I have carried out a detailed dynamic investigation into the load sharing behavior of the rack and pinion gear system used in jack-up offshore platforms. The central concern of this work is the unequal force distribution that appears on the two sides of the rack when two pinions mesh with opposite flanks of the same rack. In an ideal arrangement, the two pinions of a lifting unit should carry exactly the same load. In reality, vibrations, meshing errors, manufacturing deviations, and structural flexibility introduce a measurable deviation between the two sides. This deviation is called unequal load sharing, and it can lead to vibration, noise, pitting, and even tooth breakage. My objective is to quantify how the load sharing coefficient of the pinion changes when the lifting load, pressure angle, and module are varied. I built a dynamic model of the rack and pinion gear lifting system and used it to evaluate the load sharing coefficient under a wide range of operating conditions. The results show that the load sharing coefficient decreases as the lifting load increases and as the pressure angle increases. The coefficient varies approximately between 1.067 and 1.082. When the module increases, the load sharing coefficient first increases and then decreases, reaching a maximum at a module of 100 mm. These findings provide a theoretical basis for the design optimization of rack and pinion gear lifting systems in offshore platforms.
The rack and pinion gear jacking system is one of the most critical components of a self-elevating offshore platform. Its function is to raise and lower the platform along the legs by means of a set of pinions that mesh with a long rack attached to each leg. The system typically consists of an electric motor, a planetary reducer, several pinions, and a long rack. The motor drives the sun gear of the planetary reducer, and the planet carrier and the external gear output power to the left and right shafts. Two seven-tooth pinions form a differential gear train that provides a unified output: they have the same module and the same rotational speed. During normal operation, each lifting unit must support a load of no less than 400 t, and each pinion is designed to carry about 200 t. In the complete system, six pinions mesh with one rack. To simulate the platform load, I applied a force of 12 000 kN to the center of mass of the rack. The object of my study is one lifting unit of a single leg, specifically the seven-tooth pinions that mesh with the rack and pinion gear arrangement.
The geometry and material parameters of the main components are summarized in the table below. These values were used consistently throughout the dynamic simulations.
| Component | Number of teeth | Module (mm) | Pressure angle (°) |
|---|---|---|---|
| Central sun gear | 12 | 20 | 25 |
| Planet gear | 15 | 20 | 25 |
| Internal ring gear | 42 | 20 | 25 |
| External ring gear | 30 | 38 | 25 |
| Idler gear | 8 | 38 | 25 |
| External gear | 38 | 38 | 25 |
| Seven-tooth pinion | 7 | 100 | 25 |
I created a three-dimensional model of the rack and pinion gear lifting system in a CAD environment and then exported it as a Parasolid neutral file. The model was imported into a multibody dynamics software package. I applied constraints, defined contact forces, set the relevant parameters, and imposed a driving motion on the sun gear. The constraints included revolute joints between the ground and the sun gear, planet gears, pinions, external gear, and ring gear, with the rotation centers located at their respective centers of mass. For the three planet gears, the pinion was used as the reference body when defining the revolute joints with the ground. A translational joint was defined between the rack and the ground. The contact stiffness was calculated using Hertzian theory. The contact stiffness coefficient is given by:
$$K = \frac{4 R^{1/2} E^*}{3}$$
where the equivalent elastic modulus \(E^*\) and the equivalent radius \(R\) are defined as:
$$\frac{1}{E^*} = \frac{1-\mu_1^2}{E_1} + \frac{1-\mu_2^2}{E_2}$$
$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$
In these expressions, \(\mu_1\) and \(\mu_2\) are the Poisson’s ratios of the two contacting materials, \(E_1\) and \(E_2\) are their elastic moduli, and \(R_1\) and \(R_2\) are the equivalent radii of curvature at the contact point. Substituting the material data for the rack and pinion gear components gives a stiffness coefficient of \(3.98 \times 10^6\) N/mm\(^2\). The driving speed applied to the sun gear was 180 °/s, which corresponds to 30 r/min. The simulation time was set to 20 s with a step size of 100 steps. These settings allowed me to capture the dynamic contact forces between the pinions and the rack with sufficient resolution.
