Tooth contact analysis and manufacturing of dual lead worm ...

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.12, No.1, 2018

Paper No.17-00515© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0026]

1. Introduction

A worm gear set is essential in motion and power transmission between skew axes which do not intersect due to a

large speed reduction ratio and a high road carrying capacity (Townsend, 1991; DAVIS, 2005; Stephen, 2012). This worm gear set is composed of a driving worm and a driven worm wheel. The self-lock function is one of the main features of worm gear set. Therefore, a worm gear set is used in the index table of machine, the hoist of elevator, and others. On the other hand, the sliding velocity on the meshing of worm gear set is higher than those of other gear sets, so it is easier to burn tooth flank and transfer efficiency is lower. The material of the worm wheel is usually high strength brass in order to avoid burn-in and the material of the worm is usually Chromium molybdenum steel. When the worm and worm wheel are meshed each other under the loaded condition, the lubrication between the worm and worm wheel flank is also very important to avoid burn-in. Setting up the entry and exit gap on the tooth contact pattern is necessary to make an oil film between the worm and worm wheel flanks.

In this paper, a dual lead worm gear set whose profile is ISO type I (JIS B 0102, 1999) is targeted. Since the axial modules of right and left flanks of dual lead worm gears in ISO type I are different, the tooth thickness of the worm is

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Tooth contact analysis and manufacturing of dual lead worm gears in ISO type I

Shigenori HAMADA*, Kazumasa KAWASAKI** and Isamu TSUJI*** *Ogic Co., Ltd.

1-15-13 Hishie, Higashi-Osaka-shi, Osaka 578-0984, Japan E-mail: [email protected]

**Niigata University Ikarashi 2-nocho, Nishi-ku, Niigata 950-2181, Japan

***Iwasa Tech. Co., Ltd. 62-4 Takase-cho, Funabashi-shi, Chiba 136-0082, Japan

Received: 2 October 2017; Revised: 12 November 2017; Accepted: 8 February 2018

Abstract In this paper, the tooth contact pattern and transmission errors of dual lead worm gears in ISO type I are analyzed using tooth contact analysis (TCA) and the worm gears are manufactured. The setting of the hobbing machine for worm wheel manufacturing and the amounts of modification in profile and lead directions of worm grinding machine for worm manufacturing are determined so that the expected meshing conditions can be obtained based on TCA results. The dual lead worm gears in ISO type I were designed and manufactured

based on this method, and the actual tooth contact patterns and transmission errors were compared with analyzed ones. Compering with TCA results, actual tooth contact patters were similar to TCA results for both flanks. However, the measured transmission errors were slightly larger than TCA results. After manufacturing worm gears, the profile errors of a worm hob cutter for tool were measured using a gear measuring machine and these measured results were fed back to TCA results in order to improve the accuracy of TCA. The influence of the tooth profile errors of the hob cutter was investigated. TCA results with feedback of tooth

profile errors of the hob cutter were closer to the actual transmission errors than TCA results without feedback. As a result, the validity of this method was confirmed.

Keywords : Dual lead, Worm gears, Tooth contact analysis, Manufacturing, Tooth contact pattern, Transmission error

2© 2018 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2018jamdsm0026]

Hamada, Kawasaki and Tsuji, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.12, No.1 (2018)

changed by moving the worm axis position. It is possible to adjust the back-lash of the worm gear set using this function. In the meshing under the loaded condition for a long term, the tooth flank of the worm wheel is worn away and the back-lash increases gradually. The worm gear set used in the index table that repeats C.W. and C.C.W. rotations, increasing the back-lash causes some troubles like quadrant projection. In this case, this trouble is resolved by adjusting back-lash.

Practically, the tooth contact pattern and transmission errors are modified properly by the accumulated experiences and the instinct of disciplined operator. The mathematical model, design, tooth contact analysis (TCA), and stress analysis of the worm gear set have been developed (Litvin et al., 2007; Fang and Tsay, 2000; Sohn and Park, 2016; Simon, 2006). Localizing of the tooth contact for such a gear set is required to prevent edge contact and separation of tooth surfaces that may occur in presence of alignment errors. However, the analyzed tooth contact patterns and transmission errors have not been compared with experimental ones, and the influence of the tooth profile errors of the hob cutter for tool on tooth contact patterns and transmission errors, and its feedback to TCA have not been considered the in the previous papers.

