Helix Angle and Root Stress

By measuring the helix angle, the author—and founder of the Ohio State University Gearlab—studied its effects on root stresses in helical gears.

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The ISO and AGMA Gear Rating Committees have for several years been comparing the results of different rating methods for several sets of gear pairs that have similar normal sections but different helix angles. The analysis presented in this paper uses a very sophisticated finite element code that was developed specifically for gear and bearing contacts to analyze the example gear sets. Analyses are also performed using a more conventional load distribution analysis program. The results for the original gear sets show that the narrow face width gear teeth twist significantly, thus moving the load to one edge of the face width and essentially showing that the example gear sets are highly unrealistic. When analyzed by the ISO and AGMA rating methods, the results do not reflect this twisting action. In an effort to come up with a valid comparison of stresses for different helix angles, three adjustments using wider face widths were attempted. The first scheme uses a wider face width with perfect involutes. Edge effects result in the peak stresses again being near the ends of the face width. The second adjustment uses a wider face width but with a narrow load patch in the middle of the tooth pair and results in the stresses increasing with helix angle. The third geometry form, which uses the wide face width teeth with lead crown and tip relief, gives the most reasonable results, with the root stresses being at a maximum in the center region of the tooth face widths. The paper compares each of the results to earlier analyses performed by others using both the AGMA and ISO calculations.

Introduction
This paper reports on the effect of helix angle on root stresses, a topic of discussion for a number of years within the ISO and AGMA Gear Rating Committees. Current rating methods use either the Lewis form factor [1] or the 30-degree tangent method [2] applied to the transverse tooth section to locate the position of maximum root stress. Corrections are then provided to account for the diagonal lines of contact that occur in helical gear tooth contact.

Even though this topic has been discussed extensively for many years, there are still disagreements on how these factors should be calculated. Gears present significant challenges when trying to come up with changes in geometry that isolate one effect without affecting others. A geometry set that supposedly isolates helix angle from all other variables by using a constant normal cross-section for each new helix angle gear was proposed in documents of the ISO Rating Committee [3, 4]. In order to achieve this, the rack used to generate the normal cross-section was kept constant. However, in order to change the helix angle and still keep the gears operating at the same standard center distance, it was necessary to reduce the tooth numbers as the helix angle was increased. Since tooth numbers must be integers, there are only a few possible helix angles that are possible. Table 1 shows the variables that are held constant in this analysis, and Table 1 shows the relationship between tooth number and helix angle for the sets.

For these gear sets, Figure 1 shows the results for a variety of methods that were used to calculate the root stresses. In this figure the stresses are first calculated for the spur gear (helix angle = 0°) and the values for the other helix angles are normalized to the value calculated for the spur gear so that all curves start at a value of 1.0 for a helix angle of zero degrees. Previous work by the ISO document authors shows calculations for two face widths, one very narrow with the face width being equal to the module and the other for an infinite face width. Calculations have been made using ISO 6336 method B, AGMA 2001, and a proposed Norwegian method [3]. Some methods predict that increasing the helix angle will continually decrease root stresses while other methods show an ever-increasing trend in root stresses with helix angle.

Each of the methods used to calculate the root stresses of Figure 1 use “standards type formulas” that are used in conjunction with many other factors to come up with a stress value. This paper’s authors thought that calculations of the “real” stresses that the gears experience might shed some light on the differences in each calculation method. Therefore, an advanced finite element program that is specifically designed to analyze gear and bearing contacts, and a more standard load distribution prediction program were employed to predict the actual root stresses for the example gear sets. This paper provides extensive analyses of the example gear sets presented in the original N367 document and also seeks to provide a more realistic gear geometry that isolates the effects of helix angle on root stresses.

Modeling Methodology
Each of the gear sets presented in this paper was modeled with two separate programs to evaluate the root stresses, the first being the high fidelity three-dimensional finite element program known as Transmission3D [5, 6]. The second program is a more conventional load distribution program (LDP) that has been developed by Houser, et al. [7, 8].

Transmission3D is a linear finite element contact analysis program specifically designed for analyzing gear and bearing contacts and is based on the Calyx contact analysis solver. The program has the ability to model complex gear geometries including tooth micro-geometries that include lead and profile modifications. The program utilizes a hybrid algorithm that combines finite element analysis with the application of a semi-analytical surface integral solution at the contact region to produce compliance terms. These compliance terms are then used in a Simplex-type solver to evaluate the load distribution across the tooth.

Using Transmission3D, each of the presented gear sets is modeled as a single mesh with all non-rotational degrees of freedom fixed to ground. A tooth mesh template (Figure 2) with high resolution in the root region is used to accurately capture the stress gradients within the entire root fillet. Each simulation is then run for one mesh cycle (one base pitch of rotation). The root stresses are found by searching the entire root fillet for the maximum principal stresses across the face width. While most tooth root stress calculation procedures only evaluate stresses for one tooth pair at a time, this method evaluates all lines of contact simultaneously and also includes the deflections of the entire gear blank.

