In this work, I focus on the design, modeling, and load-capacity analysis of heavy-load large-modulus seven-tooth rack and pinion gears used in jack-up platform lifting systems. The rack and pinion gears in such systems operate under extremely high contact loads, low rotational speeds, and severe offshore environmental conditions. Because the pinion has only seven teeth, the tooth profile is far shorter than the undercut limit of standard gears, and the contact ratio is very low. Conventional standards for gear load capacity, including those based on the plane-section method, are not directly applicable to large-modulus rack and pinion gears with fewer than ten teeth. Therefore, I combined orthogonal experimental design with transient finite element dynamics to evaluate the influence of module, pressure angle, and profile shift coefficient on tooth root stress. Using this approach, I established a practical design and optimization procedure for heavy-load large-modulus seven-tooth rack and pinion gears. The final output of this work is a set of five optimized rack and pinion gears that can be combined modularly for different platform loads.
The motivation for this study is straightforward. As offshore platforms move toward deeper water and heavier payloads, the lifting system must transmit greater loads through a compact transmission. The rack and pinion gears must therefore achieve high bending strength, low stress concentration, and sufficient tooth tip thickness for heat treatment. However, the design of large-modulus seven-tooth rack and pinion gears is complicated by several factors: the number of teeth is much lower than the standard undercut limit; the theoretical contact ratio is low; the manufacturing process often uses flame cutting, milling, and form grinding rather than standard hobbing; and the actual installation may use a non-standard center distance to avoid jamming. These factors mean that the load is often carried by a single tooth pair, and the tooth root stress can become very high. My objective is to provide a reliable computational and experimental framework for designing heavy-load large-modulus seven-tooth rack and pinion gears.
The materials used for large-modulus rack and pinion gears are also important. In my analysis, I selected a high-strength alloy steel for the pinion and a high-strength low-alloy steel for the rack. The pinion material has higher surface hardness and fatigue resistance after heat treatment, while the rack material must balance strength and weldability. The material parameters I used are summarized in the following table.
| Material property | Pinion | Rack |
|---|---|---|
| Material designation | 40CrNiMo (AISI 4340) | Q690 |
| Density (kg/m³) | 7850 | 7850 |
| Elastic modulus (GPa) | 210 | 210 |
| Poisson’s ratio | 0.3 | 0.3 |
| Friction coefficient | 0.15 | 0.15 |
| Yield strength (MPa) | 1040 | 690 |
| Ultimate tensile strength (MPa) | 1200 | 850 |
| Contact fatigue limit (MPa) | 1500 | 900 |
| Bending fatigue limit (MPa) | 529 | 381 |
Tooth profile design is the first critical step. I followed several principles. First, the tooth tip must not be too thin, because during carburizing and quenching a very thin tip may be hardened through and then fracture under heavy load. In general, the tooth tip thickness should be at least 0.25 times the module. Second, the pinion must not be undercut. For a seven-tooth pinion, undercut is a serious risk because the number of teeth is far below the standard minimum. Third, the fillet radius should be as large as possible without causing interference, because a larger fillet reduces the stress concentration at the tooth root. Fourth, the addendum should be as large as possible while still satisfying the tip thickness and undercut constraints, so that the contact ratio can be improved and impact can be reduced.
The undercut condition for large-modulus rack and pinion gears can be expressed by the position of the interference point. Let the line of action be \(B_1B_2\), and let \(N_1\) be the limit point of the involute. If \(B_2\) lies below \(N_1\), the entire meshing process is involute. If \(B_2\) lies above \(N_1\), the pinion will have a non-involute transition curve meshing with the rack, which leads to undercut. The condition to avoid undercut is:
$$ N_1K = \frac{1}{2} m z \sin^2 \alpha + x m – \frac{1}{2} m z \sin^2 \alpha \geq h_a^* m $$
where \(m\) is the module, \(z\) is the number of teeth, \(\alpha\) is the pressure angle, \(x\) is the profile shift coefficient, and \(h_a^*\) is the addendum coefficient. This condition guides the selection of the minimum profile shift coefficient for a given module and pressure angle.
