reference circle, the trajectory drawn represents a shortened cycloid。 The same curve ξ, connected without moving with the rack-cutter, can be considered also as a conjugated curve of the starting point b of the involute profile。 This means that if the real rack-cutter fillet coincides with ξ, it will contact with point b at each moment and will not cut the involute profile  bq。

Table 1

⁎

Maximum fillet radius coefficient of the rack-cutter, ρmax = ρ/m。

Number of teeth, z Profile angle of the rack-cutter, α

14°30′ 15° 17°30′ 20° 22°30′ 25 28° 30°

5 1。252 1。294 1。504 1。710 1。913 2。113 2。347 2。5

10 2。504 2。588 3。007 3。420 3。827 4。226 4。695 5。0

20 5。008 5。176 6。014 6。840 7。654 8。452 9。389 10。0

40 10。015 10。353 12。028 13。681 15。307 16。905 18。779 20。0

80 20。030 20。706 24。056 27。362 30。615 33。809 37。558 40。0

160 40。061 41。411 48。113 54。723 61。229 67。619 75。115 80。0

Fig。 9。 Quantitative indices of the undercutting: a) radial and tangential; and b) tangential。

Therefore curve ξ allows us to define the following boundary condition: undercutting — type IIb is avoided if the real rack-cutter fillet AF is placed internally regarding the boundary fillet ξ。

In the case shown in Fig。 3b this condition is not satisfied and as a result the gear teeth are undercut — type IIb。

3。Еquations of boundary fillet curves

3。1。Boundary fillet — type IIа (curve η)

The equations of this curve are found using the theory of plane meshing [1], where one of the two conjugated profiles is set and the other one is obtained as an envelope of the relative positions which the specified profile occupies in the plane of the searched one。 In this case (Fig。 4) the specified profile is the radial line l (axis OlYl) of the gear and the searched profile is the boundary fillet η of the rack-cutter。