When the platform operates under the design lifting load of 12 000 kN, the contact force curves between the pinions and the rack show a clear periodic pattern. The theoretical tangential force on each pinion is 2 000 kN. The theoretical contact force \(F_0\) is obtained from the tangential force \(F_X\) and the pressure angle \(\alpha\):
$$F_0 = \frac{F_X}{\cos \alpha}$$
For a pressure angle of 25°, the theoretical contact force is approximately 2 200 kN. The actual contact forces obtained from the dynamic simulation deviate from this theoretical value because of the unequal load sharing between the two sides of the rack. To quantify this deviation, I define the load sharing coefficient \(\beta\). If the load on the first pinion is \(F_1\), the load on the second pinion is \(F_2\), and the theoretical load on each pinion is \(F\), then the load sharing coefficient for the pinion is:
$$\beta = 1 + \frac{|F – F_1| + |F – F_2|}{2F}$$
This definition provides a direct measure of how evenly the total load is distributed between the two pinions that mesh with opposite sides of the rack. A value of \(\beta = 1\) would indicate perfect load sharing. Any value greater than 1 indicates a certain degree of unequal load sharing. In my simulations, I extracted the contact force histories from the dynamic solver and processed them numerically. Because the rack has three pinions on each side, I selected one contact pair from each side for the calculation. I then computed the load sharing coefficient as a function of time and identified its maximum value as the representative load sharing coefficient for each operating condition. This maximum value is the most conservative measure of load sharing behavior.
The time history of the load sharing coefficient exhibits a periodic variation with a period of approximately 2.5 s. This period corresponds to the time required for a pinion to rotate through one tooth. The load sharing coefficient changes noticeably when the gear shifts between single-tooth contact and double-tooth contact. During double-tooth meshing, the load sharing coefficient is relatively large. During single-tooth meshing, it is relatively small. This behavior is consistent with the change in contact stiffness and load distribution as the number of contacting tooth pairs changes. The periodic fluctuation is an inherent feature of the rack and pinion gear meshing process and must be considered in any dynamic analysis of the lifting system.
I first investigated the effect of the lifting load on the load sharing coefficient. I performed simulations for lifting loads of 12 000 kN, 18 000 kN, 24 000 kN, 30 000 kN, and 36 000 kN. The resulting load sharing coefficients are summarized in the table below. The coefficient decreases as the lifting load increases. The values range from approximately 1.074 to 1.082. This trend suggests that a higher lifting load leads to a more stable platform operation from the perspective of load sharing. In other words, the larger the lifting tonnage, the smaller the relative deviation between the two sides of the rack and pinion gear system.
| Lifting load (kN) | Maximum load sharing coefficient | Observed trend |
|---|---|---|
| 12 000 | 1.082 | Highest among the tested loads |
| 18 000 | 1.079 | Decreasing |
| 24 000 | 1.077 | Decreasing |
| 30 000 | 1.075 | Decreasing |
| 36 000 | 1.074 | Lowest among the tested loads |
The decreasing trend can be explained by the fact that a higher load increases the contact deformation and the effective contact area between the rack and pinion gear teeth. This tends to equalize the load distribution because the system becomes less sensitive to small geometric errors and vibrations. At lower loads, the contact forces are smaller, and the relative influence of manufacturing and assembly errors becomes more pronounced. Therefore, the load sharing coefficient is larger at lower loads. For the design of rack and pinion gear lifting systems, this means that the system should be evaluated at its minimum operating load to ensure that the load sharing coefficient remains within acceptable limits.
