On the Issue of Measuring the Pitch of Worm Hobs

A. A. Asmus

(Presented by the scientific seminar of the departments of machine tools and metal cutting and of mechanical-engineering technology)

To determine the actual accuracy of worm hobs, it is sufficient to carry out measurement according to one of the three groups of checks provided for by GOST 9324-60.

The choice of one or another group of parameters to be checked depends to a considerable extent on the measuring instruments available at the enterprise.

The most complete picture is given by the results of measurements of the parameters of the first group. In this case, inspection of the meshing error can be carried out only on special instruments of the BB-5005, PWF-250 type, etc.

At factories, the third group is very often used; it provides for measurement of a number of parameters, including measurement of the maximum pitch error \(\Delta t\) and the maximum accumulated pitch error over the length of three adjacent pitches \(\Delta t_{\Sigma}\).

As a rule, what is checked is not the deviation of the axial pitch, but the deviation of the projected pitch of the hob (see item 5, Table 5 of GOST 9324-60).

The magnitude of the projected pitch is determined from the formulas:

\[ t_{\mathrm{pr}} = t_n \cdot \cos \omega_d \tag{1} \]

or

\[ t_{\mathrm{pr}} = t \cdot \frac{H}{H + t}, \tag{2} \]

where \(t_{\mathrm{pr}}\) is the projected pitch,
\(t_n\) is the normal pitch,
\(\omega_d\) is the angle of inclination of the helical chip grooves on the pitch cylinder,
\(t\) is the axial pitch,
\(H\) is the pitch of the helical chip grooves.

As is evident from formulas (1) and (2), the value of \(t_{\mathrm{pr}}\) is affected by the error in the pitch of the helical chip grooves \(\Delta H\).

In addition, the error \(\Delta H\) causes a displacement of the cutting edge from the surface of the basic worm (in Figs. 1 and 2, the quantity \(a\)).

Figure 1 shows a diagram for measuring the projected pitch for the right-hand side of the profile, and Fig. 2 for the left-hand side of the profile.

The deviations of the projected pitch from the nominal value are:

Fig. 1

Fig. 2

a) when measuring on the right side of the profile, \(\Delta t_{\mathrm{pr}} = b + a\),
b) when measuring on the left side of the profile, \(\Delta t_{\mathrm{pr}} = b - a\),

where \(a\) is the displacement of the cutting edge of the worm hob from the surface of the basic worm, caused by the pitch error of the helical chip grooves;

\(b\) is the amount of distortion of the projected pitch caused by the pitch error of the helical chip grooves.

The upper signs in the formulas refer to right-hand worm hobs, and the lower signs to left-hand ones.

With accuracy sufficient for practice, the quantities \(a\) and \(b\) can be determined from the formulas:

\[ a = \kappa \cdot z \cdot \sin^2 \omega_{\mathrm{d}} \cdot \cos^2 \omega_{\mathrm{d}} \cdot \sin \alpha \cdot \frac{\Delta H}{H}, \]

\[ b = \frac{t^2 \cdot \Delta H}{(H+t)(H+t+\Delta H)} \quad [1], \]

where \(\kappa\) is the amount of backing-off of the hob,

\(z\) is the number of hob teeth,

\(\alpha\) is the hob profile angle.

For right-hand worm hobs of class A, type II according to GOST 9324-60, when measuring the projected pitch on the right side of the profile:

\(\delta H\) is the permissible pitch error of the helical chip grooves according to GOST 9324-60.

The same when measuring on the left side of the profile:

The data show that, for hobs of module 5 mm and larger, the pitch errors of the helical chip grooves \(\delta H\) permitted by GOST 9324-60 cause inadmissible errors when measuring the projected pitch on the right side of the profile of right-hand hobs.

When measuring the projected pitch of right-hand hobs along the left side of the profile, the measurement results give understated error values.

From the measurement results it is difficult to judge the error of the hob helix, since when measuring the projected pitch along the right side of the profile, the quantity \(b\) and the quantity \(a\) are added, whereas when measuring along the left side of the profile they are subtracted.

When calculating tolerances for the pitch of helical chip grooves [4], only the influence of \(\Delta H\) on the greatest error of the hob helix over one revolution, \(\Delta t_b\), was taken into account. (In source [4], the quantity \(\Delta t_b\) corresponds to \(\Delta t_c\)—the cyclic pitch error.)

Significantly larger errors arise when measuring the projected pitch of multi-start worm hobs.

