Evaluation of load capacity of gears with an asymmetric tooth profile
- Tomoya Masuyama^{1}Email author and
- Naoki Miyazaki^{1}
DOI: 10.1186/s40712-016-0064-0
© The Author(s). 2016
Received: 21 August 2016
Accepted: 27 October 2016
Published: 11 November 2016
Abstract
Background
An ISO standard tooth profile has a symmetric pressure angle of 20°. However, the load capacity can be increased with respect to bending and contact pressure by increasing the pressure angle on the meshing side of an asymmetric tooth. Accordingly, we analyzed the torque transmission capacity of asymmetric gears with various pressure angles.
Methods
We calculated the deflection and bending stress of teeth by the finite element method and found the root stress taking into account the load-sharing ratio. Hertzian contact stress was calculated with respect to contact pressure. Normal vector load was converted into a torque, and torque capacity was evaluated when the stress reached the allowable stress for each case.
Results
Reduced bending stress because of an increase in tooth thickness and decreased transmission torque because of a reduction in the base circle radius work together to maximize the load capacity for bending at a pressure angle of around 30°. Maximum load capacity with respect to contact pressure is achieved when the pressure angle is made 45° by increasing the radius of the contact surface.
Conclusions
Both strength with respect to bending and contact pressure are found, and the torque transmission capacity of the gear is determined by the lower value of the two. For low-strength materials such as flame-hardened steel, damage due to contact pressure is expected for all forms of gears and the greatest torque capacity was at a pressure angle of 45°. In the case of assuming 800 Hv and an inclusion size \( \sqrt{\boldsymbol{A}}=50 \) μm for a high-strength material, the greatest torque transmission capacity is obtained at a pressure angle of 30°. In the case of assuming a moderate-strength material such as case-hardened steel, an optimal form exists at which strength with respect to bending and strength with respect to contact pressure are equal.
Keywords
Gear Pressure angle Asymmetric tooth profile Strength Torque capacity StressBackground
The ISO standards specify a symmetric form and a pressure angle of 20° for a standard rack tool. However, as the pressure angle increases past 20°, the bending strength can be expected to increase because of the increase in the tooth root thickness, and the contact pressure strength can be expected to increase because of the larger profile radius of curvature. Furthermore, although a pressure angle sufficiently greater than the limit of the tooth tip cannot be achieved, this can be solved by making the pressure angle on each side of the tooth asymmetric. Unidirectional torque transfer is the principal factor in general gearing, so there may be some issues with the strength on the low-pressure side being weaker.
Kapelevich and co-workers conducted a series of studies on asymmetric gears. In 2000, they indicated that an asymmetric gear with a higher pressure angle on the drive side would allow for an increase in load capacity while reducing weight (Kapelevich, 2000). In the same year, Litvin et al. (2000) analyzed the bending and contact stresses of an asymmetric gear tooth. They demonstrated the mitigating bending stress through the optimization of tooth profile (Kapelevich and Shekhtman, 2009). Kapelevich has integrated his exemplary work in book form (Kapelevich, 2013). A recent study by the same group has calculated both bending stress and contact pressure using three-dimensional finite element (FE) analysis to compare with a symmetric tooth profile (Kapelevich and Shekhtman, 2016).
Deng and co-workers carried out a stress analysis of an asymmetric gear with the aim of increasing the bending load capacity by increasing the tooth thickness. They achieved an increase in tooth thickness by increasing the pressure angle on the reverse side (Deng et al. 2003). Kruger et al. (2013) used a boundary element method to calculate the stress distribution of an asymmetric tooth, the stress state when an adjacent tooth is loaded, and the influence of plastic deformation due to the generation of a high compressive stress in the deep section of the load point. We found a bending stress distribution by using the finite element method (FEM) and used the result to conduct a strength simulation focusing on an inclusion distribution (Masuyama et al., 2015).
Nevertheless, there are fewer papers that focus on the between-normal vector load and the transmission torque according to changes in pressure angle in a wider range. Accordingly, in the present study, we conducted detailed strength evaluations on an asymmetric gear, performed an FEM analysis, and found the change in load-sharing ratio as a result of rotation. We then calculated the bending stress and Hertzian contact stress and evaluated the difference in torque transmission capacity according to materials.
Methods
Asymmetric tooth profiles
Finite element model
Results and discussion
Load-sharing ratio
Stress analysis
Failure of a gear tooth is generally divided into bending fracture and tooth surface damage. The former is caused by bending stress on the root and the latter by contact pressure. Accordingly, we calculated the root stress and Hertzian contact stress by the tooth surface normal vector load, taking into account the load-sharing ratio.
Bending stress
As the position of the load moves from the start point of meshing to the tooth tip, the bending moment increases with respect to unit-concentrated load, and the bending stress of the tooth root increases correspondingly. However, the present study evaluated the root stress taking into account the load-sharing ratio.
Contact stress
Estimation of allowable stress
Assuming the manufacture of asymmetric tooth forms with various materials, we determined the allowable stress using values stipulated in ISO6336-5 (ISO6336-5, 2003) and a strength evaluation formula that we proposed.
