CHAPTER 5 DESIGN OF SERRATIONS ON GEAR PLUNGE SHAVING CUTTER
5.2 Numerical examples and discussion
cp ce
r φ =M φ r (5.4)
In the next section, a numerical example is provided for design optimization of shaving cutter serration.
5.2 Numerical examples and discussion
The trace of the plunge shaving cutter’s cutting edge on the work gear during the cutting process is illustrated schematically in Fig. 5.4 (Hsu, [10]). At the beginning of a shaving cycle, the cutting edge presses into the work gear surface, moves on its tooth surface due to the relative motion, and then removes chips when contact stress reaches local ultimate stress.
Whereas the front cutting edge removes chips as it moves, the top land of the serration (i.e.
the tooth surface of the shaving cutter), rather than removing chips, plows into the work gear tooth surface without a cutting action and extrudes material in the direction opposite to the cutting direction. Therefore, the front cutting edge produces the desired cutting action, while the top land of the serration produces unwanted friction and material flow. The following
example illustrates the design optimization of cutter serrations.
Example 5.1
The basic data for the work gear and shaving cutter are listed in Table 5.1, while the data for the serrations are listed in Table 5.2. Fig. 5.4 shows the simulations of the first and the subsequent cutting marks in the mid-tooth height of the work gear. The distance between the cutting marks between succeeding cuts, termed the “cutting mark displacement”, is directly proportional to the serration displacement. The abscissa in Fig. 5.4 represents the cutting depth (unit:μm), while the ordinate represents the cutting mark displacement in the lengthwise direction of the work gear (unit:mm) where the cutting marks are repeated within a certain range.
Fig. 5.4 (b) and (c) show the end of the first and second cutting cycle simulation, whose higher peak are termed the “first peak” and “second peak”. The ratio between the height of the second peak and that of the first peak in the first cutting cycle is known as the “cutting-down ratio”, which is used as an index for cutting efficiency. In this example, the height of the second peak is 1.590 mμ and the cutting-down ratio is 0.318.
According to the basic data of work gear and shaving cutter, the cutting-down ratio is optimized. The optimum design problem is formulated:
find serration shift SF
that minimizes cutting-down ration r c subject to SF N× 1≥ ps
, in which SF =n p Ns s / 1 originally, and it is directly selected as the design variable rather than n or s p for larger design space. The minimum and maximum searching steps are s 0.001 and 0.1 mm. The cutting-down ratio has been improved from 0.318 to 0.242 (1st peak=3.1 mμ , 2nd peak=0.7 mμ ) by 23.9%. The scaled views of 1st peak and 2nd peak are shown in Fig. 5.5.
5.3 Concluding remarks
In this chapter, the design concepts of shaving cutter serrations are presented, and design optimization is also achieved. The advantages of the proposed model are listed:
1. The proposed mathematical model considers the tooth modification of shaving cutter tooth derived in chapter 4.
2. The proposed mathematical model can be adopted for design optimizations because the design variables are changed to be independent of each other.
3. The cutting-down ratio has been reduced by 23.9%, which means the shaving time can be reduced by almost 1/4 per piece.
Table 5.1 Basic data for the work gear and corresponding shaving cutter.
Work gear
Number of teeth (N2) 13 Normal module in pitch circle (mpn) 1.750 mm Normal circular tooth thickness (spn2) 3.270 mm Normal pressure angle in pitch circle (αpn) 20∘
Outside radius (ro2) 27.600 mm Form radius (rf2) 21.490 mm
Face width 24 mm
Helix angle in pitch circle (βp2) 5∘R.H.
Plunge gear shaving cutter
Number of teeth (N1) 137 Helix angle in pitch circle (βp1) 10∘R.H.
Face width 30 mm
Normal circular tooth thickness (spn1) 0.529 mm Operating data of shaving procedure
Operating center distance (Eo) 130.612 mm Operating crossed angle (γo) 14.722∘
Cone grinding wheel
Cone angle (θc) 2.961∘
Pressure angle (α) 13.882∘
Rc 350 mm
Table 5.2 Basic data for the serrations.
