The influence of the geometric parameters and cutting conditions on the theoretic deviations of worms by whirling thread device.

Cretu, Gheorghe ; Nedelcu, Dumitru ; Cretu, Dan 等

1. INTRODUCTION

The cutting of cylindrical worms by the whirling thread cutting principle is thought now as one of the most productive ways of processing. In this process the piece turns with slow speed and it is eccentric wrap up by the whirling thread cutting devices tools, which turn with high speed entering periodically into the splitter. Also for the correct achieving of the worm profile, the device's head inclines with an angle equal to that of the worm's helix reference.

The surfaces of flanks and bottom of the channels are cinematic generated, as successive wrap upping positions of the whirling thread cutting device tools. Because of this reason, the resulted surface presents successive deviations from the theoretical form of the profile (Nedelcu, D., et al, 2002). For the size determination of this deviation it is need to determine the surface equations described by two successive tools of the whirl head. Knowing the processed worm flanks equations it will be determined the relations that determine the theoretical deviation size resulted in cutting by this process.

2. TECHNOLOGY USE

For the determination of the surface equations described by the finishing tools it will be considered more reference systems (Bojan, L. 2006). First are written the surface equations described by the tools in the [O.sub.0][X.sub.0][Y.sub.0][Z.sub.0] reference system, in which [O.sub.0][X.sub.0] represents the symmetry axis of the disposal face of the whirling tools, and [O.sub.0][Y.sub.0] coincides with its own rotation axis :

([x.sup.2.o] + [z.sup.2.o]) [tg.sup.2][[alpha].sub.N] = [([y.sub.0] - [R.sub.0s]tg[[alpha].sub.N] + [bar.[e.sub.n1]]/2).sup.2] (1)

In which [D.sub.vs] represents the disposal diameter of the noses of tools from the rotation axis of the device, [[alpha].sub.N] the profiles angles, [e.sub.nl] is the normal reference cord of the worm tooth space, [R.sub.os] is the circle radius described by the point form the tool edge blade which processes the reference diameter (Cretu, Gh., 1997).

A translation of the coordinate system is performed in the coordinates [O.sub.0][x.sub.0][y.sub.0][z.sub.0] in the length of the [O.sub.0][X.sub.0] axis with an amount e, so the axis [O.sub.0][Y.sub.0] should coincide with the axis of processed worm axis, resulting [O.sub.1][X.sub.1][Y.sub.1][Z.sub.1] reference system. Equation no. 1 becomes:

[[([x.sub.1] + e).sup.2] + [z.sup.2.sub.1]] [tg.sup.2][[alpha].sub.N] = [([y.sub.1] - [R.sub.0s]tg[[alpha].sub.N] + [bar.[e.sub.n1]]/2).sup.2]] (2)

If the system is turning round the [O.sub.1][X.sub.1] line with an [gamma] angle (equal with the inclination angle of the worm helix on the reference diameter), it results the immobile OXYZ reference system. Knowing the link equation between the two reference systems, the equation of the conic becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

By geometrical considerations the value of ccentricity e it is determined:

e = [R.sub.0s] - [square root of ([R.sup.1.sub.2]) - [([bar.[e.sub.n1]]sin[gamma]/2).sup.2]] (4)

In which [R.sub.1] represents the radius of the reference cylinder.

[FIGURE 1 OMITTED]

Then is determined the equation of the normal plane at the average helix of tooth space:

x[R.sub.1] cos[phi] + yh - z[R.sub.1]sin[phi] - [h.sup.2][phi] + h [pi] / 2 = 0 (5)

[FIGURE 1 OMITTED]

For the determination of the theoretical deviation at worm flank processing with thread cutting devices, are considerate two successive positions of the finishing tools, offsite between them with an a angle, dependent on the parameters of the cutter conditions. The relation gives this angle:

[alpha] = 2[pi][n.sub.m] / [n.sub.s][z.sub.f] (6)

In which [N.sub.m] represents the worm rotation, [N.sub.s] the rotation of the whirl head and [Z.sub.m] is the number of finishing tools. The maxim deviations will be a half of this angle per each rotation. (figure 1). Their size is determined in the normal plan at the average helix of the tooth space, spine with a half of [alpha] angle around the axis Oy. For this the plan is intersected with the theoretic flank surface (resulting the curve [C.sub.1]) and a surface generated by the tools (resulting the curve [C.sub.2]).

The relation gives the [C.sub.1] curves equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

To evaluate the amount of this deviation, on this curve we consider m+1 equidistant points Ti (i=0,m), between the head and bottom cylinders, situated on equal distance with [R.sub.i] to the worm axis. In this points normal are layout to the helical surface of the worm and they are intersected with the surface described by the tools of the whirling thread cutting device, resulting [V.sub.i] points. The resulting relations from the determinations of the coordination points [T.sub.i] and [V.sub.i] are transcendental and they are solved with the computer aid. Knowing the coordinates of the [T.sub.i] and [V.sub.i] points it can be determined the amount of the deviation with the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The calculated deviation with this relation depends on many factors: the module, number of worm teeth, locating diameter of the whirling head teeth, diametric coefficient worm number revolutions, and the number of the finishing tools. For the determination of the deviation amount [DELTA]hi (i=0,m) and for the graphic representation of this dependence a special program was realized in QBASIC. The program is first receiving input data concerning the geometric parameters of the processed worm and of the cutting conditions. Solving the transcendent equation is achieved using the fractioned range method. After the coordinate determination of the [T.sub.i] and [V.sub.i], points the amount of deviations [DELTA]hi is determined. The obtained data can be graphically represented depending on point position on flank. The graphics have in their abscissa the deviation position in a perpendicular plan on the medium helix of the flank (the point 0 corresponds the exterior diameter, and the point 18 on the bottom diameter), and in the ordinate the amount of the deviation. On the same graphic are represented the deviation results for 6 values of the variable specified parameter.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

In the figures 2 and 3 are represented the deviation evolution on the worm flank for different values of module (figure 2) and diameter coefficient (figure 3).

3. CONCLUSIONS

Studying the evolution of the theoretical deviation to the cylinder worm processing with whirl processing, made with this program, a series of conclusions must be made: -the deviations are having higher values at the nose of tooth and lower at the pitch diameter

-the deviations are higher with increase of the module and the piece rotation, with diminishing the tool number of revolutions and that of the diametric coefficient

-the disposal diameter of the tools influences very little the amount of the deviation.

4. REFERENCES

Cretu, Gh., (1997). Contributii privind tehnologia de fabricare a melcilor folosind principiul filetarii in vartej, The contributions concerning the manufacturing technology of worm using the wriling process, Technical University of Iasi Publishing, pp. 40-68, Ph.D. Thesis, Iasi

Bojan, L., (2006). Advanced Technologies, Pro Literatur Verlag Publishing Germany, pp. 33-56, ISBN 3-86611-197-5, Mammendorf

Nedelcu, D., et al, (2002) Aspecte comparative privind deformatiile termice ale cutitului de strung, The comparative aspects concerning of temperature strain for tool lathe, Bulletin of Technical Institute of Iasi, pp. 165-171, ISSN 1011-2855, Iasi

Nedelcu, D., (2001) The technological parameters influence upon over rolls dimension at forming exterior grooves using Taguchi method, CITAF 2001, Technical Publishing Bucharest, pp.326-332, ISBN 973-31-2049-9, Bucharest