From ISO standards for the allowable stress, we selected a flame-hardened MQ class steel with σ _{Flim} = 360 MPa for bending and σ _{Hlim} = 1150 MPa for surface damage. Similarly, σ _{Flim} = 525 MPa and σ _{Hlim} = 1650 MPa were adopted for the allowable stress of a case-hardened ME class steel.
Here, H _{ v } is the Vickers hardness of the material and A is the projected area of defects contained in the material. In this paper, we assume hardened steel as the gear material. Accordingly, the allowable stress was found for two types of material of hardness 600 and 800 Hv, assuming an inclusion size \( \sqrt{A}=50 \) μm.
Material properties and allowable stresses
Material | Hv | σ _{Flim} (MPa) | σ _{Hlim} (MPa) |
Flame-hardened MQ class | 500 | 360 | 1150 |
Case-hardened ME class | 650–800 | 525 | 1650 |
Inclusion size \( \sqrt{\boldsymbol{A}} \) (μm) | Hv | σ _{Flim} (MPa) | σ _{Hlim} (MPa) |
50 | 600 | 600 | 2270 |
50 | 800 | 760 | 2900 |
Evaluation of load capacity
If T _{Hal} > T _{Fal} for a given gear, then the tooth can be expected to be fractured by bending. If T _{Fal} > T _{Hal}, then surface damage can be expected to occur. Accordingly, for a gear manufactured from flame-hardened MQ class steel, surface damage will probably occur with all forms. Also, the greatest torque transmission capacity can be expected to be demonstrated when the pressure angle on the meshing side is 45°. In the case of case-hardened ME class material, T _{Fal} ≈ T _{Hal} at a pressure angle of 40°. At pressure angles lower and higher than this, surface damage and bending fracture are expected to occur, respectively. If this material is used, the greatest torque transmission capacity will be at a pressure angle of 40°.
Similarly, for a material with \( \sqrt{A}=50 \) μm and hardness = 600 Hv, using a tooth for which the pressure angle becomes approximately 21° puts the load capacity for bending and contact pressure in opposition. However, even if the pressure angle becomes larger, T _{Fal} increases and a maximum torque transmission capacity can be expected at around α _{1} = 30°. In the case of \( \sqrt{A}=50 \) μm and hardness = 800 Hv, T _{Hal} is larger than T _{Fal} for any tooth profile. So, a maximum torque transmission capacity can be expected at around α _{1} = 30°, for which T _{Fal} takes its maximum value.
Conclusions
In this paper, the strength of gears with asymmetric tooth profiles is discussed considering the load-sharing ratio, and the gear performance is evaluated in relation to the torque transmission capacity. Tooth pressure angles of 20° to 45° were analyzed.
The load-sharing ratio was determined from the calculation results of deflection by FEM analysis for ISO standard tooth forms and asymmetric tooth forms. No significant difference was found for the higher pressure angle tooth.
Bending stress analysis was also performed by FEM analysis. The root stress in higher pressure angle teeth against unit load decreased because of the thick root of the tooth. However, the highest torque capacity for bending was expected for the tooth profile with a 30° pressure angle because an increase of the pressure angle leads to a smaller base circle.
The durability against surface damage was evaluated in relation to Hertzian contact stress. In the case of a higher pressure angle tooth, the contact stress decreases with increasing relative radius of curvature. Furthermore, the Hertzian stress is proportional to the square root of the normal load. Therefore, the tooth with a pressure angle of 45° has the maximum torque capacity for surface damage.
Assuming that the load capacity of a gear is determined by load capacity with respect to bending or load capacity with respect to contact pressure—whichever is lower—the greatest torque load capacity can be expected at α _{1} = 45° for all forms in the case of flame-hardened MQ class steel. The damage mode will probably be contact pressure damage.
For case-hardened ME class steel, the load capacity with respect to bending and the load capacity with respect to contact pressure are equivalent when α _{1} = 40°, in which case the greatest load capacity can be expected.
For a material based on an original strength evaluation formula with \( \sqrt{A}=50 \) μm and hardness = 600 Hv, the strength with respect to contact pressure is low at α_{1} = 20°. However, the load capacity with respect to contact pressure and the load capacity with respect to bending are reversed as α _{1} > 21°. At \( \sqrt{A}=50 \) μm and hardness = 800 Hv, the load capacity with respect to bending is low for all pressure angles. This means that the greatest load capacity can be expected when α _{1} = 30°, both with respect to bending and with respect to contact pressure.
This result indicates that a large load surface pressure angle is good for low-strength materials and a small pressure angle is good for high-strength materials. For materials with moderate strength, an optimal pressure angle is assumed to exist that realizes both load capacity with respect to bending and load capacity with respect to contact pressure.
Declarations
Acknowledgements
The authors would like to gratefully thank Dr. Yasunari Mimura for helping us to perform the FEM analysis. We also acknowledge the financial support from the Promotion Association of Tsuruoka College.
Authors’ contributions
Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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