Numerical Simulation of Contact Ellipses of Circular-Arc Curvilinear-Tooth Gears

Table 6-1 Some major design parameters of the circular-arc curvilinear-tooth gear pair

Design parameters Pinion Gear

Number of Teeth (T⁽ⁱ⁾) 18 36

Normal Module (M) 3mm/tooth

Normal Pressure Angle (α ) 20∘

Radius of the Disk-Type Cutter (Rab) (Fig. 3-7) 30mm Radius of Rack Cutter Normal Section (R⁽ⁱ⁾)

(Fig. 3-6 & Fig. 3-8) 40mm 40mm

Face Width (W) (Fig. 3-7) 30mm

The major design parameters of the proposed gear set are listed in Table 6-1. The contact patterns under different assembly conditions are acquired by the computer programs based on TCA. Fig. 6-3 depicts the contact path and the contact ellipses on the gear tooth surfaces under ideal and error assembly conditions. The contact ellipses are plotted when the gear rotates every 1˚ from -5˚ to 5˚. It is found that the bearing contact points of the gear set are localized in the middle region of the tooth flank due to the curvilinear tooth trace. The contact ellipse moves upward with assembly error of Δ =C 0.1mm. The contact ellipses under axial misalignments Δ and γ Δ are also shown in Fig. 6-4 and Fig. 6-5, γ

respectively. It is noted that the positions of the contact points of the gear set are located near the middle region of the tooth flank even under the axial misalignments.

(a) Ideal assembly condition

(b) With assembly error Δ =C 0.1mm

Fig. 6-3 Contact patterns of the gear tooth surface under ideal and error assembly conditions (Example 6-1)

(a) Horizontal axial misalignment Δ =γ_h 0.1

(b) Horizontal axial misalignment Δ =γ_h 0.5

Fig. 6-4 Contact patterns of the gear tooth surface under horizontal axial misalignments (Example 6-1)

(a) Vertical axial misalignment Δ =γ_v 0.1

(b) Vertical axial misalignment Δ =γ_v 0.5

Fig. 6-5 Contact patterns of the gear tooth surface under

Example 6-2

The instantaneous contact of tooth surfaces at a point is spread over an elliptical area, as shown in Fig. 6-6. Symbol a and b represent the semi-major and semi-minor length of the ellipse. The relationship between the radii R of the disk-type tool (Fig. 3-7) and _ab semi-minor axis of the contact ellipse b will be discussed in this example.

2a

2b

X Y

T

Fig. 6-6 Orientation and dimension of the contact ellipse

The major design parameters of the gear pair are the same as list in Table 6-1 except the value of R . Fig. 6-7 shows the contact ellipses of the gear surface with _ab R_ab =20mm,

ab 30

R = mm and R_ab =40mm under ideal assembly condition. The contact points of the gear pair are all located on the middle region of the gear flank no matter R_ab =20mm,

ab 30

R = mm or R_ab =40mm, and the length of the major axis of contact ellipses increases evidently with the increase of the R . This is due to the fact that a large radius _ab R induces _ab a smaller crowning effect on the tooth flank. If the radius R tends to infinity, the _ab curvilinear-tooth gear becomes a spur gear, and the ellipse becomes a line contact. Fig. 6-8 shows the effects of design parameters R and the semi-major axis length _ab b of the contact ellipse. It is found that the semi-major axis length b is proportional to the R . Gear _ab designers can select the proper gear design parameters for their needs.

Gear with R_ab =20mm under ideal assembly condition

Gear with R_ab =30mm under ideal assembly condition

Gear with R_ab =40mm under ideal assembly condition

Fig. 6-7 Contact patterns of the gear tooth surfaces under ideal assembly condition with

20 40 60 80 100 120 140 160 180 200 0

1 2 3 4 5 6 7 8 9 10 11

Radius of the Disk-Type Tool (mm)

Semi-Major Length of the Ellipse (mm)

Fig. 6-8 Effects of design parameters R versus _ab the semi-major axis length a of contact ellipse

Example 6-3

The main advantage of the circular-arc gear pair is efficiently to reduce the contact stress by a larger contact area on the contact tooth surfaces. However, contact ellipse is the key index to predict the contact situation. The area of the contact pattern depends on the lengths of major and minor axes. The gear parameters are chosen to investigate their effects on the corresponding contact patterns, while the radius of the disk-type cutter is fixed as

ab 30

R = mm.

As the tooth undercutting condition of the gear generated by rack cutter Σ is severer _g than the pinion generated by rack cutter Σ discussed in CHAPTER 2, because the profiles _p

of the circular-arc cutters Σ and _g Σ are different. According to the analysis results of _p Example 4-3, a larger number of teeth can effectual prevent the tooth undercutting. Therefore, the gear with a higher number of teeth is suggested and the gear pair should be generated by a concave rack cutter as shown in Fig. 3-6 to prevent tooth undercutting.