Next, I examined the influence of the pressure angle on the load sharing coefficient. I varied the pressure angle from 25° to 29° in increments of 1°. The results are presented in the following table. The load sharing coefficient decreases as the pressure angle increases. The coefficient ranges from 1.067 to 1.082 over the tested range. A larger pressure angle reduces the radial force component and increases the tangential force component for a given torque. This changes the contact geometry and the stiffness characteristics of the rack and pinion gear mesh. The result is a more uniform load distribution between the two sides of the rack. Therefore, within the feasible range of pressure angles, a larger pressure angle is preferable for improving load sharing in the rack and pinion gear system.
| Pressure angle (°) | Maximum load sharing coefficient | Observed trend |
|---|---|---|
| 25 | 1.082 | Highest among the tested angles |
| 26 | 1.078 | Decreasing |
| 27 | 1.074 | Decreasing |
| 28 | 1.070 | Decreasing |
| 29 | 1.067 | Lowest among the tested angles |
The time history of the load sharing coefficient for different pressure angles also shows the same periodic behavior with a period of about 2.5 s. The double-tooth and single-tooth meshing phases are clearly visible in the coefficient curve. The reduction in the maximum coefficient with increasing pressure angle is consistent across all time steps. This indicates that the pressure angle has a systematic effect on the load sharing behavior of the rack and pinion gear system. In practical design, increasing the pressure angle may require adjustments in other parameters such as the tooth profile and the contact ratio. Nevertheless, the results provide a clear direction for optimization.
I then studied the effect of the module on the load sharing coefficient. The module is a fundamental parameter that determines the size of the teeth and the overall dimensions of the rack and pinion gear. I varied the module from 90 mm to 110 mm in increments of 5 mm. The results are shown in the table below. The load sharing coefficient first increases and then decreases as the module increases. The maximum value occurs at a module of 100 mm. At this module, the load sharing coefficient reaches its highest value among the tested cases. When the module is smaller than 100 mm, the coefficient increases with the module. When the module is larger than 100 mm, the coefficient decreases with the module. This non-monotonic behavior indicates that there is an optimal module for load sharing, at least within the range considered. However, the overall variation in the load sharing coefficient with the module is relatively small, suggesting that the module has a limited influence on the load sharing performance of the rack and pinion gear system.
| Module (mm) | Maximum load sharing coefficient | Observed trend |
|---|---|---|
| 90 | 1.068 | Increasing with module |
| 95 | 1.075 | Increasing |
| 100 | 1.082 | Maximum |
| 105 | 1.076 | Decreasing |
| 110 | 1.069 | Decreasing |
The periodic variation of the load sharing coefficient remains evident for all modules. The period is still approximately 2.5 s, which is consistent with the pinion geometry and the rotational speed. The change in the maximum coefficient with the module is gradual. This suggests that the module is not a dominant factor in load sharing compared with the lifting load and the pressure angle. Nevertheless, the existence of a maximum at 100 mm is an interesting finding. It may be related to the interplay between contact stiffness, tooth bending stiffness, and the contact ratio. When the module is too small, the teeth are more flexible and the contact pattern may be less uniform. When the module is too large, the teeth are stiffer but the contact ratio may change, leading to different load distribution characteristics. The optimal module of 100 mm provides a balance between these effects.
To provide a comprehensive summary of the parameter study, I have compiled the ranges of the load sharing coefficient for each parameter in the following table. The table also includes the direction of the trend and the recommended design direction based on my findings.
| Parameter | Range of load sharing coefficient | Trend | Design recommendation |
|---|---|---|---|
| Lifting load (12 000–36 000 kN) | 1.074–1.082 | Decreases with increasing load | Evaluate at minimum load |
| Pressure angle (25°–29°) | 1.067–1.082 | Decreases with increasing pressure angle | Choose a larger pressure angle if feasible |
| Module (90–110 mm) | 1.068–1.082 | First increases, then decreases | Module has limited effect; 100 mm gives maximum |
The overall range of the load sharing coefficient in my study is from 1.067 to 1.082. This means that the maximum deviation from perfect load sharing is between 6.7% and 8.2%. For a rack and pinion gear system in a jack-up offshore platform, this level of unequal load sharing is generally acceptable, but it should be minimized to reduce vibration and extend fatigue life. The periodic nature of the load sharing coefficient implies that the system experiences alternating load conditions. This can lead to fatigue damage over time. Therefore, designers should consider the dynamic load sharing behavior in addition to static strength calculations.
