point P1 is given by:
$$ R_{a2} \cos(\alpha_1 – \beta_1) = a + Y’_0 + R \sin(\alpha_1 + \gamma) $$
where Ra2 is the addendum circle radius of the internal gear, calculated as:
$$ R_{a2} = \frac{m Z_2 – 2 m h_a^*}{2} $$
Here, m is the module, Z2 is the number of teeth on the internal gear, h_a* is the addendum coefficient, a is the center distance (a = m(Z2 – Z1)/2), R is the radius of the arc profile of the external gear, γ is the half-angle of the internal gear’s rectilinear tooth profile, and β1 is derived as:
$$ \beta_1 = \arcsin\left(\frac{R_b}{R_{a2}}\right) – \gamma $$
The tangent circle radius Rb is determined from the geometry at the pitch point:
$$ R_b = r_2 \sin\left(\gamma + \frac{\pi}{2 Z_2}\right) $$
where r2 is the pitch radius of the internal gear (r2 = m Z2 / 2). Similarly, for α2, the vertical coordinate of point P2 is expressed as:
$$ R_{a1} \cos\left(\frac{Z_2}{Z_1} \alpha_2 + \beta_2\right) = R \sin(\gamma – \alpha_2) + Y’_0 $$
with Ra1 as the addendum circle radius of the external gear:
$$ R_{a1} = \frac{m Z_1 + 2 m h_a^*}{2} $$
and β2 computed using:
$$ \beta_2 = \arccos\left(\frac{B Y_0 + \sqrt{B^2 Y_0^2 – (B^2 – 4 X_0^2 R_{a1}^2)(X_0^2 + Y_0^2)}}{2 R_{a1} (X_0^2 + Y_0^2)}\right) $$
where B = R_{a1}^2 + X_0^2 + Y_0^2 – R^2, and X0, Y0 are the coordinates of the arc center in the external gear’s coordinate system at the midpoint C. Solving these equations numerically—for instance, using MATLAB’s fzero function—yields α1 and α2. The重合度 ε is then:
$$ \epsilon = \frac{\alpha_1 + \alpha_2}{2\pi / Z_2} $$
To illustrate, consider two computational examples typical for internal gear manufacturers. In Example 1, parameters are: Z1 = 24, Z2 = 60, m = 3, γ = 30°, h_a* = 0.9, X0 = -25.09, Y0 = 21.51, R = 31. Calculations give Rb = 47.03, β1 = 2.59°, α1 = 5.168°, β2 = 1.054°, α2 = 3.008°, and ε = 1.363. In Example 2, with Z1 = 40, Z2 = 127, m = 10, γ = 28°, h_a* = 0.9, X0 = -172.32, Y0 = 110.25, R = 201.22, we obtain Rb = 305.026, β1 = 1.163°, α1 = 2.585°, β2 = 0.826°, α2 = 1.178°, and ε = 1.327. Both examples satisfy ε > 1, ensuring continuous meshing.
The addendum coefficient h_a* influences重合度 significantly. For internal gear manufacturers, optimizing this parameter is essential. As h_a* increases from 0.8 to 1 in steps of 0.01,重合度 ε rises approximately linearly, as shown in the table below for Example 1 parameters:
| h_a* | ε |
|---|---|
| 0.80 | 1.320 |
| 0.81 | 1.325 |
| 0.82 | 1.330 |
| 0.83 | 1.335 |
| 0.84 | 1.340 |
| 0.85 | 1.345 |
| 0.86 | 1.350 |
| 0.87 | 1.355 |
| 0.88 | 1.360 |
| 0.89 | 1.365 |
| 0.90 | 1.370 |
| 0.91 | 1.375 |
| 0.92 | 1.380 |
| 0.93 | 1.385 |
| 0.94 | 1.390 |
| 0.95 | 1.395 |
| 0.96 | 1.400 |
| 0.97 | 1.405 |
| 0.98 | 1.410 |
| 0.99 | 1.415 |
| 1.00 | 1.420 |
This linear relationship aids internal gear manufacturers in selecting appropriate h_a* values, though balancing重合度 with other factors like sliding rate is crucial; we recommend h_a* = 0.9 for optimal performance.