Major design parameters for this example are shown in Table 6-2. According to Fig. 6-9, the length of the minor-axis of contact ellipse is inverse proportional to the radius R^{( )g} . However, the dimension of the contact pattern is insensitive to the variation of R^{( )g} . Relationships between the design parameters R^{( )g} to the length of semi-major axis and semi-minor axis under ideal assembly condition are shown in Fig. 6-10. Length of the semi-minor axis of contact ellipse increases significantly conspicuously with a smaller value of R^{( )g} , but the length of semi-major axis of contact ellipse nearly takes no effect with the variation of R^{( )g} . However, as stated in Example 6-2, the length of semi-major axis can be adjusted by the radii R of the disk-type cutter. _ab

Table 6-2 Some major design parameters of the circular-arc curvilinear-tooth gear pair

Design parameters Pinion Gear

Number of Teeth(T⁽ⁱ⁾) 18 72

Normal Module(M) 3mm

Normal Pressure Angle(α ) 20∘

Radius of the Disk-Type Cutter

(R_ab) (Fig. 3-7) 30mm

Face Width (W) (Fig. 3-7) 30mm

Fig. 6-9 Contact ellipses on the gear tooth surface (Example 6-3)

20 40 60 80 100 120 140 160 180 200

0.28 0.29 0.3 0.31 0.32 0.33

Radius of the Circular-Arc Profile of the Cutter (mm)

Semi-Minor Length of the Elipse (mm)

20 40 60 80 100 120 140 160 180 200

1.62 1.64 1.66 1.68 1.7 1.72 1.74 1.76 1.78

Semi-Major Length of the Elipse (mm)

Minor axis Major axis

6.4 Remarks

In this chapter, the contact patterns are obtained by surface topology method. The instantaneous contact of tooth surface at a point is spread over an elliptical area. The effects of the assembly errors and major gear design parameters to the shape of contact ellipses are acquired by the developed computer simulation programs. According to the analysis results of numerical examples, some conclusions can be drawn as follows:

1. The contact patterns of the circular-arc curvilinear-tooth gear pair are localized near the middle region of the tooth flank due to the curvilinear tooth trace even the gear pair is meshing under axial misalignments.

2. Length of the major axis of the contact ellipse is highly related to the radius of the disk-type rack cutter.

3. Length of the minor-axis of the contact ellipse is related to the radii of the circular-arc profile of rack cutters.

CHAPTER 7

Conclusions and Future Works

Gears are widely used in industry for power transmissions. A new type of gear called circular-arc curvilinear-tooth gear, which is generated by a disk-type cutter with the normal cross section of circular-arc profiles, has been proposed in this thesis.

The mathematical model of the circular-arc curvilinear-tooth gear is developed in this study. Tooth undercutting is investigated based on the mathematical model of the gear.

Besides, the contact characteristics of the gear pair such as KE and contact ellipses can also be investigated by computer simulation programs based on the developed mathematical model and the TCA.

7.1 Conclusions

Based on the analysis results obtained in the previous chapters, some conclusions can be made as follows:

1. The circular-arc curvilinear-tooth gear is generated by a disk-type cutter with the normal cross section of the circular-arc profiles. However it can be considered that the gear is generated by an imaginary rack cutter. The mathematical model of circular-arc curvilinear-tooth gear can be derived based on the theory of gearing. The transverse gear chordal thickness measured at the middle section is larger than those of other sections.

2. The tooth undercutting condition is developed based on the mathematical model of

the occurrence of tooth undercutting at the middle sections of face width of the curvilinear-tooth gear is much easier than other sections. Compared with the concave tooth surfaces, convex tooth surfaces are mush easier to undercut. Tooth undercutting can be avoided by the design of gears with a larger number of teeth, pressure angle and radii of the circular-arc profile of the rack cutter.

3. Gear pairs with a higher contact ratio can be achieved by designing a smaller radius of the circular-edge of the cutter. The circular-arc curvilinear-tooth gear pair is insensitive to axial misalignments. However, it is sensitive to the center distance assembly error. The KE of a gear pair can be pre-design to a parabolic type by properly selecting the design parameters RΔ .

4. The bearing contact of a gear pair is localized near the middle region of the tooth flank by means of the curvilinear tooth trace, and the edge contact efficient be avoided. The instantaneous contact of tooth surfaces at a point is spread over an elliptical area. Besides, the shape of the contact ellipse can be adjusted by the radii

R of the disk-type cutter and the radii ab R of circular-arc profile of the cutter. ^{( )i} Length of the minor-axis of the contact ellipse can be increased by the design of a smaller radius R of the circular-arc profile. ^{( )i}

5. Gear pairs generated by cutters with smaller radii of the circular-arc profiles can efficiently enhance the gear strength because a larger contact area is induced.

However, the cutter with a concave normal section is much easier to undercut the gear with a smaller radius of the circular-arc profile than a convex one. It is suggested that the gear with a higher number of teeth of the gear pair should be generated by a concave rack cutter.

7.2 Future Works

The mathematical model, undercutting condition and tooth contact analysis of the circular-arc curvilinear-tooth gear have been studied in this thesis. In the future, some future works can be studied are listed as follows:

1. The proposed theorems and computer simulated results can be verified by setting up suitable experiments by the following researchers.

2. When calculating TCA, the tooth surfaces are assumed to be rigid in the developed mathematical model. The effects of loads and elastic deformations of tooth surfaces are all neglected. The finite element analysis (FEA) may be implemented to attain more realistic TCA results.

3. The mathematical model is developed based on the consideration of an imaginary rack cutter with the circular-arc profile on its normal section and the rack cutter is moving along a circular trace. However, in practice manufacturing, the gears are usually generated CNC hobbing process. Therefore, the mathematical model of circular-arc curvilinear-tooth gear cut by a hob cutter may be developed in the future.

4. The tooth profile and trace of the circular-arc curvilinear-tooth gear are all in the form of circular-arc. It can be further extended to derive the mathematical model with a noncircular-arc curve, e.g. parabolic or elliptical curves.

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