The gear tooth strength of high pressure angle gears is studied and compared with that of conventional pressure angle gears. The comparison is carried out regarding contact pressure, contact and bending stresses, and the loaded function of transmission errors. The gear geometric models are generated by the computerized simulation of the manufacturing process, and the corresponding finite element models consider a large number of contact positions along two cycles of meshing. In this way, the load sharing between different pairs of contacting teeth is considered, and the evolution of contact pressure, and contact and bending stresses all over the cycle of meshing are obtained. In general, high pressure angle spur and helical gears do not show a better behavior regarding contact pressure, contact stresses, and bending stresses than spur and helical gears with traditional pressure angle of 25 degrees. Based on the obtained results, high pressure angle gears are not expected to show better mechanical behavior in high power transmissions than gears with conventional designs. The results gathered from the finite element analysis are not in agreement with the improvement on the pitting and bending behavior of the gears that were anticipated by using analytical models as provided by international standards.
1: Introduction
The use of lower pressure angles in cylindrical gears has been decreasing over recent decades in favor of using higher pressure angles. Nowadays, it is uncommon to find new designs of cylindrical gear drives considering a pressure angle of 14.5 degrees because it is broadly accepted that gears with higher pressure angles yield better mechanical behavior regarding contact and bending stresses. Therefore, new designs of cylindrical gears consider pressure angles of either 20 degrees or 25 degrees.
The design of cylindrical gear drives with high pressure angles has attracted the attention of researchers in the past. In [1, 2], the results of a study to determine the feasibility of using high pressure angle gears in aeronautics and space applications were presented. The NASA GRC Spur Gear Test Facility was used for that purpose. Pressure angles of 20, 25, and 35 degrees were considered in three different designs of spur gears. The face width and center distance were deemed to be constant. However, the test specimens had a different number of teeth and different modules. The conclusions of this study mentioned that high pressure angle spur gears running at high speed provided performance with similar bending and contact stresses over more traditional gear pressure angles. Also, it was stated that high pressure angle gears appeared to be better suited to the low-speed, high load, and grease-lubricated conditions.
In [3], a method for specifying gear teeth with higher pressure angles was presented. The idea of this work was to achieve higher bending and surface contact strength or, in other words, to reduce bending and surface contact stresses by using the highest as possible pressure angle. Miller’s work [3] was directed to isolate the influence of the pressure angle on the results so that he kept the number of teeth and module of all designs as constant. The proposed method in [3] allowed for the determination of proper high-pressure angles for which the gear met conditions on the desired top land, contact ratio, or hob tip radius. Four designs were evaluated having the traditional 25-degree pressure angle, and three other designs having pressure angles of 33.5, 35, and 36 degrees. In the mentioned work, all data and stress calculations were performed using the AGMA GRS 3.1.7 Gear Rating Suite program. The application of high pressure angle gearing in low speeds, coarse pitch, and lower quality level was mentioned in [3] as typical applications for these gears. It was suggested that an alternative FEA (Finite Element Analysis) should be performed on the gear teeth and the results compared with those published in the mentioned work. Miller’s work inspired the research work presented in this paper.
In [4], it is stated that the lower the pressure angle, the higher the surface compressive and bending stresses become. In that work, high pressure angle gears are referred to as not as quiet as low pressure angle gears. The main reason is that the tooth deflection under load is very small for high pressure angle gears. Noise is generated when the load is transferred from one tooth to another upon impact [4]. Again, high pressure angle gears are considered to have a higher power density and are recommended for high horsepower transmissions [4].
The design of high pressure angle gears is challenging because of the obtained geometry for the gear teeth. The pointing of the gear teeth on the one hand and having enough tip edge radius for the cutting tools on the other hand are limiting factors for gear designs considering high pressure angles. To avoid those problems, asymmetric gears were proposed as a solution to increase the load capacity of gear drives while reducing their weight and dimensions [5]. In [6], the results of the comparison of several designs of asymmetric gear drives were presented. The application of high pressure angles, not only for the driving side of the gear teeth as traditionally is done but also for the coast side of the teeth was investigated. In that work, it was mentioned that the maximum contact stresses and contact pressures on the gear teeth depend only on the pressure angle of the contacting side (driving side) of the gear drive, no matter what the pressure angle of the coast side is. It was mentioned that bending stresses were reduced when higher pressure angles are used, not only for the coast side as stated in other works [7, 8] but also for the driving side.
