In my research on modified planar double-enveloping worm gear pairs, I discovered that the contact line types are no longer restricted to the conventional Type I or Type II forms. A third type, Type III, also appears, making the actual tooth surface structure of the worm gear extremely complex. Using phase angle analysis and the first limiting curve, I have conducted a detailed dissection of the worm gear tooth surface construction for all three contact types. I divided the worm gear tooth surface into working and non-working regions and determined the composition of each part. This paper presents a comprehensive, first-person account of my analytical methods and findings, supported by extensive mathematical formulations and tabular summaries.
My study focuses on the modified planar double-enveloping worm gear pair, which is widely used in heavy-duty transmission systems due to its high load capacity and favorable lubrication conditions. The worm gear tooth surface is generated by a secondary enveloping process, where a cutting tool (worm hob) replicates the motion of the worm. Understanding the structure of the worm gear tooth surface – particularly the distinction between working and non-working zones – is critical for optimizing contact patterns and reducing undesirable interference. In the following sections, I will describe the first limiting curve, the morphological characteristics of contact lines, and the structural analysis of the worm gear tooth surface for each type.
1. First Limiting Curve in Modified Worm Gear Pairs
The first limiting curve (or boundary curve) for the second enveloping process in a modified planar double-enveloping worm gear pair satisfies the following conditions:
$$
\begin{cases}
\Psi^* = \mathbf{n} \cdot \mathbf{V}^{(12)} = 0 \\
\Psi^* = \mathbf{n} \cdot \mathbf{q} + \mathbf{p} \cdot \mathbf{V}^{(12)} = 0
\end{cases}
$$
In parametric form, the first limiting curve can be expressed as:
$$
u \left[ \left( \cos\theta – \frac{i_{12}}{i_{10}} \right) \cos\beta + \left( \frac{\sin\beta}{\sin\varphi_0} – i_{01}\cos\beta \cot\varphi_0 \right) \sin\theta + \cdots \right] + a_0 \left[ \sin\beta \cot\varphi_0 \sin\theta + \left( \cos\theta – \frac{a}{a_0} \right) \cos\beta \cos\varphi_0 \right] = 0
$$
where
$$
u = \frac{b_2′ c_2 – b_2 c_2′}{a_2′ b_2 – a_2 b_2′}
$$
The transformation equations from the worm gear coordinate system to the tool coordinate system are:
$$
\begin{aligned}
x_2 &= (x_1 \cos\varphi_1^* – y_1 \sin\varphi_1^* – a) \cos\varphi_2^* + z_1 \sin\varphi_2^* \\
y_2 &= -(x_1 \cos\varphi_1^* – y_1 \sin\varphi_1^* – a) \sin\varphi_2^* + z_1 \cos\varphi_2^* \\
z_2 &= -x_1 \sin\varphi_1^* – y_1 \cos\varphi_1^*
\end{aligned}
$$
$$
\begin{aligned}
x_1 &= (x_0 \cos\varphi_0 – y_0 \sin\varphi_0 + a_0) \cos\varphi_1 – z_0 \sin\varphi_1 \\
y_1 &= -(x_0 \cos\varphi_0 – y_0 \sin\varphi_0 + a_0) \sin\varphi_1 – z_0 \cos\varphi_1 \\
z_1 &= x_0 \sin\varphi_0 + y_0 \cos\varphi_0
\end{aligned}
$$
$$
\begin{aligned}
x_0 &= u \\
y_0 &= v \sin\beta – r_b \\
z_0 &= v \cos\beta
\end{aligned}
$$
In these equations, \(i_{10}\) is the first enveloping transmission ratio, \(i_{12}\) is the second enveloping transmission ratio, \(a_0\) is the center distance of the first enveloping, \(a\) is the center distance of the second enveloping, \(\beta\) is the inclination angle of the tool gear’s inclined plane, \(u\) and \(v\) are surface parameters on the first enveloping tool plane, and the angles \(\varphi_0\), \(\varphi_1\), and \(\varphi_1^*\) denote rotational parameters of the tool gear, worm, and hob, respectively. The variable \(\theta = \varphi_1^* – \varphi_1\).
