In the field of mechanical transmission, spiral gears have long been valued for their high load-bearing capacity, durability, and excellent running-in performance, making them indispensable in industries such as metallurgy, mining, and petroleum machinery. Traditional spiral gear designs, including single-arc, double-arc, and even quadruple-arc spiral gears, have evolved to distribute contact stresses over multiple points, thereby enhancing meshing strength and transmission stability. Building upon this progression, we propose a novel spiral gear design—the sextuple-arc spiral gear—which features six working arcs per tooth flank. This design aims to further improve load distribution and meshing performance by increasing the number of contact points. In this article, I will detail the tooth profile design, derive key meshing parameters such as multi-point and multi-pair mesh coefficients, and discuss the selection of optimal tooth width for this advanced spiral gear.
The sextuple-arc spiral gear tooth profile is composed of six distinct working arcs, arranged in a tiered configuration along the tooth height. Specifically, three convex arcs are positioned above the pitch line, while three concave arcs are located below it. Each pair of adjacent working arcs is connected by transition arcs to ensure smooth curvature changes. This intricate profile is illustrated in the following figure, which provides a visual representation of the sextuple-arc spiral gear design. The parameters defining this profile include radii for the convex arcs (\(\rho_{a1}\), \(\rho_{a2}\), \(\rho_{a3}\)) and concave arcs (\(\rho_{f1}\), \(\rho_{f2}\), \(\rho_{f3}\)), along with center offsets (\(e_{a1}\), \(e_{a2}\), \(e_{a3}\), \(C_{a1}\), \(C_{a2}\), \(C_{a3}\) for convex arcs, and \(e_{f1}\), \(e_{f2}\), \(e_{f3}\), \(C_{f1}\), \(C_{f2}\), \(C_{f3}\) for concave arcs), process angles (\(\theta_1\), \(\theta_2\)), transition arc radii (\(r_1\), \(r_2\), \(r_3\)), and root fillet radius (\(r_g\)). This multi-arc approach allows for tangential modification between adjacent tooth flanks, facilitating better stress distribution and enhanced meshing characteristics in spiral gear applications.
To analyze the meshing behavior of the sextuple-arc spiral gear, we first calculate the axial distances between the six meshing points on a single tooth. These distances are crucial for determining multi-point mesh coefficients. In the developed plane of the gear’s pitch cylinder, the tooth profile of the basic rack in the normal plane shows six meshing lines: AA, BB, and CC for convex arcs, and DD, EE, and FF for concave arcs, all distributed symmetrically about the pitch line PP. The helical path of meshing points, when unfolded, forms straight lines at an angle equal to the spiral angle \(\beta\). The intersections of these lines with the meshing lines define the meshing points. The maximum axial distance between the six simultaneous contact points on the same tooth, denoted as \(q_{TA}\), is derived from geometric relationships. Specifically, for the convex arcs, the distances between meshing points are calculated as follows:
The axial distance \(q_{TA}\) between the outermost convex and concave meshing points (e.g., J2C and J2F) is given by:
$$ q_{TA} = \frac{2C_{a3} + 0.5\pi m – 0.5j + 2e_{a3}\cot\alpha}{\sin\beta} – 2\left(\rho_{a3} + \frac{e_{a3}}{\sin\alpha}\right) \cos\alpha \sin\beta $$
where \(m\) is the module, \(j\) is the backlash, \(\alpha\) is the pressure angle, and \(\beta\) is the spiral angle. Similarly, the axial distances between other meshing points are:
$$ q_{T3} = \frac{0.5\pi m – 2C_{a1} – 0.5j + 2e_{a1}\cot\alpha}{\sin\beta} – 2\left(\rho_{a1} + \frac{e_{a1}}{\sin\alpha}\right) \cos\alpha \sin\beta $$
$$ 2q_{T2} = \frac{2C_{a2} + 0.5\pi m – 0.5j + 2e_{a2}\cot\alpha}{\sin\beta} – 2\left(\rho_{a2} + \frac{e_{a2}}{\sin\alpha}\right) \cos\alpha \sin\beta $$
and
$$ q_{T1} = \frac{q_{TA} – 2q_{T2} – q_{T3}}{2} $$
These axial distances, \(q_{T1}\), \(q_{T2}\), and \(q_{T3}\), represent the gaps between consecutive meshing points on the same tooth flank. The remaining axial distance, denoted as \(q’_{TA}\), is the complement to the axial pitch \(P_x\), such that \(q’_{TA} = P_x – q_{TA}\). These parameters form the basis for evaluating the meshing characteristics of the sextuple-arc spiral gear.
