The mathematical study of the profile of toothing knives.

Pantea, Ioan ; Stanasel, Iulian ; Blaga, Florin 等

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1. INTRODUCTION

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The working profile of the knives used for toothing conical gears with curved teeth is made up of complex surfaces, surfaces that compose helixes.

A conical or cylindrical rectification tool cannot perfectly wrap around this kind of surface (Maros,, 1959).

The chipping edge of the knife is part of this surface, and the maximum permissible deviation is 0.08 mm. The large number of knives mounted on the head implies that the relieving process needs a large number of double strokes of the sled on one rotation of the toothing head

The paper presents the author's research regarding the execution of these surfaces through continuous processing following a cylindrical propeller and by using a conical abrasive tool (Pantea, 2004).

2. THE STUDY OF THE TECHNOLOGY OF TOOTHING KNIVES THROUGH THE SPATIAL GEARING METHOD

On profiling the toothing knives, the following issues must be addressed:

--determining the contour of the abrasive stone (the theoretical profile);

--determining the equations of the relieved surfaces.

Resolving these problems is done concomitantly in a calculus section using the spatial gearing laws and the coordinate system's transformation laws (Litvin, 1994). We go from the generating curve of the face gear, we determine the profile of the knife's chipping edge and the profile of the abrasive tool used for relieving. The chipping edge must have minimal errors after resharpening.

The generating curved is given under the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where p is the independent parameter of the curve.

[FIGURE 2 OMITTED]

The parametric equations and the limits of the independent parameter p used result from fig. 2

The parametric equations:

[x.sub.p] = p x sin([alpha])

[z.sub.p] = p x cos([alpha]) (2)

The limits of the independent parameters

[p.sub.s] = h' x m/cos([alpha])

[p.sub.i] = -h' x m/cos([alpha]) (3)

Particularizing (fig. 3) the arrangement of the systems for knives used for toothing conical gears with curved teeth arranged in a circular arc that have their directing curve a cylindrical propeller, the matrix product with the help of which the passing for the [x.sub.i][y.sub.i][z.sub.i] system to the [x.sub.k][y.sub.k][z.sub.k] system is achieved, is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[FIGURE 3 OMITTED]

Analogously, the passing formulas from the [x.sub.k][y.sub.k][z.sub.k] system to the [x.sub.i][y.sub.i][z.sub.i] system are deduced with the help of the reciprocal matrixes.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The (5) equations express the trajectory of the relieving motion of the disk's surface points in relation to the [x.sub.i][y.sub.i][z.sub.i] system that is considered to be immobile.

The normal line to the chipping edge is determined from the condition of perpendicularity of vectors the tangent to the chipping edge and the relative speed in the relieving process (Litvin, 1994):

[bar.N] = [bar.t] x [bar.v] (6)

As for determining the chipping edge's profile in different degrees of resharpening, the speed vector of the relative motion's projections in the [x.sub.k][y.sub.k][z.sub.k] system must be determined (fig.4).

The normal lines N are described by following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The generated surface from the rectification with a conic disc will be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The [S.sub.d] relieved positioning surface is defined as the geometric location of the chipping edges [M.sub.a], [epsilon] positions the [M.sub.a] chipping edge in relation to the technological device's [x.sub.i][y.sub.i][z.sub.i] system and for [epsilon] variable the [X.sub.det] radius vector of the relieved surfaces is determined (Stetiu et al., 1994).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

In order to determine the theoretical profile, the surface of the abrasive tool is considered the kinematic hull of the [S.sub.d] relieved positioning surface.

The position vector's expression is:

[X.sub.k] = [M.sub.ki] [X.sub.d] (10)

[FIGURE 4 OMITTED]

If the p parameter is given discreet values the coordinates of the characteristic curves plotted on the abrasive tool can be determined.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

By using the [M.sub.ik] (5) transformation matrix, the real positioning surface is determined with the relationship that depends on [theta].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. CONCLUSIONS

1. Using the algorithm, a research methodology of the geometry of toothing knives was constructed by using the spatial gearing method of research.

2. The mathematical modeling of the technology of rectification of toothing knives using the spatial gearing method implies establishing a response in regard to the optimization of the technology.

3. Through particularizing the research method for the I tool and the II tool and by using a Archimedes's spiral as a directing curve, the evaluation and optimization model of the profile of these knives is obtained through comparing the deviations in the grinding sections of the knives.

4. REFERENCES

Litvin, F.L. (1994). Gear Geometry and Applied Theory. University of Illinois, Chicago. PTR Prentice Hall, Englewood Cliffs, 0763. ISBN-13-211095-4, New Jersey.

Gramescu, Tr. s.a. (1993). Toothing technologies. Guide project book. Ed.Universitas, ISBN 5-362-01009-3, Chisinau.

Stetiu, G. s.a. (1994). Practice and theory of cutting tools, vol. I, II, III. Editura Universitatii Sibiu, ISBN 973-95604-3-1, Sibiu.

Maros, D. (1959). Kinematics of tooth wheels. Editura Tehnica Bucuresti.

Pantea, I. (2004). Contributions regarding the technology of the tools used for bent teeth bevel gears teething. Doctorate Thesis, University of Oradea.