Gear design has multiple parameters from which the gear designer can choose – while some are beneficial, some can negatively affect the system; therefore, a structured approach needs to be taken in order to define a custom pinion and rack for a liftboat.

Liftboats or jack-up oil rigs are essential for the oil and gas industry. The subsystem (rack, pinion, gearbox, and brake) within the system lift boat has specific requirements that need to be customized appropriately. This paper shall outline the design process of a rack-and-pinion system and show how the design can be optimized.

The nature of gear design is that it has multiple parameters from which the gear designer can choose. While some are beneficial, some can negatively affect the system. Therefore, a structured approach needs to be taken in order to define a custom pinion and rack for a liftboat. The influence of major gear parameters, such as modul, profile shift coefficient, as well as the pressure angle, will be analyzed, and their impact on the design discussed. Based on the previous discussion, a gear design will be selected and analyzed. As a next step in the design process, the optimization phase will show how to reduce the contact stress in the system, as well as change from an involute profile into a multi-radii pinion design. Both pinion designs will be compared and validated with FEA (finite element analyses) calculation to show and validate the improvement. The final design will not only extend the life of the product, it will save the customer a great deal of time and financial resources, as it will last longer in service. This paper can be used as a guideline and starting point to design a rack and pinion for offshore jacking applications.

1 Introduction rack and pinion design

Offshore liftboat applications are used, and their specifics are long- known in the oil and gas industry. The typical operation modes, as well as the details for the materials used for the rack and pinion, are explained in detail for these applications [1]. This paper will pay attention to the design of the pinion and show potential to optimize the system and increase the life in regard to contact stress. In these applications, the main focus pays attention to maintaining sufficient margin for root-bending stress.

For all designs shown herein, the assumption can be made that this condition is met. The nature of gear design is that the gear engineer has multiple parameters that influence each other and improve the design toward:

  • Increased life expectation.
  • Reduction in root bending stress and contact stress.
  • Reduced cost.
  • Increase power density.
  • Weight reduction.

To start a design, the loads at the pinion need to be known. The naval architect can provide the weight of the legs and the hull to get the pinion basic layout. In the next sections, this paper will discuss how parameters such as profile shift, diametral pitch, pressure angle, and number of teeth will affect the contact stress and what strong design characteristics are influenced by these. All parameters discussed in the next section are defined and calculated per standard calculation [2, 3].

1.1 Influence of module (Diametral pitch)

The diametral pitch (module), in combination with the number of teeth, are the most important basic parameter in gear design, as shown in section 1.4, Figure 4, the correlation between module, pitch diameter, and gearbox output torque. The weight of hull and legs are given by the naval architect and translating into the pinion load in metric tons. The pinion load is held constant, a higher module (smaller diametral pitch) is selected, and the gearbox output torque will increase.

Vice versa, a smaller module (higher diametral pitch) gearbox torque will decrease. With a higher module (smaller diametral pitch), these general conclusions apply:

  • Increased root bending strength.
  • Reduced contact stress.
  • Increase of pitch diameter (pinion size).
  • Increased rack width and cost.

Typically, it is intended to keep the gearbox torques low, to reduce weight and cost in the system. Therefore, the selection of the diametral pitch (module) will determine the gearbox output torque significantly. To define the diametral pitch (module), it is important to meet the root-bending stress requirements for this rack and pinion. Doing so will establish the baseline for this design. If the system is designed too conservatively, the cost and weight will increase drastically and will affect the sizing of the rack, pinion, gearbox, brake, and motor.

1.2 Influence of profile shift coefficient

The profile shift applied to the pinion will be adjusted in the rack through the calculated center distance, as well as the length (or factor) of the dedendum and addendum definition. While modifying the profile of the pinion positive, the designer needs to keep the minimum tooth thickness at the outside diameter in mind. That sets the limit to the maximum applicable profile shift coefficient at the pinion. If the profile is shifted positively on the pinion, the following conclusions can be taken:

  • Smaller tooth thickness at top (sharp tooth tip).
  • Increased contact ratio.
  • Extension of center distance.
  • Reduction in contact stress.
Figure 1: Pinion comparison in profile shift.
Table 1: dNF active root diameter.

In Figure 1, we see the gear profile changing as specified with the profile shift factor. Table 1 points out that the root form diameter is not linear, growing as the outside diameter increases as well as the tooth thickness decreases with a higher positive profile shift modification. The designer shall keep in mind that the same characteristics seen on the pinion will apply to the rack. The literature [3, 4] recommends maintaining the minimum tooth thickness at the outside diameter according to Equation 1.


san is normal tooth thickness at tip circle.

m is module.

