## Undercut

The undercut during hobbing has always kept the gear manufacturers busy. When the tooth number of a hobbed gear according to the hobbing system gets below the limit number (32 for pressure angle 14°30′ and 17 for pressure angle 20°) one can see that the tooth foot displays a pocket; the smaller the tooth number, the greater the pocket.

In the common cases, it is only considered as a nuisance for tooth numbers < 17 for pressure angle 14°30 ‘ and < 12 for pressure angle 20°, because from this limit a noticeable weakening of the tooth foot occurs and an important part of the involute flank has been hobbed away below the pitch circle; this part of the flank has no longer contact during the meshing.

The result, especially for larger circumferential speeds, is a less good operation of the gear pair, characterized by noise, vibration, a faster wear and of course a lesser resistance to fraction.

This problem can be solved with the possibilities below either by:

- increasing the pressure angle;
- decreasing of the tooth depth;
- shifting the hobbing zone during milling;
- or a combination of these different possibilities.

In addition the wear factors, load capacities, tooth forms, coëfficient of contact and tip-root clearance were taken into account. In this chapter we will not get too deep into this complex matter. We will stick to simplified formulas, which allow us to design gears.

For clarification, you will find on the right a gear with z = 10 teeth, a pressure angle of a = 20 ° in three different designs. There was no profile correction applied in figure B, but there is a slight undercut to notice, regarding the small number of teeth. This undercut was adjusted in figure A by applying a correction factor x = + 0.5. In figure C the undercut was emphasized by applying a negative correction factor x =-0.5.

## Tooth correction for cylindrical gears

Tooth correction consists of an addendum modification i.e. at undercut a displacement outward of tip circle and dedendum circle with in between the involute profile, applied to the base circle. In this way, one can bring the foot circle closer to the base circle and at the base a stronger tooth combined with flanks that participate over their entire height in the meshing (see figure).

The addendum modification changes neither the pitch diameter nor the pitch, there where tip circle and dedendum circle enlarge.

When on a pinion with small teeth number an addendum modification to outside, i.e. a positive tooth correction is applied, the accompanying wheel can be corrected in the reverse sense. So in this case the tip circle and dedendum circle of the wheel decrease and one speaks about a negative tooth correction.

A gear set of which the pinion is corrected positive and the wheel negative with the same value for x, is called a X-zero pair. The centre distance of such a set remains the same as of a similar set, without addendum modification. See image below.

When only the pinion is corrected (+), the centre distance increases. As the pitch circles have to roll without slipping on each other, one assumes that this increases the pitch circle of the wheel (without addendum modification) in order to remain running correctly on the pitch circle of the pinion. This new pitch circle is called the actual pitch circle.

## Tooth correction for pinion and rack:

In many cases of conversion of a rotation into a rectangular movement by pinion and rack, one searches to achieve a solid tooth on a small pinion; as a result, one gets mainly a small number of teeth and to obtain a favourable tooth shape one must apply positive tooth profile correction on the pinion. Therefore one shifts the axis of the pinion away from the pitch line of the rack; this shift is equal to the applied addendum modification. The pitch line of the gear rack, which normally coincides with the reference line, moves in order to stay in touch / contact with the pitch circle of the pinion which was not changed by the addendum modification.

## Notes

The tip circle of a corrected gear is different from a gear that is not corrected. The correction consists of applying an addendum modification of the hob in positive or negative sense compared with reference to the gear. The correction is positive when one moves away the tooth profile from the gear center and negative if one moves it to the center.

- In most cases, the price for the toothing is independent of the correction.
- The correction admits to limit the specific glide and improves the load capacities of the gear set. On the other hand, a wrong correction choice will lead to a reduction of the coefficient of contact quotient and thickness of the tooth tip.
- At a positive adjustment on the pinion and an equally large negative adjustment of the wheel center distance don’t change.
- The correction coefficient “x” is the quotient that one obtains by the addendum modification (measured on the radius) divided by the module.
- In the below formulas, we use the following notes: Zv = = virtual teeth number (Zv = Z for straight teeth)

Σx = x 1 + x 2 = sum of the addendum modifications

λ = divided coefficient of the addendum modifications

Σ Zv = Zv1 + Zv2 total number of teeth

k = tip correction coefficient

## Calculations

- Ratio of the corrected coefficients of gear set: Pinion: x1 = λ.Wheel: x2 = Σx – x1 With: 0,5 < λ < 0,75 for a speed decreasing gearbox 0 < λ <0,5 for a speed increasing gearbox
- A chart with the limiting values of the sum of the tooth corrections
- A chart with the limiting values of the tooth corrections for external and internal gears in function of the number of (virtual) teeth and the addendum modification factor.
- Tip correction factor k x ≤ 0,6 k = 0,01.(50x – 3.Z
_{v}+ 6) - x > 0,6 k = 0,01.(70x – 3. Z
_{v}– 6) where d_{a}= m_{n}.if k is calculated negative, one takes k=0 - Value of centre distance and operating tangential pressure angle ()

Z_{m}= .(Z_{1}+

d_{m}= .(d_{1}+ d_{2})

tg_{at}=

= (inva = tana – a)

a.cos = d_{m}.cosa_{t}

## Tooth corrections for internal gears

In the design of internal gears and defining the tooth corrections six conditions should be taken into account:

- The undercut of the pinion (see DIN3960).
- Primary meshing failure (see DIN3993): this failure occurs within the meshing area. Here two meshing failures can occur, either between pinion root and wheel tip or between pinion tip and wheel root.
- Secondary meshing failure (see DIN3993): this meshing failure occurs outside the meshing area. This failure occurs when the diameter of the pinion approaches the diameter of the wheel. The limit for gears is with a pressure angle of 20° = zn1 zn2-8 without change centre distance. When the centre distance may change the difference limit can become z2-z1=1.
- Coefficient of contact limit: ea = 1,10.
- Possible limit of tip thickness= e = 0,20mn (see DIN3960).
- Tip-root clearance ≥ 0,2mn (see DIN3960). These six conditions are displayed in graphs per gear pair identical to the graph below, which can be found in DIN 3993. We point out further on possible meshing failures as a result of the production of the internal gear with the cutter. For this we refer to the chart and DIN 3993.