For the rack, the tooth tip thickness must also be checked. The allowable rack addendum can be written as:
$$ h_{a2} \leq \frac{2m – s}{8 \sin \alpha} $$
where \(s\) is the tooth thickness on the rack. For the pinion, the tip thickness condition is:
$$ 2 r_a \left( \frac{s}{2r} + 2 x \tan \alpha + \operatorname{inv}\alpha – \operatorname{inv}\alpha_a \right) \geq 0.25 m $$
where \(r_a\) is the tip radius, \(r\) is the pitch radius, \(\alpha_a\) is the pressure angle at the tip circle, and \(\operatorname{inv}\) is the involute function. These equations ensure that the tooth tip remains thick enough for heat treatment and that the tooth does not become sharp.
In addition to undercut and tip thickness, I considered the influence of module, pressure angle, profile shift coefficient, contact ratio, and fillet radius. The module determines the tooth size and the bending strength. In general, a larger module increases the tooth thickness and the load capacity, but it also increases the overall size and manufacturing cost. The pressure angle affects both the tooth root thickness and the curvature radius of the involute. A larger pressure angle increases the root thickness and reduces the contact stress, but it also reduces the contact ratio and may make the tooth tip thinner. The profile shift coefficient is especially important for a seven-tooth pinion. A positive profile shift can avoid undercut and increase the root thickness, but an excessive positive shift reduces the rack tip thickness and the contact ratio. The contact ratio must be kept above 1 to ensure continuous transmission. The fillet radius should be as large as possible to reduce stress concentration.
Because standard gear load-capacity formulas are not suitable for large-modulus seven-tooth rack and pinion gears, I reviewed the available calculation methods. The plane-section method, which is the basis of many standards, assumes that the tooth is a cantilever beam and calculates the nominal bending stress at the root. The AGMA and ISO methods use empirical correction factors obtained from tests on small and medium gears. The DNV guideline considers marine transmission conditions, but its empirical coefficients are also limited for modules greater than 50. The折截面法, or folded-section method, is more consistent with actual tooth breakage, but it introduces a stress penetration factor that is difficult to determine for new designs. The elastic method, such as conformal mapping, can give accurate stress distributions but is computationally complex. The finite element method is therefore the most practical approach for large-modulus rack and pinion gears because it can model the actual tooth profile, contact conditions, and transient meshing process.
For preliminary module estimation, I used a simplified formula derived from the standard bending stress equation. The standard formula for tooth root stress is:
$$ \sigma_{F0} = \frac{F_t}{b m_n} Y_{Fa} Y_{Sa} Y_{\beta} $$
where \(F_t\) is the tangential force, \(b\) is the face width, \(m_n\) is the normal module, \(Y_{Fa}\) is the tooth form factor, \(Y_{Sa}\) is the stress correction factor, and \(Y_{\beta}\) is the helix angle factor. For large-modulus rack and pinion gears, this formula can be simplified by grouping the tooth form and stress correction effects into a single coefficient \(Y’\). The simplified stress is:
$$ \sigma_{F0} = Y’ \frac{F_t}{b m} $$
The module can then be estimated from:
$$ m \geq \sqrt[3]{\frac{F_t Y’}{\phi_m z_1^2 \sigma_{FP}}} $$
where \(\phi_m\) is the face width coefficient, \(z_1\) is the number of pinion teeth, and \(\sigma_{FP}\) is the allowable bending stress. Using this approach, I estimated the initial module for five load levels: 100 t, 125 t, 160 t, 200 t, and 250 t. The results are shown in the following table.
| Load level | Estimated module | Rounded module set for orthogonal design |
|---|---|---|
| 100 t | 49.834 | 49, 50, 51 |
| 125 t | 55.716 | 55, 56, 57 |
| 160 t | 63.036 | 62, 63, 64 |
| 200 t | 70.476 | 70, 71, 72 |
| 250 t | 78.794 | 78, 79, 80 |
After the preliminary module estimation, I designed an orthogonal experiment to study the effects of module, pressure angle, and profile shift coefficient. I used three levels for the module and four levels for both the pressure angle and the profile shift coefficient. A full factorial experiment would require \(3 \times 4 \times 4 = 48\) simulations. By using an orthogonal array, I reduced the number of simulations to 16 while still capturing the main effects. The factors and levels are listed below.
| Factor | Level 1 | Level 2 | Level 3 | Level 4 |
|---|---|---|---|---|
| Module, \(m\) | Low | Medium | High | — |
| Pressure angle, \(\alpha\) | 26° | 27° | 28° | 29° |
| Profile shift coefficient, \(x\) | 0.05 | 0.10 | 0.15 | 0.20 |
The orthogonal array \(L_{16}(3 \times 4^2)\) was used because the module has three levels and the other two factors have four levels. This is a mixed-level orthogonal design. I used the pseudo-level method to balance the levels. The 16 simulations covered the most important combinations of module, pressure angle, and profile shift coefficient.
