I propose a new class of screw gears, which I call six-arc screw gears. These gears are point-meshing screw gears and are intended to increase load capacity, improve running-in behavior, and distribute contact stress over a larger number of meshing points. In conventional single-arc screw gears, a single tooth pair usually has only one meshing point within one axial pitch. In double-arc screw gears, the load is shared by two meshing points, while in four-arc screw gears, the load is shared by four meshing points. Following this logic, I design a six-arc tooth profile in which six working arcs are arranged on one side of the tooth, so that six meshing points can act simultaneously within a single axial pitch. This arrangement is expected to further increase the load capacity and improve the smoothness of transmission of screw gears.
I begin with the basic tooth profile of the six-arc screw gears. The working tooth profile consists of six circular arcs. Three convex arcs are located above the pitch line, and three concave arcs are located below the pitch line. Adjacent working arcs are connected by transition arcs. The profile is of the stepped type, and tangential modification is applied between neighboring tooth profiles along the tooth height direction. This design allows the six working arcs to engage in a sequential and distributed manner, which is the fundamental reason for the multi-point meshing behavior of these screw gears.
Basic Tooth Profile Parameters
I define the main geometric parameters of the six-arc screw gears as follows. The convex arc radii are $$\rho_{a1}$$, $$\rho_{a2}$$, and $$\rho_{a3}$$. The concave arc radii are $$\rho_{f1}$$, $$\rho_{f2}$$, and $$\rho_{f3}$$. The center offset quantities for the convex arcs are $$e_{a1}, e_{a2}, e_{a3}$$ and $$C_{a1}, C_{a2}, C_{a3}$$. The center offset quantities for the concave arcs are $$e_{f1}, e_{f2}, e_{f3}$$ and $$C_{f1}, C_{f2}, C_{f3}$$. The process angles for the convex arcs are $$\theta_1$$ and $$\theta_2$$. The transition arc radii are $$r_1, r_2, r_3$$, and the root arc radius is $$r_g$$. These parameters fully determine the basic rack profile and the subsequent meshing characteristics of the six-arc screw gears.
| Parameter | Description |
|---|---|
| $$\rho_{a1}, \rho_{a2}, \rho_{a3}$$ | Radii of the three convex working arcs above the pitch line |
| $$\rho_{f1}, \rho_{f2}, \rho_{f3}$$ | Radii of the three concave working arcs below the pitch line |
| $$e_{a1}, e_{a2}, e_{a3}$$ | Center offset quantities of the convex arcs |
| $$C_{a1}, C_{a2}, C_{a3}$$ | Additional center offset quantities of the convex arcs |
| $$e_{f1}, e_{f2}, e_{f3}$$ | Center offset quantities of the concave arcs |
| $$C_{f1}, C_{f2}, C_{f3}$$ | Additional center offset quantities of the concave arcs |
| $$\theta_1, \theta_2$$ | Process angles of the convex arcs |
| $$r_1, r_2, r_3$$ | Transition arc radii between adjacent working arcs |
| $$r_g$$ | Root arc radius |
Axial Pitch Calculations
To calculate the meshing characteristics of the six-arc screw gears, I first determine the axial distances between the six meshing points on the same tooth. I consider the basic rack profile in the normal plane and the developed plane of the pitch cylinder. The pitch line is denoted by $$PP$$. The three meshing lines of the convex arcs are $$AA$$, $$BB$$, and $$CC$$, and the three meshing lines of the concave arcs are $$DD$$, $$EE$$, and $$FF$$. These meshing lines are distributed on both sides of the pitch line. When the gear meshes, the contact points lie on cylindrical helices. After developing the pitch cylinder, these helices become straight lines that form the helix angle $$\beta$$ with the axial direction. The intersection of a contact line and a meshing line on the pitch plane is a meshing point of the screw gears.