This paper is intended to answer the following research question: Can we extend and recommend the use of pressure angles of 30 degrees or higher in symmetric gears to improve the mechanical behavior of cylindrical gear drives? The methodology to carry out this research work is based on the application of finite element analysis on models comprising of five pairs of contacting teeth, considering a very fine mesh on the contacting surfaces. Not only one point of contact will be considered for the analysis but the evolution of contact and bending stresses, contact pressure, and loaded function of transmission errors along two complete cycles of meshing will be investigated. This perspective will provide a unique overview of the mechanical behavior of high pressure angle gear drives and will be fundamental to provide further recommendations for the use of gears with high pressure angles.
2: Methodology of Evaluation of the Mechanical Behavior of Cylindrical Gears
The mechanical behavior of gear drives can be evaluated by using analytical and pseudo-empirical methods as provided by international standards (AGMA — American Gear Manufacturers Association, or ISO — International Organization for Standardization) or numerical methods as the finite element method. AGMA and ISO provide a methodology to evaluate the contact and bending stresses, and with them, the safety factors against the risk of failure of the gear drive by pitting or bending are calculated. On the other side, using the finite element method, the contact and bending stresses on the gear tooth can be determined, as well as the contact pressure, contact deformations, and tooth deflections that are taken into consideration for the determination of the loaded function of transmission errors.
Many researchers refer to contact stresses in technical publications, but many times, there is no mention to what they consider as contact stresses. Frequently, Von Mises stresses on the contact area, contact pressure, or the minimum principal stress are considered indicators of the contact stress, but they may sometimes yield to different conclusions.
Regarding Von Mises stresses as an indicator of contact stress, due to the fact that maximum Von Mises stresses occur underneath the contacting surfaces, they are difficult to capture in a finite element analysis due to the need of using a large number of finite elements under the surface in the finite element mesh. Those finite element meshes would be extremely costly for computation. Instead, the maximum value of contact pressure has been found not being so sensitive to the finite element mesh used for computations, and it is recommended as the main indicator of contact stresses for comparison between different gear designs. Besides, either the AGMA or ISO methodologies use contact pressure (Hertz pressure) as the basis for their calculations (see Section 3).
Regarding bending stresses, Von Mises stresses may be affected by some contact positions where the load is shared between two pairs of teeth and the traction due to the bending of a given tooth is low in comparison with the compression in the same fillet area due to the bending of the previous tooth. Also, contacts near the form diameter may influence the Von Mises stresses on the fillet area. For this reason, the maximum principal stress in the fillet area (the largest tensile stress) is considered for comparison of bending stresses among different gear designs.
3: AGMA’s Approach for the Evaluation
of Contact and Bending Stresses
The American Gears Manufacturers Association (AGMA) has developed and disseminated methodologies to design and analyze gear drives during the last hundred years. AGMA started in 1916 to create standards that define gear types, tooth sizes, tolerances, and in general, to set up methodologies to contribute to making gears safer and interchangeable even if manufactured in different locations or by different companies. The method presented in the AGMA 2101-D04 standard [9], Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth, has been used in this work to assess the performance of high pressure angle gears according to analytical methods. The AGMA 2101-D04 standard is the metric edition of AGMA 2001-D04. It provides the formulas for rating the pitting resistance and bending strength of spur and helical involute gear teeth.
The pitting resistance is considered a function of the Hertzian contact (compressive) stress, and for that, the well-known model of contact between two cylinders is used as a basic model. The contact stress number σH for cylindrical gear teeth according to AGMA [9] is given by:
The contact point in which the contact stress number is determined is considered through the geometry factor ZI. For spur and low axial contact ratio helical gears, the geometry factor ZI is calculated considering the radius of curvature of pinion and gear for contact at the lowest point of single tooth contact (LPSTC) for the pinion. For conventional helical gears, the geometry factor ZI is calculated considering the radii of curvature of pinion and gear for contact at the mean radius or middle radius of the working profile of the pinion [10].