Measurement of the tooth surface of the spiral bevel or hypoid gear has been carried out, and the processing error was corrected by the movement of the machine, namely, by machine settings on hypoid generator (Artoni et al., 2013; Fan et al., 2008; Kato and Kubo, 1993; Takeda et al., 2014). However, the researches with respect to the influences of the tool precision on the transmission errors have been hardly carried out. When the tool error is measured before machining and the relation between the amount of error and the transmission errors is clear, a precise gear machining can be realized. In particular, since a worm gear set is used in the index table of machine, the set of high quality is required. However, it is impossible to satisfy the requirement unless only the machine setting is corrected. Therefore, it is important to measure, analyze, and feedback the tool errors.

In this paper, before manufacturing worm gear set, the tooth contact pattern and transmission errors are analyzed to realize a good meshing performance using TCA program (Abe and Tsuji, 2004). According to TCA results, the settings of hobbing machine for the worm wheel manufacturing and worm grinding machine for worm manufacturing are decided directly. Afterwards, the high precision dual worm gear set is manufactured as based upon TCA results. After manufacturing worm gears, the profile errors of a worm hob cutter for tool are measured using a gear measuring machine and these measured results are fed back to TCA results in order to improve the accuracy of TCA. The influence of the tooth profile errors of the hob cutter is investigated. TCA results with feedback of tooth profile errors of the hob cutter were closer to the actual transmission errors than TCA results without feedback. The tooth profile of the hob cutter has not been considered in the previous studies. As a result, the validity of this manufacturing method was verified.

2. Tooth surfaces of worm and worm wheel for worm gear set 2.1 Generating line of cylindrical worm

The tooth flank form of the worm gear set with a cylindrical worm gear is defined as the generating line appearing on the worm flank. Figure 1 shows the coordinate system of a worm flank. The components for x, y and z axes of generating line are expressed as A, B and C, respectively. The variable parameter u expressing a position of generating line is introduced. Therefore, A, B, and C are expressed in Eq. (1):

� = ��, � = ��, = �� (1)

Assuming that θ is the rotation angle of the screw motion of worm, the tooth flank of the cylindrical worm is

generally expressed in Eq. (2): � = � + , + �� = � cos � − � sin �� = � sin � + � cos � (2) � = ℎ�� − �� +

where h is the reduced pitch and θ0 is the initial value of θ.

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Fig. 1 Coordinate system of worm

In the case of worm gear set in ISO type I, the generating line is directly decided by Eq. (3):

� = ��, � = �, = � tan �� (3)

where rg is base circle radius. γg is lead angle on the base circle. Substituting Eq. (3) for Eq. (2),

� = �� cos � − � sin � = �� sin � + � cos � (4) � = ℎ�� − �� + � tan��

x, y and z in Eq. (4) are the tooth flank of ISO type I. 2.2 Meshing condition of worm and worm wheel

Figure 2 shows the coordinate system expressing the axes of the worm and worm wheel whose angular velocities are ω1 and ω2, respectively. The worm axis coincides with z axis. The common perpendicular of the axes of the worm and worm wheel is y axis. The offset distance is a.

The velocity vector of the worm and worm wheel at contact point are v1 and v2, respectively. These equations are as follows:

1 = "1�× �� (5) 2 = "2 × �� − %,�

The relative velocity w between the worm and worm wheel at contact point is expressed in Eq. (6):

&��, �� = ' − ( ��= "')* − �� + *��, + � − %��+ (6) where i, j, and k are unit vectors of x, y, and z directions, respectively and ε =ω1/ω2.

The worm wheel is generated by a tool surface that is identical to the worm tooth flank in principle. Assuming the relative velocity w between the worm and worm wheel at the moment of rotation angle θ and unit surface normal vector n of the tooth flank of worm, the equation of meshing between the two gears is as follows (Sakai, 1955; Litvin and Fuentes, 2004).