For the finite element modeling, I used a combination of CAD and CAE tools. I first generated the involute profile in AutoCAD using a parametric spline. The involute points were calculated from the standard involute equations. Then I imported the two-dimensional profile into SolidWorks to create the three-dimensional pinion and rack models. The pinion and rack were assembled at the standard center distance for the initial model. The three-dimensional model was then exported as a STEP file and imported into HyperMesh for meshing. In HyperMesh, I divided the cross-section into quadrilateral elements and then extruded them into hexahedral elements. The hexahedral elements were used because they provide better accuracy for contact analysis and avoid the problems of tetrahedral elements in frictional contact. The element type was Solid186, and the contact pair was defined with CONTA174 and TARGE170. The gear tooth surface was the contact surface, and the rack tooth surface was the target surface. The contact algorithm was the extended Lagrangian method, and the normal stiffness factor was set to 0.1. The friction coefficient was 0.15. The mesh size was controlled so that the total number of elements was between 100,000 and 150,000, and the number of nodes was between 100,000 and 150,000. A mesh convergence check showed that the stress difference between two successive refinements was less than 10%, which is acceptable for engineering design.
The boundary conditions were applied as follows. For the pinion, I used an MPC184 element to create a rotating joint. The MPC184 element was connected to a beam element BEAM188, and the beam cross-section was a circle with the same radius as the pinion bore. The MPC184 element allowed rotation about the X-axis. By applying a rotational displacement to the MPC184 element, the pinion was forced to rotate. The beam element constrained the translational and other rotational degrees of freedom. For the rack, I constrained the Y and Z translations on the non-tooth surfaces and applied a distributed force on one face. The force was applied as a ramp load after the contact was established. I used two load steps: the first load step from 0 to 1 s ensured that contact was active, and the second load step from 1 to 25 s applied the full load and refined the substeps for accuracy. The transient dynamics solver was used with the HHT time integration method to include inertia effects. The large deformation option was turned on. The APDL commands were used to define the table loads and the solution controls.
After solving, I used the time-history postprocessor to extract the maximum stress on the tensile side and the compressive side of the tooth root. The results for the 100 t load are shown in the following table.
| Simulation | Module | Pressure angle | Profile shift | Tensile root stress (MPa) | Compressive root stress (MPa) |
|---|---|---|---|---|---|
| 1 | 49 | 26° | 0.05 | 341.112 | 381.732 |
| 2 | 49 | 27° | 0.10 | 371.835 | 370.601 |
| 3 | 49 | 28° | 0.15 | 345.877 | 343.984 |
| 4 | 49 | 29° | 0.20 | 358.951 | 328.341 |
| 5 | 50 | 26° | 0.10 | 388.392 | 361.506 |
| 6 | 50 | 27° | 0.05 | 377.996 | 370.846 |
| 7 | 50 | 28° | 0.20 | 357.479 | 317.813 |
| 8 | 50 | 29° | 0.15 | 349.508 | 324.910 |
| 9 | 51 | 26° | 0.15 | 333.328 | 352.409 |
| 10 | 51 | 27° | 0.20 | 333.929 | 337.921 |
| 11 | 51 | 28° | 0.05 | 333.988 | 363.884 |
| 12 | 51 | 29° | 0.10 | 367.408 | 331.290 |
| 13 | 50 | 26° | 0.20 | 365.464 | 351.745 |
| 14 | 50 | 27° | 0.15 | 340.612 | 352.414 |
| 15 | 50 | 28° | 0.10 | 350.504 | 354.424 |
| 16 | 50 | 29° | 0.05 | 344.081 | 358.003 |
Similar tables were generated for the 125 t, 160 t, 200 t, and 250 t load levels. The general trend is that the compressive root stress is often higher than the tensile root stress. This is important because the standard method usually uses the tensile stress as the design criterion. For large-modulus seven-tooth rack and pinion gears, the compressive side must also be checked. The maximum stress values for all five load levels are summarized in the following table.