The effective allowable number σHPeff is obtained as a function of the allowable contact stress number of the material, σHP, modified by the stress cycle factor for pitting resistance, ZN, the hardness ration for pitting resistance, ZW, the temperature factor, Yθ, and the reliability factor, YZ. The formula for the effective allowable stress number σHPeff is:
The bending resistance is based on a model that considers a cantilever plate and the tensile stresses obtained on the side of application of the load. When we extrapolate it to gears, the bending model gives the tensile stress in the fillet area of the driving side of the gear teeth. The bending stress number σF in a gear tooth is given by:
The effective allowable bending stress number σHPeff , depends on the allowable bending stress number of the material σHP , modified by the stress cycle factor for bending strength, YN, the temperature factor, Yθ, and the reliability factor, YZ, similar to the case of contact stress. The formula to calculate the effective allowable stress number σHPeff is:
4: Finite Element Analysis
Before the application of finite element analysis, tooth contact analysis (TCA) has to be performed to obtain the angular positions of pinion and gear along one or two cycles of meshing. A general-purpose algorithm for simulation of meshing and contact of pinion and gear tooth surfaces is used. This algorithm is based on a numerical method that takes into account the position of the tooth surfaces and minimizes the distance until contact is achieved [11]. The algorithm assumes rigid body behavior and has been extended to two cycles of meshing in [12]. It is valid for linear, point, or edge contacts. A grid of 61 × 61 points on each gear tooth surface, a total of three pairs of tooth surfaces, and a virtual compound thickness of 0.0065 mm have been considered here for determination of contact patterns and functions of unloaded transmission errors.
Finite element models are built automatically from the gear tooth surfaces and following a procedure that is extensively illustrated in [13]. Node coordinates are determined on the gear tooth surfaces as a function of the chosen numbers of nodes in longitudinal, profile, and fillet directions. The finite element model considers five pairs of teeth to keep the boundary conditions, represented by a rigid surface, far enough from the contact areas. A reference node on the axis of the pinion (respectively, the gear) controls the body motion of the rigid surface. Whereas the rigid surface of the gear is held at rest at each contact position by blocking its reference node, a torque about the pinion axis is applied to the reference node of the pinion rigid surface.
A model as the one shown in Figure 1 has been considered for the analysis. Sixty-one contact positions distributed along two cycles of meshing are investigated to capture every detail of the evolution of contact and bending stresses all over two cycles of meshing.
5: Comparison of the Mechanical Behavior
of Cylindrical Spur Gears with High
Pressure Angles
Three cylindrical gear sets with different pressure angles are analyzed and compared regarding contact and bending stresses, as well as the loaded function of transmission errors. Figure 2 shows the gear tooth geometry of the evaluated gear sets. Table 1 provides the general design parameters of the evaluated spur gear sets. For all cases, the center distance and number of teeth of pinion and gear are kept constant. The addendum and dedendum coefficients of the generating rack cutters are adjusted to avoid pointing of the gear teeth for all cases. The root radius coefficient is also chosen to allow for a suitable geometry of the generating tool. The profile shift coefficients for each gear set were selected for optimal specific sliding.
The most important derived parameters of the evaluated gear sets are shown in Table 2. The transverse contact ratio decreases as the pressure angle increases, as shown in Table 2.
The rating of the three designs presented in Table 1 is performed according to the AGMA 2101-D04 standard. A torque of 900 Nm is applied to the pinion of the gear set. A speed of 1 rpm is considered to neglect the dynamic effects on the gear performance. Most of the factors of influence on the contact and bending stress numbers have been taken into account equal to 1.0, to isolate the effect of the geometry on the stresses (see Table 3). The stress cycle factor for all cases of design corresponds to a service life of 20,000 hours.
Table 3 shows the factors of influence and main parameters affecting the calculation of the contact and bending stress numbers according to AGMA 2101-D04. Steel, grade 2, carburized and hardened, having an allowable contact stress number of 1,550 MPa and an allowable bending stress number of 450.0 MPa was selected for the calculations of the safety factors.]