,��, �� ∙ &��, �� = 0 (7) where n is represented by

��,��, �� =/�/� ×

∂�∂�

1/�/� ×/�/�1

��8��

Worm axis

x

O yθ

z

3



O y

x

ω1

Worm axis

Axis ofworm wheelω2

Fig. 2 Coordinate system expressing axes of worm and worm wheel

Substituting Eq. (6) and (8) for Eq. (7), the locus surface ��, ��of contact line between the worm and worm wheel is represented by Eq. (7):

��, �� = � + , + �� = �� cos � − � sin � = �� sin � + � cos � (9)

� = − + �% − *ℎ�tan �� cos �

2.3 Tooth contact analysis

The worm and worm wheel mesh in coordinate system in O-xyz as shown in Fig. 2 in order to analyze the tooth contact pattern and transmission errors of worm gear set. The analysis method is the same as that in the case of other kinds of gears (Tsuji et al., 2013; Kawasaki et al., 2015). That is, the position vector of the tooth surfaces of the worm and worm wheel must coincide and the direction of two unit surface normals at these positions must also coincide in order to contact two surfaces. The tooth contact pattern and transmission errors of worm gear set were analyzed under no-loaded condition based upon mentioned above.

Table 1 shows the dimensions of dual lead worm gear set manufactured in this study. The tooth profile is ISO type I. The center distance is 200 mm. The reduction ratio is 50. The axial module of flank A is 6.2466 and that of flank B is 6.4960. This means that the high precision dual lead worm gear set is used in the index table of machining center.

The circular arc modification of worm tooth flank is set up in order to control the tooth contact pattern for tooth profile direction as shown in Fig. 3. The amount of circular arc profile modification of worm is 0.02 mm at tip and root. In this case, the shape of worm tooth flank should be convex.

Figure 4 shows the analyzed results of tooth contact pattern of both flanks A and B. The entry gap to keep good lubrication condition and the exit gap to avoid vibration and noise are set up for both flanks. In this paper, the pitch correction method is applied to control the tooth contact pattern for tooth width direction. The axial leads of both flanks of worm are extended 0.1% for flank A and 0.15% for flank B of original leads. In addition, the pressure angle of worm is modified from original pressure angle to meshing for both flanks.

Figure 5 shows the analyzed transmission errors. The shape of the transmission errors of one meshing is parabolic as shown in Fig. 5. When the circular arc profile modification of worm is set up, the shape of the transmission errors of one meshing becomes parabolic. This shape is very important to have the point of intersection between adjacent transmission error curves. The amount of transmission errors is decided by distance between the top of this parabolic curve and the point of intersection. The transmission errors are 3.9 to 6.7 second for both flanks.

The tooth flank modifications mentioned above make the worm gear insensitive for manufacturing and alignment errors in some cases. Therefore, the application of an appropriate tooth flank modification is very important to create practical high precision worm gear set. Table 2 shows the tooth flank modifications of worm shaft applied in TCA. Both the axial leads and pressure angles in flanks A and B are different. The arc modifications in flanks A and B are the same and are 0.02 mm.

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Table 1 Dimensions of dual lead worm gear

Flank A Flank B

Center distance 200 mm

Starts of worm 1 Right hand

Wheel num. of teeth 50

Outside dia. of worm 92 mm

Mean dia. of worm 79.2 mm

Tooth width of wheel 60 mm

Axial module 6.2466 6.4960

Pressure angle 14.8828 deg. 19.9033 deg.

Flank A Flank B

Fig. 3 Arc modifications in tooth profile direction

Flank A Flank B

Fig. 4 Results of analyzed tooth contact patterns

Profile line without modification

Profile line without modification

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Transmission errors (sec.)

3.9sec Rotation angle

of worm（deg.）

0 4 -4


Rotation angle

of worm（deg.）

0 4 -4

6.7sec

Flank A Flank B

Fig. 5 Results of analyzed transmission errors

Table 2 Tooth profile modification of worm shaft

Fig. 6 Tooth grinding of worm shaft

Fig. 7 Gear cutting of wheel

Flank A Flank B

Axial lead 19.6439 mm 20.4384 mm

Pressure angle 15.0428 deg. 20.1033 deg.

