a b s t r a c t  Article history:

In this work a generalized approach for defining the phenomenon undercutting of involute teeth is proposed, where besides the traditional boundary case, called undercutting — type I, additionally two more boundary cases, defined as undercutting of type IIa   and type IIb   are described。 According to this approach, the traditional undercutting — type I is caused by the 83295

Available online 21 February 2013

rectilinear profile of the rack-cutter and the non-traditional undercutting — type IIa and type

Keywords: Spur gear Undercutting Involute profile Gear fillet Rack-cutter fillet Boundary fillet

1。Introduction

IIb is caused by the rack-cutter fillet。 The parametric equations of the so-called boundary fillets of the types IIa and IIb defining the area of existence of the rack-cutter fillet are drawn。 Explicitly, an additional boundary condition for avoiding the undercutting of type IIa and type IIb is drawn。 The maximum value of the radius of the rack-cutter fillet, at which there is no undercutting, is specified。 Two types of quantitative indices for the estimation of the extent of undercutting of the teeth in a radial and tangential direction are  defined。

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The issues connected with tooth undercutting have a significant importance for the design and manufacture of gears and due to this they are widely published in the technical literature。 The conditions for avoiding of the traditional undercutting in different types of gears are defined by Litvin [1,2], Colbourne [3] and many other authors [4–11]。 Assuming the properties of the involute curve, Chen [4] presents a proof of the condition for the undercutting of tooth profiles of cylindrical gears。 By the use of cylindrical gears with curvilinear shaped teeth Tseng et Tsay [5] propose a mathematical model of the undercutting, based on the differential geometry and a numerical method for calculation。 When the lateral surfaces of the spur-gear teeth are modified, their undercutting is studied by Chao et Tsay [6] – for spherically concave and convex teeth and by Wang et Fong [7] – when generating longitudinally concave and convex teeth by the dual face-hobbing method。 The undercutting of elliptical gears, generated by rack-cutters [8] is also studied in the literature and different methods for defining the undercutting of the rotor of a cycloid pump of internal meshing [9–11] are proposed。

The problem connected with the use of gears of undercut teeth is especially topical when designing cylindrical gears of asymmetric teeth [12–15]。 In this case it is admissible that the coast side of the symmetric tooth is undercut。 The undercut gears are also used when the number of teeth is very small [16] and, in some cases, the undercutting influences positively the loading among simultaneously meshing gear couples [17]。

In the traditional theory of involute meshing [1–3,18,19] the condition which restricts the undercutting of gears (generated using a rack-cutter) is defined, when considering the meshing of the rectilinear profile of the rack-cutter of an involute profile of the produced gear。 In this case, the undercutting of teeth, defined by the authors as an “undercutting — type I”, is avoided when the trajectories of all points of the rectilinear profile of the rack-cutter cross the actual line of action。

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E-mail address: oalipiev@uni-ruse。bg (O。 Alipiev)。

0094-114X/$ – see front matter © 2013 Elsevier Ltd。 All rights reserved。 http://dx。doi。org/10。1016/j。mechmachtheory。2013。01。012

Nomenclature

c bottom clearance (mm)