The major concern is that the small contact ratio in the rack and pinion design will lead to high plastic deformation at the rack tip. It also can be taken into consideration that a smaller profile modification can account for more plastic deformation on the pinion, and it allows for adjusting the center distance accordingly during a scheduled maintenance after five or 10 years of service. In practice, after significant hours of service, it is often seen that tooth tips are as sharp as “razor blades,” which will lead to an immediate replacement of the pinion. Since the racks are made out softer material, they will wear into a much more optimal shape during operation, rack teeth are seen to be deformed with two significant concave deformations over the line of action.

1.3 Influence of pressure angle α

In the industry and on all kinds of applications, higher pressure angles are widely discussed [5, 6] and used. For this specific application, a higher pressure angle is common to achieve higher root-bending strength and reduced contact stress. For the purpose of evaluating the impact on contact stress for a given load, pressure angles of 25˚, 28˚, 30˚, and 32˚ are investigated while maintaining the same profile shift, as well as diametral pitch. According to involute generation theory, a gear profile is covering pressure angles from 0˚ to 60˚. It is the authors understanding that for this specific application, the definition of the base diameter db is more important in the initial design phase, when the pressure angle is part of Equation 2. As pointed out, it is preferred to have a small pinion pitch diameter within the system to reduce the output torque at the gearbox in the system. (Figure 2)


α is the pressure angle.

m is the module.

z is the number of teeth.

Figure 2: Generation of the involute function.

As shown in Equation 2, the impact of a higher-pressure angle generates a smaller-base diameter and vice-versa. A smaller pressure angle generates a bigger base diameter. Typically with higher-pressure angles the following characteristics are benefiting rack and pinion:

  • Reduction in root bending stress.
  • Reduction in contact stress.
  • Reduction in contact ratio.
  • Higher radial loads.

As shown in Figure 3, below the reduction in root-bending and contact stress is slowing at a pressure angle of 28 degrees. Therefore it can be concluded that higher-pressure angles above 30˚ tend to have more technical disadvantages than the gain in root bending-strength, as well as pitting resistance.

Figure 3: Root bending and contact stress depending on pressure angle.

1.4 Influence of number of teeth at the pinion

As broadly explained, the rack and pinion design will determine the gearbox output torque and therefore will influence the system in size and cost. Figure 4 shows the linear relation of pitch diameter, number of teeth on the pinion, and gearbox torque needed to operate at a constant pinion load. As illustrated, the pinion torque for a 7-tooth pinion starts ≈ 121,697 ft-lbs (165 kNm) and increases for a 9-tooth pinion up to 165,951 ft-lbs (225 kNm). In the liftboat fleet, there are rack and pinions known with more than 10 teeth. Typically at a given load and module, a higher number of teeth at the pinion reduces bending — and contact — stress significantly. This enhanced life for pinions with more than 8 teeth can be explained with a much longer radius of curvature as well as associated a better contact ratio compared to a 7-tooth pinion. The trade-off for that extended pinion life is a heavier jack-up gearbox and pinion. In the industry, pinions made with 7 or 8 teeth are the most common. The downside of a low number of teeth for the pinion is the increased number of load cycles. For example, a pinion with 8 teeth would need 12.5 percent fewer revolutions, and a 9-tooth pinion would need 22.2 percent fewer revolutions compared to a 7-tooth pinion to achieve the same travel distance on the rack. On the contrary, let’s say a 10-tooth pinion will be able to last longer, and the plastic deformation is less significant, compared to a 7-tooth pinion, assuming the same amount of service hours. It would also be possible to adjust the center distance after an x amount of service to accommodate the plastic deformation on the pinion. It can be concluded in this discussion that it is rather a decision within the system on the number of teeth the pinion should have.

Figure 4: Required Gearbox output torque.

1.5 Rack and Pinion Geometry

Based on the discussion in sections 1.1 to 1.4, the following gear design has been chosen for the pinion and rack. Due to the manufacturing process and this specific application, the rack and pinion need to account for sufficient backlash and tip clearance. Table 2 and Figures 5 and 6 illustrate the chosen gear geometry. In the next step, this rack and pinion system will be analytically evaluated and further optimization potential discovered.

Figure 5: Involute Pinion design.
Figure 6: Involute Rack design.
Table 2: Specified rack and pinion design.

