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Journal of Mechanical Design | Volume 131 | Issue 6 | Research Paper
Research Papers

Analysis of General Characteristics of Transmission Error of Gears With Convex Modification of Tooth Flank Form Considering Elastic Deformation Under Load

[+] Author and Article Information
Edzrol Niza Mohamad, Hiroaki Murakami, Aizoh Kubo

Department of Mechanical Engineering and Science, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto-shi, Kyoto 606-8501, Japan

Masaharu Komori

Department of Mechanical Engineering and Science, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto-shi, Kyoto 606-8501, Japankomorim@me.kyoto-u.ac.jp

Suping Fang

State Key Laboratory for Manufacturing System Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinafangsuping@hotmail.com

J. Mech. Des 131(6), 061015 (May 21, 2009) (9 pages) doi:10.1115/1.3116261 History: Received March 30, 2008; Revised February 23, 2009; Published May 21, 2009
Article
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The vibration/noise of power transmission gears is a serious problem for vehicles including automobiles, and therefore many studies on gear vibration have been reported. These studies, however, were carried out by investigation using numerical simulations in which gears with specific dimensions and tooth flank modifications under specific loading were considered. Therefore, the general characteristics of the transmission error of gears have not been clarified theoretically. In this report, a general model for the tooth meshing of gears is proposed; in which a quasi-infinite elastic model composed of springs with stiffness peculiar to the gear is incorporated. The transmission error of gears is formulated by theoretical equations. An investigation on the factors affecting the general characteristics of transmission error is accomplished using the formulated equations. The qualitative characteristic of the transmission error of gears with convex tooth flank form deviation is determined by the actual contact ratio and qualitative elements of gears, i.e., tooth flank form deviation and the distribution of stiffness. Even if the amplitude of torque, the amount of tooth flank form deviation, and other quantitative elements are not determined, the qualitative characteristic of transmission error can be derived. The peak-to-peak value of transmission error increases proportionately to the amount of tooth flank form deviation.
Copyright © 2009 by American Society of Mechanical Engineers
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References

1
Umeyama, M., Kato, M., and Inoue, K., 1998, “Effects of Gear Dimensions and Tooth Surface Modifications on the Loaded Transmission Error of a Helical Gear Pair,” ASME J. Mech. Des.  [CrossRef], 120 , pp. 119–125.
2
Litvin, F. L., 1994, "Gear Geometry and Applied Theory", Prentice-Hall, Englewood Cliffs, NJ, pp. 258–287.
3
Litvin, F. L., Chen, N. X., Lu, J., and Handschuh, R. F., 1995, “Computerized Design and Generation of Low-Noise Helical Gears With Modified Surface Topology,” ASME J. Mech. Des.  [CrossRef], 117 , pp. 254–261.
4
Litvin, F. L., Gonzalez-Perez, I., Fuentes, A., Hayasaka, K., and Yukishima, K., 2005, “Topology of Modified Surfaces of Involute Helical Gears With Line Contact Developed for Improvement of Bearing Contact, Reduction of Transmission Errors, and Stress Analysis,” Math. Comput. Modell., 42 , pp. 1063–1078.  [CrossRef]
5
Umeyama, M., 1994, “Effects of Modified Tooth Surface of a Helical Gear Pair on the Transmission Error and its Optimal Design,” "Proceedings of the International Gearing Conference", UK, pp. 377–382.
6
Kahraman, A., and Blankenship, G. W., 1999, “Effect of Involute Contact Ratio on Spur Gear Dynamics,” ASME J. Mech. Des.  [CrossRef], 121 , pp. 112–118.
7
Wagaj, P., and Kahraman, A., 2002, “Impact of Tooth Profile Modifications on Transmission Error Excitation of Helical Gear Pairs,” Sixth Biennial Conference on Engineering Systems Design and Analysis , Paper No. ESDA2002/DES-005.
8
Wagaj, P., and Kahraman, A., 2002, “Influence of Tooth Profile Modification on Helical Gear,” ASME J. Mech. Des.  [CrossRef], 124 , pp. 501–510.
9
Sundaresan, S., Ishii, K., and Houser, D. R., 1994, “A Parametric Study on the Effect of Geometric Gear Design Variables on Static Transmission Error,” "Proceedings of the International Gearing Conference", UK, pp. 383–388.
10
Kubo, A., NonakaT., Kato, N., Shogo, S., and Ohmori, T., 1992, “Representative Form Accuracy of Gear Tooth Flanks on the Prediction of Vibration and Noise of Power Transmission,” AGMA Paper No. 92FTM9.
11
de Vaujany, J. -P., Guingand, M., Remond, D., and Icard, Y., 2007, “Numerical and Experimental Study of the Loaded Transmission Error of a Spiral Bevel Gear,” ASME J. Mech. Des.  [CrossRef], 129 , pp. 195–200.
12
Saiki, K., and Watanabe, T., 2000, “Transmission Error Analysis of Helical Gears for Any Load Condition,” "Proceedings of the Eighth International Power Transmission and Gearing Conference", Paper No. PTG-14421.
13
Umeyama, M., Kato, M., and Inoue, K., 1996, “Transmission Error of a Helical Gear Pair With Modified Tooth Surfaces: 1st Report, Actual Contact Ratio and the Effects of the Load on the Transmission Error,” Trans. Jpn. Soc. Mech. Eng., Ser. C, 62 (603), pp. 4332–4340.
14
Komori, M., Kubo, A., and Kawasaki, Y., 2000, “Design Method of Vibrationally Optimum Tooth Flank Form for Involute Helical Gears With Scattering in Pressure Angle and Helix Angle Deviation,” Trans. Jpn. Soc. Mech. Eng., Ser. C, 66 (646), pp. 1959–1966.
15
Komori, M., Kubo, A., Fujino, H., and Suzuki, Y., 2003, “Simultaneous Optimization of Tooth Flank Form of Involute Helical Gears in Terms of Both Vibration and Load Carrying Capacity,” JSME Int. J., Ser. C, 46 (4), pp. 1572–1581.  [CrossRef]
16
Shibata, Y., Kondou, N., and Ito, T., 1995, “Optimization of Tooth Profile Design for Hypoid Gear,” "Proceedings of the 73rd JSME Fall Annual Meeting (IV)", pp. 232–233.
17
Maatar, M., and Velex, P., 1997, “Quasi-Static and Dynamic Analysis of Narrow-Faced Helical Gear With Profile and Lead Modifications,” ASME J. Mech. Des.  [CrossRef], 119 , pp. 474–480.
18
Kato, S., 1996, “A Study on Technology of Measurement Accuracy in Hypoid,” Ph.D. thesis, Kyoto University, Japan.
19
Wang, Z., 1998, “A Fundamental Study on Analysis and Simulation of Running Performance of Hypoid Gear,” Ph.D. thesis, Kyoto University, Japan.
20
Kubo, A., Tarutani, I., Gosselin, C., Nonaka, T., Aoyama, N., and Wang, Z., 1996, “General Characteristics of Vibration of Gears With Convex Form Modification of Tooth Flank (1st Report, General Model of Meshing Condition of Gears),” Trans. Jpn. Soc. Mech. Eng., Ser. C, 72 (715), pp. 960–968.
21
Kubo, A., and Umezawa, K., 1977, “On the Power Transmitting Characteristics of Helical Gears with Manufacturing and Alignment Errors: 1st Report, Fundamental Consideration,” Trans. Jpn. Soc. Mech. Eng., 43 (371), pp. 2771–2783.

