Rigid and flexible association of kinematic chains in gear processing machines.

Predincea, Nicolae ; Constantin, George ; Ghionea, Adrian 等

1. INTRODUCTION

For the generating of a surface on machine tools, the generatrix G or the director curve D are flat analytical curves (involute, ellipse, Archimedean spiral, polygonal curves, closed contour, resulting from combinations of simple flat curves, etc.) or three-dimensional curves (cylindrical helix, cone-shaped helix, combinations of these with certain flat curves, etc.). The generation of these curves requires the quantitative composing of at least two simple generating motions, of rotation (R) or of translation (T); the composition laws are expressed by kinematic conditioning function resulting from the kinematics of generating the paths G or D (Predincea & Constantin, 1993). These functions, defined in relation to the parameters of the simple generating motions ([y.sub.s], [y.sub.p]--tool S or workpiece P speeds) express, at the same time, the kinematic connection of the output values [y.sub.e1] and [y.sub.e2] of the two kinematic chains.

The kinematic connection as rigid kinematic link imposed to the outputs [y.sub.e1] and [y.sub.e2], can be obtained by means of a fictitious mechanism W (Fig. l) that connects the two output ends of the two associated chains with each other (kinematic loop or closed kinematic chain). The transfer ratio of the fictitious mechanism is expressed by the kinematic conditioning function itself that is essential for achieving the paths G or D.

[i.sub.w] = [w.sub.e]/[w.sub.i] = [y.sub.e1]/[y.sub.e2] = [Y.sub.i]/[Y.sub.e] = 1/[i.sub.T] = [f.sub.12]([y.sub.s], [y.sub.p]) = [f*.sub.12]([x.sub.s], [x.sub.p]), (1)

where [w.sub.i] and [w.sub.e] are the inputs/outputs of the Botez mechanism (Predincea et al., 1995); [Y.sub.i], [Y.sub.e]--inputs/outputs of the closed kinematic chain; [i.sub.T]--total transfer ratio of the closed kinematic chain.

The Botez mechanism is also an ideal mechanism characterized by the following features: ideal accuracy, infinite rigidity, neglecting the weight and friction between its elements; the geometrical and motion parameters represent theoretic values for comparison purposes, having in view the study of the cinematic and dynamic accuracy.

For closed kinematic chain, the adjusting equation derives from the transfer equation (Predincea & Constantin, 1993):

[i.sub.R] = [A.sub.i] / [B.sub.i] = [K.sub.i] (1 / [i.sub.w])([i.sub.D1] / [i.sub.D2]), (2)

where [K.sub.i] = [i.sub.11][i.sub.12] / [i.sub.21][i.sub.22] is the kinematic chain constant; i becomes F for treading, R for rolling and d for relieving.

2. RIGID ASSOCIATION OF CLOSED KINEMATIC CHAINS

In many cases, the generating of complex-shaped surfaces requires the kinematic obtaining of both generating curves, G and D. For each of these there are kinematic conditioning functions of the form (1). Sometimes, however, the three-dimensional director curves and the complex shaped curves are obtained kinematically, as resultants of two or more paths of the simple generating motions.

The same degree of complexity is also specific to the situation when the kinematic generating conditions of rigid character are added conditions of technological (worm gear processing with tangential feed) and/or economic nature (spur and helical cylindrical gear processing).

Consequently, the achievement of the paths G and D with the above mentioned features requires either the existence of two or more closed kinematic chains in mixed association, or that of a single kinematic chain of closed type associated with one or several technological kinematic chains. Frequently, closed kinematic chains are mixed associated by means of a differential mechanism [M.sub.[SIGMA]] (Figs. 1 and 2).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The general association structure is characterized by the following features:

--the output values of the two kinematic chains in mixed association in parallel-series are summed up algebraically by a differential mechanism M;

--there is a branch common to both closed kinematic chains comprised between the differential mechanism and workpiece; at the output end of this branch there are the values [w.sub.i], and [w.sub.e] of the fictitious mechanisms W' and W", identical to the generating parameters of the curves G and D;

--each Botez mechanism correlates the motion parameters that are necessary for achieving the curves G and D, so that the adjustment function of the closed kinematic chains becomes