| Load | Module | Pressure angle | Profile shift | Tensile stress (MPa) | Compressive stress (MPa) |
|---|---|---|---|---|---|
| 100 t | 51 | 28° | 0.15 | 312.04 | 390.10 |
| 125 t | 55 | 28° | 0.20 | 250.55 | 319.23 |
| 160 t | 64 | 29° | 0.20 | 298.25 | 376.16 |
| 200 t | 72 | 29° | 0.20 | 292.47 | 373.89 |
| 250 t | 79 | 28° | 0.20 | 300.04 | 382.80 |
To analyze the influence of each factor, I used both range analysis and variance analysis. Range analysis, or the直观分析法, compares the mean response at each level of a factor. For a factor with \(k\) levels, the range \(R\) is the difference between the maximum and minimum mean values. A larger range indicates a stronger effect. Because the factors have different numbers of levels, I used a correction coefficient \(h\) to adjust the range. The corrected range is:
$$ R’ = R \cdot h $$
where \(h\) depends on the number of levels. The values of \(h\) for different level numbers are given in the following table.
| Number of levels | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| \(h\) | 0.71 | 0.52 | 0.45 | 0.40 | 0.37 | 0.36 | 0.34 | 0.32 | 0.31 |
For the 100 t load, the range analysis for the tensile and compressive root stresses is shown below.
| Level | Module tensile | Module compressive | Pressure angle tensile | Pressure angle compressive | Profile shift tensile | Profile shift compressive |
|---|---|---|---|---|---|---|
| Level 1 | 354.444 | 356.165 | 357.074 | 361.848 | 349.294 | 368.616 |
| Level 2 | 344.081 | 358.003 | 356.093 | 357.945 | 369.535 | 354.455 |
| Level 3 | 342.163 | 346.376 | 346.962 | 345.026 | 342.331 | 343.429 |
| Level 4 | — | — | 354.987 | 335.635 | 353.955 | 333.955 |
| \(R\) | 12.281 | 11.627 | 10.112 | 26.213 | 27.204 | 34.661 |
| \(R’\) | 12.772 | 12.092 | 9.101 | 23.592 | 24.484 | 31.195 |
For the 125 t load, the corrected range values show that the module has the largest effect on the tensile stress, while the profile shift coefficient has a strong effect on both sides. The pressure angle has a moderate effect on the compressive stress. The detailed range analysis is given in the following table.
| Level | Module tensile | Module compressive | Pressure angle tensile | Pressure angle compressive | Profile shift tensile | Profile shift compressive |
|---|---|---|---|---|---|---|
| Level 1 | 304.325 | 351.624 | 339.404 | 366.517 | 332.195 | 366.978 |
| Level 2 | 342.306 | 352.270 | 333.356 | 357.388 | 329.329 | 358.645 |
| Level 3 | 360.453 | 349.620 | 336.500 | 345.996 | 351.103 | 348.548 |
| Level 4 | — | — | 339.917 | 335.884 | 336.764 | 331.615 |
| \(R\) | 56.128 | 2.650 | 6.561 | 30.633 | 21.774 | 35.363 |
| \(R’\) | 58.373 | 2.756 | 5.905 | 27.570 | 19.597 | 31.827 |
For the 160 t load, the profile shift coefficient has the largest corrected range, followed by the module and the pressure angle. The module has a stronger effect on the tensile side than on the compressive side. The pressure angle affects both sides almost equally. The range analysis is shown below.
