In modern marine propulsion, the azimuth thruster stands out for its exceptional maneuverability and dynamic positioning capabilities. A critical component within its power transmission system is the set of bevel gears, responsible for redirecting power flow and providing the necessary speed reduction from the prime mover to the propeller. The selection of teeth numbers for these bevel gears is a foundational design step, influencing gear size, load capacity, meshing quality, and ultimately the thruster’s performance and noise characteristics. This article presents a comprehensive, systematic methodology developed to efficiently navigate the vast combinatorial space of possible teeth numbers and identify optimal sets for both single-stage and two-stage bevel gear reductions in azimuth thrusters.
The core challenge in designing these transmissions lies in the interdependency of numerous constraints. The primary input is the overall speed reduction ratio, $u_{total}$, defined by the prime mover’s input speed and the propeller’s required output speed. For a single-stage reduction, this ratio is the simple quotient of the driven bevel gear’s teeth to the driving bevel gear’s teeth. For two-stage reductions, it is the product of the ratios of the two stages: $u_{total} = u_1 \cdot u_2 = (z_2 / z_1) \cdot (z_4 / z_3)$, where $z_1$ and $z_3$ are the pinion teeth numbers, and $z_2$ and $z_4$ are the gear teeth numbers for the first and second stage, respectively. The designer must select integer teeth numbers from practical ranges (e.g., pinions from 11 to 25, gears from 17 to 46) that satisfy this ratio within a tight tolerance while simultaneously meeting a host of other critical geometrical and manufacturing constraints for spiral bevel gears.
The fundamental geometrical relationship for a 90-degree shaft angle spiral bevel gear pair, with the pinion not offset, forms the basis for several key constraints. The pitch cone angle for the gear, $\delta_2$, is determined by the teeth ratio $u$:
$$
\delta_2 = \arctan(u) = \arctan\left(\frac{z_2}{z_1}\right)
$$
The mean normal module, $m_{mn}$, is a critical parameter linking teeth numbers to physical gear dimensions like face width $b$ and mean spiral angle $\beta_m$. It can be derived from the geometry of the gear. The mean pitch diameter of the gear, $d_{m2}$, is given by:
$$
d_{m2} = (2 R_{e2} – b) \sin \delta_2
$$
where $R_{e2}$ is the outer cone distance. The mean normal module is then:
$$
m_{mn} = \frac{d_{m2}}{z_2} \cos \beta_m = \frac{(2 R_{e2} – b) \sin \delta_2}{z_2} \cos \beta_m
$$
Substituting $\sin \delta_2 = z_2 / \sqrt{z_1^2 + z_2^2}$, we obtain a foundational relationship:
$$
m_{mn} = \frac{(2 R_{e2} – b) \cos \beta_m}{\sqrt{z_1^2 + z_2^2}}
$$
For hardened, heavy-duty spiral bevel gears, such as those conforming to the Klingelnberg Cyclo-Palloid system, empirical guidelines dictate that the face width $b$ is a fraction of the outer cone distance ($b \approx 0.29R_{e2}$ to $0.33R_{e2}$) and that the ratio $m_{mn}/b$ should fall within a specific range, typically 0.10 to 0.14. This imposes a direct constraint on the chosen teeth numbers, the spiral angle, and the gear size. The expression can be rearranged to check this condition:
$$
\frac{m_{mn}}{b} = \frac{(2(R_{e2}/b) – 1) \cos \beta_m}{\sqrt{z_1^2 + z_2^2}}
$$
Given standard ranges for $R_{e2}/b$ and $\beta_m$ (e.g., 30° to 35°), this formula allows for the calculation of the theoretical minimum and maximum $m_{mn}/b$ value for any proposed teeth combination $(z_1, z_2)$. Only combinations yielding a value within the 0.10-0.14 band are considered geometrically feasible for a robust design.
Beyond geometry, meshing quality requirements introduce further filters. A paramount rule is that the teeth numbers of a meshing pair should be coprime (i.e., their greatest common divisor is 1). This ensures even tooth wear over time, as each tooth of the pinion meshes with every tooth of the gear before the cycle repeats, distributing machining errors and promoting uniform load distribution. For two-stage reductions, an additional manufacturing consideration arises if the gear cutting process uses a specific number of blade groups. To ensure even wear on the cutting tools, it is often recommended that the teeth numbers of all gears are also coprime with the number of blade groups (e.g., 5).
Manually evaluating all possible teeth combinations against this matrix of constraints is prohibitively time-consuming. The number of possibilities is staggering. For a single-stage reduction with 15 possible pinion teeth and 30 possible gear teeth, there are 450 raw ratio combinations. For a two-stage reduction, considering both stages independently leads to $450 \times 450 = 202,500$ possible overall ratio combinations, not accounting for symmetry. A manual, trial-and-error approach is inefficient and risks overlooking superior solutions.
To overcome this, we developed an algorithmic selection process implemented using a combination of spreadsheet functions and VBA (Visual Basic for Applications) programming. This method automates the exhaustive search and constraint checking. The process begins by defining the target overall ratio $u_{total}$ and an allowable deviation $\Delta u$ (e.g., ±0.02 for single-stage, ±0.01 for two-stage). Acceptable ranges for pinion and gear teeth numbers are also defined.