I also examined the relationship between the contact force and the load sharing coefficient more closely. The contact force between a pinion and the rack can be expressed as a function of the applied torque, the pitch radius, and the pressure angle. For a single pinion, the tangential force \(F_t\) is:
$$F_t = \frac{T}{r_p}$$
where \(T\) is the torque applied to the pinion and \(r_p\) is the pitch radius. The radial force \(F_r\) is:
$$F_r = F_t \tan \alpha$$
The total contact force \(F_n\) is then:
$$F_n = \frac{F_t}{\cos \alpha}$$
These relationships show that the pressure angle directly influences the ratio between the tangential and radial force components. A larger pressure angle increases the radial force and decreases the tangential force for a given torque. This changes the contact stress distribution and the deformation of the teeth. In the rack and pinion gear system, the two pinions on opposite sides of the rack experience forces in opposite directions. The unequal load sharing arises because the two pinions do not deform identically. The difference in deformation leads to a difference in contact force. The load sharing coefficient quantifies this difference. My simulations show that increasing the pressure angle reduces this difference, which is beneficial for the rack and pinion gear system.
Another important factor is the contact stiffness. The Hertzian contact stiffness given earlier assumes smooth, elastic contact between two cylinders. In reality, the contact between rack and pinion gear teeth is more complex because of the involute profile, the finite number of teeth, and the changing contact point along the line of action. Nevertheless, the Hertzian formula provides a useful first estimate. The dynamic simulations capture the actual contact behavior more accurately because they include the full geometry and the inertial effects. The contact stiffness affects the load sharing because it determines how the load is distributed among the contacting teeth. When the stiffness is high, the load is concentrated on fewer teeth. When the stiffness is low, the load is distributed over more teeth. The optimal load sharing occurs when the stiffness distribution is balanced between the two sides of the rack.
The module influences the contact stiffness because it determines the size of the teeth. A larger module means larger teeth, which generally have higher bending stiffness but lower contact stiffness per unit width. The interplay between these effects leads to the non-monotonic trend observed in my simulations. At a module of 100 mm, the combination of bending stiffness and contact stiffness appears to produce the most uneven load sharing. This is somewhat counterintuitive because one might expect a larger module to improve load sharing. However, the dynamic effects and the contact ratio must be considered. The maximum load sharing coefficient at 100 mm indicates that this module is not optimal for load sharing, even though it may be optimal for other criteria such as strength or manufacturability. Designers should therefore consider multiple objectives when selecting the module for a rack and pinion gear system.
The load sharing coefficient also depends on the number of pinions that mesh with the rack. In the full system, six pinions mesh with one rack. My study focused on one lifting unit with two pinions. The interaction among multiple pinions can further affect the load distribution. When more pinions are engaged, the total load is shared among more contact points, and the individual load sharing coefficient may change. However, the fundamental trends observed in my study are expected to remain valid. The effect of the number of pinions is a topic for future research.
I also considered the effect of manufacturing errors and assembly errors. Although I did not explicitly introduce random errors in this study, the dynamic model inherently includes the effects of geometric deviations because the CAD model represents the nominal geometry. In reality, errors such as pitch deviation, profile deviation, and misalignment will increase the load sharing coefficient. Therefore, the values reported here represent a lower bound for the actual load sharing coefficient. Designers should apply a safety factor to account for manufacturing and assembly errors. The load sharing coefficient should be kept as close to 1 as possible to ensure reliable operation of the rack and pinion gear system.
The simulation results also provide insight into the vibration behavior of the lifting system. The periodic fluctuation of the load sharing coefficient is a source of vibration. The frequency of this fluctuation is determined by the pinion rotational speed and the number of teeth. For a pinion with 7 teeth rotating at 30 r/min, the tooth meshing frequency is:
$$f_m = \frac{n z}{60}$$
where \(n\) is the rotational speed in r/min and \(z\) is the number of teeth. Substituting \(n = 30\) and \(z = 7\) gives a meshing frequency of 3.5 Hz. The period of 2.5 s observed in the load sharing coefficient corresponds to a frequency of 0.4 Hz, which is the frequency of the single-tooth/double-tooth alternation. This is lower than the meshing frequency because the load sharing coefficient is influenced by the contact ratio and the phase of the meshing cycle. Understanding these frequencies is important for avoiding resonance in the platform structure. If the natural frequency of the platform or the lifting system coincides with the meshing frequency or its harmonics, large vibrations can occur. The load sharing coefficient can be used as an indicator of the dynamic excitation level.
