⁎ ⁎ ⁎

Fig。 12。 Undercutting — type IIb: z =6; x = xmin = 0。449; δr = 0。068; δt = 0。114; α=20°; ha = 0。8; c* = 0。7; ρ*= 10。

When on the middle-line m-m the tooth thickness sm is equal to the width em of the tooth space (sm = em = p/2), sg is calculated by the formula

sg  ¼ m。0:5π−2hm tanα。  ; ð14Þ

and the depths hη and hξ of the respective boundary areas ACE and ADE are derived by the parametric Eqs。 (4) and (10) of the curves η and ξ, taking into consideration the equations

Xη  ¼ Xξ  ¼ r ðα   − cosα    sinαÞ þ 0:5sg    ; ð15Þ

hη   ¼ −Yη   ; hξ   ¼ −Yξ    : ð16Þ

On Fig。 8 is shown the variation of the geometric shape of the boundary areas of the fillet curve (the areas ACE and ADE on Fig。 7) at different values of the pressure angle α of the rack-cutter and different number of teeth z of the gear。 It is immediately seen that by increasing the number of teeth z, the depths hη and hξ of the respective boundary areas increase as well。 The same depths, with one and the same number of teeth, decrease when increasing the pressure angle α。

In the case where the rack-cutter fillet is profiled on an arc from a circle, the following boundary condition is defined: the undercutting — type II (IIa and IIb) is avoided if the radius of the fillet is smaller or equal to the radius of the curve ρη,A (Fig。 7) of the boundary fillet — type IIa  (curve η) in point A。

Since in point A of the curve η (Fig。 6) the value of the angular parameter is φ= α, for the radius of the curve ρη,A in this point according to Eq。 (9), it is obtained

ρη;A  ¼ mz sinα  : ð17Þ

Then the boundary condition for a non-undercutting of type II, when the fillet curve of the rack-cutter is an arc оf a circle of a radius ρ, is defined by the inequality

ρ≤ρmax  ¼ ρη;A ð18Þ

and is finally written as the following type(19)

where ρ*= ρ/m is a coefficient of the radius of the circle, on which the rack-cutter fillet is profiled。 The condition (19) shows that the undercutting — type II depends only on the number of teeth z of the gear and the profile angle α of the rack-cutter。

In Table 1 the maximum values of the dimensionless coefficient ρmax⁎, are shown, corresponding to profile angles used in

practice for a different number of cut teeth。 In order to avoid the undercutting — type II, it is enough for the radius ρ, on which the rack-cutter fillet is profiled, to be smaller or equal to the respective value ρmax⁎, multiplied by the module m of the gear (ρ≤ρmax⁎m)。 From Table 1 it is seen that ρmax⁎ increases when the pressure angle α and number of teeth z are increased。 Besides, from the specified values it becomes clear that if a standard rack-cutter is used, for which α=20° and ρ* = 0。38, the teeth cut are not undercut — type IIa  and type IIb  (at z = 5 and α=20° → ρmax⁎ = 1。71)。

Here it is important to note that if the teeth cut are not undercut — type IIa, they are also not undercut — type IIb。 Therefore

satisfying the boundary condition (19) guarantees the non-undercutting of teeth of type IIa, as well as of type IIb。