Arc modification 0.02 mm 0.02 mm

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(a) Worm (b) Worm wheel Fig. 8 Measurement of worm and worm wheel

3. Manufacturing of worm and worm wheel 3.1 Generation of worm wheel and grinding of worm

Figure 6 shows the situation of the tooth flank of the worm ground using a NC worm grinding machine. This

machine has NC functions which can process various types of worms and corresponds to tooth flank modifications. Moreover, it can manufacture the tooth profile of worm in ISO type I, and it automatically shapes the grinding wheel and processing worm. As mentioned above, this is applied to the tooth flank modifications of the worm in Table 2.

Figure 7 shows the situation of the generation of worm wheel using a specialized high precision hobbing machine. In worm wheel manufacturing, the profile of the cutting edge of the dual lead worm hob cutter is the same as the original profile of worm. The center distance of hobbing machine is 200 mm that is the same as worm gear set and tilting angle of the cutter head is 0 degree. The material of worm wheel is high strength brass. Since the thermal expansivity of brass is higher than that of the steel, it is necessary to pay attention to keep temperature as cutting.

3.2 Measurement of worm gear set The lead and profile errors of the worm were measured using a gear measuring machine as shown in Fig. 8. The

profile errors of the worm wheel were also measured. As a result, it was confirmed that both the worm and worm wheel members were manufactured properly and the requirement was satisfied. Both the profile error and lead error were less than 1µm. This means that the designed tooth surface using TCA can be reproduced. The measured results are shown in Table 3.

Table 3 Measured results of lead and profile

Worm [µm]

Worm Wheel [µm]

Absolute value of total amount error [µm]

Flank A Flank B Flank A Flank B Flank A Flank B

Angular deviation of profile 9.1 1.9 9.6 2.9 0.5 1.0

Profile total deviation 14.5 9.3 13.5 8.9 1.0 0.4

Lead form deviation 2.6 1.5 1.8 2.5 0.8 1.0

Lead angular deviation 0.8 -0.3 1.7 -1.2 0.9 0.9

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4. Measuring tooth contact pattern and transmission errors

4.1 Tooth contact pattern

The actual tooth contact patterns and transmission errors of manufactured dual worm gear set were checked on the meshing tester as shown in Fig. 9. Figure 10 shows the results of TCA and actual tooth contact patterns. Compering with TCA results, actual tooth contact patterns are similar to TCA results for both flanks.

Fig. 9 Meshing test of worm gears

Fig. 10 Comparison of results of TCA and actual tooth contact patterns

Fig. 11 Measuring instrument of transmission errors

Flank A Flank B

TCA Actual TCA Actual

Worm wheel

Worm

Output encoder (worm wheel)

Input encoder (worm)

8



-10-505

10Transmission errors (Flank A)

One revolution of worm wheel

Fig.12 Results of measured transmission errors

Table 4 Comparison of amplitude of analyzed and measured transmission errors

Flank A Flank B

Results of TCA 3.90 sec. 6.70 sec.

Measured results 6.33 sec. 7.91 sec.

Difference 2.43 sec. 1.21 sec.

4.2 Transmission errors

Figure 11 shows the measuring instrument of the transmission errors. The transmission errors are calculated by dividing the pulse of encoders between input (worm) and output (worm wheel) based upon the reduction ratio.

Figure 12 shows the measured transmission errors. The horizontal axis shows revolution angle of worm wheel and the vertical axis shows transmission errors. The transmission errors are combined with short waves caused by one meshing of tooth and the long wave caused by one revolution of the worm wheel.

Table 4 shows the comparison of the amplitudes of the analyzed using TCA and measured transmission errors. There are slight differences between the amplitudes of the transmission errors (short wave) of flanks A and B. Comparing between TCA and measured results, the differences were 2.43 seconds at the flank A and 1.21 seconds at the flank B, respectively.

It is considered that the cause is the accuracy of gear blank, the runout and profile errors of the hob cutter, the profile errors of gears, the accuracy of machine and others, but the profile errors of the hob cutter has the largest influence.

5. Worm wheel hob cutter measurement

The consistencies of the tooth contact pattern and transmission errors between TCA and the measured results are recognized each other. However, the subtle differences of transmission errors between them exist, caused by some manufacturing and measuring errors. In this section, it is considered how transmission errors are influenced by tooth profile errors of hob cutter.