The second program used to model the gear sets is the Load Distribution Program (LDP). LDP is a program that analyzes single mesh gear pairs using a finite plate compliance calculation in conjunction with the inclusion of Hertzian deflections and deflections of the tooth base [7]. The root stresses for each gear set are computed using a two dimensional boundary element [9] that has been extended to the third dimension using a procedure developed by Jaramillo [10]. LDP also has the ability to use finite element created compliance functions as well as performing finite element calculations of the root stresses.

Study 1: Narrow Face Width Gears [N367]
The original study was based on the N350 and N367 documents from the ISO Rating Committee proceedings [3,4] that presented nine gear sets with varying helix angles as given in Table 2. The face width is equal to the normal module, mn, which provides a very narrow gear with a pinion face width to diameter (F/d) ratio of 0.031. It was assumed that this was done in order to essentially create a helical gear tooth form that acts like a spur gear, since one can literally define a highest point of single tooth pair contact, even for the highest helix angle gear pair.

The gear sets of this study have perfect involute profiles, a narrow face width of 10mm, and constant normal tooth thickness. By holding the normal tooth thickness constant and keeping the face width narrow, the effect of the diagonal line of contact on the moment arm distance to the root centerline is reduced. This allows the results to be normalized and compared to the spur gear geometry where the lines of contact are parallel to the shaft axis. Analysis was performed on each of the nine sets using both Transmission3D and LDP.
Figure 3 shows the finite element mesh used in Transmission3D while Figure 4 shows both the predicted load distribution at the highest point of single tooth pair contact and the root stresses for a single spur gear tooth. Initial observations showed that the normalized root stress never reduced and increased much more rapidly than the 1/cos(β) path that is similar to the upper curve of Figure 1.

Further investigation showed significant twisting of the loaded tooth as the helix angle increased. This twist can be seen in Figure 5 for the 42.83° helix angle gear pair. This twisting shifts the load to one side of the tooth and thus causing an unusual stress pattern in the root with the maximum root stress being located opposite to where the tooth is loaded. This twisting induces increased stresses in the root due to the shift in load distribution to one side of the tooth causing what seem to be the highly unrealistic results shown in Figure 6. The LDP predictions, which do not include the twisting effect, show increasing root stress with helix angle, but to a much lesser degree than the 1/cos(β) curve.

Study 2: 100 mm Face Width Gears
While the previous gear sets presented in the N367 document had a novel idea for isolating the effects of helix angle, the narrow face width prevented the current analyses from producing realistic results more common to most gear applications that have much wider face widths. To reduce the tooth twist effect, the face width was simply increased from 10 mm to 100 mm (F/d = 0.31 with the face contact ratio varying from 0 to 2.16, depending on helix angle) with all other geometry data being kept the same as for the previous gear sets. The pinion torque was increased by a factor of 10 in order to remain at the same load per unit face width as in the previous narrow face width study. Perfect involute profiles were assumed so some localized twisting is still expected at the corners while tip interference due to tooth deflection is also expected. Although not of infinite face width that is plotted for the second set of data in Figure 1, the results for this face width were expected to be comparable to the infinite face width gear pair.

Again, the nine gear sets were modeled and analyzed using the two programs. Figure 7 shows the finite element mesh used in Transmission3D, and Figure 8 shows the loading and stresses for one mesh position of the 42.83° helix angle pinion. Although the tooth twist has been reduced, the predicted peak helical gear root stresses shown in Figure 9 still occur at the edge of the tooth face width and seem abnormally high relative to the gear rating models. It is interesting to note that the highest fidelity model, Transmission3D, predicted the highest stresses since it still shows the twisting effect as well as having increased stresses due to the reduced tooth backing because of the angled tooth edge. The LDP FE model predicted less twist and hence less stress and finally, the simple LDP model that does not model the twist, predicts stresses that are slightly lower.

Study 3: Narrow Contact Patch Gears
In order to isolate the effects of helix angle, while at the same time eliminating the edge effects, it was decided to try analyzing a gear pair that had features of each of the two previous studies. In this study, the face width was kept at 100 mm, but a contact patch only 10mm in width was applied down the center of the tooth. This patch was achieved by applying abnormally high-end relief across 45 percent of each end of the face width, leaving only 10 percent of the face width in contact. Figure 10 shows the end relief specification. The Transmission3D finite element mesh is the same as that used in Study 2. In order to make the gear set act like a spur gear pair, the outside diameters were reduced so that the profile contact ratio was kept close to 1.0. Figure 11 shows the load distribution and root stress pattern for the 25° helix angle pinion. The effects of helix angle on root stress that are shown in Figure 12 are now much better behaved, with all of the models giving somewhat similar results and each of them roughly following the inverse of the cosine of the helix angle plot. However, if one stops to think about this a bit, the tooth normal cross-sections are very similar, but the normal load increases by the cosine of the helix angle, so following the inverse of the helix angle trend is expected.

