Work piece
3.4 Secondary Cutting
The proposed tooth surfaces of the curvilinear-tooth gear are generated by a hob cutter. It is clear that the minimum nominal radius of circular arc tooth trace of the gear must be limited. Otherwise, some regions of the tooth surfaces will be cut again by the hob cutter.
(mm) Z2
-30 -20 -10 0 10 20 30 18T22T
14T
l
El
S34T30T26T (mm)
Fig. 3.2 Location of singular points under different numbers of teeth
-30 -20 -10 0 20 (mm) R =100 mm
R =110 mm R =120 mm
10 30 Z2 lE
lS
c c c
(mm)
Fig. 3.3 Location of singular points under different nominal radii
The tooth surfaces shown in Fig. 3.4, denoted by Σ and 2a , are two successive concave surfaces of the curvilinear-tooth gear. Axis is the rotational axis of the hob cutter. The curve is one of the helixes on the hob tooth surface with radius . Figure 3.4 shows that the tip of hob tooth penetrates into the gear tooth surface when the gear tooth surface
2b
Σ
Zh
Li
ri 2b
Σ Σ is generated. The helix 2a
intersects the tooth surface and the point M is an intersection point.
Li 2b
Σ
To check for possible secondary tooth cutting, the following algorithm is used for hobbing process:
Step 1. Represent the hob cutter surface Σ1, gear tooth surface , and the unit normal of the gear tooth surface in the fixed coordinate system by vector functions
Σ2
) , , ( f f f
f X Y Z
S
) , , , ,
)(
1 (
o
f l φ θ ψ r
r , r(f2)(l,φ, jδ ,Rc), n(f2)(l,φ, jδ,Rc), (3.30) Where δ =2π /T2 is the pitch angle, and j is the tooth number. Vector function
represents the tooth surface )
, , ,
)(
2 (
c
f l φ jδ R
r Σ when j =1. Symbols and 2b l φ
are the surface parameters of the hob cutter while θ and ψ are the parameters of motion for hobbing process.
Step 2. Determine the parameters, θi and ψi, expressed in equations of meshing (2.22) and (2.23). θi and ψi depict the motion parameters of the hob cutter when the point on the tooth surface is generated by the hob cutter. Index represents the last point to be check for possible tooth secondary cutting.
i i=iE
Step 3. Check secondary tooth cutting. The secondary tooth cutting occurs if the following condition is satisfied.
r
Z Li
h
2a M
i
2b
Enlargement 4:1
Fig. 3.4 Secondary cutting occurs on the tooth surface Σ when the tooth 2b surface Σ2a is generated
ε
>
− ⋅
− (2)
) 2 ( ) 1 (
) 2 ( ) 1
( )
(
f f f
f
f n
r r
r
r , (3.31)
where ε is a positive value smaller than 1. The vectors and are shown in Fig 3.5. Equation (3.31) shows that the angle between the unit vector of vector and unit normal vector is less than 90 degrees. It implies that the helix intersects the tooth surface.
) 2 ( ) 1 (
f
f r
r − n(f2)
) 2 ( ) 1 (
f
f r
r − n(f2)
Li
Step 4. Determine and if the solution of equation (3.31) exists. The secondary tooth cutting for the curvilinear-tooth gear can be avoid when the , where is the minimum nominal radius of circular arc tooth trace allowed for secondary cutting. Figure 3.6 illustrates the corresponding flowchart for the discussed algorithm.
ro Rcmin
cmin
c R
R > Rcmin
An example for secondary tooth cutting is shown in Fig. 3.7 (a). The design data are: =3.5 mm, normal pressure angle =25 deg., lead angle =3.567 deg.,
=65.3 mm, numbers of teeth =14, face width =80 mm, and =80 mm. The regions with green or red color on the tooth surfaces are the secondary cutting regions. Figure 3.7 (b) depicts the analysis result for secondary tooth cutting. It is found the secondary cutting region with red color is near the middle section of the tooth flank.
Mn Do
Rc
The ZA worm-type hob cutter with single thread and normal module =3 mm is used to generated the curvilinear-tooth gears with numbers of teeth= 35 and face width= 60 mm. The analysis results for secondary tooth cutting are shown in Fig. 3.8.
and are the outside diameter of hob cutter and nominal radius of circular arc tooth trace, respectively. If the outside diameter of hob cutter and normal pressure angle are 67.5 mm and 20 deg., respectively, the minimum nominal radius of
Mn
Do Rc
Do
circular arc tooth trace allowed for secondary tooth cutting is 103 mm. When the tooth surface of the curvilinear-tooth gear are generated by a hob cutter with = 118 mm and normal pressure angle= 20 deg., the outside diameter without secondary tooth cutting must be smaller than 79.5 mm. According to Fig. 3.8, there are two methods to avoid secondary tooth cutting. One method is to decrease the outside
Rc
Do
L
if
r
(1)- r
f(2)f
n
(2)Fig. 3.5 Relations between the vector r(f1)−r(f2) and unit normal vector n(f2)
Check START
Determine vector functions r(f1)(l,φ,θ,ψ,ro), )
, , ,
)(
2 (
c
f l φ jδ R
r , n(f2)(l,φ, jδ,Rc) and set i=1 Input design parameters of the hob cutter and
the curvilinear-tooth gear: r and o R c
END
Determine the motion parameters, θi and ψi, by using Eqs. (2.22) and (2.23)
ε
>
− ⋅
− (2)
) 2 ( ) 1 (
) 2 ( ) 1
( )
(
f f f
f
f n
r r
r r
YES NO
Secondary cutting occurs Update YES
+1
= i i
Check i=iE NO
Fig. 3.6 Flowchart for determination of secondary cutting
diameter of the hob cutter, and the other method is to increase the normal pressure angle of the hob cutter. The Figure 3.8 can be considered for the hobbing process of the curvilinear-tooth gear to avoid secondary tooth cutting.
3.5 Remarks
The tooth surface of the proposed curvilinear-tooth gear is the envelope to the two-parameter family of surfaces. The kinematic method to find the differentiated equations of meshing has been developed for analyzing tooth undercutting. According to the undercutting analysis results, the occurrence of tooth undercutting at both-end sections of the face width of the curvilinear-tooth gear is much easier than other sections. The convex tooth surfaces compared with the concave tooth surfaces are much easier to be undercut. The tooth undercutting of the curvilinear-tooth gear can be avoided with a larger number of teeth or a larger pressure angle. Besides, the tooth undercutting may be reduced by increasing the nominal radius of circular arc tooth traces.
Owning to the geometric character of the hob cutter, the hob cutter with a larger outside diameter or a curvilinear-tooth gear with a smaller nominal radius of circular arc tooth trace will result in secondary tooth cutting when the curvilinear-tooth gears are generated by a hob cutter. Increasing the normal pressure angle or decreasing the outside diameter of the hob cutter can avoid secondary tooth cutting under the same nominal radius of circular arc tooth traces.
Fig. 3.7 (a) An example for secondary tooth cutting
(Courtesy of Professor Ariga, Nippon Institute of Technology, Japan)
Z =0 mm 2
Fig. 3.7 (b) The analysis result for secondary tooth cutting region on tooth surfaces
0 20 40 60 80 100 120 140 160
40 50 60 70 80 90 100 110
Outside diameter of hob cutter Do (mm)
Nominal radius Rc (mm)
Normal Pressure Angle=20 deg. Normal Pressure Angle=25 deg.
Fig. 3.8 Relationship between the outside diameter of hob cutter and nominal radius of circular tooth trace for secondary cutting