When designing a herringbone gear, it is critical to reserve a sufficient relief groove width to ensure proper cutting tool clearance during hobbing. The standard Q/ZB 135-73 provides a table of minimum relief groove width bmin for hobbing herringbone gears, but this table prescribes the same value regardless of the gear tooth count. In reality, for different tooth numbers of the large herringbone gear, the minimum relief groove width bmin can vary. In this paper, we develop a calculation model for the relief groove width of herringbone gears based on simplifying the hob and the herringbone gear geometry. Using this model, we compute and tabulate bmin as a function of module mn, helix angle β, and the large gear tooth number Z. The resulting tables are convenient for practical engineering use.
Mathematical Model
We establish a coordinate system as shown in the conceptual diagram (not reproduced here for brevity). Let the gear axis be the z-axis, and the N-end face of the relief groove contain the x-axis and y-axis, intersecting perpendicularly. The hob axis lies in the y-z plane, intersecting the x-axis at a right angle, and forming an angle α with the y-axis. Thus, the unit vector along the hob axis is n = (0, cosα, sinα). The angle between the hob axis and the gear axis (z-axis) is γ = 90° – α. During hobbing of a herringbone gear, the hob axis is inclined such that γ = 90° – (λ + β), where λ is the hob rake angle and β is the gear helix angle. For simplicity, we directly use α in the derivation.
Define the following geometric parameters:
- D1: gear root circle diameter
- D2: gear tip circle diameter
- D: hob tip circle diameter
- 2L: total hob length
- x0: distance between hob axis and gear axis in the x-direction, equal to (D1 + D)/2
The hob tip cylindrical surface is tangent to the gear root cylindrical surface at a point P. We consider two cases depending on the hob length relative to the gear geometry.
Case 1: Short hob length
When the hob is relatively short, the tip circle of the hob end face M1 intersects the gear tip cylindrical surface at two points in the first octant. The larger z-coordinate of these intersection points gives bmin.
The equation of the hob end face M1 (passing through point (x0, L cosα, L sinα) with normal vector n) is:
$$ \cos\alpha (y – L\cos\alpha) + \sin\alpha (z – L\sin\alpha) = 0 \tag{1} $$
The hob tip cylindrical surface equation is:
$$ (x – x_0)^2 + (-y\sin\alpha + z\cos\alpha)^2 = \frac{D^2}{4} \tag{2} $$
The intersection curve of (1) and (2) defines the tip circle of the hob end face. The gear tip cylindrical surface is:
$$ x^2 + y^2 = \frac{D_2^2}{4} \tag{3} $$
We solve the system (1), (2), (3) for the first octant. Introduce a parameter t by setting:
$$ x = \frac{D_2}{2} t, \quad y = \frac{D_2}{2}\sqrt{1 – t^2} \tag{4} $$
Substituting into (1) gives:
$$ z = \frac{L}{\sin\alpha} – \frac{D_2\sqrt{1 – t^2}}{2\tan\alpha} \tag{5} $$
Then substitute (4) and (5) into (2) and simplify to obtain:
$$ C_{12} t^2 + C_{11} t + C_{10} = \sqrt{1 – t^2} \tag{6} $$
where the coefficients are:
$$ \begin{aligned}
C_{12} &= -\frac{D_2 \cos\alpha}{4L} \\
C_{11} &= -\frac{x_0 \tan\alpha \sin\alpha}{L} = -\frac{(D_1 + D) \tan\alpha \sin\alpha}{2L} \\
C_{10} &= \left[ \frac{D_1^2 + 2D_1 D}{4} + \frac{L^2}{\tan^2\alpha} + \frac{D_1^2}{4\sin^2\alpha} \right] \frac{\tan\alpha \sin\alpha}{D_2 L}
\end{aligned} \tag{7} $$
Squaring both sides of (6) yields a quartic equation in t:
$$ e_4 t^4 + e_3 t^3 + e_2 t^2 + e_1 t + e_0 = 0 \tag{8} $$
with
$$ \begin{aligned}
e_4 &= C_{12}^2 \\
e_3 &= 2C_{12} C_{11} \\
e_2 &= C_{11}^2 + 2C_{10} C_{12} + 1 \\
e_1 &= 2C_{10} C_{11} \\
e_0 &= C_{10}^2 – 1
\end{aligned} \tag{9} $$
Solving (8) for t and substituting into (5) gives two possible z values; the larger one is the required bmin for this case.
Case 2: Medium to long hob length
When the hob is longer, the hob end face tip circle may intersect the gear tip cylinder at only one point or not at all. In this situation, bmin corresponds to the maximum z-coordinate of the intersection curve between the hob tip cylinder and the gear tip cylinder, which occurs at a point where the derivative of z with respect to t vanishes.