2. Morphology of the First Limiting Curve for the Second Enveloping
To solve the first limiting curve equations numerically, I employed an optimization procedure with the following algorithmic steps:
- Define the range of the parameter \(\varphi_1^*\).
- Determine an initial sequence for the optimization variable \(\varphi_0\).
- Substitute the values of these two rotational parameters into one of the first limiting curve equations to solve for \(u\).
- Using the other equation of the first limiting curve as the objective function, perform optimization to solve for the remaining variable.
- Substitute all determined values back to generate the first limiting curve.
For Type I contact line patterns (as shown conceptually in the original paper, but without referencing figure numbers), the first limiting curve appears as a single branch on the right side of the worm gear tooth surface. For Type II patterns, the curve also has a single branch but with a different topology. However, for Type III patterns, the contact line family on the worm gear tooth surface contains both Type I and Type II lines. Consequently, the first limiting curve splits into two distinct branches: one corresponding to the envelope of the Type I contact lines on the right side, and the other corresponding to the envelope of the Type II contact lines on the right side. These branches are denoted as \(\Gamma_1^{(2)}\) and \(\Gamma_2^{(2)}\), respectively.
In the following table, I summarize the key parameters and characteristics of the first limiting curves for the three contact types.
| Contact Type | Number of Branches | Location Relative to Tooth Surface | Associated Contact Line Family |
|---|---|---|---|
| Type I | 1 | Right side of worm gear tooth | Single V-shaped family (Σ₂(1) and Σ₂(21)) |
| Type II | 1 | Right side of worm gear tooth | Two interleaved families (Σ₂(1) and Σ₂(21)) |
| Type III | 2 | Both from right side but distinct envelopes | Mixed families (Σ₂(11), Σ₂(12) and Σ₂(21), Σ₂(22)) |
3. Principle of Worm Gear Tooth Surface Structure Analysis
The entire worm gear tooth surface is generated by enveloping the tool worm (hob). The working region of the worm gear tooth is the area where contact lines actually exist and bear the load. The non-working regions are formed by the front and rear transition surfaces. The front transition surface C₁ is produced by the first cutting edge of the tool worm, while the rear transition surface C₂ is produced by the last cutting edge. In my analysis, I approximate the positions of the first cutting edge H and the last cutting edge K by the intersection of the effective working length \(L_w\) of the tool worm with the worm helical surface.
As shown in the analysis, the contact lines on the right side of the worm gear tooth surface exhibit crossing phenomena, leading to interference. Moreover, the front and rear transition surfaces may cut off part of the contact area. In general, the envelope surface formed by the contact line family that is farther from the worm gear solid body will be trimmed by the envelope surface that is closer to the solid body. Based on this principle, I performed a positional analysis of the multiple envelope surfaces appearing on the worm gear tooth.
For a cross-section perpendicular to the worm gear axis at a given \(z_2 = z_0\), I project the section onto the \(xOy\) plane. The solid worm gear tooth and the cross-sectional curves \(\Gamma_1\) and \(\Gamma_2\) of the envelope surfaces are shown conceptually. For any radius \(R\) between the tooth root and the tooth tip circle, the intersection points of \(\Gamma_1\) and \(\Gamma_2\) with this arc are denoted by A and B. The corresponding phase angles measured from the tooth center are \(\theta_1\) and \(\theta_2\). If \(\theta_1 > \theta_2\), then envelope surface 1 is closer to the worm gear solid body at that radius; conversely, if \(\theta_1 < \theta_2\), envelope surface 2 is closer.
By applying this method for multiple radii \(R\) and multiple axial sections \(z_0\), I can determine which envelope surface actually remains on the final worm gear tooth surface after the cutting process.
