In the field of marine propulsion, azimuth thrusters have become increasingly vital due to their exceptional maneuverability and operational versatility. These systems integrate steering and propeller functions, allowing full 360-degree rotation to provide thrust in any direction. A critical component within azimuth thrusters is the transmission bevel gear system, which transfers torque and power from the prime mover—such as diesel engines, electric motors, or hydraulic motors—to the propeller. The design of these bevel gears, particularly the selection of tooth numbers, poses significant challenges due to the myriad of possible combinations and stringent operational requirements. This article presents a comprehensive methodology for selecting optimal tooth numbers for bevel gears in azimuth thruster transmissions, leveraging computational tools like Excel and VBA to streamline the process and enhance design efficiency.
The complexity of bevel gear tooth number selection arises from multiple constraints, including gear ratio targets, tooth count ranges, and geometric and material considerations. For instance, in azimuth thrusters, the transmission often involves one or two stages of bevel gear reduction to achieve the necessary speed reduction from high-speed prime movers to lower-speed propellers. The total gear ratio is typically known based on input and output speeds, but the individual tooth numbers for each bevel gear must be chosen from vast possibilities. Manual selection is tedious and prone to oversight, potentially missing optimal configurations. Therefore, a systematic approach is essential to navigate this design space effectively.
Bevel gears in azimuth thrusters are commonly spiral bevel gears, such as Klingelnberg cycloidal bevel gears, with a shaft angle of 90 degrees and no pinion offset. These gears are case-hardened for heavy-duty applications, and their tooth numbers must satisfy several criteria. Key requirements include the gear ratio, the need for tooth numbers to be coprime to avoid uneven wear, and relationships between geometric parameters like module, face width, and cone distance. The midpoint normal module \(m_{mn}\) is a critical parameter, and for hardened bevel gears under heavy loads, it should fall within a specific range relative to the face width \(b\). The relationship between tooth numbers and these parameters can be derived from fundamental gear geometry.
For a bevel gear pair with a 90-degree shaft angle, the gear ratio \(u\) is defined as the ratio of the number of teeth on the larger gear \(z_2\) to that on the smaller gear \(z_1\):
$$ u = \frac{z_2}{z_1} $$
The pitch angle of the larger gear \(\delta_2\) is given by:
$$ \delta_2 = \arctan(u) = \arctan\left(\frac{z_2}{z_1}\right) $$
This leads to:
$$ \sin \delta_2 = \frac{z_2}{\sqrt{z_1^2 + z_2^2}} $$
The midpoint pitch diameter of the larger gear \(d_{m2}\) is:
$$ d_{m2} = (2R_{e2} – b) \sin \delta_2 $$
where \(R_{e2}\) is the outer cone distance and \(b\) is the face width. For Klingelnberg bevel gears, the recommended face width is \(b = (0.29 \text{ to } 0.33) R_{e2}\). The midpoint normal module \(m_{mn}\) is consistent for both gears and is expressed as:
$$ m_{mn} = \frac{d_{m2}}{z_2} \cos \beta_m $$
where \(\beta_m\) is the midpoint spiral angle, typically between 30° and 35°. Substituting the expressions yields:
$$ m_{mn} = \frac{(2R_{e2} – b) \cos \beta_m}{\sqrt{z_1^2 + z_2^2}} $$
For hardened bevel gears under heavy loads, \(m_{mn}\) should be in the range of \(0.1b\) to \(0.14b\). This constraint links tooth numbers to geometric parameters, guiding the selection process.
To address the tooth number selection for single-stage reduction, we developed a method using Excel and VBA. The pinion tooth number \(z_1\) is allowed to vary from 11 to 25, and the gear tooth number \(z_2\) from 17 to 46, resulting in 450 potential gear ratios. In Excel, we list these tooth numbers and compute the gear ratio \(u = z_2 / z_1\), setting \(u = 0\) for non-coprime pairs or speed increases. Given a target gear ratio, such as \(u = 2.21\), and an allowable deviation of 0.02, VBA code scans through the 450 ratios to identify matches within the tolerance. For each valid pair, we calculate the theoretical minimum and maximum values of \(m_{mn}/b\) to ensure they fall within 0.1 to 0.14. The results are summarized in a table for clarity.
| Target Gear Ratio | Allowable Deviation | Valid Gear 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 |
From Table 1, only three bevel gear pairs meet the initial criteria. Based on the \(m_{mn}/b\) constraint, the pair with \(z_2 = 46\) and \(z_1 = 21\) may be less favorable. Further refinement can involve selecting a higher spiral angle \(\beta_m\), such as 35°, to improve contact ratio. For \(\beta_m = 35°\), only the pair with \(z_2 = 31\) and \(z_1 = 14\) satisfies all conditions, demonstrating how this method enables quick preliminary selection for single-stage bevel gear systems.
For two-stage reduction, the process becomes more complex due to the multiplication of gear ratios. The total gear ratio \(u\) is the product of two individual ratios \(u_1\) and \(u_2\):
$$ u = u_1 \times u_2 $$
where \(u_1 = z_2 / z_1\) and \(u_2 = z_4 / z_3\). Assuming similar tooth number ranges for each stage—pinions \(z_1\) and \(z_3\) from 11 to 25, and gears \(z_2\) and \(z_4\) from 17 to 46—each stage has 450 possible ratios. Using Excel, we compute all combinations of \(u_1\) and \(u_2\), generating 202,500 total ratios (accounting for duplicates by setting some to zero). For a target total ratio, such as \(u = 5.8\), with an allowable deviation of 0.01, VBA code searches for matches. Additional constraints include coprime tooth numbers and avoidance of common factors with cutter blade groups (e.g., assuming 5 blade groups). The results are filtered to list valid configurations, along with \(m_{mn}/b\) values for each bevel gear pair.