From (2) and (4) we can express z explicitly as a function of t (taking the positive root):
$$ z(t) = \frac{D_2 \tan\alpha}{2} \sqrt{1 – t^2} + \sqrt{\frac{D^2}{4} – \left( \frac{D_2 t}{2} – x_0 \right)^2} \frac{1}{\cos\alpha} \tag{10} $$
Differentiate (10) with respect to t and set dz/dt = 0:
$$ -\frac{D_2 \tan\alpha}{2} \frac{t}{\sqrt{1 – t^2}} – \frac{D_2}{\cos\alpha} \frac{ \left( \frac{D_2 t}{2} – x_0 \right) }{ 2 \sqrt{ \frac{D^2}{4} – \left( \frac{D_2 t}{2} – x_0 \right)^2 } } = 0 \tag{11} $$
After simplification, we obtain a quartic equation:
$$ A_4 t^4 + A_3 t^3 + A_2 t^2 + A_1 t + A_0 = 0 \tag{12} $$
where
$$ \begin{aligned}
A_4 &= \frac{D_1^2 \cos^2\alpha}{4} \\
A_3 &= -D_2 x_0 \cos^2\alpha \\
A_2 &= \frac{D^2 \sin^2\alpha}{4} – \frac{D_1^2}{4} + x_0^2 \cos^2\alpha \\
A_1 &= D_2 x_0 – \frac{D_2 x_0^2}{D} \\
A_0 &= -x_0^2 \left(1 – \frac{x_0^2}{D^2}\right)
\end{aligned} \tag{13} $$
Solving (12) yields the value of t that maximizes z. Substituting this t into (10) gives the maximum z, which is bmin for this case.
The actual bmin for a given herringbone gear is taken as the larger value obtained from Case 1 and Case 2, ensuring the relief groove is wide enough for the hob.
Influence of Addendum Modification Coefficient
The root and tip diameters of the herringbone gear depend on the normal addendum modification coefficient xn:
$$ \begin{aligned}
D_1 &= m_n \left( \frac{Z}{\cos\beta} – 2h_a^* – 2c^* + 2x_n \right) \\
D_2 &= m_n \left( \frac{Z}{\cos\beta} + 2h_a^* + 2x_n \right)
\end{aligned} \tag{14} $$
Thus xn affects both x0 = (D1+D)/2 and D2. For typical values (|xn| ≤ 0.2), the change in D2 and x0 is small compared to terms involving Z/ cosβ. Our numerical tests for xn = 0, 0.1, and 0.2 show that the resulting bmin values differ by less than 0.2 mm for modules up to 10 mm. Hence the influence of xn on the relief groove width of herringbone gears can be safely neglected.
Results: Tabulated Minimum Relief Groove Widths
Using the above model, we computed bmin for a range of modules mn = 4 to 10 mm, helix angles β = 30°, 35°, 40°, and large gear tooth numbers Z = 25 to 120. The computations assume standard hob geometry (hob tip diameter D = 2.5 mn + 10 mm, hob length 2L = 80 mm for mn ≤ 6, and 2L = 100 mm for larger modules). The results are organized in the tables below. Two columns are provided: “I” for gears of grade A, B, C (coarse precision) and “II” for AA grade (high precision). The difference arises from different hob design parameters specified in the standard.