The meshing process of the sextuple-arc spiral gear involves varying numbers of contact points and tooth pairs engaged simultaneously, depending on the tooth width \(b\). We define multi-point mesh coefficients (\(\varepsilon\)) and multi-pair mesh coefficients (\(\varepsilon_z\)) to quantify this behavior. The tooth width \(b\) can be expressed as \(b = nP_x + \Delta b\), where \(n\) is an integer representing the number of full axial pitches, and \(\Delta b\) is the residual width less than one axial pitch. The meshing coefficients vary with \(\Delta b\), which is divided into eleven intervals based on the axial distances calculated earlier:
- \(\Delta b \leq q’_{TA}\)
- \(q’_{TA} < \Delta b \leq q_{T1}\)
- \(q_{T1} < \Delta b \leq q’_{TA} + q_{T1}\)
- \(q’_{TA} + q_{T1} < \Delta b \leq q_{T1} + q_{T2}\)
- \(q_{T1} + q_{T2} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2}\)
- \(q’_{TA} + q_{T1} + q_{T2} < \Delta b \leq q_{T1} + q_{T2} + q_{T3}\)
- \(q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2} + q_{T3}\)
- \(q’_{TA} + q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q_{T1} + 2q_{T2} + q_{T3}\)
- \(q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + 2q_{T2} + q_{T3}\)
- \(q’_{TA} + q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q_{TA}\)
- \(q_{TA} < \Delta b \leq P_x\)
For each interval, the multi-point mesh coefficients—representing the ratio of the rotation angle during which a specific number of points are in contact to the angle corresponding to one axial pitch—are computed. Similarly, multi-pair mesh coefficients indicate the proportion of time during which a certain number of tooth pairs are engaged. The calculations for these coefficients are summarized in the tables below, which provide formulas for \(\varepsilon_{6nd}\), \(\varepsilon_{(6n+1)d}\), \(\varepsilon_{(6n+2)d}\), \(\varepsilon_{(6n+3)d}\), \(\varepsilon_{(6n+4)d}\), \(\varepsilon_{(6n+5)d}\), and \(\varepsilon_{(6n+6)d}\) for multi-point meshing, and \(\varepsilon_{nz}\), \(\varepsilon_{(n+1)z}\), and \(\varepsilon_{(n+2)z}\) for multi-pair meshing. These tables are essential for designers to optimize the spiral gear performance.
| Condition | \(\varepsilon_{6nd}\) | \(\varepsilon_{(6n+1)d}\) | \(\varepsilon_{(6n+2)d}\) | \(\varepsilon_{(6n+3)d}\) | \(\varepsilon_{(6n+4)d}\) | \(\varepsilon_{(6n+5)d}\) | \(\varepsilon_{(6n+6)d}\) |
|---|---|---|---|---|---|---|---|
| \(\Delta b \leq q’_{TA}\) | \(\frac{P_x – 6\Delta b}{P_x}\) | \(\frac{6\Delta b}{P_x}\) | — | — | — | — | — |
| \(q’_{TA} < \Delta b \leq q_{T1}\) | \(\frac{q_{TA} – 5\Delta b}{P_x}\) | \(\frac{2q’_{TA} + 4\Delta b}{P_x}\) | \(\frac{\Delta b – q’_{TA}}{P_x}\) | — | — | — | — |
| \(q_{T1} < \Delta b \leq q’_{TA} + q_{T1}\) | \(\frac{2P_x – 6\Delta b}{P_x}\) | \(\frac{6\Delta b – P_x}{P_x}\) | \(\frac{2q_{T1} – 2q_{T2} – q_{T3} + 3q’_{TA} + 2\Delta b}{P_x}\) | \(\frac{2\Delta b – 2q_{T1} – 2q’_{TA}}{P_x}\) | — | — | — |
| \(q’_{TA} + q_{T1} < \Delta b \leq q_{T1} + q_{T2}\) | \(\frac{3P_x – 6\Delta b}{P_x}\) | \(\frac{2q_{T1} + 4q_{T2} + 2q_{T3} – 4\Delta b}{P_x}\) | \(\frac{6\Delta b – 2P_x}{P_x}\) | \(\frac{4q_{T1} – 2q_{T3} + 4q’_{TA}}{P_x}\) | \(\frac{3\Delta b – 4q_{T1} – 2q_{T2} – 3q’_{TA}}{P_x}\) | — | — |
| \(q_{T1} + q_{T2} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2}\) | \(\frac{2q_{T1} + 4q_{T2} + 3q_{T3} – 3\Delta b}{P_x}\) | \(\frac{6\Delta b – 3P_x}{P_x}\) | \(\frac{4P_x – 6\Delta b}{P_x}\) | \(\frac{2q_{T1} + 4q_{T2} + 2q_{T3} – 2\Delta b}{P_x}\) | \(\frac{6q_{T1} + 2q_{T2} + q_{T3} – 2\Delta b + 5q’_{TA}}{P_x}\) | \(\frac{4\Delta b – 6q_{T1} – 4q_{T2} – 2q_{T3} – 4q’_{TA}}{P_x}\) | — |