The maximum axial distance between six simultaneous contact points on the same tooth is denoted by $$q_{TA}$$. For the three convex meshing points, the axial distances between adjacent points are $$q_{T1}$$ and $$q_{T2}$$. The axial distance between the convex and concave meshing points is $$q_{T3}$$. The axial distance between the contact point $$J_{1F}$$ and the next contact point $$J_{2C}$$ is denoted by $$q’_{TA}$$, and it satisfies
$$q’_{TA} = P_x – q_{TA}$$
From the geometric relations of the developed profile, I obtain
$$O_C O_F = \frac{2C_{a3} + 0.5\pi m – 0.5j + 2e_{a3}\cot\alpha}{\sin\beta}$$
$$O_B O_E = \frac{2C_{a2} + 0.5\pi m – 0.5j + 2e_{a2}\cot\alpha}{\sin\beta}$$
$$O_A O_D = \frac{0.5\pi m – 2C_{a1} – 0.5j + 2e_{a1}\cot\alpha}{\sin\beta}$$
Here, $$m$$ is the module, $$j$$ is the backlash, and $$\alpha$$ is the pressure angle. I also define the axial correction terms
$$\Delta_{b3} = \left(\rho_{a3} + \frac{e_{a3}}{\sin\alpha}\right)\cos\alpha\sin\beta$$
$$\Delta_{b2} = \left(\rho_{a2} + \frac{e_{a2}}{\sin\alpha}\right)\cos\alpha\sin\beta$$
$$\Delta_{b1} = \left(\rho_{a1} + \frac{e_{a1}}{\sin\alpha}\right)\cos\alpha\sin\beta$$
Therefore, the maximum axial distance between the six meshing points is
$$q_{TA} = \frac{2C_{a3} + 0.5\pi m – 0.5j + 2e_{a3}\cot\alpha}{\sin\beta} – 2\Delta_{b3}$$
The axial distance between the first concave meshing point and the adjacent convex meshing point is
$$q_{T3} = \frac{0.5\pi m – 2C_{a1} – 0.5j + 2e_{a1}\cot\alpha}{\sin\beta} – 2\Delta_{b1}$$
The distance between the second and third convex meshing points is obtained from
$$2q_{T2} = \frac{2C_{a2} + 0.5\pi m – 0.5j + 2e_{a2}\cot\alpha}{\sin\beta} – 2\Delta_{b2}$$
Finally, the remaining axial distance between the first and second convex meshing points is
$$q_{T1} = \frac{q_{TA} – 2q_{T2} – q_{T3}}{2}$$
These axial distances are the foundation for the interval division and for the calculation of the multi-point mesh coefficient and the multi-pairs tooth mesh coefficient of the six-arc screw gears.
Meshing Interval Division
During operation, the number of meshing points of the six-arc screw gears changes with the rotation of the gear. The variation law depends on the tooth width $$b$$. I write the tooth width as
$$b = nP_x + \Delta b$$
where $$n$$ is an integer representing the number of complete axial pitches contained in the tooth width, and $$\Delta b$$ is the remaining width that is less than one axial pitch. The multi-point mesh coefficient and the multi-pairs tooth mesh coefficient of the six-arc screw gears depend on the range in which $$\Delta b$$ lies. I divide $$\Delta b$$ into the following eleven intervals:
| Interval number | Range of $$\Delta b$$ |
|---|---|
| 1 | $$\Delta b \leq q’_{TA}$$ |
| 2 | $$q’_{TA} < \Delta b \leq q_{T1}$$ |
| 3 | $$q_{T1} < \Delta b \leq q’_{TA} + q_{T1}$$ |
| 4 | $$q’_{TA} + q_{T1} < \Delta b \leq q_{T1} + q_{T2}$$ |
| 5 | $$q_{T1} + q_{T2} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2}$$ |
| 6 | $$q’_{TA} + q_{T1} + q_{T2} < \Delta b \leq q_{T1} + q_{T2} + q_{T3}$$ |
| 7 | $$q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2} + q_{T3}$$ |
| 8 | $$q’_{TA} + q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q_{T1} + 2q_{T2} + q_{T3}$$ |
| 9 | $$q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + 2q_{T2} + q_{T3}$$ |
| 10 | $$q’_{TA} + q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q_{TA}$$ |
| 11 | $$q_{TA} < \Delta b \leq P_x$$ |
These intervals determine how many meshing points and how many tooth pairs are active at each instant. For the six-arc screw gears, the meshing point number can vary from six to twelve. The corresponding multi-point mesh coefficients are denoted by $$\varepsilon_{6nd}$$, $$\varepsilon_{(6n+1)d}$$, $$\varepsilon_{(6n+2)d}$$, $$\varepsilon_{(6n+3)d}$$, $$\varepsilon_{(6n+4)d}$$, $$\varepsilon_{(6n+5)d}$$, and $$\varepsilon_{(6n+6)d}$$. The subscript $$n$$ represents the integer number of axial pitches contained in the tooth width.
Meshing Characteristic Calculation
I now calculate the meshing characteristics of the six-arc screw gears for the first case, namely $$\Delta b \leq q’_{TA}$$. In this case, the lower end face of the gear is located on the line $$1-1$$. During meshing, the number of contact points changes as follows.