4. Structural Analysis for Type I Contact Line Transmission
In Type I transmission, the contact lines on the worm gear tooth surface form a V-shaped pattern. These lines constitute one leaf of the worm gear surface family, denoted \(\Sigma_2^{(1)}\). The other leaf, \(\Sigma_2^{(21)}\), is formed by the inverted V-shaped pattern. The two leaves overlap on the right side of the worm gear tooth. In \(\Sigma_2^{(1)}\), there are no singular points. However, in \(\Sigma_2^{(21)}\), each contact line contains a singular point (the first limiting curve), dividing each line into two parts: \(\Sigma_2^{(21)}\) (the valid part) and \(\Sigma_2^{(22)}\) (the invalid part) . The sign of the first limiting function on \(\Sigma_2^{(22)}\) is negative, indicating that this part protrudes into the worm solid body and is cut away during manufacturing. Thus, only \(\Sigma_2^{(1)}\) and the valid part \(\Sigma_2^{(21)}\) remain on the worm gear tooth surface.
Now, both \(\Sigma_2^{(1)}\) and \(\Sigma_2^{(21)}\) exist on the right side. To determine which one actually forms the worm gear tooth surface, I performed a phase angle analysis. The following table presents a sample of my computed results.
| \(z_2\) (mm) | \(R\) (mm) | \(\theta_2^{(1)}\) (rad) | \(\theta_2^{(21)}\) (rad) | \(\Delta\theta = \theta_2^{(1)} – \theta_2^{(21)}\) (rad) | \(\Delta L\) (mm) |
|---|---|---|---|---|---|
| 1.9826 | 147.3 | 1.2289 | 1.1197 | 0.1092 | 0.2807 |
| 4.436 | 147.7 | 1.2345 | 1.2180 | 0.0156 | 0.0401 |
| -0.2703 | 145.13 | 1.2191 | 1.1074 | 0.1117 | 0.2872 |
| -1.1516 | 144.24 | 1.2143 | 1.1146 | 0.0997 | 0.2563 |
| -3.8256 | 141.14 | 1.2009 | 1.1082 | 0.0927 | 0.2383 |
From Table 2, it is clear that \(\theta_2^{(1)}\) is always greater than \(\theta_2^{(21)}\) for all tested points; \(\Delta\theta\) is consistently positive. This means that \(\Sigma_2^{(1)}\) is closer to the worm gear solid body, and thus \(\Sigma_2^{(21)}\) is cut away. The small values of \(\Delta L\) (in the order of 0.01–0.3 mm) indicate that the two surfaces are very close, but \(\Sigma_2^{(1)}\) prevails.
Consequently, the worm gear tooth surface for Type I transmission consists of three distinct regions, as summarized in the table below.
| Region | Type | Composition | Description |
|---|---|---|---|
| I | Non-working | Front transition surface C₁ | Machined by the first cutting edge of the worm hob |
| II | Working | Σ₂(1) (contact line family) | Concave area with dual-line contact, good lubrication |
| III | Non-working | Rear transition surface C₂ | Machined by the last cutting edge of the worm hob |
In the working region (Region II), the worm gear tooth surface achieves high load capacity and favorable lubricant entrapment due to the double-line contact.
5. Structural Analysis for Type II Contact Line Transmission
For Type II transmission, the contact lines on the worm gear tooth surface consist of two interleaved families: \(\Sigma_2^{(1)}\) and \(\Sigma_2^{(21)}\). Both families contain a first limiting curve on the right side. Following the same phase angle analysis, I determined that the surface \(\Sigma_2^{(1)}\) is closer to the solid body than \(\Sigma_2^{(21)}\) in some areas, but the relative positions vary. Additional analysis with the front and rear transition surfaces reveals that the front transition surface C₁ intersects \(\Sigma_2^{(1)}\) and \(\Sigma_2^{(21)}\) along distinct tangent lines. The rear transition surface C₂ only contacts \(\Sigma_2^{(1)}\) within the tooth surface; its intersection with \(\Sigma_2^{(21)}\) lies outside the actual tooth boundary. This leads to a segmentation of the worm gear tooth surface into five regions, as described in the following table.
| Region | Type | Composition | Description |
|---|---|---|---|
| I | Non-working | Front transition surface C₁ | Machined by first cutting edge |
| II | Working | Σ₂(1) | Primary working area from V-shaped contact lines |
| III | Non-working | Rear transition surface C₂ | Machined by last cutting edge (part of it) |
| IV | Working | Σ₂(21) | Secondary working area from inverted V-shaped lines |
| V | Non-working | Front transition surface C₁ (extended) | Another portion machined by first cutting edge |
Regions II and IV together constitute the working area, but they are not contiguous. This discontinuity can negatively affect load distribution and lubrication. In practice, designers should aim to maximize the size of the working zones and minimize the non-working zones to improve transmission performance.