In addition to the above-mentioned errors associated with the measurement scheme (the quantities \(a\) and \(b\)), when measuring multi-start worm hobs errors are permitted that are associated with calculating the nominal value of \(t_{\mathrm{pr}}\). The quantity \(t_{\mathrm{pr}}\) is determined by formulas (1) and (2). In inspection practice it was observed that, for multi-start hobs, formulas (1) and (2) give different numerical values of \(t_{\mathrm{pr}}\).

Many believe that formula (2) gives more accurate values. Thus, D. I. Semenchenko [3] states that “at a helix angle of the helical chip grooves \(\omega_d > 7^\circ\), formula (1) must not be used, since it gives significant deviations connected with the fact that in reality we have a helix (and not a straight line) located on a cylindrical surface and projected onto the axial plane.”

This assertion is erroneous.

S. P. Kartsev [1] rightly indicated that the method of measuring the projection of the normal pitch onto the axial plane has no theoretical errors, but he did not consider the inspection of multi-start hobs.

Fig. 3

From Fig. 3 it is evident that, for different values of the angles \(\omega_d\) and \(\tau_d\), \(\triangle ABC\) and \(\triangle KLM\) are similar:

\[ \frac{AO}{KN}=\frac{AC}{KM}, \quad \text{i.e.} \quad \frac{t_{\mathrm{pr}}}{H}=\frac{t}{H+t}, \]

then

\[ t_{\mathrm{pr}}=t\cdot\frac{t}{H+t}, \]

i.e., formula (2) is valid for various values of the angles \(\omega_{\mathrm{d}}\) and \(\tau_{\mathrm{d}}\).

Formula (1) is valid only when \(\omega_{\mathrm{d}}=\tau_{\mathrm{d}}\); indeed, when \(\omega_{\mathrm{d}}=\tau_{\mathrm{d}}\), from Fig. 3 we have:

\[ KL=(H+t)\cdot\cos\omega_{\mathrm{d}} \quad\text{and}\quad KL=\frac{H}{\cos\omega_{\mathrm{d}}}, \]

then

\[ (H+t)\cdot\cos\omega_{\mathrm{d}}=\frac{H}{\cos\omega_{\mathrm{d}}} \quad\text{or}\quad \frac{H}{H+t}=\cos^2\omega_{\mathrm{d}}. \]

Substituting this value into formula (2), we obtain:

\[ t_{\mathrm{pr}}=t\cdot\cos^2\omega_{\mathrm{d}}; \]

bearing in mind that

\[ t=\frac{t_n}{\cos\omega_{\mathrm{d}}}, \]

we obtain

\[ t_{\mathrm{pr}}=t_n\cdot\cos\omega_{\mathrm{d}}, \]

i.e., formula (1).

Consequently, calculation by formulas (1) and (2), for any values of \(\omega_{\mathrm{d}}\), gives the same value of \(t_{\mathrm{pr}}\).

Indeed, when calculating the value \(t_{\mathrm{pr}}\) for multi-start cutters by formulas (1) and (2), different results are obtained even when \(\omega_{\mathrm{d}}=\tau_{\mathrm{d}}\).

But in this case formula (1), and not formula (2), gives the more correct values, since formula (1) is also valid for multi-start cutters, whereas formula (2) is valid only for single-start cutters.

The value \(t_{\mathrm{pr}}\) for multi-start cutters can be determined from the following considerations. Suppose that only one thread start is cut on the cutter. Then the value of the projection of the lead onto the axial plane of the cutter can be determined by formula (2):

\[ \kappa_{\mathrm{pr}}=\frac{K_0H}{H+K_0}, \]

where \(K_0\) is the axial lead of the cutter turns;
\(\kappa_{\mathrm{pr}}\) is the projection of the lead onto the axial plane of the cutter.

Substituting \(K_0=t\cdot z_{\mathrm{st}}\) and \(\kappa_{\mathrm{pr}}=t_{\mathrm{pr}}\cdot z_{\mathrm{st}}\), where \(z_{\mathrm{st}}\) is the number of starts of the worm cutter, we obtain:

\[ t_{\mathrm{pr}}=t\frac{H}{H+t\cdot z_{\mathrm{st}}}. \tag{3} \]

It follows from formula (3) that formula (2) is a special case when \(z_{\mathrm{st}}=1\).

When \(\omega_{\mathrm{d}}=\tau_{\mathrm{d}}\), which occurs in most cases, the values of \(t_{\mathrm{pr}}\) calculated by formulas (1) and (3) coincide.

Attention should be drawn to one further feature.