Table 1: β = 30°
| Z | Grade I (A/B/C) | Grade II (AA) | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| m=4 | 5 | 6 | 7 | 8 | 9 | 10 | m=4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
| 25 | 40 | 37 | 45 | 50 | 55 | 63 | 71 | 78 | 33 | 40 | 45 | 51 | 57 | 64 | 70 |
| 60 | 37 | 46 | 55 | 60 | 68 | 77 | 85 | 36 | 43 | 47 | 53 | 59 | 66 | 72 | |
| 80 | 41 | 50 | 57 | 61 | 70 | 79 | 88 | 38 | 44 | 48 | 54 | 60 | 67 | 73 | |
| 100 | 43 | 51 | 58 | 62 | 71 | 81 | 89 | 38 | 44 | 49 | 54 | 60 | 67 | 73 | |
| 120 | 45 | 52 | 60 | 63 | 72 | 81 | 90 | 39 | 45 | 49 | 55 | 60 | 68 | 74 | |
Table 2: β = 35°
| Z | Grade I (A/B/C) | Grade II (AA) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| m=4 | 5 | 6 | 7 | 8 | 9 | 10 | m=4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 40 | 41 | 43 | 56 | 63 | 72 | 81 | 89 | 38 | 45 | 50 | 56 | 62 | 69 | 75 |
| 60 | 45 | 36 | 61 | 65 | 75 | 84 | 94 | 39 | 46 | 51 | 57 | 63 | 71 | 77 |
| 80 | 46 | 53 | 63 | 65 | 76 | 86 | 95 | 41 | 47 | 51 | 58 | 64 | 71 | 77 |
| 100 | 47 | 56 | 64 | 66 | 78 | 79 | 96 | 41 | 47 | 52 | 58 | 64 | 72 | 78 |
| 120 | 48 | 57 | 65 | 67 | 77 | 87 | 97 | 41 | 48 | 52 | 58 | 64 | 72 | 78 |
Table 3: β = 40°
| Z | Grade I (A/B/C) | Grade II (AA) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| m=4 | 5 | 6 | 7 | 8 | 9 | 10 | m=4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 40 | 51 | 61 | 70 | 73 | 83 | 94 | 107 | 45 | 51 | 56 | 62 | 69 | 77 | 84 |
| 60 | 53 | 63 | 71 | 74 | 84 | 95 | 107 | 45 | 52 | 56 | 63 | 69 | 78 | 84 |
| 80 | 53 | 64 | 71 | 74 | 85 | 96 | 108 | 45 | 52 | 56 | 63 | 69 | 78 | 84 |
| 100 | 54 | 64 | 72 | 74 | 85 | 96 | 108 | 46 | 52 | 57 | 63 | 69 | 78 | 84 |
| 120 | 54 | 64 | 72 | 74 | 85 | 96 | 108 | 46 | 52 | 57 | 63 | 70 | 78 | 84 |
Instructions for Using the Tables
- Select the appropriate table according to the herringbone gear helix angle β (30°, 35°, or 40°).
- Locate the row corresponding to the tooth number Z of the large herringbone gear.
- For a given normal module mn, read the minimum relief groove width bmin from the relevant column:
- Use the “Grade II (AA)” columns for herringbone gears of AA precision grade.
- Use the “Grade I (A/B/C)” columns for grades A, B, or C.
- The values given are already the minimum required; the designer may add a small margin if desired.
- The influence of addendum modification coefficient xn is negligible for typical values (|xn| ≤ 0.2).
Discussion
The tabulated results clearly show that the minimum relief groove width bmin for a herringbone gear increases with both module and tooth number. For a fixed module and helix angle, increasing the tooth number Z generally leads to a larger bmin, because the gear tip circle diameter grows, requiring the hob to retract further to avoid interference. Similarly, a larger helix angle β reduces the effective axial clearance for the hob, thus demanding a wider relief groove. The difference between grade I and grade II values reflects the tighter tolerances and different hob geometry used for high-precision herringbone gears.
Our model assumes ideal cylindrical surfaces and perfect alignment. In practice, additional factors such as hob runout, gear blank eccentricity, and thermal expansion may require a safety margin. Nevertheless, the computed bmin values provide a reliable lower bound for the relief groove width in herringbone gear design.
The proposed calculation method has been implemented in a computer program that rapidly evaluates bmin for arbitrary combinations of mn, β, Z, and hob parameters. The program solves the quartic equations (8) and (12) using robust numerical methods. The tables presented here cover the most common ranges used in industrial herringbone gear applications. For non‑standard parameters, the program can be rerun to obtain custom values.
Conclusion
We have developed a mathematical model for determining the minimum relief groove width required when hobbing herringbone gears. The model accounts for the hob and gear geometry, the helix angle, and the tooth number of the large gear. Through systematic computation, we generated compact tables that relate bmin to module, helix angle, and tooth number for two common precision grades. These tables overcome the limitation of the standard Q/ZB 135‑73, which gave a single fixed value irrespective of tooth number. Using our results, designers of herringbone gears can now select the most economical relief groove width without risking interference, thereby improving machining efficiency and tool life.
We emphasize the importance of relief groove width in herringbone gear manufacturing. A herringbone gear with an insufficient relief groove will cause the hob to strike the opposite helix during cutting, leading to chipped teeth and shortened tool life. Conversely, an overly generous groove weakens the gear blank and wastes material. The tables provided here offer a rational basis for choosing the optimal bmin for any given herringbone gear design.
Future work could extend the model to include the effects of hob wear, finishing cuts, and non‑standard rake angles. Additionally, experimental validation of the computed bmin values on actual herringbone gear hobbing operations would be valuable. However, the theoretical foundation presented here is sound and has been verified by numerous industrial examples.
In summary, the relief groove width of a herringbone gear is not a constant but depends on the gear’s module, helix angle, and tooth count. The tables in this paper enable engineers to quickly and accurately determine the required bmin for their specific herringbone gear parameters, facilitating more efficient design and manufacturing processes.