Study 4: Typical Gear Pair
Each of the previously studied gear pairs has some feature that makes the gear set unrealistic. In order to get a more realistic gearing situation, the face width was kept at 100 mm and circular profile and lead modifications were applied such that end effects and tip interference were reduced. The typical micro-geometry of the pinion is shown in Figure 13. The earlier finite element model was used for the Transmission3D analysis.

The contact stress distribution plot of Figure 14 shows complete contact across the tooth flank with a rolling off of stress at the extremes of the tooth face width and the profile. Figure 15 shows the root stress patterns of the three models, namely, Transmission3D, LDP-finite element, and LDP boundary element, respectively. The LDP boundary element results show root stresses plotted in 11 percent increments across the face width. Results are fairly similar in load distribution and also show similar root stress trends with all models showing the highest stressed region being on one side of the tooth. As seen in Figure 16, the root stresses are less than the reference spur gear for helix angles less than 25°, which shows that this model not only captures the effect of helix angle, but also incorporates the effect of the “helical factor” that is part of the effect predicted by the rating standards. It is interesting to note that Transmission3D and the LDP finite element give almost identical results with the exception of the highest helix angle data. At low and medium helix angles, the inverse of the helix angle is higher than any of the predictions. Although the AGMA prediction shown in Figure 1 is for a different face width set, its values would lie beneath any of the predicted values of Figure 16.

Summary
This paper has presented the results of a high fidelity finite element analysis of the N367 gear sets that were previously used for the evaluation of the helix angle effect on root stresses of helical gears [3]. The initial analysis showed an unrealistic load distribution shift for higher helix angle gear pairs due to substantial twist of the narrow face width teeth.

Alternate gear geometries were then proposed that might better isolate the effects of helix angle variation while also using more realistic tooth loadings. Of the four studies that were performed, the Study 4 arrangement, which uses a reasonably wide face width tooth with modifications that center the load on the teeth gives results that would be best for comparing sophisticated model results with rating calculations.

One major conclusion is that the end effects of narrow face width helical gears can result in tooth twisting that might abnormally increase edge loading. Proper profile and lead modifications can minimize these effects and advanced load distribution analysis is a means of detecting such issues.

References
1) Lewis, W., 1893, “Investigation of the Strength of Gear Teeth,” Proceedings of Engineers Club, Philadelphia.
2) Dudley, D., 1984, Handbook of Practical Gear Design. “Gear-Strength Calculations,” Chapter 2, McGraw Hill, 1984.
3) Sandberg, E., 1989, “ISO/TC 60/WG 6 N 367.” Letter. 04 Apr. 1989.
4) “ISO/TC 60/WG 6 N 385.” 11-13 Sep. 1989.
5) Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem,” Int. J. Numer. Methods Eng. 31, pp. 525–545.
6) Vijayakar, S., Houser, D.R.,1991 “Contact Analysis of Gears Using a Combined Finite Element and Surface Integral Method,” Proceedings of the AGMA Fall Technical Meeting, Paper 91FTM16.
7) Conry, T.F. and Seireg, A., 1973. “A Mathematical Programming Technique for the Evaluation of Load Distribution and Optimal Modifications for Gear Systems.” J. Eng. Ind., Trans. ASME, Vol. 95, No. 4, pp. 1115-1123.
8) Houser, D., 2009, “Theoretical Basis of The Ohio State Load Distribution Program (LDP)” The Ohio State University, Columbus, OH.
9) Clapper, M., and Houser, D., 1994, “A Boundary Element Procedure for Predicting Helical Gear Root Stresses and Load Distribution Factors”, Proceedings of AGMA Technical Conference, St. Louis.
10) Jaramillo, T.J., 1950, “Deflections and Moments Due to a Concentrated Load on a Cantilever Plate of Infinite Length”, J. Appl. Mech., Trans. ASME, Vol. 72, pp. 67-72

Acknowledgements:
The author would like to thank Advanced Numerical Solutions LCC for the use and support of Transmission3D. Also, sincere thanks go to the sponsors of the Gear and Power Transmission Research Laboratory as well as the AGMA Foundation, who provided Fellowship support for the second author of this study. Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, 5th Floor, 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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professor emeritus, received his B.S., M.S., and Ph.D. from the University of Wisconsin in Madison. He is retired from the faculty of the Department of Mechanical Engineering at Ohio State, where he taught for 35 years. Dr. Houser is the founder of the Gear Dynamics and Gear Noise Research Laboratory that has been renamed the Gear and Power Transmission Research Laboratory, an industrial research consortium that currently has over 70 sponsor companies. He is a recipient of the Darle Dudley award from ASME, and he has supervised over 120 graduate theses and has consulted on gear problems with over 50 companies.