Figures

Grahic Jump Location
+Distribution of angle-distance conversion factor on the tooth flank of the sample hypoid gear
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Distribution of angle-distance conversion factor on the tooth flank of the sample hypoid gear
Figure 6

Distribution of angle-distance conversion factor on the tooth flank of the sample hypoid gear

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+Spring model used to express elastic deformation of the tooth pair in contact
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Spring model used to express elastic deformation of the tooth pair in contact
Figure 7

Spring model used to express elastic deformation of the tooth pair in contact

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+Relationship between actual contact length of contact line and unit stiffness under different loads investigated using the conventional simulation (x=0 denotes the contact line at the middle of the plane of action)
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Relationship between actual contact length of contact line and unit stiffness under different loads investigated using the conventional simulation (x=0 denotes the contact line at the middle of the plane of action)
Figure 8

Relationship between actual contact length of contact line and unit stiffness under different loads investigated using the conventional simulation (x=0 denotes the contact line at the middle of the plane of action)

Grahic Jump Location
+Definition of functions of unit stiffness
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Definition of functions of unit stiffness
Figure 9

Definition of functions of unit stiffness

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+Flow for calculating normalized transmission error by the general model
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Flow for calculating normalized transmission error by the general model
Figure 12

Flow for calculating normalized transmission error by the general model

Grahic Jump Location
+Definition of length of actual contact pattern
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Definition of length of actual contact pattern
Figure 1

Definition of length of actual contact pattern

Grahic Jump Location
+CTFFD, curve on contact line, ridge curve, and their extended part expressed on the plane of action of a helical gear
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CTFFD, curve on contact line, ridge curve, and their extended part expressed on the plane of action of a helical gear
Figure 2

CTFFD, curve on contact line, ridge curve, and their extended part expressed on the plane of action of a helical gear

Grahic Jump Location
+Definition of ridge curve
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Definition of ridge curve
Figure 3

Definition of ridge curve

Grahic Jump Location
+Effect of s on the form of ridge curve
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Effect of s on the form of ridge curve
Figure 4

Effect of s on the form of ridge curve

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+Definition of curve on contact line at position x
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Definition of curve on contact line at position x
Figure 5

Definition of curve on contact line at position x

Grahic Jump Location
+Sample of unit stiffness expressed using its representative, where the conditions in Fig. 8 are simulated
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Sample of unit stiffness expressed using its representative, where the conditions in Fig. 8 are simulated
Figure 10

Sample of unit stiffness expressed using its representative, where the conditions in Fig. 8 are simulated

Grahic Jump Location
+Relationship among actual contact length lcl, deformation D, and deformed area S on the contact line
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Relationship among actual contact length lcl, deformation D, and deformed area S on the contact line
Figure 11

Relationship among actual contact length lcl, deformation D, and deformed area S on the contact line

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+Absolute amount of transmission error for helical gear calculated using general model
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Absolute amount of transmission error for helical gear calculated using general model
Figure 13

Absolute amount of transmission error for helical gear calculated using general model

Grahic Jump Location
+Relationship between actual contact ratio and peak-to-peak value of transmission error for helical gear calculated using conventional simulation and general model
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Relationship between actual contact ratio and peak-to-peak value of transmission error for helical gear calculated using conventional simulation and general model
Figure 14

Relationship between actual contact ratio and peak-to-peak value of transmission error for helical gear calculated using conventional simulation and general model

Grahic Jump Location
+Distribution of actual contact ratio corresponding to the maximum positions of peak-to-peak value of transmission error
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Distribution of actual contact ratio corresponding to the maximum positions of peak-to-peak value of transmission error
Figure 15

Distribution of actual contact ratio corresponding to the maximum positions of peak-to-peak value of transmission error

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