5.1 Theoretical hob design

The tooth flank of the worm in ISO type I is an involute helicoid, and this involute helicoid is generated by a rack cutter. A tool with a cutting edge on the worm is called a hob tool. The rack shape is projected to hob movement. The worm wheel is made by pushing in synchronism with this rack cutter.

The outer diameter of the hob cutter has dimensions designed using TCA so as to achieve optimal tooth contact, and it is usually set about 10% larger than the outside diameter of the worm. The lead γ is determined from the basic relationship of the gear geometry in Eq. (10).

-10

-5

0

5

10Transmission errors (Flank B)

9



� = tan3( 456�(7( 8��10�

The tooth thickness is determined by subtracting the amount of backlash. The numbers of hob cutter and worm

threads should be designed with the same number.

5.2 Hob shape measurement principle using probe

The backing off of the hob cutter is designed by Archimedes curve. Even when the cutting face of the hob cutter is

reground, the same cutting performance as a new one can be obtained without changing the tooth profile shape. The tooth profile of the hob cutter is measured using a gear measuring machine. The measured position is inclined by 1 degree from the lake face of the cutting edge as shown in Fig 13. From the design principles of the hob cutter, this measuring method can be accurately evaluated (Klingelnberg GINA Hob, 2012).

Fig. 13 Measuring position of hob tooth profile

Fig. 14 Measurement of tooth profile errors of hob cutter

Archmedes curve

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Transmission errors (sec.) Transmission errors (sec.)

Rotation angle of worm（deg.）

0 4 -4

7.9sec

Flank A Flank B

Fig. 15 Results of measured tooth profile errors

Table 5 Results of approximations for tooth profile errors of hob cutter

9 = %�: + ;�' + <� + 7 Standard

deviation

[µm]

a b c d

Flank A -0.01 1.5009 -70.85 1068 0.503

Flank B -0.0092 1.3398 -62.56 945.57 0.733

Flank A Flank B

Fig. 16 Results of amplitude of analyzed transmission errors with feedback of tooth profile errors of hob cutter

Table 6 Comparison of analyzed, feedback, and measured transmission errors

Flank A Flank B

TCA(Analyzed) 3.9 sec. 6.7 sec.

TCA(Feedback) 5.9 sec. 7.9 sec.

Measured results 6.33 sec. 7.91 sec.


5.9sec Rotation angle of worm（deg.）

0 4 -4

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5.3 Influence of hob cutter error on TCA

At first, the tooth profile errors of hob cutter are measured using a gear measuring machine as shown in Fig. 14. The measured results of tooth profile errors for both flanks are shown in Fig. 15. The horizontal axis shows the radius

of hob cutter and the vertical axis shows tooth profile errors. The tooth profile errors of the hob cutter are 4 µm on the root and 14 µm on the tip for flank A. The tooth profile errors of the hob cutter are 8 µm on the root and 6 µm on the tip for flank B. Due to these profile errors, it was considered that the difference between theoretical TCA result and actual measured one occurred.

Then, these errors δ are approximated by polynomial function of three degree of radius u of hob cutter using method of least squares (C. R. Wilie). The results of approximations are listed in Table 5. The standard deviation of both flanks is less than 1µm.

Based on the results, transmission errors are re-calculated and the influence of the tooth profile errors of the hob cutter is investigated. These results were reflected to TCA program.

The analyzed transmission errors with feedback of the tooth profile errors of the hob cutter to TCA are shown in Fig. 16. The comparison of the values of amplitude of analyzed, fedback, and measured transmission errors are listed in Table 6. The value with feedback of the tooth profile errors of the hob cutter to TCA is 5.9 sec. for flank A and 7.9 sec. for flank B, respectively. These values are closer to the actual ones than theoretical TCA without any errors.

From these results, the influence of tooth profile errors of hob cutter to transmission errors was confirmed.