Table 4 shows the safety factors for pitting and bending according to AGMA 2101-D04 for the three designs of spur gear sets. According to the standard, the pressure angle improves both the safety factor for pitting and that for bending. As shown in Table 4, the maximum contact stress for a spur gear drive with 30-degree pressure angle is reduced 4.38 percent with respect to a similar design having a pressure angle of 25 degrees. Bending stresses are also reduced 12.4 percent for the design having a pressure angle of 30 degrees. For the design with the higher-pressure angle (35 degrees), the contact stresses are reduced by 7.30 percent with respect to the design having the pressure angle of 25 degrees. The corresponding bending stresses are reduced by 25.6 percent for the design having a pressure angle of 35 degrees. Generally speaking, and based on the results obtained, by increasing the pressure angle as much as geometrically is possible, the safety factor for pitting and bending is improved, so that the transmissible power can be increased accordingly.
Regarding the results obtained from the application of the finite element method, Figure 3 shows the evolution of the maximum contact pressure along 61 contact positions distributed in two cycles of meshing. The contact pressure is obtained at the middle tooth of a total of five teeth of the pinion model. It is observed, for each case of design, that the maximum contact pressure occurred, certainly, at the LPSTC point. A minor reduction of contact pressure is observed from the design with 25 degrees to the design with 35 degrees as AGMA 2104-D04 predicts. The reduction of the transverse contact ratio for high pressure angle gears is also noted on the shorter portion along the sixty-one contact positions for which a gear tooth is in contact under pressure (see Figure 3).
Figure 4 shows the evolution of maximum Von Mises stress on the contacting surfaces of the pinion teeth. The highest value of Von Mises stress occurs underneath the contacting surfaces, as Hertz theory predicts. Considering the applied finite element model as a reference for comparison without further judging the accuracy of the obtained results (since a higher amount of finite elements would be required to get a better estimation of the real Von Mises stresses), Figure 4 shows the same tendency on the evolution of the maximum Von Mises stresses on the contacting surfaces of the pinion, experiencing the highest value at the LPSTC. A slight reduction of the contact stresses is also observed for the design with higher pressure angles, and it is in agreement with the results obtained for the contact pressure and predicted by the analytical method.
Regarding bending stresses, Figure 5 shows the evolution of the maximum principal stress at the pinion fillet during 61 contact positions. The maximum value occurs at the highest point of single tooth contact (HPSTC), as AGMA predicts. However, the results of maximum stresses and relative comparison are different from those obtained through the application of the analytical model. In fact, an increment of the maximum principal stress is observed as the pressure angle increases. It is clearly seen in Figure 5 how the portion of meshing for single tooth contact is increased as the pressure angle increases. Although the maximum bending stress is well predicted by the analytical model for the pressure angle of 25 degrees, it also predicts that the bending stresses should be decreasing, but on the contrary, in Figure 5, it is observed that the maximum principal stress is increasing for the pinion as the pressure angle increases.
Figure 6 shows the evolution of the maximum principal stress at the fillet area of the middle tooth of the gear for the three cases of design. Although some reduction of bending stresses is obtained for the design with 30 degrees at the fillets of the gear teeth, design with 35 degrees shows an increment on the maximum bending stress with respect to the previous case. Therefore, the idea that higher pressure angles will reduce bending stresses cannot be generalized. One of the main influences on the bending stresses for the evaluated cases is the different values of the root radius coefficient used for each design.
Figure 7 shows the loaded function of transmission errors for the three cases of design evaluated in this work. The maximum peak-to-peak value of transmission errors is decreasing when the pressure angle increases. The Discrete Fourier Transform (DFT) of these functions should be evaluated before drawing final conclusions in terms of noise and vibration excitation.
6: Comparison of the Mechanical Behavior of Cylindrical Helical Gears with High Pressure Angles
Most of the published literature regarding the application of high pressure angle cylindrical gears has dealt with spur gears. Not much work has been found related to the design of high pressure angle helical gears. In this section, three designs of helical gears with pressure angles of 25 degrees, 30 degrees, and 35 degrees will be compared regarding contact and bending stresses, and loaded functions of transmission errors. Table 5 shows the general design parameters for the helical gear designs, which were chosen similar to those of the previously analyzed designs of spur gear sets. A helix angle of 20 degrees was chosen for all designs.
Table 6 shows the most important derived geometric parameters of the evaluated helical gear sets. The transverse contact ratio, the overlap ratio, and the total contact ratio have been shown for reference.
Table 7 shows the most important factors of influence on the calculation of the contact and bending stress numbers. Any factor of influence not listed in Table 7 was kept equal to those shown in Table 3.
