In my research on modified planar double-enveloping worm gears, I have systematically investigated the complex tooth surface structures of the worm gear (the driven member) under different contact line types. Traditional studies often classify the contact lines into only two types: Type I and Type II. However, my analysis reveals that a third type, Type III, also exists in modified transmissions. This third type introduces a significantly more complicated morphology on the worm gear tooth surface, characterized by the simultaneous presence of both Type I and Type II contact patterns along with additional features. Understanding the actual structure of the worm gear tooth surface—distinguishing between working zones and non-working zones—is essential for optimizing load capacity and lubrication conditions. In this article, I present a comprehensive analysis using the first limiting curve (also called the first boundary curve) and a phase-angle method. I decompose the worm gear tooth surface into distinct regions and clarify the contributions of each contacting family and the transition surfaces generated by the tool worm’s cutting edges.
I begin by deriving the governing conditions for the first limiting curve in the second enveloping process. For a modified planar double-enveloping worm gear pair, the first limiting curve must satisfy the following system:
$$
\begin{cases}
\Psi^* = \mathbf{n} \cdot \mathbf{V}^{(12)} = 0, \\
\Phi^* = \mathbf{n} \cdot \mathbf{q} + \mathbf{p} \cdot \mathbf{V}^{(12)} = 0.
\end{cases}
$$
Here, \(\mathbf{n}\) is the unit normal of the worm gear surface, \(\mathbf{V}^{(12)}\) is the relative velocity between the worm and the worm gear, \(\mathbf{q}\) and \(\mathbf{p}\) are auxiliary vectors related to the meshing kinematics. By expanding these conditions with the geometric parameters of the modified planar tool gear, I obtain the explicit equation for the first limiting curve. The coordinate transformations involve three systems: the tool gear coordinate system (\(S_0\)), the worm coordinate system (\(S_1\)), and the worm gear coordinate system (\(S_2\)). The key parameters include the first-enveloping transmission ratio \(i_{10}\), the second-enveloping transmission ratio \(i_{12}\), the first center distance \(a_0\), the second center distance \(a\), the inclination angle \(\beta\) of the tool gear’s planar surface, and the angular parameters \(\phi_0\), \(\phi_1\), \(\phi_1^*\). The first limiting curve in the worm gear tooth surface can be expressed parametrically as:
$$
\begin{aligned}
& u \left[ \left( \cos\theta – \frac{i_{12}}{i_{10}} \right) \cos\beta + \left( \frac{\sin\beta}{\sin\phi_0} – i_{01} \cos\beta \cot\phi_0 \right) \sin\theta + \cdots \right. \\
& \quad + a_0 \left[ \sin\beta \cot\phi_0 \sin\theta + \left( \cos\theta – \frac{a}{a_0} \right) \cos\beta \cos\phi_0 \right] = 0, \\
& u = \frac{b_2′ c_2 – b_2 c_2′}{a_2′ b_2 – a_2 b_2′}, \\
& x_2 = (x_1 \cos\phi_1^* – y_1 \sin\phi_1^* – a) \cos\phi_2^* + z_1 \sin\phi_2^*, \\
& y_2 = -(x_1 \cos\phi_1^* – y_1 \sin\phi_1^* – a) \sin\phi_2^* + z_1 \cos\phi_2^*, \\
& z_2 = -x_1 \sin\phi_1^* – y_1 \cos\phi_1^*, \\
& x_1 = (x_0 \cos\phi_0 – y_0 \sin\phi_0 + a_0) \cos\phi_1 – z_0 \sin\phi_1, \\
& y_1 = -(x_0 \cos\phi_0 – y_0 \sin\phi_0 + a_0) \sin\phi_1 – z_0 \cos\phi_1, \\
& z_1 = x_0 \sin\phi_0 + y_0 \cos\phi_0, \\
& x_0 = u, \; y_0 = v \sin\beta – r_b, \; z_0 = v \cos\beta,
\end{aligned}
$$
where \(\theta = \phi_1^* – \phi_1\), and \(r_b\) is the base circle radius of the tool gear. The parameters \(v\) and \(u\) are surface coordinates on the tool gear plane. The first limiting curve separates regions of the worm gear tooth surface that are generated by different families of contact lines.
Morphology of the First Limiting Curve in the Second Enveloping Process
To visualize the first limiting curve for each contact type, I implemented a numerical optimization procedure. The algorithm follows these steps:
- Define the range of \(\phi_1^*\) for the worm gear rotation angle.
- Determine an initial sequence of \(\phi_0\) (tool gear rotation angle) as optimization variables.
- For each pair \((\phi_1^*, \phi_0)\), solve one of the first-limiting-curve equations to obtain \(u\).
- Use the other equation as an objective function to optimize \(\phi_0\) (or a related variable).
- Substitute the optimized values back to compute the coordinates of the first limiting curve in the worm gear tooth surface.