| Level | Module tensile | Module compressive | Pressure angle tensile | Pressure angle compressive | Profile shift tensile | Profile shift compressive |
|---|---|---|---|---|---|---|
| Level 1 | 351.015 | 356.867 | 323.776 | 363.178 | 352.675 | 373.896 |
| Level 2 | 319.432 | 353.193 | 330.715 | 359.398 | 326.824 | 361.470 |
| Level 3 | 343.604 | 347.714 | 335.696 | 350.488 | 325.504 | 345.252 |
| Level 4 | — | — | 343.295 | 337.901 | 328.481 | 330.346 |
| \(R\) | 31.583 | 9.153 | 19.519 | 25.277 | 27.171 | 43.550 |
| \(R’\) | 32.846 | 9.519 | 17.567 | 22.749 | 24.454 | 39.195 |
For the 200 t load, the profile shift coefficient dominates the compressive stress, and the pressure angle also has a notable effect on the compressive side. The module has the smallest effect. The range analysis is given below.
| Level | Module tensile | Module compressive | Pressure angle tensile | Pressure angle compressive | Profile shift tensile | Profile shift compressive |
|---|---|---|---|---|---|---|
| Level 1 | 344.653 | 345.079 | 338.681 | 359.995 | 348.377 | 365.078 |
| Level 2 | 342.209 | 346.255 | 340.499 | 350.818 | 344.399 | 351.676 |
| Level 3 | 340.705 | 345.510 | 346.037 | 341.385 | 339.363 | 341.198 |
| Level 4 | — | — | 344.557 | 330.941 | 337.636 | 325.146 |
| \(R\) | 3.948 | 1.176 | 7.356 | 29.054 | 10.741 | 39.932 |
| \(R’\) | 4.106 | 1.223 | 6.620 | 26.149 | 9.667 | 35.939 |
For the 250 t load, the profile shift coefficient has the largest effect, followed by the pressure angle. The module has a moderate effect. The range analysis is shown below.
| Level | Module tensile | Module compressive | Pressure angle tensile | Pressure angle compressive | Profile shift tensile | Profile shift compressive |
|---|---|---|---|---|---|---|
| Level 1 | 325.276 | 347.190 | 325.467 | 359.481 | 353.977 | 368.233 |
| Level 2 | 324.096 | 335.535 | 327.779 | 354.615 | 330.185 | 353.473 |
| Level 3 | 338.075 | 346.443 | 321.116 | 319.005 | 328.407 | 340.824 |
| Level 4 | — | — | 337.183 | 331.603 | 298.974 | 302.173 |
| \(R\) | 13.979 | 11.655 | 16.067 | 40.476 | 55.003 | 66.060 |
| \(R’\) | 14.538 | 12.121 | 14.460 | 36.428 | 49.503 | 59.454 |
The variance analysis provides a more rigorous assessment of significance. I calculated the total sum of squares, the between-group sum of squares, and the within-group sum of squares. The F-statistic is the ratio of the mean square between groups to the mean square within groups. The results for the five load levels are summarized below.
| Load | Factor effect on tensile root stress | Factor effect on compressive root stress |
|---|---|---|
| 100 t | Module: not significant; Pressure angle: not significant; Profile shift: not significant | Module: not significant; Pressure angle: not significant; Profile shift: significant |
| 125 t | Module: significant; Pressure angle: not significant; Profile shift: not significant | Module: not significant; Pressure angle: not significant; Profile shift: significant |
| 160 t | Module: not significant; Pressure angle: not significant; Profile shift: not significant | Module: not significant; Pressure angle: not significant; Profile shift: significant |
| 200 t | Module: not significant; Pressure angle: not significant; Profile shift: significant | Module: not significant; Pressure angle: not significant; Profile shift: significant |
| 250 t | Module: not significant; Pressure angle: not significant; Profile shift: not significant | Module: not significant; Pressure angle: not significant; Profile shift: not significant |
The variance analysis shows that the profile shift coefficient is the most influential factor on the compressive root stress. It is significant at the 100 t, 125 t, 160 t, and 200 t load levels. The module and pressure angle are generally not significant, except for the module effect on the tensile stress at 125 t and the profile shift effect on the tensile stress at 200 t. This confirms that the compressive side must be considered carefully in the design of heavy-load large-modulus seven-tooth rack and pinion gears. It also shows that the standard approach, which uses only the tensile stress, is insufficient.