Table 8 shows the safety factors for pitting and bending according to AGMA 2101-D04 for the three designs of helical gear sets. The analytical model does not predict any benefits on the increment of the pressure angle for helical gears. Actually, the safety factor for the case of higher pressure angle is lower than that for the designs with lower pressure angles. However, there is a substantial improvement on the safety factor for bending in designs with higher pressure angles according to the analytical model.
Figure 8 shows the evolution of the contact pressure (CPRESS in Abaqus) for the middle pair of contact teeth. It corresponds with the third tooth in a model of five pairs of teeth. Pressure angle of 30 degrees yields slightly lower maximum contact pressure than the reference design with a pressure angle of 25 degrees. As shown in Figure 8, the design with the pressure angle of 35 degrees shows a substantial increment on the contact pressure, which is not entirely predicted by the analytical model for the determination of the contact stress number.
Figure 9 shows the evolution of the maximum Von Mises stress on the contacting surfaces of the pinion, no matter in which tooth it appears. Notice that the evolution is a periodic function as expected, due to the evaluation of two cycles of meshing. The design with the higher-pressure angle of helical gears shows the higher Von Mises stresses, also in agreement with Figure 8, which shows higher values of contact pressure for that design. Designs with 25 and 30 degrees of pressure angle show similar maximum levels of contact stresses. In general, as shown in Figures 8 and 9, no advantages are observed for using high pressure angles in helical gear drives in terms of contact stresses and contact pressure on the gear tooth surfaces.
Figure 10 shows the evolution of the maximum principal stress (larger tensile stress) at the fillet surface of one tooth of the pinion. The maximum bending stress for higher pressure angle gears, contrary to what is predicted by the analytical models, is higher for the design with 35 degrees of pressure angle. Designs of 25 and 30 degrees yield similar levels of maximum principal stresses at the fillet. No advantage is observed for the design with higher pressure angles in terms of bending stresses. Therefore, not only has the safety factor not increased, but according to the finite element method, it has been reduced due to higher values of bending stresses. The analytical model to calculate the bending stress seems not to capture the increment of stresses due to larger stress concentration factors in designs with higher pressure angles, where small root radius coefficients are used.
Figure 11 shows the loaded function of transmission errors for helical gears with different pressure angles. It can be observed that the peak-to-peak maximum transmission errors are lower for the design with higher pressure angles. Noise and vibration excited by transmission errors might be lower for high pressure angle helical gear designs. However, the difference is slight and might not be noticeable. In Figure 11, it can be observed that the loaded function of transmission errors is not exactly periodic near the initial and final contact points. This is a characteristic of finite element models with boundary conditions slightly affecting the obtained results. The use of finite element models of seven pairs of contacting teeth for helical gears would improve the periodicity of the loaded function of transmission errors. However, it is not expected that different results would be obtained by using those models.
7: Conclusions
Based on the results obtained in this work, the following conclusions can be drawn:
High pressure angle spur gears show a minor reduction in contact stresses and contact pressure. The maximum values for both contact stresses and the contact pressure are obtained at the lowest point of single tooth contact, as considered by the analytical methods.
Maximum bending stresses for spur gears with a high-pressure angle might not only be reduced, but in some cases, might be increased. An example of design in which the maximum bending stresses at the pinion fillet increased was shown. The influence of small root radius coefficients on bending stresses for high pressure angle gears should be further evaluated.
High pressure angle helical gears yield higher maximum contact pressure, higher maximum Von Mises contact stress, and higher maximum principal stresses at the fillet area. No advantage was found for using or recommending high pressure angle helical gears. Only the loaded function of transmission errors showed a peak-to-peak value smaller than that obtained for designs with lower pressure angle helical gears.
Whereas the analytical models yielded lower bending and contact stresses for the designs with higher pressure angles, the numerical analysis of stresses by the finite element method applied to two cycles of meshing yielded higher values of stresses. The causes of this difference should be further investigated and incorporated into the rating methods.
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ABOUT THE AUTHORS Dr. Alfonso Fuentes-Aznar is with the Rochester Institute of Technology, and Dr. Ignacio Gonzalez-Perez is with the Polytechnic University of Cartagena. This article published by permission of the American Gear Manufacturers Association, copyright © 2017, 17FTM03, ISBN: 978-1-55589-537-2. The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the AGMA.