6. Structural Analysis for Type III Contact Line Transmission
Type III contact line transmission is the most complex. As shown by the two branches of the first limiting curve (Γ₁ and Γ₂), the contact line families are further subdivided. Let \(\Sigma_2^{(11)}\) and \(\Sigma_2^{(12)}\) correspond to one family (derived from Type I lines), and \(\Sigma_2^{(21)}\) and \(\Sigma_2^{(22)}\) to the other (derived from Type II lines). The first limiting curve Γ₁ divides \(\Sigma_2^{(12)}\) into \(\Sigma_2^{(121)}\) and \(\Sigma_2^{(122)}\), while Γ₂ divides \(\Sigma_2^{(22)}\) into \(\Sigma_2^{(221)}\) and \(\Sigma_2^{(222)}\). Using the sign of the first limiting function, I find that \(\Sigma_2^{(222)}\) has a negative sign (protruding into the worm), and is cut away. Similarly, both \(\Sigma_2^{(121)}\) and \(\Sigma_2^{(122)}\) also have negative signs, so they are also removed. The only valid subsurfaces that remain on the worm gear tooth after machining are \(\Sigma_2^{(11)}\) and \(\Sigma_2^{(221)}\) – the portions with positive first limiting function.
Further analysis with the front and rear transition surfaces shows that the front transition surface C₁ contacts the remaining surfaces along a single tangent line, and the rear transition surface C₂ does not contact them at all within the tooth boundary. Therefore, the worm gear tooth surface for Type III consists of three regions, summarized below.
| Region | Type | Composition | Description |
|---|---|---|---|
| I | Non-working | Front transition surface C₁ | Machined by first cutting edge |
| II | Working | Composite of Σ₂(11), Σ₂(221), and an “original” straight line segment | Mixed contact line pattern: contains Type I, Type II, and original-type straight lines |
| III | Non-working | Front transition surface C₁ (another portion) | Additional non-working area machined by first cutting edge |
The working region (II) in Type III is unique because it incorporates multiple contact line types simultaneously. This mixed pattern may offer advantages in terms of contact ratio and vibration damping, but further experimental validation is needed.
7. Summary and Conclusions
Through my comprehensive analysis using the first limiting curve and phase angle methods, I have successfully delineated the working and non-working regions on the worm gear tooth surface for all three contact types in modified planar double-enveloping worm gear pairs. The key findings are:
- For Type I transmission, the worm gear tooth surface has a central working zone formed by a single smooth envelope, flanked by two non-working zones from the transition surfaces.
- For Type II transmission, the working zone is split into two separate areas, which may lead to uneven load distribution.
- For Type III transmission, the working zone is a hybrid containing Type I, Type II, and original-type straight contact lines, offering a complex but potentially beneficial contact pattern.
The phase angle comparison tables (Tables 2 and similar) provided quantitative evidence for which envelope surface survives the machining process. The classification tables (Tables 3–5) offer clear guidance for optimizing the design of the worm gear tooth surface — for instance, by adjusting hob geometry to enlarge working regions and reduce non-working regions.
My analysis lays a solid foundation for further studies on the performance of modified planar double-enveloping worm gear pairs, including load capacity, efficiency, and lubrication characteristics. Future work should focus on experimental validation of these theoretical findings and explore the practical implications of the Type III contact line pattern for high-torque applications.
I believe that the structured approach presented here will be of great value to engineers and researchers working with worm gear drives, especially when dealing with modified double-enveloping geometries. The clear identification of working and non-working zones allows for targeted modifications to the worm gear tooth surface to improve contact performance and extend service life.