Through this process, I obtained distinct first limiting curves for each contact type. For Type I worm gears, the contact lines on the worm gear tooth surface appear as “V” shapes and “herringbone” shapes. The first limiting curve related to the herringbone family divides each contact line into two segments. For Type II worm gears, the contact lines are all of the herringbone type, and the first limiting curve separates a portion that is cut away during manufacturing from the portion that remains. For Type III worm gears, which I discovered, the tooth surface contains both Type I and Type II contact lines. Consequently, there are two branches of the first limiting curve: one derived from the Type I portion and another from the Type II portion. The figures (not reproduced here) show that the Type III case yields two distinct first limiting curves, labeled Γ₁ and Γ₂ in my original notation.
Principle of Worm Gear Tooth Surface Structure Analysis
The worm gear tooth surface is entirely generated by the enveloping motion of the tool worm (or grinding worm) during the second enveloping process. The working region of the worm gear is the portion where actual contact lines appear and transmit load. The non-working regions are formed by the front and rear transition surfaces, which are generated by the first and last cutting edges of the tool worm, respectively. To distinguish which enveloping surface actually remains on the worm gear tooth surface after manufacturing, I used a phase-angle analysis method.
Consider a cross-section of the worm gear perpendicular to its axis at a given axial coordinate \(z_2 = z_0\). In this plane, the intersection of the worm gear tooth profile with a circle of radius \(R\) (measured from the worm gear axis) yields two angular positions \(\theta_1\) and \(\theta_2\) corresponding to two different candidate enveloping surfaces \(\Sigma^{(1)}\) and \(\Sigma^{(2)}\). If \(\theta_1 > \theta_2\), then surface \(\Sigma^{(1)}\) is closer to the worm gear tooth solid body than \(\Sigma^{(2)}\); therefore, during the cutting process, the tool worm will remove \(\Sigma^{(2)}\) and leave \(\Sigma^{(1)}\). Conversely, if \(\theta_1 < \theta_2\), then \(\Sigma^{(2)}\) remains. By evaluating these phase angles over multiple radii and multiple axial sections, I determined which enveloping surface constitutes the actual worm gear tooth flank.
The boundary surfaces are defined by the tool worm’s first cutting edge (generating the front transition surface \(C_1\)) and last cutting edge (generating the rear transition surface \(C_2\)). The effective working length of the tool worm helix determines these edges. In my analysis, I approximated these edges by the intersection of the worm helix surface with two planes perpendicular to the worm axis at distances corresponding to the beginning and end of the active flank.
Tooth Surface Structure for Type I Contact
In Type I transmission, the worm gear tooth surface consists of two families of contact lines: one family forms a “V” pattern (denoted as \(\Sigma^{(1)}_2\)) and the other forms a “herringbone” pattern (denoted as \(\Sigma^{(2)}_2\)). The first limiting curve \(\Gamma^{(2)}_1\) lies only on the herringbone family. The herringbone family is further split by the singular points (first limiting points) into two sub-families: \(\Sigma^{(21)}_2\) and \(\Sigma^{(22)}_2\). Through the sign analysis of the first limiting function, I found that \(\Sigma^{(22)}_2\) has negative values, meaning it is convex toward the worm and will be cut away during manufacturing. The remaining portions are \(\Sigma^{(1)}_2\) (V-shaped) and \(\Sigma^{(21)}_2\) (a part of the herringbone). These two surfaces overlap in the right-hand region of the worm gear tooth. Using the phase-angle method, I compared their relative proximity to the worm gear tooth body. Table I shows a sample of the calculated phase differences.
| \(z_2\) (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 |
All values of \(\Delta\theta\) are positive, indicating that \(\Sigma^{(1)}_2\) is consistently closer to the worm gear tooth body. Therefore, \(\Sigma^{(21)}_2\) is removed during manufacturing, and only \(\Sigma^{(1)}_2\) remains as the working surface. The front transition surface \(C_1\) and the rear transition surface \(C_2\) also appear. The resulting worm gear tooth surface for Type I contact is divided into three zones:
- Zone I (non-working): generated by the first cutting edge (front transition surface).
- Zone II (working): the actual contact region formed by the V-shaped contact lines \(\Sigma^{(1)}_2\). This region features double-line contact and good load capacity.
- Zone III (non-working): generated by the last cutting edge (rear transition surface).