For a single-stage reduction, the algorithm proceeds as follows:
1. Generate all integer pairs $(z_1, z_2)$ within the specified ranges.
2. Calculate the ratio $u = z_2 / z_1$ for each pair.
3. Discard pairs where $z_1$ and $z_2$ are not coprime.
4. Retain pairs where $|u – u_{target}| \le \Delta u$.
5. For each retained pair, calculate the theoretical min and max $m_{mn}/b$ using the extreme values of $R_{e2}/b$ and $\beta_m$.
6. Present the filtered list, allowing the designer to choose based on optimal $m_{mn}/b$ and preferred spiral angle $\beta_m$.
The results for a target ratio of $u=2.21$ with $\Delta u=0.02$ are shown below. Only three pairs meet the ratio and coprimeness constraints. Further selection is guided by the $m_{mn}/b$ criterion.
| Overall Ratio Target | Allowed Deviation | Matched Ratio | Gear Teeth ($z_2$) | Pinion Teeth ($z_1$) | Min $m_{mn}/b$ | Max $m_{mn}/b$ |
|---|---|---|---|---|---|---|
| 2.21 | 0.02 | 2.214 | 31 | 14 | 0.122 | 0.150 |
| 2.21 | 0.02 | 2.211 | 42 | 19 | 0.090 | 0.111 |
| 2.21 | 0.02 | 2.190 | 46 | 21 | 0.082 | 0.101 |
Given the $m_{mn}/b$ range of 0.10–0.14, the pair (31, 14) is the most promising. If a higher spiral angle (e.g., 35°) is preferred for increased contact ratio, this pair becomes the definitive choice as it is the only one whose maximum $m_{mn}/b$ value remains within the limit at the upper spiral angle bound.
The methodology for two-stage reductions is more complex but follows the same logical structure. The algorithm must find quadruplets $(z_1, z_2, z_3, z_4)$ such that $(z_2/z_1) \cdot (z_4/z_3) \approx u_{total}$. To avoid redundancy where the stages are simply swapped, symmetry-breaking rules are applied. The steps are:
1. Generate all valid, coprime pairs for the first stage $(z_1, z_2)$ and their ratios $u_1$.
2. Generate all valid, coprime pairs for the second stage $(z_3, z_4)$ and their ratios $u_2$.
3. Compute the Cartesian product, calculating $u_{total} = u_1 \cdot u_2$ for each combination of a first-stage pair and a second-stage pair.
4. Apply symmetry filter (e.g., discard if the quadruplet is a simple swap of another).
5. Retain quadruplets where $|u_{total} – u_{target}| \le \Delta u$.
6. For each retained quadruplet, calculate the $m_{mn}/b$ range for both gear pairs.
7. Apply an additional filter: all teeth numbers should be coprime with the cutter blade group number (e.g., 5).
8. Present the final filtered list for designer evaluation.
Applying this to a target $u_{total}=5.8$ with $\Delta u=0.01$ yields numerous initial matches. The table below shows a subset of the results after applying the ratio and coprimeness filters. The final selection is made by enforcing the $m_{mn}/b$ criterion (0.10–0.14) and the blade group coprimeness rule.
| Total Ratio Target | Allowed Deviation | Matched Total Ratio | $z_2$ | $z_1$ | $z_4$ | $z_3$ | Min $m_{mn}/b$ (Stage 1) | Max $m_{mn}/b$ (Stage 1) | Min $m_{mn}/b$ (Stage 2) | Max $m_{mn}/b$ (Stage 2) |
|---|---|---|---|---|---|---|---|---|---|---|
| 5.8 | 0.01 | 5.802 | 18 | 11 | 39 | 11 | 0.197 | 0.242 | 0.102 | 0.126 |
| 5.790 | 18 | 11 | 46 | 13 | 0.197 | 0.242 | 0.087 | 0.107 | ||
| 5.802 | 31 | 11 | 39 | 22 | 0.126 | 0.155 | 0.093 | 0.114 | ||
| 5.792 | 32 | 13 | 40 | 17 | 0.120 | 0.148 | 0.095 | 0.117 |
After applying all constraints, including $m_{mn}/b$ and blade group compatibility, a manageable set of optimal candidate quadruplets remains. For instance, the set (31, 11, 37, 18) and (32, 13, 33, 14) are strong candidates. The final choice can be fine-tuned by selecting a specific spiral angle $\beta_m$ within its range to optimize contact ratio and strength.
In conclusion, the systematic design of bevel gears for azimuth thrusters requires a methodical approach to navigate a high-dimensional constraint space. This methodology, leveraging computational search algorithms, transforms an intractable manual selection problem into a efficient, deterministic process. By codifying the essential constraints—overall ratio tolerance, teeth coprimeness, geometric compatibility via the $m_{mn}/b$ ratio, and manufacturing considerations—we enable the rapid identification of optimal teeth numbers for both single-stage and complex two-stage bevel gear transmissions. This not only significantly improves design efficiency but also ensures that no superior combination is overlooked, leading to more robust, reliable, and performance-optimized bevel gear sets for demanding marine propulsion applications. The principles and algorithmic framework are universally applicable and can be extended to transmissions with even more stages or to other gear types where multiple interacting parameters must be optimized simultaneously.