Work piece
2.7 Surface Deviations of Curvilinear-Tooth Gear
The gear design parameters are chosen the same as those listed in Table 2.1.
Figure 2.7 shows the tooth profiles of the curvilinear-tooth gear generated by nominal radii of circular arc tooth trace =100 mm, 120 mm, and 200 mm at the cross section =0 mm and =30 mm, respectively. The tooth profiles generated by a hob cutter with different nominal radii of circular arc tooth trace are the same at the cross-section =0 mm as illustrated in Fig. 2.7(a). The proposed curvilinear-tooth gear can be viewed as the gear with a varying helical angle across the whole face width, and the helical angle at the cross section =0 mm is zero. Therefore, the tooth profiles of the curvilinear-tooth gear generated by different at cross section
=0 mm is the same as that of spur gears. The results illustrated in Fig. 2.7(a) verify that the mathematical models proposed herein are correct. Figure 2.7(b) shows the tooth profiles generated by a hob cutter with different are not the same at the cross section =30 mm.
Rc
Z2 Z2
Z2
Z2
Rc
Z2
Rc
Z2
Figure 2.8 reveals the transverse chordal thickness deviations of the curvilinear-tooth gear generated by nominal radii of circular arc tooth trace =100 mm, 120 mm, and 200 mm, respectively. It is found that the transverse chordal thickness at the middle section of face width, i.e., =4.709 mm, is larger than those
Rc
tc
at other sections. A smaller results in a smaller transverse chodal thickness at the ends of face width as shown in Fig. 2.8. Figure 2.9 illustrates the relationship between the component of the tooth cross section and the tooth thickness at the addendum circle, . It reveals that the tooth thickness at the middle section of face width, i.e., =1.599 mm, is smaller than those at other sections, and a smaller induces a larger tooth thickness at the ends of face width.
Rc
Z2
ta
ta Rc
ta
The tooth thickness of the curvilinear-tooth gear generated by hob cutters with different nominal radii of circular arc tooth trace are the same at the middle section of the tooth flank, i.e., =0 mm. The analysis results shown in Figs. 2.8 and 2.9 indicated that the deviation of tooth thicknesses, and , at the both ends of the face width decreased when the nominal radius of circular arc tooth trace is increased.
Z2
tc ta
Rc
It is noted that the herringbone gears are in line contact, however, the curvilinear-tooth gear pair proposed herein is in point contact. Therefore, the advantage of the proposed gear pair is not sensitive to the axial misalignments.
According to the analysis result shown in Fig. 2.8, the curvilinear shapes of the teeth are similar to those of crowned gear tooth surfaces. Thus, the phenomenon of gear edge contact of the curvilinear-tooth gears can be avoided.
2.8 Remarks
The mathematical model of curvilinear-tooth gears has been developed on the basis of the CNC hobbing machine cutting mechanism and the gear theory. The model is represented as a function of hob cutter design parameters and generating motion parameters. The developed mathematical model provides the industry with an
R =200 mm R =120 mm R =100 mmc
c c
0 2 4 6 8 (mm)
4
2 6
8 (mm)
Pitch circle
(a) Tooth profile at cross section Z2=0 mm
Pitch circle
c c
R =100 mmc
R =120 mm R =200 mm
8 (mm)
0 2 4 6
2 4 6
8 (mm)
(b) Tooth profile at cross section Z2=30 mm
Fig. 2.7 Different tooth profiles of the curvilinear-tooth gear generated by R =100 mm, 120 mm and 200 mm
Z t
c (mm)0 10
-10 20 30
-30 -20
4.468 4.709
0
R =120 mm R =200 mm R =100 mm
2
(mm)
c c c
Fig. 2.8 Transverse chordal thickness on different cross section
1.599
Z
2.004
t
a (mm)2
30(mm)
-30 -20 -10 0 10 20
R =200 mm R =120 mm R =100 mmc
c c
Fig. 2.9 Thickness of tooth at addendum circle on different cross section
efficient method to design and manufacture curvilinear-tooth gears. The illustrated approach can be further extended to derive the mathematical model of non-circular face width gears, for example, parabolic or elliptical curved tooth traces.
The tooth surface deviations induced by different nominal radii of circular arc tooth traces are also investigated. The transverse gear chordal thickness measured at the middle section is larger than those of other sections, but the tooth thickness at the addendum circle in the middle section of face width is smaller than those of other sections. The developed tooth mathematical model helps to explore the possibility for further investigations, such as sensitivity, kinematic errors and contact stress analyses.
CHAPTER 3
Tooth Undercutting and Secondary Cutting Analysis
3.1 Introduction
Tooth undercutting is an important issue for gear manufacturers. It is known that gears with tooth undercutting may result in a lower load capacity of a mating gear pair and the gear mismatch during gear engagement. The undercutting points on tooth surfaces are indeed singular points. According to the concept of differential geometry, a surface point is defined as a singular point if its tangent vector is equal to zero.
Mathematically, the problem to find the undercutting points on tooth surface is the problem to obtain the points on tooth surface whose tangent vector is zero. The points on cutter surface generating the singular points on tooth surface will determine a limited line on the cutter surface. Undercutting of tooth surface can be avoided if the limited line is out of the working dimensions of cutter surfaces.
The secondary tooth cutting is the special phenomenon in hobbing process. It means that some regions of tooth surface are cut again by the surface of hob cutter although the same region of tooth surface has been generated by other surfaces of hob cutter. The phenomenon of secondary cutting may appear when the cutting path of hob cutter is under a special cutting condition. When the curvilinear-tooth gears are generated by the hob cutter, the hob cutter with a larger outside diameter or the generated curvilinear-tooth gear with a smaller nominal radius of circular arc tooth trace may induce the secondary tooth cutting.
In this chapter, tooth undercutting is investigated by using the method proposed by Litvin, and the relationships between the outside diameter of hob cutter and nominal radius of circular arc tooth trace as secondary cutting appeared on tooth
surface are also studied.