Comparing rectilinear-arc internal gears with involute profiles highlights differences in重合度. For involute internal gears,重合度 is calculated as:
$$ \epsilon_\alpha = \frac{1}{2\pi} \left[ Z_1 (\tan \alpha_{a1} – \tan \alpha) – Z_2 (\tan \alpha_{a2} – \tan \alpha) \right] $$
where α is the pressure angle (typically 20°). Using the same parameters as earlier, for Example 1 (Z1=24, Z2=60), εα = 1.972, and for Example 2 (Z1=40, Z2=127), εα = 1.916. The table below summarizes the comparison:
| Gear Pair Type | Example 1 ε | Example 2 ε |
|---|---|---|
| Involute Internal Gears | 1.972 | 1.916 |
| Rectilinear-Arc Internal Gears | 1.363 | 1.327 |
Thus, involute internal gears offer higher重合度, implying better smoothness, but rectilinear-arc profiles compensate with other advantages, such as improved machining for hard surfaces, which is vital for internal gear manufacturers.
Next, we analyze meshing efficiency, a critical performance metric for internal gear manufacturers seeking to minimize energy losses. Meshing efficiency η is defined as the ratio of output to input power, accounting for friction losses. Assuming a constant friction coefficient μ = 0.1 and focusing on geometric and meshing parameters, the efficiency for internal gear pairs with 1 < ε < 2 is given by:
$$ \eta = 1 – \frac{W_f}{W_1} = 1 – \mu \pi \left( \frac{1}{Z_1} – \frac{1}{Z_2} \right) (\epsilon_1 + \epsilon_2 + 1 – \epsilon_1 – \epsilon_2) $$
where W_f is the friction loss work, W_1 is the driving work per tooth pair, ε1 is the重合度 after the pitch point, ε2 is the重合度 before the pitch point, and ε = ε1 + ε2. The angles φ2 and ε2 are derived from the geometry: φ2 = π/(2Z2) + α2, and ε2 = φ2 / (2π/Z2). Then, ε1 = ε – ε2. For Example 1, φ2 = 4.508°, ε2 = 0.751, ε1 = 0.612, and η = 99.55%. For Example 2, φ2 = 1.887°, ε2 = 0.666, ε1 = 0.661, and η = 99.70%. These high efficiencies stem from the balanced distribution of ε1 and ε2,接近 ε/2, which minimizes losses.
For involute internal gears, meshing efficiency uses the same formula, with ε1 and ε2 calculated as:
$$ \epsilon_1 = \frac{Z_1}{2\pi} (\tan \alpha_{a1} – \tan \alpha) $$
$$ \epsilon_2 = \frac{Z_2}{2\pi} (\tan \alpha – \tan \alpha_{a2}) $$
For Example 1, ε1 = 1.171, ε2 = 0.801, η = 99.18%; for Example 2, ε1 = 0.857, ε2 = 1.059, η = 99.49%. The comparison table below shows the advantage of rectilinear-arc profiles:
| Gear Pair Type | Example 1 η | Example 2 η |
|---|---|---|
| Involute Internal Gears | 99.18% | 99.49% |
| Rectilinear-Arc Internal Gears | 99.55% | 99.70% |
The higher efficiency in rectilinear-arc internal gears is attributed to the more equitable split of ε1 and ε2, reducing friction losses around the pitch point. This is particularly beneficial for internal gear manufacturers designing high-efficiency transmission systems, such as those used in automotive or aerospace applications where energy conservation is paramount.
In practical terms, internal gear manufacturers must consider additional factors like lubrication, material properties, and surface roughness, which can affect overall efficiency. However, our analysis focuses solely on geometric and meshing parameters to isolate their impact. The rectilinear-arc profile not only simplifies hard tooth surface grinding for internal gears but also enhances meshing efficiency, making it a superior choice in many scenarios. For instance, in gear pumps—often referred to as quiet pumps—this profile reduces noise and eliminates oil trapping, further boosting performance. As internal gear manufacturers adopt these optimized profiles, they can achieve longer service life and lower operational costs.
In conclusion, our study demonstrates that rectilinear-arc internal gear pairs, while having a lower重合度 than involute pairs, exhibit significantly higher meshing efficiency when considering geometric parameters. This makes them ideal for applications where efficiency outweighs slight reductions in smoothness. Internal gear manufacturers can leverage these findings to optimize designs, selecting parameters like h_a* = 0.9 for a balance between重合度 and efficiency. Future work should explore dynamic effects, thermal behavior, and real-world testing to validate these results under varied operating conditions. Ultimately, the rectilinear-arc profile represents a significant advancement for internal gear manufacturers aiming to produce high-performance, durable gear systems.