CALCULATION OF THE PROFILING ERROR IN GEAR CUTTING WITH DISC-TYPE MODULE CUTTERS

A. A. ASMUS

(Presented by the scientific seminar of the Department of Mechanical-Engineering Technology, Machine Tools, and Metal Cutting)

The profile of a disc-type module cutter of a given number is calculated for the smallest number of teeth of the gears cut by the cutter.

When cutting, with the cutter, a gear having the largest number of teeth that can be cut by the given cutter number, profiling errors \(\Delta f\) arise. The error \(\Delta f\) can be calculated by the following method.

The coordinates of the gear profile are calculated for the smallest and largest numbers of teeth cut by the given cutter number. The scheme for calculating the coordinates of the profile points is shown in Fig. 1.

Fig. 1

Values of \(R_x\) are assigned within the limits from \(R_i\) (in the case where \(R_i > R_0\)) and from \(R_0\) (in the case where \(R_i < R_0\)) to \(R_e\).

The value of \(\alpha_x\) is determined from the relation

\[ \cos \alpha_x = \frac{R_0}{R_x}. \]

Knowing the width of the tooth-space \(S_{\text{vp.d}}\) along the arc of the pitch circle of radius \(R_d\), the angle \(\varphi_x\) is determined:

\[ \varphi_x=\frac{S_{\text{vp.d}}}{2R_d}+ inv\,\alpha_x - inv\,\alpha_0 . \]

Then the coordinates of the points of the involute will be:

\[ x=R_x\cdot \sin \varphi_x, \]

\[ y'=R_x\cdot \cos \varphi_x - R_i . \]

The greatest profiling error will occur at the profile points located on the radius \(R_e\) of the outside circle of the gear. In connection with this, it is sufficient to determine the coordinates of one point when \(R_x=R_e\).

Figure 2 shows two profiles, one of which, I, corresponds to the tooth profile of a gear with \(Z_{\min}\), and the other, II, to that with \(Z_{\max}\) for the given number of the disk cutter. Points \(A\) and \(B\) are located on the outside diameter of the gears.

Fig. 2

For determining the profile error (according to position 10 of item 9 of GOST 1643-56), through point \(A\) one must draw theoretical profile I, and through point \(B\) draw the normal to profile I (straight line III). Then the profile error is

\[ \Delta f_1=MB \simeq AB\cdot \cos(\alpha_{xB}+\varepsilon_{xB}-\Theta). \]

Or, reasoning analogously, through point \(B\) one may draw theoretical profile II and the normal to the profile at point \(A\) (straight line IV). Then

\[ \Delta f_2=AN' \simeq AB\cdot \cos(\alpha_{xA}+\varepsilon_{xB}-\Theta), \]

where

\[ AB=\sqrt{(x_B-x_A)^2+(y_A-y_B)^2}. \]

As the profiling error \(\Delta f\), one may take

\[ \Delta f=\frac{\Delta f_1+\Delta f_2}{2}. \]

or

\[ \Delta f=\sqrt{(x_B-x_A)^2+(y_A-y_B)^2}\cos(\alpha_x+\varphi_x-\Theta), \]

where \(x_A;\ y_A\) and \(x_B;\ y_B\) are, respectively, the coordinates of points \(A\) and \(B\).

\[ \alpha_x=\frac{\alpha_{xA}+\alpha_{xB}}{2} \quad\text{and}\quad \varphi_x=\frac{\varphi_{xA}+\varphi_{xB}}{2}, \]

\[ \operatorname{tg}\Theta=\frac{y_A-y_B}{x_B-x_A}. \]

Using the method described above, the author calculated the profiling errors for disk-type module cutters according to GOST 10996-64. The calculation of the coordinates \(x\) and \(y\) was carried out on a Minsk-1 electronic computer. The errors \(\Delta f\) obtained as a result of the calculation were compared with the permissible errors of the profile of gear wheels \(\delta f\) according to GOST 1643-56. The results of the comparison for wheels of the 9th and 10th degrees of accuracy are given in Table 1. The table gives the values

\[ \frac{\Delta f}{\delta f}\cdot 100\%. \]

The values of \(\delta f\) for the 9th and 10th degrees of accuracy were obtained from the values of \(\delta f\) for the 8th degree of accuracy by multiplying them by coefficients: 1.6 for the ninth degree of accuracy, and \(1.6^2\) for the tenth degree of accuracy.

From an analysis of the data in Table 1 it follows that, with disk-type module cutters according to GOST 10996-64, it is not possible in all cases to cut wheels of the 9th and 10th degrees of accuracy.

In individual cases the errors exceed the permissible deviations even for the 11th degree of accuracy.

It is also noteworthy that the quantity

\[ \frac{\Delta f}{\delta f}\cdot 100\% \]

changes sharply for different cutter numbers.

Thus, in the case of cutting wheels with module 14 mm of the 10th degree of accuracy by cutter No. 4, the value

\[ \frac{\Delta f}{\delta f}\cdot 100\% \]

will be 85%, whereas for wheels cut by cutter No. \(6^{1}/_{2}\), it is correspondingly 200%.

A reduction in the difference between the quantities

\[ \frac{\Delta f}{\delta f}\cdot 100\% \]

can be achieved by changing the range of numbers of teeth cut by each cutter number.

Conclusion

The method proposed in the article makes it possible to calculate the profiling errors arising when gear wheels are cut with disk-type module cutters.