Analysis of the influence of gear root chamfer radius on the strength of climbing gears

For self-elevating units, the lifting locking system is one of the most critical equipment and has important applications in the field of marine engineering. The reliability of the jacking and fixation system is crucial for the safety of the platform, and the jacking and fixation system, the strength of the climbing gear and rack directly affects the reliability of the system. Therefore, optimizing the structural parameters of the climbing gear and rack to enhance their strength has always been a research hotspot in the field of marine engineering. This article uses ANSYS simulation software to model the climbing gear and rack, in order to analyze the impact of gear root chamfer radius on the strength of the climbing gear.


Introduction
Self-elevating units have the advantages of low cost, low steel consumption, continuous operation under various sea conditions, and high efficiency, thus playing a huge role in the development of offshore oil and gas resources both domestically and internationally [1].The jacking system, as a key equipment in the design and manufacturing of self-elevating units, has always been highly valued [2].Compared to hydraulic cylinder jacking, gear and rack jacking operates more smoothly and is easier to operate, resulting in a higher usage rate of gear and rack jacking on offshore platforms [3].The jacking device usually consists of an electric motor or hydraulic motor, a gearbox, a brake, a small gear, etc [4].
The super-large module gear rack has a strong load-bearing capacity, which can achieve continuous platform lifting, fast speed, and flexible operation [5].Therefore, this super large module gear rack Jacking device is often used on self-elevating units.The gear and rack jacking device consists of a climbing gear, a gearbox, an electric motor, and a brake.It adopts a single-line transmission and is installed on a fixed pile frame, meshing with the rack to bear the vertical load between the platform and the pile legs.During jacking and operation support, the meshing of gears and racks bears the weight of the entire platform and the environmental loads generated by wind, waves, and currents.Therefore, gears and racks are the main load-bearing components of the jacking system.It is necessary to analyze the load-bearing capacity of gears during the jacking process to ensure the safety of the system [6].Figure 1 shows the lifting gear of a certain self-elevating platform.

Figure 1.
Climbing Gear of a Self-elevating unit.As shown in Figure 1, which has the characteristics of a large modulus and a small number of teeth.Engineering experience has shown that the strength failure of the pinion and rack is one of the most common faults on the platform [7], which has an extremely important impact on the safety and economic performance of the platform during service.Generally speaking, the tooth root circle is the point where stress concentration occurs and the strength is relatively weak [8].Therefore, reducing the bending stress and contact stress of the gear and rack through structural optimization and improvement is of great significance for improving the load-bearing level of the gear and rack, and improving the service life and work safety of the self-elevating unit gear and rack [9].
The calculation formula for tooth root bending stress is as follows [10]: where bending stress; nominal tangential force of the dividing circle; represents tooth width; represents modulus; represents correction coefficient related to gears, related to engineering experience.
The calculation formula for tooth contact stress is as follows: where represents correction coefficient related to gears; represents basic values of gear contact stress.
In addition, gears are prone to tooth surface wear and failure.This article also verifies the relevant state of tooth surface contact stress, which has certain significance for the design, construction, and reinforcement measures of climbing gears.

Model parameter
Simulation is carried out using Ansys APDL.Taking a certain platform as an example, the module of the climbing gear is 97.02, the number of teeth is 7, the modification coefficient is 0.3, and the pressure angle is 30°.The specific parameters of the model are shown in Table 1.The tooth root chamfers are set to non-chamfered, 15°, and 30° climbing gears, and the models are shown in Figures 2-4.

Mesh
Due to the need to use 3D solid units for calculation in this modeling, the Solid45 unit in Ansys was used to establish the overall model of the gear and rack.For the target surface in contact, TARGE 170 units are used for unit generation.This enables the transmission of contact force between gears and racks.Considering the actual lubrication situation of the gears and racks used on the self-elevating drilling platform, sliding friction coefficients of 0.1, 0.2, 0.3, 0.4, and 0.5 were taken in the calculation.

Boundary
Boundary conditions are applied to the model based on the actual working conditions of the gear and rack.Due to the vertical lifting of the gear, three translational degrees of freedom in the horizontal direction of the gear should be limited, namely the x, y, and z directions.In addition, considering that the current analysis is a static analysis mainly used to solve the static behavior of the structure under manager load, it is necessary to restrict three rotational degrees of freedom for gears, namely the x, y, and z axes.To facilitate the setting of gear degrees of freedom, a node is established at the center of the gear, and the gear circumferential grid node is rigidly coupled to this point, limiting the translational and rotational degrees of freedom in three directions at this point.For a gear rack, its rotational degrees of freedom in three directions and translational degrees of freedom in the y and z directions should be limited to simulate the gear lifting state.Gear torque is converted into rack climbing force, thereby adding rack end face load.Under storm conditions, the single tooth force is 444.288T, which is converted into a surface load of 78 N/mm 2 .