Based on the range and variance analyses, I selected the best parameter combinations for each load level. The optimization criteria were to minimize the maximum stress on both the tensile and compressive sides, to keep the tooth tip thickness above the minimum, and to maintain a contact ratio greater than 1. The optimized parameters and the corresponding stresses are given in the following table.
| Load | Module | Pressure angle | Profile shift | Tensile stress (MPa) | Compressive stress (MPa) |
|---|---|---|---|---|---|
| 100 t | 51 | 28° | 0.15 | 312.04 | 390.10 |
| 125 t | 55 | 28° | 0.20 | 250.55 | 319.23 |
| 160 t | 64 | 29° | 0.20 | 298.25 | 376.16 |
| 200 t | 72 | 29° | 0.20 | 292.47 | 373.89 |
| 250 t | 79 | 28° | 0.20 | 300.04 | 382.80 |
These five sets of rack and pinion gears form a modular family. The pinion and rack for each load level can be combined to build a single pile leg transmission system. In the final part of this work, I simulated a 600 t single pile leg system with three pinions on each side of the rack. The pinions were arranged symmetrically, and the center distance was set to be 4 mm larger than the standard value to avoid jamming. The pinion parameters are listed below.
| Parameter | Pinion | Rack |
|---|---|---|
| Module (mm) | 51 | 51 |
| Number of teeth | 7 | — |
| Pressure angle | 28° | 28° |
| Face width (mm) | 245 | 280 |
| Profile shift coefficient | 0.15 | — |
| Material | 40CrNiMo | Q690 |
The finite element model of the single pile leg system had 561,840 elements and 638,608 nodes. The mesh quality was checked, and the element size was uniform. The pinions were constrained using MPC184 joints, and the rack was loaded with a distributed force on the middle nodes. The total load was 600 t, distributed over 54,096 nodes, giving an average force of 108.695 N per node. The solution was run for 15 s, with the first second used to establish contact and the remaining time used for the full load. The stress results showed that the compressive root stress was about 371 MPa, and the tensile root stress was about 301 MPa for the most heavily loaded pinion. These values are lower than the stresses for a single pinion under the same total load. This is because the load is shared among multiple pinions, and the tooth deformation allows the load to redistribute. Therefore, the single-pinion design can be used as the basis for a modular system. By combining the five optimized rack and pinion gears, a wide range of platform loads can be accommodated.
The results of this work lead to several conclusions. First, the design of heavy-load large-modulus seven-tooth rack and pinion gears must consider both the tensile and compressive root stresses. The compressive stress is often higher and is strongly influenced by the profile shift coefficient. Second, the profile shift coefficient is the most significant factor for the compressive root stress, and its effect becomes more pronounced as the load increases. Third, a pressure angle of 28° or 29° is generally favorable, but the optimal choice depends on the load level. Fourth, the module has a smaller effect than the profile shift coefficient, but it still influences the stress distribution. Fifth, the fillet radius and tooth tip thickness must be carefully controlled to avoid undercut and heat-treatment cracking. Sixth, the modular design approach allows a small number of optimized rack and pinion gears to cover a wide range of loads, which shortens the design cycle and reduces cost.
For future work, I recommend several extensions. First, experimental validation should be performed on the optimized rack and pinion gears to measure the actual tooth root stress. Strain gauges can be placed at the root fillets, and the measured values can be compared with the finite element results. Second, the simplified module estimation formula should include a profile shift correction factor and a pressure angle correction factor. This would improve the preliminary design stage. Third, the effects of face width, addendum coefficient, and fillet radius should be studied more systematically. In this work, I selected the maximum fillet radius and a suitable addendum coefficient, but further optimization may be possible. Fourth, the transient dynamics simulation can be extended to include more realistic load spectra, such as wind and wave loads, to assess fatigue life. Fifth, the manufacturing tolerances and assembly errors should be included in the finite element model to study their influence on load sharing and stress concentration.
In summary, I have developed a practical design and optimization method for heavy-load large-modulus seven-tooth rack and pinion gears. The method combines orthogonal experiment design, transient finite element dynamics, range analysis, and variance analysis. It produces five optimized rack and pinion gears that can be used as modular building blocks for jack-up platform lifting systems. The results show that the profile shift coefficient is the dominant factor for the compressive root stress, and that the compressive side must be considered in addition to the tensile side. The modular approach allows the same rack and pinion gears to be combined for different loads, which is beneficial for standardization and cost reduction. I believe this work provides a useful reference for the design of large-modulus rack and pinion gears in offshore engineering and other heavy-load applications.