ρη   ¼ 8r2 sin2φ cos2φ−4r2 sin2φ cos2φÞ ð8Þ

wherein, after a transformation, appears as follows:

ρη  ¼ 2r sinφ ¼ mz sinφ: ð9Þ

3。2。Boundary fillet — type IIb  (curve ξ)

It is obtained as a trajectory of the point b of the coordinate system XbObYb connected with the gear (Fig。 5), drawn in a coordinate system XξOξ

Fig。 11。 Undercutting — type IIa: z =6; x = xmin = 0。449; δr =0; δt = 0。045; α=20°; ha = 0。8; c* = 0。7; ρ*= 3。2。

The obtained trajectory ξ, as it was already explained, represents a shortened cycloid, whose parametric equations can be written as follows

where rb  is the radius of the base circle, defined by the formula

rb  ¼ r cosα  ¼ 0:5mz cosα  : ð11Þ

In the initial position of the coordinate systems, when at φ= 0, point b coincides with point b", and at φ= α – with point b' ≡ A。 In that case, point A appears as an inflection point, which pides the curve ξ into two parts: a concave segment NA and a convex segment AM。 In this case it should be taken into account that only segment AM is the real rack-cutter fillet — type IIb。

The equation of the radius of the curvature of the curve ξ is obtained in an analogous way as the curve η, with the use of Eq。 (6)。 In this case the first and second derivatives of Xξ and Yξ to φ, after differentiating Eq。 (10), are defined by the equations。

and the curvature radius ρξ = ρξ(φ) of the curve ξ is obtained by the formula

b −rrb  cosφ 2 cosαð cosα− cosφÞ

4。Conditions of non-undercutting — type II

In order to clarify why the rack-cutter fillet undercuts the gear teeth, in Fig。 6a the two boundary curves η and ξ are drawn simultaneously in the current position, where their common contact point coincides with the starting point A of the line of action AB (the position where φ= α)。 In this position the radial line OE (Fig。 6a), representing simultaneously a rectilinear profile of the rack-cutter, crosses the curve ξ in its inflection point, which, in this case, coincides with point A and simultaneously appears as a contact point of OE with the curve η。

From Fig。 6b it becomes clear that if the rack-cutter fillet is an arc of a small radius ρ1 there exists no undercutting — type IIa and type IIb, because in this case the arc AF1 lies on the internal side of curves η and ξ。 When the rack-cutter fillet is positioned between both curves η and ξ (the arc AF2 of a radius ρ2 > ρ1) an undercutting — type IIa appears, and when the rack-cutter fillet is placed between the curve ξ and the line OE (the arc AF3 of a radius ρ3 > ρ2) besides an undercutting — type IIa, an undercutting — type IIb  is derived。

The areas in which the rack-cutter fillet AFE (representing a circle, ellipse, trochoid, parabola, etc。), can be inscribed, without provoking an undercutting of the involute teeth of type IIa and type IIb, are defined by the boundary areas ACE and ADE, shown in Fig。 7。 In this case, the boundaries AC and CE of the area ACE, constructed by Eq。 (4), limit the undercutting of type IIa, and the boundaries AD and DE of the area ADE, constructed by Eq。 (10), limit the undercutting of type IIb。 The length of the common boundary AE of both boundary areas is equal to the tooth thickness sg  over the tip straight line g-g of the rack-cutter。