A/B = [f'.sub.1,2]([i'.sub.w],[i.sub.D],[i'.sub.D]) = K x [f.sub.1',2]([x.sub.s], [x.sub.p]); (3)

A"/B" = [f.sub.1']([i".sub.w],[i.sub.D],[i".sub.D]) = [+ or -] K" x [f.sup.*.sub.1",2]([x".sub.s], [x.sub.p"], [x.sup.*.sub.p]). (4)

--equation (3) and (4) allow for an easy and highly accurate determination of the change gears; calculation difficulties appear in case of the geometric parameters x, resulting from the closing condition of the closed kinematic chain, is a trigonometric function (e.g. [x.sup.*.sub.p]);

--the existence of the two closed kinematic chains requires in their structure a relatively large number of mechanisms, which contributes to the substantial increasing of the errors of kinematic and dynamic nature, the differential gear and change gears having one of the most influential contributions.

3. FLEXIBLE ASSOCIATION OF KINEMATIC CHAINS

The flexible CNC association (Figs. 3 and 4) refers to:

--the kinematic structure of the machine tool is constituted only by independent generating kinematic chains (main and feed chains) (Kief & Roschiwal, 2007);

--simple structures of the kinematic chains due to the adjustment achieved only electromechanically or electrically,

--the kinematic closing condition imposed by the generation of the curves G and D is easily achieved through CNC;

--if in rigid association the inputs [Y.sub.i] of the two associated kinematic chains are subject of a rigid connection ([L.sub.R], [L.sub.F]) in association with flexible program, the connection is pseudoelastic (Weck & Brecher, 2005);

--possibility of identification, reducing and compensating for in real time the errors caused by static, dynamic and thermal behavior of the mechanisms in the kinematic chain structure.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

4. CONCLUSIONS

The conditions imposed by Botez mechanism are available both for rigid and flexible association. Although the flexible type programming, by NC, has penetrated in this field of the machine-tools generating surfaces by thread cutting and. rolling, the rigid program-carrier of the machine-tools in this field is still present.

The advantages of the machines with rigid program-carrier are known, i.e.: accuracy, reliability, availability to any producer, low costs, absence of difficulties with respect to the supply and operating of certin components, that are specific to the equipment required for the flexible program-carrier.

Those with flexible program-carrier have the following advantages: simple structures of the kinematic chains, and consequently a high accuracy, possibility of easy integration in flexible production cells and systems, reduced costs in fabrication preparing and maintenance, easiness in adaptation to any production type. But acquiring the processing precision (profile and pitch errors) is limited due to the errors resulting in gear wheel calculation.

5. REFERENCES

Kief, H., Roschiwal, H. (2007). NC/CNC Handbuch 2007/2008 (NC/CNC Handbook 2007/2008), Hanser Verlag, ISBN-10: 3-446-40943-2.

Predincea, N. & Constantin, G. (1993). Asocierea mixta a lanturilor cinematice inchise la masini de danturat (Mixed association of the closed kinematic chains in gear processing machines), Scientific Bulletin of University Cluj-Napoca, Series Machine Building, pp. 315-318.

Predincea, N.; Ispas, C., Minciu, C. & Ghionea, A. (1995). Teoria asocierii lan^urilor cinematice inchise (Theory of Association of Clsoed Kinematic Chains), T.C.M.M., No. 11, Edit. Tehnica, Bucharest, pp. 24-36.

Sandu, C.; Predincea, N. & Balan, E. (2000). CNC-Freiformmaschinen zum Zyklo-palloid-verzahn von Kegelradern (CNC Machines for processing the cyclo-palloid teeth of bevel gears), Scientific Bulletin of North University of Baia Mare, Series C, Vol. XIV, Fascicle: Tribology, Machine Manufacturing Technology, pp. 234-244.

Weck, M., Brecher, C. (2005). Werkzeugmaschinen l-Maschinenarten und Anwendungsbereiche (Machine tools 1--Machine types and fields of use), Springer-Verlag, Berlin, Heidelberg, ISBN: 978-3-540-22504-1.