6. Conclusion

The tooth contact pattern and transmission errors of the worm and worm wheel in ISO type I was set up using TCA program. Then, the worm and worm wheel were manufactured based upon TCA result. The consistency of tooth contact patterns and transmission errors between TCA and measured results was recognized. After manufacturing worm gears, the profile errors of a worm hob cutter for tool were measured using a gear measuring machine and these measured results were fed back to TCA results in order to improve the accuracy of TCA. The influence of the tooth profile errors of the hob cutter was investigated. TCA results with feedback of tooth profile errors of the hob cutter were closer to the actual transmission errors than TCA results without feedback. As a result, the validity of this manufacturing method was confirmed.

References Abe, Y. and Tsuji, I., Tooth contact analysis for worm gear, The JSME Symposium on Motion and Power Transmissions

2004 (in Japanese) (2004), pp. 371-373.

Artoni, A., Gabiccini, M. and Kolivand, M., Ease-off based compensation of tooth surface deviations for spiral bevel and hypoid gears: Only the pinion needs corrections, Mechanism and Machine Theory, Vol .61 (2013), pp. 84-101.

Davis, J. R., Gear Materials, Properties, and Manufacture, ASM International Technical Books Committee (2005), pp. 8-9.

Fang, H. S. and Tsay, C. B., Mathematical model and bearing contacts of the ZN-type worm gear set cut by oversize hob cutters, Mechanism and Machine Theory, Vol. 35 (2000), pp.1689-1708.

Fan, Q., DaFoe, RS., and Swanger, JW., Higher order tooth flank form error correction for face-milled spiral bevel and hypoid gears, Transactions of the ASME, Journal of Mechanical Design (2008), Vol. 130, pp. 072601-1-7.

Involute helicoid, Vocabulary of gear terms - Definitions related to geometry, JIS B 0102 (1999) (in Japanese), 1.4.2.1, p.12.

Kato, S. and Kubo, A., Proper setting of form-cutting machine of hypoid gears using measured tooth flank form, Transactions of the JSME (in Japanese), Vol. 59, No. 563 (1993), pp. 2245-2250.

Kawasaki, K., Tsuji, I., Gunbara, H. and Houjoh, H., Method for remanufacturing large-sized skew bevel gears using CNC machining center, Mechanism and Machine Theory, Vol. 92 (2015), pp. 213-229.

Klingelnberg GINA Hob, 403502A, 03-003en-US (2012). Litvin, F. L. and Fuentes, A., Gear Geometry and Applied Theory. 2nd Ed (2004), pp. 98-101, Cambridge University

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Press. Litvin, F. L., Gonzalez-Perez, I., Yukishima, K., Fuentes, A. and Hayasaka, K., Design, simulation of meshing, and

contact stresses for an improved worm gear drive, Mechanism and Machine Theory, Vol. 42 (2007), pp.940–959. Sakai, T., A study on the tooth profile of hypoid gears, Transactions of the JSME (in Japanese), Vol. 21, No. 102 (1955),

pp. 164-170. Simon, V., Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears,

Mechanism and Machine Theory, Vol. 41 (2006), pp. 707-724.

Sohn, J. and Park, N., Geometric interference in cylindrical worm gear drives using oversized hob to cut worm gears, Mechanism and Machine Theory, Vol. 100 (2016), pp. 83-103.

Stephen, P. R., Handbook of Practical Gear Design and Manufacture. 2nd Ed., CRC Press (2012), pp. 34-39.

Takeda, R., Komori, M. and Nishino, T., Performance analysis of generated hypoid gear based on measured tooth flank form data, Mechanism and Machine Theory (2014), Vol.72, pp. 1-16.

Townsend, D. P., Dudley’s Gear Handbook. 2nd Ed., the Design, Manufacture, and Application of Gears (1991), pp. 2.40-2.46, McGraw-Hill.

Tsuji, I., Kawasaki, K., Gunbara, H., Houjoh, H. and Matsumura, S., Tooth contact analysis and manufacture on multitasking machine of large-sized straight bevel gears with equi-depth teeth, Transactions of the ASME, Journal of Mechanical Design, Vol. 135, No. 2 (2013), pp. 034504-1-8.

Wilie, C. R., Advanced Engineering Mathematics Third Ed. (1988) (in Japanese), pp.123-139, Brain Co. Publishing.

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Tooth contact analysis and manufacturing of dual lead worm ...

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