2 Analytical evaluation and Optimization

The basic hertzian stress calculation according to gear standard calculations [7, 8] is done based on the geometry and materials given. As typical for these applications, the calculation will be separated between static loads as well as combined loads, according to certification body requirements [9].

2.1 ISO 6336 vs. AGMA 2001-D04 Contact stress assessment

As expected, the high-loaded tooth flank results in high contact stress. Based on these results and a service life of more than 20 years, the life expectation per gear calculation is zero due to exceeding the permissible contact stress. (Table 3)

Table 3: Pinion load – contact stress.

2.2 Hertzian contact theory

Based on the hertzian contact theory [10], the rack-and-pinion contact stress can be analyzed in more detail. In order to take a deeper look into the involute profile of the pinion, an approximation of the radius in the dedendum must be taken and used. As shown in Figure 7, the radius is starting at the SAP (dNf) and will meet the involute profile tangentially approximately at the pitch diameter d. This model is an analog to a cylinder in contact with a flat surface.

Figure 7: Involute Pinion design analogue to cylinder on flat surface.

This method enables the designer to explore stress-reduction potentials, as well as analyzes the principle stress in each direction of this system. Figure 8 shows the comparison between an involute profile pinion and a multi-radii pinion. For the load case “preload jacking,” the nominal contact stress sc is with the involute profile design 396 ksi (2,732 N/mm2). It is intended to increase the dedendum radii up to 4.13 inches (105 mm) while the nominal contact stress drops to 347 ksi (2,395 N/mm2). In the daily business of gearbox manufacturing, we use the same principle when the bearing life is not met. The first typical iteration of the bearing supplier is to increase the number of rollers or to increase the roller diameter.

Figure 8: Involute profile vs. multi radii contact stress.

Equations 3 to 5 as mentioned in [10] illustrate how the contact stress is calculated. Based on Equation 4, changes in rPinion are most sensitive to the result of the contact width a of the cylinder in contact. In a system where loads are given and face width is determined by the steel plate, rPinion is the variable with the most leverage to reduce the contact stress.


E is modulus of elasticity.

L is length of cylinder.

a is 1/2 of the contact width.

F is Force.

rPinion is radius of the dedendum.

2.3 Pinion detail design

As discussed in the previous section, the pinion design will have an increased dedendum radius, compared to the approximation of the involute profile. In theory, this will reduce the nominal contact stress by 10 percent or more depending on the pinion design and tooth form. Subsequently, the involute form will be replaced by 3 different radii, plus tip- and root- radius. All curvatures will mate tangentially with each other. Figure 9 shows the transformed and stress-optimized pinion design. The involute pinion design will only be used as a theoretical step to get to the multi radii design, where the pitch diameter is only a theoretical reference from the involute profile.

Figure 9: Optimized radii pinion design.

2.4 System influence

This article will also cover the system aspect of pinion and rack. It is important for the life of the rack and pinion that the pinion is well aligned relative to the rack to maintain the predicted contact stress. As typical in the shipbuilding industry and a marine environment, the precision is limited to the capability of handling big machinery pieces and welding tolerances. Therefore the legs are typically guided as shown in Figure 10 [12]. The guides are adjusted during installation and custom fit for each leg. It is important to monitor the wear of the guide plates as well as the rack deformation and backlash between rack and pinion. The backlash shall be defined with sufficient tolerance with respect to the system tolerance. The maximum backlash tolerance shall be defined to maintain a minimum contact ratio of εα = 1.015. It also needs to be mentioned that the whole jacking tower strains within the elastic limits of the material during jacking operation. This enables
the rack to move relative to the pinion until an equilibrium of pinion forces is in place within the tower structure. All these aspects need to be taken into consideration for the pinion designer and the naval architect.

Figure 10: General arrangement leg guides.

3 Numerical evaluation

Linear FEA analyses were carried out using ANSYS R19.0 to analyze the stress of each pinion design. The rack and pinion was modeled so the contact of rack and pinion is in the lowest point of single tooth contact (equal to point B per DIN and ISO) of the pinion. The mesh was discretized in the area of contact to achieve the most accurate results. The load cases leg operation, pre-load operation and maximum storm hold were used to analyze and compare both designs and results.

3.1 Linear FEA at storm holding comparison Involute vs. radii pinion

The results of the FEA contact stress is showing typically 5 percent to 10 percent lower than calculated stress based on hertzian contact theory. Comparing the results with AGMA/ISO contact stress calculation, the difference is typically 18 percent to 23 percent lower. Figures 11 and 12 illustrate the results for the load case of preload jacking for both pinion designs.

