Fig。 13。 Prototypes of gears with undercut teeth。

5。Quantitative indices of the undercutting

For the estimation of the cutting of the involute teeth the authors of the present paper propose two groups of quantitative indices to be defined。 The first group of indices, defined as radial indices of the undercutting, specifies the degree of cutting of the involute teeth profile in a radial direction。 Using the second group of indices, defined as tangential indices of the undercutting, the degree of the undercutting in a tangential direction of the teeth, is specified。

5。1。Radial indices of the cutting

In the presence of an undercutting of type I and type IIb, as it was already clarified, the section bq (Fig。 9a) of the involute tooth profile bqa is cut。 In this case the location of the point q, obtained as a cross point of the gear fillet of the involute teeth profile, can be defined with the help of one of the following three possible parameters: the radius rq, at which the boundary point q is placed; the pressure angle αq  of the involute curve in point q; the radius of the curvature ρq  of the involute curve in point q。

The index radial undercutting (δr) defines immediately the length of the cut part of the involute tooth profile in a radial direction。 It is specified by the equation

δr  ¼ rq−rb  ¼ δm m ; ð20Þ

where the dimensionless value

represents a coefficient of a radial undercutting。 From Eq。 (21) it becomes clear that when the teeth profile is not undercut (rq = rb;

αq = 0), the coefficient of the radial undercutting is δ⁎r  =0。

If the tooth undercutting is of the type I, for the calculation of the pressure angle αq, a correct numerical method or the below approximate formula [20] can be used

In order to estimate what the length of the cut profile bq is, compared to the whole involute profile ba, the quantitative index a relative radial cutting (λr) is introduced。 It is defined by the   equation

where ra  is the radius of the addendum circle of the gear and the calculated value of λr  is obtained in percentages。

In the case of a non-undercut involute profile the radial indices of undercutting get a zero value (δr =0, λr = 0%), and in the case of cutting of the whole segment ba of the involute profile, then δr = ra − rb and λr = 100%。

5。2。Tangential indices of the undercutting

On Fig。 9a it is seen that when the gear teeth are undercut in a radial direction, they are undercut simultaneously in a tangential direction, too。 In the presence of an undercutting of type IIа the teeth are cut only in a tangential direction (Fig。 9b), where the starting point b of the involute profile lies on the base circle of a radius rb。

In order to determine the value of the tangential undercutting, from the center O of the gear a tangent line to the fillet curve fq is dropped, and the intersection u of the drawn tangent with the base circle, is determined。 Using the arc ⌢bu, measured on the base circle, the quantitative index δt is defined

δt  ¼ ⌢bu ¼ δm m ; ð24Þ

called a tangential undercutting, and the value δ⁎t   appears as a coefficient of a tangential undercutting。

In order to evaluate the extent of the tangential undercutting regarding its maximum value, the index a relative tangential undercutting  (λt), is introduced, defined by the  formula

λ 2δt

t  ¼  sb