In terms of design optimization, my findings suggest that the lifting load should be considered at its minimum operating value when evaluating load sharing. The pressure angle should be selected as large as practical within the constraints of the tooth profile and the contact ratio. The module should be chosen based on a compromise between load sharing and other design requirements. A module of 100 mm gives the maximum load sharing coefficient in my study, but the variation is small, so other modules may be acceptable if they provide better performance in other areas. The rack and pinion gear system should be designed with sufficient stiffness and damping to reduce the dynamic amplification of the load sharing coefficient. The use of flexible pinions or floating supports can also help to equalize the load between the two sides of the rack.
I also performed a sensitivity analysis to determine which parameter has the greatest effect on the load sharing coefficient. The results are summarized in the following table. The sensitivity is expressed as the change in the load sharing coefficient per unit change in the parameter. The pressure angle has the largest sensitivity, followed by the lifting load, and then the module. This means that small changes in the pressure angle can have a significant effect on load sharing. Therefore, the pressure angle should be tightly controlled during manufacturing and assembly. The lifting load is also important, but it is determined by the platform operation and cannot be adjusted easily. The module is the least sensitive parameter, which confirms that it has a limited influence on load sharing.
| Parameter | Sensitivity of load sharing coefficient | Rank |
|---|---|---|
| Pressure angle | Approximately -0.00375 per degree | 1 |
| Lifting load | Approximately -3.3 × 10⁻⁷ per kN | 2 |
| Module | Non-monotonic; maximum variation about 0.014 | 3 |
The load sharing coefficient is a dimensionless measure, but its interpretation depends on the application. In some standards, a load sharing coefficient greater than 1.1 is considered unacceptable for planetary gear systems. For the rack and pinion gear system studied here, the maximum coefficient is 1.082, which is below this threshold. This indicates that the system has reasonably good load sharing characteristics. However, the coefficient is not constant, and it can increase if manufacturing errors are present. Therefore, a safety margin should be included in the design. A target load sharing coefficient of 1.05 or lower would be desirable for critical applications.
The dynamic model I used includes the flexibility of the gear teeth through the contact stiffness. It also includes the inertia of the rotating components and the translational inertia of the rack. The model does not include the flexibility of the platform structure or the leg. In a full system simulation, the platform and leg flexibility would add additional degrees of freedom and could affect the load sharing. The leg is a long slender structure that can bend under load. This bending can cause misalignment between the rack and the pinions, which would increase the load sharing coefficient. Therefore, the actual load sharing coefficient in a real platform may be higher than the values reported here. Future work should include the flexibility of the platform and leg in the dynamic model.
I also note that the load sharing coefficient is affected by the phase relationship between the two pinions. In the differential gear train, the two pinions are connected through the gear train. The phase of their meshing with the rack depends on the initial assembly and the accumulated rotation. If the two pinions are not perfectly synchronized, the load sharing will be worse. The dynamic model captures the phase relationship because it simulates the full gear train. The results show that the load sharing coefficient oscillates with time, which indicates that the phase relationship changes continuously. This is a natural consequence of the meshing process. The maximum coefficient occurs when the phase relationship is least favorable. Therefore, the maximum value is the most important design parameter.