| Target Gear Ratio | Allowable Deviation | Valid Total Ratio | First Stage: \(z_2 / z_1\) | Second Stage: \(z_4 / z_3\) | 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 | 31 / 14 | 39 / 11 | 0.126 | 0.155 | 0.102 | 0.126 |
| 5.790 | 42 / 19 | 46 / 13 | 0.090 | 0.111 | 0.087 | 0.107 | ||
| 5.810 | 46 / 21 | 37 / 11 | 0.082 | 0.101 | 0.107 | 0.132 | ||
| 5.790 | 23 / 11 | 36 / 13 | 0.163 | 0.200 | 0.108 | 0.133 | ||
| 5.802 | 26 / 11 | 27 / 11 | 0.147 | 0.181 | 0.142 | 0.175 | ||
| 5.798 | 28 / 11 | 41 / 18 | 0.138 | 0.170 | 0.093 | 0.114 | ||
| 5.802 | 31 / 11 | 35 / 17 | 0.126 | 0.155 | 0.107 | 0.131 | ||
| 5.793 | 31 / 11 | 37 / 18 | 0.126 | 0.155 | 0.101 | 0.124 | ||
| 5.802 | 36 / 11 | 39 / 22 | 0.110 | 0.136 | 0.093 | 0.114 | ||
| 5.804 | 38 / 11 | 42 / 25 | 0.105 | 0.129 | 0.085 | 0.104 | ||
| 5.804 | 25 / 12 | 39 / 14 | 0.149 | 0.184 | 0.100 | 0.123 | ||
| 5.799 | 28 / 13 | 35 / 13 | 0.134 | 0.165 | 0.111 | 0.137 | ||
| 5.802 | 32 / 13 | 33 / 14 | 0.120 | 0.148 | 0.116 | 0.142 | ||
| 5.792 | 32 / 13 | 40 / 17 | 0.120 | 0.148 | 0.095 | 0.117 | ||
| 5.810 | 35 / 13 | 41 / 19 | 0.111 | 0.137 | 0.092 | 0.113 | ||
| 5.802 | 36 / 13 | 44 / 21 | 0.108 | 0.133 | 0.085 | 0.105 | ||
| 5.803 | 41 / 13 | 46 / 25 | 0.096 | 0.119 | 0.079 | 0.098 | ||
| 5.808 | 34 / 15 | 41 / 16 | 0.112 | 0.137 | 0.094 | 0.116 | ||
| 5.804 | 37 / 15 | 40 / 17 | 0.104 | 0.128 | 0.095 | 0.117 | ||
| 5.801 | 33 / 16 | 45 / 16 | 0.113 | 0.139 | 0.087 | 0.107 | ||
| 5.790 | 35 / 16 | 45 / 17 | 0.108 | 0.133 | 0.086 | 0.106 | ||
| 5.799 | 41 / 16 | 43 / 19 | 0.094 | 0.116 | 0.088 | 0.109 | ||
| 5.803 | 39 / 17 | 43 / 17 | 0.097 | 0.120 | 0.090 | 0.110 | ||
| 5.798 | 45 / 17 | 46 / 21 | 0.086 | 0.106 | 0.082 | 0.101 |
From Table 2, 24 configurations meet the initial criteria. Applying the \(m_{mn}/b\) constraint and requiring tooth numbers to be coprime with cutter blade groups (e.g., 5), we narrow down to seven promising bevel gear sets. For example, one set is \(z_1 = 28, z_2 = 11, z_3 = 41, z_4 = 18\). Further refinement by setting \(\beta_m = 35°\) reduces the options to two sets: \(z_1 = 31, z_2 = 11, z_3 = 37, z_4 = 18\) and \(z_1 = 32, z_2 = 13, z_3 = 33, z_4 = 14\). This demonstrates the efficacy of the method in handling the complexity of two-stage bevel gear systems.
The overall design methodology involves several steps. First, we gather requirements for bevel gear tooth numbers, including gear ratios, geometric constraints, and material specifications. These are translated into programmatic rules using Excel formulas and VBA scripts. For single-stage reduction, we enumerate tooth number combinations, filter based on gear ratio tolerance and coprime condition, and validate using the \(m_{mn}/b\) range. For two-stage reduction, we extend this to multiple stages, considering product ratios and additional factors like cutter blade compatibility. The process is iterative, allowing designers to adjust parameters such as allowable deviation or spiral angle to optimize results. This approach is not limited to azimuth thrusters but can be adapted to other multi-stage bevel gear transmissions, showcasing its versatility.
In conclusion, the selection of tooth numbers for bevel gears in azimuth thruster transmissions is a critical yet challenging task. By leveraging computational tools like Excel and VBA, we have developed a systematic method that efficiently explores the design space, ensuring optimal bevel gear configurations. This method accounts for key constraints such as gear ratios, coprime tooth numbers, and geometric parameters, significantly reducing manual effort and minimizing the risk of overlooking superior designs. The use of tables and formulas, as demonstrated, provides clear insights into the selection process. Future work could extend this methodology to three or more stages of reduction or integrate it with advanced optimization algorithms for further enhancement. Ultimately, this approach contributes to the advancement of marine propulsion systems by improving the design efficiency and performance of bevel gear transmissions.
The importance of bevel gears in azimuth thrusters cannot be overstated, as they directly impact power transmission efficiency, noise levels, and operational reliability. Through meticulous tooth number design, we can achieve smoother operation, longer service life, and better overall system integration. The methodology outlined here serves as a practical guide for engineers, enabling them to navigate the complexities of bevel gear design with confidence. By embracing digital tools and analytical frameworks, the marine industry can continue to innovate and optimize propulsion technologies for diverse applications, from harbor tugs to dynamic positioning vessels.