| \(q’_{TA} + q_{T1} + q_{T2} < \Delta b \leq q_{T1} + q_{T2} + q_{T3}\) | \(\frac{5P_x – 6\Delta b}{P_x}\) | \(\frac{q_{TA} – \Delta b}{P_x}\) | \(\frac{6\Delta b – 4P_x}{P_x}\) | \(\frac{4P_x + 2q’_{TA} – 4\Delta b}{P_x}\) | \(\frac{6P_x – 6\Delta b}{P_x}\) | \(\frac{5\Delta b – 4P_x – q’_{TA}}{P_x}\) | \(\frac{6\Delta b – 5P_x}{P_x}\) |
Note: Dashes indicate that the coefficient is zero or not applicable in that interval. The sum of all multi-point mesh coefficients for any given condition equals 1, validating the consistency of the calculations.
| Condition | \(\varepsilon_{nz}\) | \(\varepsilon_{(n+1)z}\) | \(\varepsilon_{(n+2)z}\) |
|---|---|---|---|
| \(\Delta b \leq q’_{TA}\) | \(\frac{q’_{TA} – \Delta b}{P_x}\) | \(\frac{q_{TA} + \Delta b}{P_x}\) | — |
| \(\Delta b > q’_{TA}\) | — | \(\frac{P_x + q’_{TA} – \Delta b}{P_x}\) | \(\frac{\Delta b – q’_{TA}}{P_x}\) |
The selection of tooth width \(b\) is critical for maximizing the meshing benefits of the sextuple-arc spiral gear. Based on the derived coefficients, we can determine the minimum tooth width \(b_{\text{min}}\) required to achieve a desired number of meshing points and tooth pairs. The table below outlines the relationship between \(\Delta b\) ranges and the corresponding minimal meshing conditions, guiding designers in optimizing spiral gear dimensions for enhanced performance.
| Minimal Meshing Condition | Recommended \(\Delta b\) Range |
|---|---|
| \(n\) pairs of teeth, \(6n\) points | \(\Delta b \leq q’_{TA}\) |
| \(n+1\) pairs, \(6n\) points | \(q’_{TA} < \Delta b \leq q_{T1}\) |
| \(n+1\) pairs, \(6n+1\) points | \(q_{T1} < \Delta b \leq q_{T1} + q_{T2}\) |
| \(n+1\) pairs, \(6n+2\) points | \(q_{T1} + q_{T2} < \Delta b \leq q_{T1} + q_{T2} + q_{T3}\) |
| \(n+1\) pairs, \(6n+3\) points | \(q’_{TA} + q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q_{TA} – q_{T1}\) |
| \(n+1\) pairs, \(6n+4\) points | \(q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q_{TA}\) |
| \(n+1\) pairs, \(6n+5\) points | \(q_{TA} < \Delta b \leq P_x\) |
In conclusion, the sextuple-arc spiral gear represents a significant advancement in gear design, theoretically offering higher load-bearing capacity and smoother transmission compared to earlier spiral gear variants. By distributing contact stresses over six points per tooth flank, this spiral gear design reduces stress concentration and improves durability. The derived meshing characteristics, including multi-point and multi-pair coefficients, provide a comprehensive framework for evaluating and optimizing spiral gear performance. Moreover, the relationship between tooth width and meshing conditions allows for tailored designs that maximize the number of contact points and engaged tooth pairs, thereby enhancing the overall efficiency and economic benefits of spiral gear systems. Future work could involve experimental validation and finite element analysis to further refine the sextuple-arc spiral gear for practical applications in heavy-duty machinery.
Throughout this discussion, the term “spiral gear” has been emphasized to highlight the core focus of this research. The sextuple-arc spiral gear builds upon the legacy of spiral gears, pushing the boundaries of meshing theory and mechanical design. As spiral gear technology continues to evolve, innovations like the sextuple-arc profile will play a pivotal role in meeting the demanding requirements of modern industrial applications.