Serration pitch 1.85 mm R.H.
Number of starts 62 Serration displacement -0.216 mm
Figure 5.1 Diagram of a shaving cutter and a work gear (Hsu, 2006).
Figure 5.2 Diagram of a shaving cutter with helix-staggered serrations (Hsu, 2006).
Serration pitch Radial infeed only
X1
Y1
Z1
ps
Figure 5.3. Design parameters of plunge shaving cutter serration (a) serration pitch (b) serration displacement.
Unit: μm
Depth
Repeating range Unit: mm Unit: μm
Depth
Repeating range Unit: mm cutter
cutter cutter work gear work gear work gear
(a)
(b)
(c)
First peak
Second peak
Figure 5.4 Cutting path calculations (Hsu, 2006) (a) path of the cutting edge on the work gear (b) end of the first cutting cycle simulation (c) end of the second cutting cycle simulation.
Figure 5.5 Optimized cutting-down ratio of Example 5.2 (a)1st cut (b) 1st peak (c) 2nd peak.
(a)
(b)
(c)
First peak
Second peak
CHAPTER 6
CONCLUSIONS AND FUTURE WORKS
6.1 Conclusions
This dissertation has investigated gear tooth crowning induced by transverse and plunge gear shaving. In the past, these are very time consuming because the machine setting and the shaving cutter need to be modified back and forth by trial and error. In this dissertation, mathematical models for analyses and design optimization are proposed to solve this problem.
For transverse shaving, influences of machine setting parameters and cutter assembly errors have been observed. Design optimization for robustness of gear transmission error has also been accomplished. For plunge shaving, the analytical descriptions of crowned gear and hence plunge shaving cutter have been constructed so that the grinding wheel can be optimized to minimized the topographic error. The cutting trace of plunge shaving cutter has also been analyzed so that the final real tooth forms can be predicted. Besides, the shaving efficiency has also been improved. Based on the results of the numerical examples, the following conclusions have been drawn:
1. For the crowning mechanism, the crowning effect is shown to be sensitive to the angle θ between guideway and horizontal as well as the horizontal distance d between h pivot and pin. The horizontal and the center distance errors Δh and ΔE0 are also proved significant to gear tooth crowning.
2. To have a double crowned gear shaved by traditional gear shaving machine for better performance in transmission error, four categories of parameter need to be considered:
modification of shaving cutter, assembly errors of shaving cutter, assembly errors of gears and machine setting parameters. Among the 11 selected parameters, the coefficient
a concerning the modification of shaving cutter and the angle c θ between the guide
way and horizontal on the shaving machine contribute the most to transmission error.
3. To manufacture the double crowned gear by transverse shaving, the machine setting parameter and cutter profile coefficient can be calculated and optimized beforehand without trial and error, which reduces the required time for development.
4. To interpolate gears with both lead and profile crownings (double crowned), more sampling points are needed in the radial direction. In manufacturing a shaving cutter, the lead crowning can be compensated by adjusting the cone angle, and the profile crowning can be implemented by modifying the profile of the grinding wheel.
5. The proposed mathematical model of shaving cutter serration considers the tooth modification of shaving cutter tooth.
6. The proposed mathematical model of shaving cutter serration can be adopted for design optimizations because the design variables are changed to be independent of each other.
7. To manufacture the double crowned gear by plunge shaving, the required profile of cone grinding wheel and hence the shaving cutter tooth surface can be calculated and optimized beforehand without trial and error, which reduces the required time for development; the cutter serration can also be optimized for better cutting efficiency.
6.2 Future works
Based on the mathematical models provided in this thesis for analyzing the shaving process, the following relevant research needs to be undertaken in the future:
1. New geometry and new displacement for the shaving cutter serrations should be investigated to improve the finishing effect of the plunge shaving cutter.
2. As regards the tool life of the shaving cutter, further study is needed on workpiece quality as it affects the wear of the serration.
3. Because the shaving cutter can be reground several times, future study should use the acceptable variation ratio of the contact length in the shaving cutter design to find the
regrinding range.
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