Figure 11: Radii pinion FEA results on rack.
Figure 12: Involute pinion FEA results on rack.

The FEA results agree with the analytical approach of modifying the involute pinion design to a multi radii design. Furthermore, we have reduced the pinion stress by approximately 11 percent of what will increase the pinion and rack life. It also confirms that a custom design for rack and pinion is essential for these applications. The naval architect can now find a good compromise between rack widths and pitch diameter to keep the cost for the jacking system as low as possible.

4 Conclusion

Table 4 summarizes all results of each pinion design and calculation method. It can be concluded that the contact stress calculation per AGMA and ISO, with their allowable stress sac/σHP are not fully applicable to this specific application. However, the contact stress needs to be carefully evaluated and analyzed. The difference between the FEA results of the involute profile and the standard calculation method is not accurate enough (20% – 23%) difference. This may lead to an overly cautious design with higher cost and weight. On the other hand, the difference between the hertzian contact theory and FEA results of the involute pinion is 5 percent — 3 percent close enough. Interestingly, the FEA results from the radii design are much farther apart to the results of the hertzian contact theory
(12.5% – 8.5%). Overall, it can be concluded that a custom pinion design will reduce the contact stress drastically and extend the expected life of the system.

Table 4: Contact stress comparison.

Under the first operational loads, the rack and pinion will plastically deform. It is the author’s understanding that residual stresses will be induced during that run-in period in the material and prevent further deformation while elastic. This phenomenon is called elastic shakedown [11] or strain hardening. It is widely used in pressure vessel analyses and evaluation. Since all influences are not fully understood, it is recommended to monitor the deformation of rack and pinion in regular intervals. Reasonable means could be backlash or pinion tip clearance to the rack. A specified gauge could determine the acceptance criteria, and the service could be performed in a timely manner. It is up to the liftboat operator and government requirements as to what kind of measures are taken to maintain safe and sound jacking operation during the life of a liftboat. 


  1. Nowoisky, Adrian, 2017 “Predicting Life on Through Hardened Steel Rack and Pinion for Jacking Applications in the Offshore Industry,” AGMA FTM17, Dana Incorporated, p. 2-3.
  2. DIN 3960: 1987, Definitions, parameters and equations for involute cylindrical gear pairs.
  3. AGMA 913 – A98: 1998, Method for specifying the geometry of spur and helical gears.
  4. Matek, Wilhelm., 2001, Roloff/ Matek Maschinen- elemente, edition 15, VIEWEG Verlag, Page 673, Chap. 21.1.4 – 3.
  5. Dr. Fuentes-Aznar, Alfonso, 2017 “Gear Tooth Strength Analysys of High Preassure Angle Cylindrical Gears,” AGMA FTM17, Rochester Institute of Technology, p. 2.
  6. Miller, Rick, 2016 “Designing Very Strong Gear Teeth by Means of High Pressure Angles,” AGMA FTM16, Innovative Drive Solutions LLC, p. 2.
  7. ANSI/AGMA 2101-D04, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth (Metric Edition).
  8. ISO 6336-2:2006, Calculation of load capacity of spur and helical gears – Part 2: Calculation of surface durability (pitting) + Corrigendum ISO 6336-2/AC1:2008.
  9. ABS Mobile Offshore Drilling Units 2016, Part 6-Chapter 1 -Section 9 “Jacking and Associated Systems,” p. 79.
  10. Ansel C. Ugural, Saul K. Fenster; “Advanced Mechanics of materials and applied elasticity,” Fifth edition, p.163.
  11. Williams, John A.; “Contact between solid surfaces”; 2001, p. 36.
  12. Picture courtesy of Allriggroup.

Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented September 2018 at the AGMA Fall Technical Meeting in Oakbrook, Illinois. 18FTM03

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Adrian Nowoisky is a senior product engineer at Dana Incorporated specializing in designing custom gearboxes for industrial applications. He earned a Dipl. Ing. (FH) degree from the Anhalt University of Applied Sciences in Koethen/ Germany in 2005. In his professional experience of more than 15 years, Nowoisky took part of developing a dual clutch transmission at ZF Getriebe GmbH, accessory gearboxes for jet engines at Rolls-Royce, as well as jack up and industrial gearboxes. At the ThyssenKrupp Marine Systems shipyard in Kiel/ Germany, he gained experience designing machinery for submarines.