Root bending stress
Usually, the stress at the root of a gear tooth is not solely due to bending stress.Bending stress dominates.When gears are subjected to external loads, in addition to generating bending stress, they also generate other stresses such as shear stress.Some scholars have shown that the error between bending stress and combined stress in terms of quantity is less than 5%.To avoid tedious calculations, in engineering, the calculation of tooth root stress is generally only considered for bending stress calculation, and other stresses are omitted [6].
The calculation results show that the maximum stress of the gear occurs near the tooth root.The stress cloud diagram is calculated using a sliding friction coefficient of 0.2, as shown in Figures 5-7.For non-chamfered, 15°, and 30° climbing gears with root chamfers, the maximum bending stresses under the same conditions are 805 MPa, 781 MPa, and 765 MPa, respectively.
The maximum bending stress under sliding friction coefficient conditions of 0.1, 0.2, 0.3, 0.4, and 0.5 is extracted as shown in Figure 8.The horizontal axis represents the friction coefficient, and the vertical axis represents the maximum bending stress.The bending stress is observed at 0°, 15°, and 30°b ending radii.
Figure 8. Maximum bending stress under different friction coefficients.According to the calculation results, as the chamfer of the tooth root increases, the maximum bending stress of the gear decreases.Therefore, it can be considered that the selection of a suitable tooth root circle radius has a certain impact on the reduction of tooth root bending stress.
In addition, as the surface friction coefficient increases, the maximum tooth root bending stress decreases.However, an excessive friction coefficient can lead to poor lubrication and affect mechanical life.Therefore, the optimal friction coefficient needs to be determined based on actual conditions [11].

Tooth contact stress
The contact surface between gears and racks is often accompanied by local high stress.In addition, under the influence of cyclic loading of gears and racks, it is easy to yield or crack the materials of gears or racks, and cause fatigue failure.This is also a common form of failure in practical engineering.So understanding the contact and stress states of gears and racks is of great significance for the design, construction, and reinforcement measures of climbing gears.
Taking a gear with a root circle chamfer of 15° as an example, the distribution of contact stress at the tooth surface can be seen in Figure 9.The contact stress on the tooth surface is distributed in a U-shape along the z-axis direction.The maximum surface contact stress is located at the endpoint of the rack contact, and the change in contact stress is not significant as the surface friction coefficient increases.
The stress value is extracted along the x-axis (tooth) direction through the point of maximum surface contact stress, as shown in Figure 11.The horizontal axis in the figure represents the X-direction distance of the gear, and the vertical axis represents the contact stress on the tooth surface.The results are observed under a friction coefficient of 0.1-0.5.Along the x-axis (tooth) direction, the surface contact stress is distributed in an n-shape, with the maximum stress occurring at the contact center.As the surface friction coefficient increases, the change in contact stress is not significant, so it is believed that the surface friction coefficient has a small impact on the contact stress on the gear surface.

Conclusion
(1) The tooth root circle has a significant impact on reducing stress concentration.As the radius of the tooth root circle increases, the maximum stress tends to decrease.However, as the radius of the tooth root circle increases, the tooth thickness at the tooth root also decreases accordingly.Therefore, obtaining the optimal tooth root circle radius based on analysis is crucial for improving the strength of climbing gears.
(2) The surface contact stress is distributed in an M-shape along the gear axis direction and in a U-shape at the contact surface with the gear rack.Therefore, local strengthening treatment can be applied to the contact endpoint between the gear surface and the rack.
(3) The increase in friction coefficient has a certain impact on the reduction of bending stress.However, an excessive friction coefficient can lead to poor lubrication and affect mechanical life.Therefore, the optimal friction coefficient needs to be determined based on actual conditions.

Figure 5 .
Figure 5. Cloud diagram of chamfer calculation without tooth root.

Figure 9 .
Figure 9. Distribution of Contact Stress.To quantitatively analyze the results, the stress values will be extracted along the z-axis (gear axis) direction through the point of maximum surface contact stress, as shown in Figure10.The horizontal axis in the figure represents the Z-direction distance of the gear, and the vertical axis represents the contact stress on the tooth surface.The results are observed under a friction coefficient of 0.1-0.5.

Figure 10 .
Figure 10.Z-axis contact stress distribution curve.The contact stress on the tooth surface is distributed in a U-shape along the z-axis direction.The maximum surface contact stress is located at the endpoint of the rack contact, and the change in contact stress is not significant as the surface friction coefficient increases.The stress value is extracted along the x-axis (tooth) direction through the point of maximum surface contact stress, as shown in Figure11.The horizontal axis in the figure represents the X-direction distance of the gear, and the vertical axis represents the contact stress on the tooth surface.The results are observed under a friction coefficient of 0.1-0.5.

Figure 11 .
Figure 11.Contact stress distribution curve in the x-axis direction.Along the x-axis (tooth) direction, the surface contact stress is distributed in an n-shape, with the maximum stress occurring at the contact center.As the surface friction coefficient increases, the change in contact stress is not significant, so it is believed that the surface friction coefficient has a small impact on the contact stress on the gear surface.