Finite Element Analysis of a Floating Planetary Ring Gear with External Splines

An investigation of the stresses and deflections of a floating ring gear with external splines working in a large planetary wheel motor of a mining truck. 

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The design of a planetary ring gear is always a challenge due to the contradictive requirements it must comply with. On one hand, the ring gear should be strong and stiff enough to successfully carry the applied load. On the other hand, it should have as small volume as possible to account for the radial restraints of a planetary gearbox imposed by the truck tires. Additionally, the ring gears are floating in many cases. Altogether, the designer job becomes very complicated.

One of the design issues of the strength calculations of the ring gear is that none of the American Gear Manufacturers Association (AGMA) standards, including ANSI/AGMA 2001-D04 [1], rate internal gears. No acceptable methodology is defined to calculate geometry factor for internal gears, following the method for external gears in ANSI/AGMA 908-B89 [2]. ANSI/AGMA 2001-D04 provides guidelines for the calculations of the ring gear rim thickness by introducing the rim thickness factor. 

Very often the ring gears have external spline teeth, which transmit the torque to the final driven member of the gearbox. ANSI/AGMA6123-B06 [3] gives a methodology for calculating splines, which include shear capacity, fretting, and wear as well as ring bursting. The standard assumes that 50% of the splines are carrying the torque. Other approaches suggest different ways to calculate splines [7].

The stresses in the gear and the spline teeth are influenced by the deflections of the gear itself and also by the deflections of the entire gearbox. AGMA does not have published codes for calculating these deflections. It stresses in different standards like ANSI/AGMA 2001-D04 the importance of determining the deflections, and provides examples of using Finite Element Analysis (FEA) methods.

Because of these difficulties in the engineering design of ring gears, more and more researchers are using modern calculation methods like FEA. One researcher showed that when properly used, FEA and AGMA methods give closer results [6]. In this study a ring gear assembly is evaluated using FEA software in order to determine the stresses and deflections in the system.

Design Model
The assembly consists of a carburized ring gear and a through hardened torque tube (Figure 1) used in the wheel motor of large electrically driven mining truck. The torque path comes from three planet gears (not shown and used in the study), whose teeth are meshed with the internal teeth of the ring gear. The ring gear transmits the torque to the torque tube through its external splines. The torque tube transmits the torque from its internal splines to the hub through a bolted joint and from there to the truck tires. 

The meshing areas between each planet and internal gear teeth are marked as zones 1, 2, and 3 (Figure 2). Each one is modeled with different root radii and crowning of the external splines. The gear and spline teeth are ground to quality 6 per ANSI/AGMA 2015-1-A01 [4] and ANSI/AGMA 20015- 2- A06 [5].

Loading and Constraints
The applied torque is assumed to be equally divided among the three planets per ANSI/AGMA6123-B06 (Figure 3), which recommends a mesh load factor of “unity” for high speed and high quality gears for the presented case. For the purpose of this study the force acting on each tooth is applied at the highest point of single tooth contact for each location (Figure 4) and equally distributed along the line of contact, which means using load distribution factor of “unity.”

The application of the force transmitted through the splines is determined by the FEA software. The ring gear is floating in all directions. The radial and circular movement of the gear is limited by the backlash in the assembly and the axial movement is restricted by axial stoppers, which are not shown.

FEA Modeling
A nonlinear static analysis of the ring gear and torque tube was conducted in ABAQUS. The ring gear teeth and the splines of the ring gear and torque tube were modeled with a fine mesh of 8-node brick elements. The model transitioned to a coarser mesh of linear tetrahedral elements away from the splines (Figure 5).

Using a cylindrical coordinate system, the loads were applied on three internal teeth of the ring gear. Contact pairs were used to capture the interaction between ring gear and torque tube splines. These contact pairs were defined between the mating surfaces of the ring gear’s external splines and the torque tube’s internal splines. The model had about 3.5 million nodes and the analysis was run on a supercomputer in order to obtain results in a reasonable time frame.

Results
FEA snapshots of only the overall displacement magnitude of the ring gear and the splines stresses are shown on Figure 6 and Figure 7. Von Mises, maximum and minimum principle root stresses in locations I, II, and III of only one tooth flank (Figure 8) of the ring gear teeth and external splines, and torque tube internal splines, are shown on the graphs of Figure 9 through Figure 17.

Conclusions
• The analysis shows heavy triangulation of the ring gear (Figure 6).

• Only about 10% of the splines teeth carry most of the load at the same time. Figure 7

• Close analysis of these stressed teeth (not shown) points out that the stresses are not evenly distributed and only few teeth at a time take the highest load. This leads to the conclusion that the fatigue calculations of splines with similar behavior are as important as the shearing and wearing calculations. Figure 8

• In many instances the principle stresses are higher than the allowable fatigue stress numbers per ANSI/AGMA 2001-D04. Figure 9

• The higher crowning increases the stresses and that is why the stresses in zone 2 are generally smaller than the other zones. Figure 10

• The smaller root radius increases the stress and this is clearly seen in the highest stresses in zone 3. Figure 11

• The results from the FEA study confirm the field feedback. Figure 12

• The FEA is an effective method of analyzing and predicting the complicated deflections of floating planetary ring gears. Figure 13

Limitations of the Study
• Only two parts of the entire wheel motor are used in this study — the ring gear and the torque tube. If other parts like the hub, the wheel bearings, the planets, etc. and stress-influencing factors like the truck load were added, the stiffness of the investigated mechanical system would change and the results would be different. Figure 14

• The force transmitted from the planets to the internal teeth of the ring gear was applied at the highest point of single tooth contact and uniformly distributed along the tooth surface. In reality the force may not be at that point for flexible systems and certainly would not be distributed evenly. Figure 15

• Only three zones of the ring gear were modeled with different crowning and root radii, which limit the understanding of their influence on the stresses. Figure 16

Acknowledgements
The authors would like to thank to Srinivas Rallabandi and Deepak Rotti who worked on the FEA modeling and analysis, as well as Nick Dame who worked on the solid models.

References
1. ANSI/AGMA 2001-D04, "Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth"

2. AGMA 908 – B89, "Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth"

3. ANSI/AGMA 6123-B06, "Design Manual for Enclosed Epicyclic Gear Drives"

4. ANSI/AGMA 2015-1-A01, "Accuracy Classification System – Tangential Measurements for Cylindrical Gears

5. ANSI/AGMA 2015-2-A06, Accuracy Classification System – Radial Measurements for Cylindrical Gears"

6. Kirov, V., "Comparison of the AGMA and FEA Calculations of Gears and Gearbox Components Applied in the Environment of Small Gear Company," 10FTM05

7. Silvers, J., Sorensen, C.D., and Chase, K.W., "New Statistical Model for Predicting Tooth Engagement and Load Sharing in Involute Splines," 10FTM07.

** 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.

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has MENg degree from the East Ukrainian National University, PhD in gearing from the University of Ruse, Bulgaria and MBA from the University of Wisconsin - Whitewater. He has more than 25 years of experience in different fields of gearing - manufacturing, design, assembly, tooling, management. Currently Dr. Kirov is Senior Engineering Specialist in Design at Caterpillar Global Mining LLC, South Milwaukee. 
has a Ph.D. in Mechanical Engineering from North Carolina State University. She has over 13 years of industrial experience, with a focus on structural analysis. Currently Dr. Wang is enngineering manager at Caterpillar Global Mining LLC, South Milwaukee.