η boundary fillet curve — type IIa of the rack-cutter

ξ boundary fillet curve — type IIb of the rack-cutter

When using the traditional approach, the influence of the rack-cutter fillet on the teeth undercut is not taken into account and the results obtained are correct only at small values of the radius of the rack-cutter fillet。 When this radius exceeds a specified value [16], it can turn out that the traditional condition for non-undercutting is satisfied and actually the teeth of the involute gear have been undercut by the rack-cutter fillet (undercutting — type II)。 That is why in the present paper a new generalized approach for defining undercutting of teeth is proposed, where simultaneously with the traditional boundary case (undercutting — type I) two more boundary cases, defined as “undercutting — type IIa” and “undercutting — type IIb”, are taken into consideration。 In these two additional cases the undercutting is done by the rack-cutter fillet but not by its rectilinear profile。 Hence the undercutting

Fig。 1。 Traditional undercutting — type I: a) undercutting; b) non-undercutting; and c) boundary case。

type IIa is characterized by the decrease of the tooth thickness in their dedendum without cutting the involute profile while in the presence of undercutting type IIb  a part of the initial section of the involute profile is cut。

2。Undercutting of teeth at meshing of an involute gear with the rack-cutter

2。1。Undercutting  — type  I  (traditional case)

The undercutting of teeth — type I of the involute gear is obtained when at its meshing with the rack-cutter (Fig。 1a), points of the rectilinear profile KE of the tooth cutter are situated under the line of action AB [1,2]。 Then the trajectories of these points (lines parallel to the rack-cutter centrode n-n) do not cross the line of action AB (they cross its extension), and the rectilinear segment КЕ of the rack-cutter becomes a non-operating profile that penetrates into the bottom of the processed tooth and cuts a part of an involute profile。

In order to avoid the undercutting — type I, it is necessary that the tip-line g-g of rectilinear profile of the rack-cutter crosses the line of action AB (Fig。 1b)。 Otherwise, when the crossing point L lies outside the line of action on the extension of the line PA (Fig。 1а), the cut teeth are undercut。 This means that the condition for non-undercutting — type I generally can be expressed by the inequality

PA≥PL; ð1Þ

and taking into account the specified distances marked on Fig。 1 。PA  ¼ r sinα  ;   PL  ¼ ðha −XÞ= sinα。  the traditional condition for

non-undercutting — type I finally is written in the following way   [2,18,19](2)

ð2Þ

In the inequality (2) x =X/m is an addendum modification coefficient of the rack-cutter, ha⁎=ha/m is an addendum coefficient of basic rack tooth, z represents the number of teeth, α is the profile angle of the rack-cutter, and m is the module of the gear (m =2r/z)。

Fig。 2。 Non-traditional undercutting — type II: a) non-undercutting; b) undercutting — type IIa; c) undercutting — type IIb。

Minimum addendum modification Xmin of the rack-cutter, eliminating the undercutting — type I, is defined by the equation

。 2  。

a −0:5z sin α

: ð3Þ