In the first interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: the six points of tooth 2 and point 1 of tooth 3. In the second interval, the axial movement distance is $$q’_{TA} – \Delta b$$, and there are six contact points: points 1 to 6 of tooth 2. In the third interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: point 6 of tooth 1 and points 1 to 6 of tooth 2. In the fourth interval, the axial movement distance is $$q_{T1} – \Delta b$$, and there are six contact points: point 6 of tooth 1 and points 1 to 5 of tooth 2. In the fifth interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: points 5 and 6 of tooth 1 and points 1 to 5 of tooth 2. In the sixth interval, the axial movement distance is $$q_{T2} – \Delta b$$, and there are six contact points: points 5 and 6 of tooth 1 and points 1 to 4 of tooth 2. In the seventh interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: points 4 to 6 of tooth 1 and points 1 to 4 of tooth 2. In the eighth interval, the axial movement distance is $$q_{T3} – \Delta b$$, and there are six contact points: points 4 to 6 of tooth 1 and points 1 to 3 of tooth 2. In the ninth interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: points 3 to 6 of tooth 1 and points 1 to 3 of tooth 2. In the tenth interval, the axial movement distance is $$q_{T2} – \Delta b$$, and there are six contact points: points 3 to 6 of tooth 1 and points 1 and 2 of tooth 2. In the eleventh interval, the axial movement distance is $$\Delta b$$, and there are seven contact points: points 2 to 6 of tooth 1 and points 1 and 2 of tooth 2. In the twelfth interval, the axial movement distance is $$q_{T1} – \Delta b$$, and there are six contact points: points 2 to 6 of tooth 1 and point 1 of tooth 2.
From this sequence, I obtain the six-point mesh coefficient and the seven-point mesh coefficient as follows:
$$\varepsilon_{6nd} = \frac{P_x – 6\Delta b}{P_x} = 1 – \frac{6\Delta b}{P_x}$$
$$\varepsilon_{(6n+1)d} = \frac{6\Delta b}{P_x}$$
For the tooth-pair meshing coefficients, I obtain
$$\varepsilon_{nz} = \frac{q’_{TA} – \Delta b}{P_x}$$
$$\varepsilon_{(n+1)z} = \frac{q_{TA} + \Delta b}{P_x}$$
I repeat this calculation for all eleven intervals and summarize the results in the following tables. The multi-point mesh coefficients are given in Table 1 and Table 2, while the multi-pairs tooth mesh coefficients are given in Table 3.
Table 1. Multi-point mesh coefficients for the first six intervals.
| Interval | $$\Delta b \leq q’_{TA}$$ | $$q’_{TA} < \Delta b \leq q_{T1}$$ | $$q_{T1} < \Delta b \leq q’_{TA} + q_{T1}$$ | $$q’_{TA} + q_{T1} < \Delta b \leq q_{T1} + q_{T2}$$ | $$q_{T1} + q_{T2} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2}$$ | $$q’_{TA} + q_{T1} + q_{T2} < \Delta b \leq q_{T1} + q_{T2} + q_{T3}$$ |
|---|---|---|---|---|---|---|
| $$\varepsilon_{6nd}$$ | $$\frac{P_x – 6\Delta b}{P_x}$$ | $$\frac{q_{TA} – 5\Delta b}{P_x}$$ | 0 | 0 | 0 | 0 |
| $$\varepsilon_{(6n+1)d}$$ | $$\frac{6\Delta b}{P_x}$$ | $$\frac{2q’_{TA} + 4\Delta b}{P_x}$$ | $$\frac{2P_x – 6\Delta b}{P_x}$$ | $$\frac{2q_{T1} + 4q_{T2} + 2q_{T3} – 4\Delta b}{P_x}$$ | 0 | 0 |
| $$\varepsilon_{(6n+2)d}$$ | 0 | $$\frac{\Delta b – q’_{TA}}{P_x}$$ | $$\frac{6\Delta b – P_x}{P_x}$$ | $$\frac{2q_{T1} – 2q_{T2} – q_{T3} + 3q’_{TA} + 2\Delta b}{P_x}$$ | $$\frac{3P_x – 6\Delta b}{P_x}$$ | $$\frac{2q_{T1} + 4q_{T2} + 3q_{T3} – 3\Delta b}{P_x}$$ |
| $$\varepsilon_{(6n+3)d}$$ | 0 | 0 | $$\frac{2\Delta b – 2q_{T1} – 2q’_{TA}}{P_x}$$ | $$\frac{6\Delta b – 2P_x}{P_x}$$ | $$\frac{4q_{T1} – 2q_{T3} + 4q’_{TA}}{P_x}$$ | 0 |
| $$\varepsilon_{(6n+4)d}$$ | 0 | 0 | 0 | $$\frac{3\Delta b – 4q_{T1} – 2q_{T2} – 3q’_{TA}}{P_x}$$ | 0 | 0 |
| $$\varepsilon_{(6n+5)d}$$ | 0 | 0 | 0 | 0 | 0 | 0 |
| $$\varepsilon_{(6n+6)d}$$ | 0 | 0 | 0 | 0 | 0 | 0 |
Table 2. Multi-point mesh coefficients for the remaining five intervals.