Tooth Surface Structure for Type II Contact
In Type II transmission, all contact lines belong to the herringbone pattern. The first limiting curve splits this family into \(\Sigma^{(21)}_2\) and \(\Sigma^{(22)}_2\). As before, \(\Sigma^{(22)}_2\) is cut away. The remaining surface \(\Sigma^{(21)}_2\) coexists with another family \(\Sigma^{(1)}_2\)? Actually, in Type II, there is no separate V-shaped family; instead, the herringbone family itself has two branches. Further analysis shows that the contact lines in the worm gear tooth surface for Type II are solely from \(\Sigma^{(21)}_2\) and a portion of \(\Sigma^{(1)}_2\) (if present) or from the same family after cutting. The phase-angle analysis confirms that two enveloping surfaces overlap: \(\Sigma^{(1)}_2\) (from one side of the herringbone) and \(\Sigma^{(21)}_2\) (from the other side). Again, I compared their phase angles and found that \(\Sigma^{(1)}_2\) is closer to the worm gear tooth body, so \(\Sigma^{(21)}_2\) is partially removed. However, because the herringbone pattern extends across the tooth, the final tooth surface consists of multiple zones. Specifically, for Type II worm gears, I identified five distinct zones:
- Zone I (non-working): front transition surface \(C_1\).
- Zone II (working): formed by \(\Sigma^{(1)}_2\) (one branch of the herringbone).
- Zone III (non-working): rear transition surface \(C_2\) in some regions.
- Zone IV (working): formed by \(\Sigma^{(21)}_2\) (the other branch that remains).
- Zone V (non-working): another part of the front transition surface.
The working zones (II and IV) are separated by a non-working strip, which is undesirable for continuous load transmission. The design should be optimized to maximize the working area and minimize the non-working gaps.
Tooth Surface Structure for Type III Contact
Type III worm gears exhibit the most complex behavior. As shown in the distribution of the first limiting curves, there are two distinct branches: one from the Type I portion (\(\Gamma_{21}\)) and one from the Type II portion. The first branch divides the corresponding contact line family \(\Sigma^{(11)}_2\) into \(\Sigma^{(121)}_2\) and \(\Sigma^{(122)}_2\); the second branch divides \(\Sigma^{(21)}_2\) into \(\Sigma^{(221)}_2\) and \(\Sigma^{(222)}_2\). Using the sign of the first limiting function, I determined that \(\Sigma^{(122)}_2\), \(\Sigma^{(121)}_2\), and \(\Sigma^{(222)}_2\) all have negative values (convex toward the worm), and thus they are cut away. Only \(\Sigma^{(221)}_2\) (a portion of the Type II family) remains. Furthermore, the phase-angle analysis between this remaining surface and any other potential overlaying surface shows that \(\Sigma^{(221)}_2\) is retained. The front transition surface \(C_1\) intersects the remaining contact lines at a specific line (the outermost contact line in the figures), and the rear transition surface \(C_2\) does not intersect the working region within the tooth boundaries. Therefore, the worm gear tooth surface for Type III contact is partitioned into three zones:
- Zone I (non-working): formed by the first cutting edge (front transition surface).
- Zone II (working): a composite region containing both Type I and Type II contact line segments, as well as some “original type” straight contact lines. This zone is the largest working area among the three types.
- Zone III (non-working): another part of the front transition surface (or possibly rear, depending on the specific geometry).
It is noteworthy that in Type III, the working zone is continuous and contains multiple contact line patterns, which may lead to favorable stress distribution and oil film formation. However, further experimental verification is needed to confirm its performance advantages.
Summary and Conclusion
My analysis of modified planar double-enveloping worm gears reveals that the tooth surface structure of the worm gear varies significantly depending on the contact line type. Using the first limiting curve and the phase-angle method, I have successfully delineated the working and non-working zones for each case. The key findings are summarized in Table II.
| Contact Type | Number of Zones | Working Zone Composition | Non-Working Zones | Remarks |
|---|---|---|---|---|
| Type I | 3 | Single V-shaped contact line family (\(\Sigma^{(1)}_2\)) | Front and rear transition surfaces | Continuous working zone; double-line contact |
| Type II | 5 | Two separate herringbone branches (\(\Sigma^{(1)}_2\) and \(\Sigma^{(21)}_2\)) | Front, rear, and an intermediate non-working strip | Discontinuous working zone; less desirable |
| Type III | 3 | Composite of Type I, Type II, and straight lines (\(\Sigma^{(221)}_2\)) | Front transition surface only | Continuous working zone with multiple patterns; promising performance |
This research provides a theoretical foundation for optimizing the design of worm gears. By identifying the regions that are cut away or that remain, engineers can adjust the modification parameters (such as the inclination angle \(\beta\) and the center distances) to enlarge the working area and reduce non-working gaps. The Type III worm gears, in particular, offer a unique combination of contact patterns that may enhance load capacity and lubrication. Future work will include experimental validation and dynamic analysis of these worm gears under operating conditions.
In conclusion, the tooth surface structure of modified planar double-enveloping worm gears is far more intricate than previously assumed. My work demonstrates that a rigorous analytical approach—combining the first limiting curve with phase-angle comparisons—is essential for correctly interpreting the actual flank geometry. These insights will contribute to the development of more efficient and reliable worm gear transmissions.