To further illustrate the effect of the parameters, I have derived an approximate expression for the load sharing coefficient based on the contact stiffness and the applied load. If the two pinions have slightly different contact stiffnesses \(K_1\) and \(K_2\), and the total load is \(F_{\text{total}}\), then the loads on the two pinions are:
$$F_1 = \frac{K_1}{K_1 + K_2} F_{\text{total}}$$
$$F_2 = \frac{K_2}{K_1 + K_2} F_{\text{total}}$$
The theoretical load on each pinion is \(F = F_{\text{total}} / 2\). Substituting these into the definition of the load sharing coefficient gives:
$$\beta = 1 + \frac{|F – F_1| + |F – F_2|}{2F} = 1 + \frac{|K_1 – K_2|}{K_1 + K_2}$$
This simplified expression shows that the load sharing coefficient depends on the relative difference in contact stiffness between the two pinions. If \(K_1 = K_2\), then \(\beta = 1\). If the stiffness difference is large, \(\beta\) is large. In reality, the contact stiffness varies with the contact position along the tooth profile and with the applied load. The dynamic simulation captures these variations. The simplified expression provides a useful qualitative understanding. It also shows that reducing the stiffness difference between the two sides is the key to improving load sharing. This can be achieved by improving manufacturing accuracy, using flexible elements, or adjusting the pressure angle and module to balance the stiffness.
The load sharing coefficient also depends on the friction between the teeth. Friction opposes the sliding motion of the teeth and can affect the contact force distribution. In my simulations, I included a friction coefficient in the contact model. The effect of friction on load sharing is secondary compared with the effects of load, pressure angle, and module. However, friction can cause additional heating and wear, which may change the contact geometry over time. For long-term operation, the load sharing coefficient may drift due to wear. Therefore, regular maintenance and inspection of the rack and pinion gear system are important.
I have also considered the effect of the rack stiffness. The rack is a long beam that supports the pinions. If the rack is flexible, it will deflect under the load, and the deflection will affect the contact force distribution. In my model, the rack was treated as a rigid body. This is a reasonable assumption for a single lifting unit because the rack segment between two pinions is short. However, for a full leg with multiple lifting units, the rack stiffness may become important. Future studies should include a flexible rack model to investigate this effect. The load sharing coefficient may increase if the rack is flexible because the deflection can cause misalignment.
The results of my study have several practical implications for the design and operation of rack and pinion gear lifting systems in offshore platforms. First, the load sharing coefficient should be evaluated at the minimum operating load, not at the maximum load. Second, a larger pressure angle is beneficial for load sharing, so the design should use the largest pressure angle that is compatible with the tooth profile and the contact ratio requirements. Third, the module should be selected based on a compromise between load sharing and other design criteria. A module of 100 mm gives the maximum load sharing coefficient, but the variation is small, so other modules may be chosen if they offer advantages in strength, manufacturability, or cost. Fourth, the system should be designed with sufficient damping to reduce the dynamic amplification of the load sharing coefficient. Fifth, manufacturing and assembly tolerances should be tight, especially for the pressure angle, because it has the largest sensitivity. Sixth, the platform and leg flexibility should be considered in the design to avoid unexpected increases in the load sharing coefficient.
In conclusion, I have established a dynamic model of the rack and pinion gear lifting system of a jack-up offshore platform and used it to study the load sharing characteristics of the pinions. The load sharing coefficient decreases with increasing lifting load and increasing pressure angle. The coefficient varies between 1.067 and 1.082. The load sharing coefficient first increases and then decreases with increasing module, reaching a maximum at a module of 100 mm. The pressure angle has the strongest influence on load sharing, followed by the lifting load, and then the module. The periodic variation of the load sharing coefficient with a period of about 2.5 s is caused by the alternation between single-tooth and double-tooth meshing. The results provide a theoretical basis for the design optimization of rack and pinion gear lifting systems. By selecting appropriate parameters and controlling manufacturing errors, the load sharing coefficient can be minimized, leading to reduced vibration, noise, and wear, and improved reliability of the offshore platform.
I believe that this work contributes to a better understanding of the dynamic behavior of rack and pinion gear systems in heavy-duty lifting applications. The methodology can be extended to other types of gear systems and other loading conditions. The findings can be used by designers to improve the performance of jack-up offshore platforms and to ensure safe and efficient operation in harsh marine environments.