| Interval | $$q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + q_{T2} + q_{T3}$$ | $$q’_{TA} + q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q_{T1} + 2q_{T2} + q_{T3}$$ | $$q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q’_{TA} + q_{T1} + 2q_{T2} + q_{T3}$$ | $$q’_{TA} + q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q_{TA}$$ | $$q_{TA} < \Delta b \leq P_x$$ |
|---|---|---|---|---|---|
| $$\varepsilon_{6nd}$$ | 0 | 0 | 0 | 0 | 0 |
| $$\varepsilon_{(6n+1)d}$$ | 0 | 0 | 0 | 0 | 0 |
| $$\varepsilon_{(6n+2)d}$$ | 0 | 0 | 0 | 0 | 0 |
| $$\varepsilon_{(6n+3)d}$$ | $$\frac{4P_x – 6\Delta b}{P_x}$$ | $$\frac{2q_{T1} + 4q_{T2} + 2q_{T3} – 2\Delta b}{P_x}$$ | 0 | 0 | 0 |
| $$\varepsilon_{(6n+4)d}$$ | $$\frac{6\Delta b – 3P_x}{P_x}$$ | $$\frac{6q_{T1} + 2q_{T2} + q_{T3} – 2\Delta b + 5q’_{TA}}{P_x}$$ | $$\frac{5P_x – 6\Delta b}{P_x}$$ | $$\frac{q_{TA} – \Delta b}{P_x}$$ | 0 |
| $$\varepsilon_{(6n+5)d}$$ | 0 | $$\frac{4\Delta b – 6q_{T1} – 4q_{T2} – 2q_{T3} – 4q’_{TA}}{P_x}$$ | $$\frac{6\Delta b – 4P_x}{P_x}$$ | $$\frac{4P_x + 2q’_{TA} – 4\Delta b}{P_x}$$ | $$\frac{6P_x – 6\Delta b}{P_x}$$ |
| $$\varepsilon_{(6n+6)d}$$ | 0 | 0 | $$\frac{5\Delta b – 4P_x – q’_{TA}}{P_x}$$ | $$\frac{6\Delta b – 5P_x}{P_x}$$ | 0 |
For each interval, I observe that the sum of all multi-point mesh coefficients is equal to 1. This confirms that the coefficients represent the fractions of the total rotation angle occupied by the different numbers of meshing points. This result is essential for understanding the meshing behavior of the six-arc screw gears.
Table 3. Multi-pairs tooth mesh coefficients.
| Parameter | $$\Delta b < q’_{TA}$$ | $$\Delta b > q’_{TA}$$ |
|---|---|---|
| $$\varepsilon_{nz}$$ | $$\frac{q’_{TA} – \Delta b}{P_x}$$ | 0 |
| $$\varepsilon_{(n+1)z}$$ | $$\frac{q_{TA} + \Delta b}{P_x}$$ | $$\frac{P_x + q’_{TA} – \Delta b}{P_x}$$ |
| $$\varepsilon_{(n+2)z}$$ | 0 | $$\frac{\Delta b – q’_{TA}}{P_x}$$ |
Again, the sum of the multi-pairs tooth mesh coefficients is equal to 1 in each case. This is a direct consequence of the fact that, during one angular pitch, the gear must pass through all possible tooth-pair engagement states. The multi-pairs tooth mesh coefficients describe how many tooth pairs share the load at any instant, and they are important for evaluating the load distribution in six-arc screw gears.
Minimum Tooth Width Determination
The multi-point mesh coefficients and the multi-pairs tooth mesh coefficients show that the tooth width directly affects the number of meshing points and the number of meshing tooth pairs. To select a reasonable tooth width for six-arc screw gears, I must determine the relationship between the desired minimum number of meshing pairs, the desired minimum number of meshing points, and the remaining width $$\Delta b$$. I summarize this relationship in Table 4.
