Mathematical model for curvature studies of the flanks of cylindrical gear with cycloid al teeth.

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

1. INTRODUCTION

Knowing the values of the curvature radius, especially in contact zone, is necessary for the strength calculations regarding the stress contact and the stress inflection of the teeth and the thickness of lubricant film between the flanks as well.

A cycloid is the geometric place of a fixed point on a circle that is rolling without sliding on a straight line. Depending on the position of the considered fix point on the circle [O.sub.s], the resulted trajectory can be:

The cycloid is generated in the [[GAMMA].sub.D] plan as a trajectory of a point which is fixed by the rolling circle [O.sub.s], that rolls on a fixed straight line (fig.1).

The cycloidal director is transposed by rolling on surface of wheel part that has radius. The generation hook that defines the cycloidal teeth has the flanks defined by straight cutting edges that are attached on the fix point on the circle [O.sub.s].

The reference profile of generation hook is settled in its median plan and the reference angle [[alpha].sub.0] is 20.

The flanks of the wheel part are simultaneously generated by rolling with straight line and continuous division. The two cycloidal directories of the flanks must be generated with different curvature radius in order to realize the camber of the teeth. The curves which define the flanks of tooth are kinematic generated simultaneously by correlated motions.

[FIGURE 1 OMITTED]

2. THE FLANKS OF THE GENERATING HOOK

For the purpose to generate the cycloid of the rolling circle it can be attached to the circle a disc, where are fixed a few equidistantly knives on, resulting a head milling tool. In order to perform the analytical study of generation motions and the geometry of the flanks generated on the wheel part you consider several coordonate systems (fig.1). The considered systems are triortogonale, each of them being attached to an element that participates at generation process: [S.sub.S]--the reference system of cutting edges [T.sub.s] and [T.sub.d]. Its origin is in the centre of the head milling tool. It is rotated simultaneously with the tools; [S.sub.CG]--the reference system of generating hook. It is a mobile one, which moves with the rolling speed correlated with rotation motion of wheel part. Its origin is in the centre of the knife located in the middle plan of generating hook (Stanasel, I., Mihaila, I., Ghionea, A. 2003). The parametric equations of the flanks of the generation hook are obtained by changing the reference system of the tool edges from SS to Sca system.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

3. THE CURVATURE RADIUS OF THE FLANKS

For the curves indicated by the parametric equations, the calculation of the curvature radius is made by the relation:

[rho] = [[square root of ([x'.sup.2] + [y'.sup.2]).sup.3]]/x' y" - y' x" (2)

In order to determine the curvature radius of the generating hook flanks the relationship (2) will be applied for (1). The derivates of equations are settled related to parameter [PHI]s. After doing the calculations you obtain (Ionescu, Gh., D 1984).:

[rho] = A/B (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

This relationship can be used in a calculation program made in Matlab (Ghinea, M., Firetanu V., 1998) to determine the curvature radius for the two points [M.sub.s] and [M.sub.d] which belong to the two flanks generated by the cutting edges

In tables 1 are summarized data obtained by the calculations mentioned before.

The geometric place of the curbure centers of a plan curve indicated parametric is exprimed by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The parametric equations of the centers of the curvature radius can be obtained by applying the relationship (5) for (1). After doing the calculations are obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The program allows determining the coordonates of the curvature radius centers for any point that belongs to the cycloidal directory by using the relationship (6).

[FIGURE 2 OMITTED]

The data obtained after using the program were used for graphical representations from fig.2, which illustrates the line of the flanks of the generating hook, the curvature radius and its centre for different points.

In the relations below were used the following notations:

[R.sub.S]--the head milling tool radius;

[i.sub.c]--the number of the groups of knives;

[R.sub.R]--the rolling circle radius;

[[PHI].sub.s]--the rotation angle of head milling tool;

m--module of the gear

[[alpha].sub.0]--presure angle of the gear

u--parameter that pointed the points on the cutting edges of the tool;

[k.sub.1,2]--tell the right or the left cutting edge

4. CONCLUSIONS

Based on the coordinating systems there were established the analytic relationships of the cycloidal trajectory of the tool.

There were determined the analytical relationships and it was elaborated a performed calculation program for the curvature radius of the flanks generated by the cutting edges.

The analytical study shows that by the proposed generating procedure is possible to obtain cylindrical gears with curved teeth which assure the localization of contact zone in the centre of the teeth.

5. REFERENCES

Ghinea, M., Firetanu V., (1998) Matlab, numerical calculus, graphics, applications, ISBN 973-601-275-1, Editura Teora, Bucuresti.

Ghionea, A., Constantin, G., Stanasel I., Ghionea, I., (2008) Milling heads for processing curved teeth in cylindrical gears, Proceedings of the Oradea University CD Rom edition, pp. 1407-1412, ISSN 1583-0691, may 2008 Editura Universitatii din Oradea, Oradea.

Ionescu, Gh., D. (1984). The differential theory of the curves and surfaces with technical applications, Editura Dacia, Cluj-Napoca.

Stanasel, I., Mihaila, I., Ghionea, A. (2003)- Contributions to the study of generation of the flanks of the generating hook of cylindrical curved gear in oblong cycloidal arc, DETC'03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference pp. 785-789, ISBN 0791837025, Chicago, Illinois, USA.

Tab. 1. The values of the curvature radius for [i.sub.c]=1.

 m=2,5 mm, Rs=50 mm,

 [i.sub.c]=1

 [absolute value
 of
 [[rho].sub.d]]-
 [absolute value [absolute value [absolute value
 of of of
[y.sub.CG] [[rho].sub.s]] [[rho].sub.d]] [[rho].sub.s]]
mm mm mm mm

15 48,864 52,729 3,8646
-10 48,605 52,489 3,8841
-5 48,345 52,249 3,9037
0 48,085 52,009 3,9233
5 47,825 51,768 3,9430
10 47,564 51,527 3,9628
15 47,303 51,286 3,9826