X min  ¼ x minm ¼ hm m

Xmin corresponds to the so-called boundary case — type I (Fig。 1c), where the tip-line g-g passes through the boundary point A (the point where the line of action contacts the base circle of a radius rb)。

Fig。 3。 Conditions of non-undercutting: a) type IIa; and b) type IIb。

Fig。 4。 Boundary fillet — type IIa of the rack-cutter。

2。2。Undercutting  — type  II  (non-traditional case)

As already mentioned, the undercutting — type II is caused by the rack-cutter fillet AF (Fig。 2) in the process of tooth cutting。 When this fillet is a circle of a small radius ρ1 (Fig。 2a) the teeth cut are not undercut。 Then the gear fillet fb does not cross the radial line Ob, passing from the center O to the starting point b of the involute profile ba (at X = Xmin point b lies on the base circle of a radius rb)。

At a comparatively larger radius ρ2 of the rack-cutter fillet AF (Fig。 2b), an undercutting — type IIa is obtained, where the fillet gear fb of the generated tooth crosses (cuts) the radial line Ob, but does not cross the involute profile ba。 This means that in the presence of undercutting — type IIa the tooth thickness at the bottom decreases without cutting an involute profile in the vicinity of its starting point b。

When the radius ρ3 (Fig。 2c) of the rack-cutter fillet increases considerably, the fillet gear fq crosses the radial line Ob, as well as the involute profile ba。 In this case (undercutting — type IIb) besides the decrease of the tooth thickness at the bottom, the segment bq of the involute profile is also cut。

Fig。 5。 Boundary fillet — type IIb of the rack-cutter。

Fig。 6。 Fillets of the rack-cutter: a) boundary fillets; and b) real fillets。

The essence of undercutting — type IIa and type IIb is explained on Fig。 3, where undercutting — type I is avoided by a positive displacement of the rack-cutter at a distance Xmin。 At this boundary displacement, the tip-line g-g of the rack-cutter crosses the line of action AB in starting point A。

In order to define the maximum radius of the rack-cutter fillet AF, corresponding to the boundary case where there is no undercutting — type IIa, the curve η, called a boundary fillet — type IIa is drawn additionally on Fig。 3a。 It is obtained as an envelope of the relative positions taken by the radial line l (the line ObE) of the gear in the plane of the rack-cutter, when realizing the meshing between the rectilinear profile AE of the rack of the involute profile ba of the gear。 In other words, the profiles l and η are also conjugated profiles at rolling without sliding of the centrode line n-n of the а rack-cutter on the reference circle of the gear of a radius r。

Knowing the curve η allows us to define the following boundary condition: the undercutting — type IIa is avoided if the real rack-cutter fillet AF is placed internally regarding the boundary fillet η (in the material of the cutter)。

In Fig。 3a the rack-cutter fillet AF is placed externally regarding the curve η, as a result of which gear teeth are undercut — type

IIa。

Analogously the condition for non-undercutting — type IIb is defined by drawing a curve ξ (Fig。 3b), called a boundary fillet — type IIb。

In this case, the curve ξ is obtained as a trajectory (drawn in the plane of the rack-cutter) of the point а from the plane of the reference circle of a radius r, rolling without sliding on a reference circle on the line n-n。 As point b lies on the internal side of the

Fig。 7。 Rack-cutter。

Fig。 8。 Boundary areas of the rack-cutter fillet。