Table 4. Selection ranges for the tooth width $$b$$.
| Minimum meshing pairs and points | Selection range for $$b$$ |
|---|---|
| $$n$$ pairs, $$6n$$ 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 | $$q_{TA} < \Delta b \leq P_x$$ |
From Table 4, I can choose the tooth width of six-arc screw gears according to the required load capacity and transmission smoothness. If a larger number of meshing points is desired, I should select $$\Delta b$$ in a range that provides more simultaneous contact points. If a larger number of meshing tooth pairs is desired, I should select $$\Delta b$$ in a range that provides more load-sharing pairs. Because the tooth width also affects the axial size and the manufacturing cost, the final choice should balance performance and economy.
Discussion
The six-arc tooth profile significantly increases the number of meshing points compared with single-arc, double-arc, and four-arc screw gears. In single-arc screw gears, the load is concentrated on one point per tooth within one axial pitch. In double-arc screw gears, the load is distributed over two points. In four-arc screw gears, the load is distributed over four points. In the six-arc screw gears that I propose, the load can be distributed over six points on one side of the tooth, and the total number of simultaneous meshing points can reach twelve in certain intervals. This is expected to substantially improve the load capacity and the running-in behavior of screw gears.
The calculation of the axial distances $$q_{TA}$$, $$q_{T1}$$, $$q_{T2}$$, and $$q_{T3}$$ is essential because these distances determine the interval boundaries for $$\Delta b$$. The interval boundaries, in turn, determine the multi-point mesh coefficients and the multi-pairs tooth mesh coefficients. I have derived these distances from the basic rack geometry and the helix angle of the screw gears. The resulting formulas are explicit and can be used directly in the design of six-arc screw gears.
The multi-point mesh coefficients in Table 1 and Table 2 show how the number of meshing points varies with $$\Delta b$$. For example, when $$\Delta b$$ is very small, the six-point and seven-point meshing states dominate. As $$\Delta b$$ increases, the eight-point, nine-point, ten-point, eleven-point, and twelve-point meshing states appear in sequence. The exact sequence depends on the relative magnitudes of $$q_{T1}$$, $$q_{T2}$$, $$q_{T3}$$, and $$q’_{TA}$$. Therefore, the designer can control the meshing point sequence by adjusting the arc radii and the center offsets of the six-arc screw gears.
The multi-pairs tooth mesh coefficients in Table 3 show how the load is shared among different tooth pairs. When $$\Delta b < q’_{TA}$$, there are usually $$n$$ or $$n+1$$ tooth pairs in mesh. When $$\Delta b > q’_{TA}$$, there are usually $$n+1$$ or $$n+2$$ tooth pairs in mesh. This means that the six-arc screw gears can maintain multiple tooth pairs in contact over a wide range of tooth widths. This is beneficial for reducing the load per tooth pair and for improving the fatigue life of the screw gears.
The minimum tooth width determination in Table 4 provides a practical guide for selecting the tooth width of six-arc screw gears. If the designer wants at least $$n+1$$ tooth pairs and at least $$6n+3$$ meshing points, the remaining width $$\Delta b$$ should lie in the range
$$q’_{TA} + q_{T1} + q_{T2} + q_{T3} < \Delta b \leq q_{TA} – q_{T1}$$
If the designer wants at least $$n+1$$ tooth pairs and at least $$6n+4$$ meshing points, the remaining width should lie in the range
$$q_{T1} + 2q_{T2} + q_{T3} < \Delta b \leq q_{TA}$$
These ranges can be used together with the axial pitch $$P_x$$ to determine the total tooth width $$b = nP_x + \Delta b$$. By choosing $$n$$ and $$\Delta b$$ appropriately, the designer can achieve the desired number of meshing points and tooth pairs for six-arc screw gears.
Conclusion
I have proposed six-arc screw gears as a new type of point-meshing screw gears with six working arcs on one side of the tooth. I have defined the basic tooth profile parameters, derived the axial distances between the six meshing points, divided the remaining tooth width into eleven intervals, and calculated the multi-point mesh coefficients and the multi-pairs tooth mesh coefficients. I have also provided tables for the selection of a reasonable tooth width. The results show that the number of meshing points and the number of meshing tooth pairs in six-arc screw gears are functions of the tooth width. By selecting the tooth width properly, the designer can achieve a larger number of meshing points and tooth pairs, which improves the load capacity and the transmission smoothness of the screw gears. The formulas and tables presented here provide a convenient basis for the design and analysis of six-arc screw gears.