It follows from formula (1) that the value \(t_{\mathrm{pr}}\) depends on \(\omega_{\mathrm{d}}\). When measuring \(t_{\mathrm{pr}}\), the measuring tip is set not exactly along the mean diameter \(D_t\), but with some deviation \(\frac{\Delta D_t}{2}\) (Fig. 4).

The values of \(\omega_{\mathrm{d}}\) at points 1 and 2 are different, which would seem to have to lead to a change in the value of \(t_{\mathrm{pr}}\).

In reality the value \(t_{\mathrm{pr}}\) remains constant.

Let us prove this proposition.

In Fig. 5, I is the development of the helical line of the surface of the chip groove passing through point 1 (Fig. 4);

II is the development of the helical line of the surface of the basic worm of the cutter passing through point 1 (Fig. 4);

III is the development of the helical line of the surface of the chip groove passing through point 2 (Fig. 4);

IV—the developed view of the helical line of the surface of the thread of the hob’s main worm, passing through point 2 (Fig. 4).

Fig. 4.

Fig. 5

Equation of straight line I in the coordinates \(XOy\):

\[ y=\frac{\pi D_t}{H}\,x. \]

Equation of straight line II:

\[ y=\pi D_t\left(1-\frac{x}{t\cdot z_{\mathrm{zakh}}}\right). \]

The abscissa of point \(A\), the intersection of straight lines I and II:

\[ \frac{\pi D_t}{H}x=\pi D_t\left(1-\frac{x}{t\cdot z_{\max}}\right), \]

\[ x=t\cdot z_{\max}\frac{H}{H+t\cdot z_{\max}}. \]

Equation of straight line III:

\[ y=\frac{\pi(D_t+\Delta D_t)}{H}\,x. \]

Equation of straight line IV:

\[ y=\pi(D_t+\Delta D_t)\left(1-\frac{x}{t\cdot z_{\max}}\right). \]

The abscissa of point \(B\), the intersection of straight lines III and IV:

\[ \pi(D_t+\Delta D_t)\frac{x}{H}=\pi(D_t+\Delta D_t)\left(1-\frac{x}{t\cdot z_{\max}}\right) \]

\[ x=t\cdot z_{\max}\frac{H}{H+t\cdot z_{\max}}. \]

The abscissas of points \(A\) and \(B\) are identical; consequently, the value \(t_{\mathrm{pr}}\) does not depend on the position of the measuring tip over the height of the cutter profile.

Conclusions

1. Measurement of the projected pitch of cutters according to GOST 9324-60 is permissible only for cutters up to module 5 mm.

2. Measurement of the projected pitch for cutters with modules from 5 mm to 12 mm is permissible provided that the tolerance for the pitch of the helical chip grooves \((\delta H)\) is reduced by a factor of two.

3. Measurement of the projected pitch for cutters with modules from 13 mm to 20 mm is permissible provided that the nominal value of \(t_{\mathrm{pr}}\) is determined by formula (2), on the basis of the actual values of \(H\) obtained when sharpening the cutter.

To determine the nominal value of \(t_{\mathrm{pr}}\) as a function of \(H\), tables or nomograms may be compiled.

1. For multiple-start worm hobs, the value \(t_{\mathrm{pr}}\) must be determined by formula (3), on the basis of the actual value of \(H\) obtained when sharpening the cutter.

2. Determination of the value \(t_{\mathrm{pr}}\) by formula (2) for multiple-start cutters leads to erroneous results, since formula (2) does not take into account the number of starts.

3. Displacement of the measuring tip in the radial direction does not introduce additional errors when measuring the projected pitch.

References

1. S. P. Kartsev. Measurement of the pitch of worm hobs. Machines and Tooling, No. 11, 1950.

2. I. I. Semenchenko, V. M. Matyushin, G. N. Sakharov. Design of Metal-Cutting Tools. Mashgiz, 1962.

3. D. I. Semenchenko. Technology for Manufacturing Worm Hobs. Central Bureau of Technical Information, Ministry of Machine-Tool and Tool Industry of the USSR, 1955.

4. All-Union Scientific Research Institute. Finishing Single-Start Worm Hobs for Cylindrical Gear Wheels with an Involute Profile. (Replacing GOST 3346-46 and GOST 2973-45.) Draft GOST (first edition). Moscow, 1959.

5. GOST 9324-60. Finishing Single-Start Worm Hobs for Cylindrical Gear Wheels with an Involute Profile.

6. A. N. Grubin, M. B. Likhchier, M. S. Polotskii. Gear-Cutting Tools. Part II. Mashgiz, 1946.