In order to solve the problem of the geometrical synthesis, two mobile coordinate systems are introduced: XlOlYl is connected with the gear (the specified profile l); while XηOηYη is connected with the rack-cutter (the searched profile η)。 As the axis OηXη (the centrode line) of the rack-cutter rolls without sliding on the reference circle (of a radius r) of the gear, the displacement s of XηOηYη is synchronized with the rotation of XlOlYl at an angle φ, where s = rφ。 The place of the contact points of profiles l and η in the motionless plane, designed with K, A, etc。 is defined as perpendicular lines to the respective positions, in which the radial line

Table 2

Geometric parameters of involute spur gears with undercut teeth。

Parameters and dimensions Symbol Equation Type of undercutting

ρmax = z sinα 2。052 2。052 2。052

Minimum addendum modification  coefficient xmin xmin = ha −0,5z sin2α 0。649

λr  =[mδr  /(ra  − rb)]。100 12。74 % 0 % 4。76 %

λt    =(2mδt  /sb)。100 20。66 % 4。82 % 12。21 %

Fig。 10。 Undercutting — type I: z =6; x = −0。2 b xmin; δr = 0。125; δt = 0。147; α=20°; ha =1; c* = 0。25; ρ*= 0。38。

l takes, are dropped from the pitch point P。 In fact the equations of the curve η are derived by defining the place of the same contact points in the rectilinear moving coordinate system XηOηYη。

After executing the respective transformations and conversions, the parametric equations of the boundary fillet — type IIa are

finally written as follows:

Xη ¼ rðφ− cosφ sinφÞ ¼ XηðφÞ ;

2 4Þ

Yη ¼ −r sin φ ¼ YηðφÞ ; ð

where φ is the angular parameter of the curve η, and r is the radius of the reference circle of the gear, calculated by the equation

r ¼ mz=2 : ð5Þ

The obtained curve η is pided by point А to two segments: AN and AM。 On Figs。 4 and 3a it is seen that only the segment AM appears as the real boundary rack-cutter fillet。 This means that when drawing the real curve η, the parameter φ gets an initial value φ= α (point A) and increases in the direction from point A to point M。

From the differential geometry it is known that the radius of a curvature ρ on each curve, specified as X = X(φ), Y = Y(φ), is defined from the equation

Y€ are the first and second derivatives to the parameter φ。

After differentiating Eq。 (4) and taking into consideration that

η  ¼ 2r sin φ;

X€ η  ¼ 4r sinφ cosφ;

Y_ η  ¼ −2r sinφ cosφ;

Y€ η  ¼ −2r cos2φ; ð7Þ

for the equation of the radius of the curvature of the curve η, the following formula is obtained

4r2 sin4φ þ 4r2 sin2φ cos2φ