⋅100

½% ]  ; ð25Þ

where the tooth thickness over the base circle sb  is calculated by the equation

sb  ¼ m cosαð0:5π þ 2x tanα þ zinvαÞ  : ð26Þ

From Eq。 (25) it is seen that when δt =0 ⇒ λt = 0%, and when the whole dedendum is cut (2δt = sb), the relative tangential undercutting is λr = 100%。 In this case, in front of δt the coefficient 2 is added, as the tangential undercutting is done on both sides of the tooth being cut。

6。Calculation results, prototypes and discussions

In order to check the theoretical dependence obtained, relating to the undercutting of teeth of type I, type IIа  and type  IIb, a computer simulation of cutting the teeth of three different gears – prototypes, is realized。 The basic parameters and geometric sizes of these gears (of undercut teeth), as well as the indices of a radial and tangential undercutting, are given in Table 2。 In the third column of the same table the formulas for the calculation of the respective values are provided。

The generation of the involute profile and fillet curve of the cut teeth is realized with the help of the approach used by Litvin [1], Fetvaci [21] and other authors, under which the relative positions of the rack-cutter are constructed (drawn) in the plane of the gear being cut。 For this purpose a software product was developed by the authors。 On Figs。 10, 11 and 12 the tooth generation of the three gears in Table 2 is visualized。

From Fig。 10 it is immediately seen that if the rack-cutter parameters are α=20°, ha⁎ =1, c* = 0。25, ρ* = 0。38, when cutting a

gear of parameters z = 6 and x = −0。2 an undercutting of type I is obtained, because the tip line g − g of the rectilinear profile of the rack-cutter does not cross the meshing line in the part A'PK。 In this case, the teeth are undercut simultaneously in a radial and tangential direction and the undercutting indices are respectively: δr = 1。25 mm; λr = 12。74%; δt = 1。47 mm; λt = 20。66%。

The presence of an undercutting of type IIа is found from Fig。 11, where the generation of the gear profiles (z =6; x =xmin =0。449), is realized by the rack-cutter of parameters: α=20°; ha⁎=0。8; c*=0。7; ρ* =3。2。 It is seen from the figure that the teeth are not undercut of type I, as the tip line g −g at x =xmin passes through point A'。 Besides, the starting point b of the involute profile ba lies on the base circle, and as a result, the teeth are not undercut in a radial direction (δr =0 mm, λr =0%)。 In this case only a tangential undercutting is obtained, caused by the rack-cutter fillet AF, where δt =0。47 mm and λt =4。82%。 This is determined by the fact that the rack-cutter fillet (profiled over a circle of a radius ρ*> ρmax⁎=2。05) is placed in the area ADE (Fig。 7), outside the boundary area ACE。

When generating the teeth, shown on Fig。 12, the parameters of the gear and rack-cutter, excluding the coefficient ρ* = 10, are the same as on Fig。 11。 And in this case, from Fig。 12 it is seen that the teeth are not undercut of type I, as the tip line g − g passes through point A'。 The undercutting obtained is of type IIb and is also provoked by the rack-cutter fillet AF。 In this case, due to the larger value of ρ*, the fillet curve AF is positioned outside both boundary areas ACE and ADE。 As a result, the teeth are undercut in a tangential, as well as in a radial direction and the undercutting indices are respectively: δr = 0。68 mm; λr = 4。76%; δt = 1。14 mm; λt = 12。21%。

Tooth undercutting is also observed on metal prototypes of gears, shown on Fig。 13, cut by a rack-cutter of a gear-cutting machine MAAG。 The geometrical parameters and sizes of the prototypes are the same as those of the computer generalized gears on Figs。 10, 11 and 12。 In this case, on Fig。 13a is shown an undercutting of type I, on Fig。 13b – an undercutting of type IIа, and on Fig。 13c – an undercutting of a type IIb。