Computer Methods in Applied Mechanics and Engineering
Generation of planar and helical elliptical gears by application of rack-cutter, hob, and shaper
Introduction
The designers have tried for many years application of non-circular gears in automatic machines and instruments. The obstacle was the lack of effective methods for generation based on enveloping theory applied for non-circular gears. Previously, the generation was based on simulation of meshing of the generating tool with master-gears that have been developed by Fellows [6], Bopp and Reuther [3]. The breakthrough has happened in 1949–1951 by developing enveloping methods based on meshing of the generating tool (rack-cutter, hob, shaper) with a non-circular gear [11], [12], [15]. Such an idea is illustrated in the case of application of a rack-cutter as follows:
- (i)
Centrode 3 of the rack-cutter is a straight line t–t that is a common tangent to centrodes 1 and 2 of mating non-circular gears 1 and 2 and rolls over 1 and 2 (Fig. 1).
- (ii)
Rolling is provided wherein the rack-cutter translates along tangent t–t and is rotated about the instantaneous center of rotation I.
The related motions of the rack-cutter and the non-circular gear may be determined considering the motions of the generating tool and one of the centrodes of the pair one as follows (Fig. 2):
- (a)
The rack-cutter centrode (notified as I) is in mesh with centrode II of a non-circular gear.
- (b)
Rolling is provided by observation of the equation
Eq. (1) is obtained considering that the rack-cutter performs only translational motion with velocity v(I) along common tangent to centrodes I and II.
The non-circular gear II performs: rotational motion about center of rotation OII, and translational motion in the direction that is perpendicular to t–t. Vectors and represent velocities of gear II of rotational and translational motions (Fig. 2a). Drawings of Fig. 2b illustrate positions of rack-cutter I and non-circular gear II in fixed coordinate system Sf (Fig. 2b).
It will be shown in Section 2.1 that functions and are nonlinear ones.
Fig. 3 shows the remodelled cutting machine (1951) applied for generation of non-circular gears by enveloping method [11], [13]. Functions and have been generated by application of two cam mechanisms. These functions have been computerized as discussed in [15].
The list of references [1], [2], [4], [7], [9], [10], [11], [12], [13], [14], [16], [17], [18], [19], [20] cover titles of works of many researchers.
Numerical examples illustrate the ideas developed in the paper.
Section snippets
Algorithm of rolling motions
The derivation of the algorithm is based on application of following planar coordinate systems (Fig. 4a and b): (i) movable coordinate systems Sc and S1 rigidly connected to the rack-cutter and the non-circular gear, (ii) an auxiliary movable coordinate system Sn and fixed coordinate system Sf. Coordinate systems Sc, S1, Sf, and Sn are planar ones, however, the developed algorithm covers the concept of rolling for generation of planar and as well helical gears. Although the derivations are
Derivation of worm thread generating surfaces represented in coordinate system Sw
Application of a grinding worm or a hob for generation of elliptical gears may result in an improvement of productivity and reliability of this type of gears.
A worm thread surface Σw, that is in imaginary meshing with rack-cutter tooth surface Σc, is being determined. Conditions of meshing between both surfaces, Σw and Σc, allows worm thread surface to be determined. The procedure is as follows:
- (1)
Worm shaft and gear shaft are crossing by angle γwg. Fig. 10a shows the installation of worm axode on
Derivation of surface Σ1 of elliptical gear generated by shaper
The derivation is based on the following procedure:
- (1)
Two coordinate systems Ss and S1 are considered rigidly connected to the shaper and the to be determined surface.
- (2)
An involute tooth surface is considered as given for a shaper with pitch radius ρs.
- (3)
Coordinate system Ss is rotated while coordinate system S1 is rotated and translated. Motions of systems Ss and S1 are defined by kinematic relation of their centrodes (see Fig. 13).where
- (4)
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