The kinematic generation of the generatrix curves of the geometrical surfaces.

Sandu, Ioan-Gheorghe ; Strajescu, Eugen

1. INTRODUCTION

The generation of the real surfaces on machine-tools is realized after the cinematic generation principle of the generation of the geometrical surfaces corresponding to their form.

In this way, the geometrical surfaces are generated by a generating curve G in it movement along the directrix curve D (Botez, 1967). The theoretic curves G and D of a geometric surface has the form gave by the geometric form of the surface and these curves are generated by materialization, by cinematic way or by programming..

In the case of the complex surfaces, (Sandu & Strajescu, 2007), (Strajescu & Sandu, 2006), (Sandu & Strajescu 2006), the curves G and D have a complex geometrical form, namely different than the circle or the right line (epicycloids, hypocycloid, involutes, circular helix etc.) These can be generated by cinematic ways using the composition of the simple movements cinematically coordinated in order to obtain their theoretical form.

2. COMPOSITIONS OF SIMPLE MOVEMENTS

In the anterior original studies (Sandu & Strajescu 2004), (Sandu & Strajescu 2007), in which was treated the generation cinematic as complexes trajectories G and D of some curves with a large utility in engineering, there was advanced the notions of simple movement, composition of simple movements and ratio RCCnn - ratio that assures the cinematic coordination of the simple movements being composed so that the generating element can generate the trajectory G or D with the given form.

On them bases, the paper presents new original aspects of cinematic generation of another two largely used curves G and D at the real complex surfaces' generation on machine tools.

The composition in the same plane of two rotation movements by interior rolling. It is considered a circular base having the radius [R.sub.B] with the center O in the origin of the co-ordinates system XOY (fig. 1). A circular rolling curve having the [R.sub.R] radius makes simultaneously a rotation movement around it center Or with the angular speed [[omega].sub.1] and a rotation movement of the center [O.sub.r] around the center O with the angular speed [[omega].sub.2], so that it is rolling on the base without sliding on the base, in it interior.

It is considered a generating point M of the rolling curve. For an instant position of the rolling curve gave by the rolling angle [PHI], the point M is angular positioned by the angle [phi]. It size can be deduced from the condition of the rolling without sliding of the rolling curve on the base, that impose the congruence of the size of the arcs AB and BM.

AB = BM, (1)

where:

AB = [R.sub.B] [PHI] and BM = [R.sub.R] [phi].

By consequence,

[R.sub.B] [PHI] = [R.sub.R] [phi], (2)

from where it results:

[phi] = ([R.sub.B]/[R.sub.R]) [PHI]. (3)

From the equality (2) it is to deduce by differentiating around the time dT

[R.sub.B] d[PHI]/dT = [R.sub.R] d[phi]/dT (4)

so it is

[R.sub.B] [[omega].sub.2] = [R.sub.R] [[omega].sub.1] (5)

Or:

[R.sub.CCIN] = [[omega].sub.2]/[[omega].sub.1] = [R.sub.R]/[R.sub.B], (6)

that impose to the two rotation movements to be composed the cinematic coordination of the rolling without slide of the rolling curve on the base, (fig. 1).

In the generating point M appear two speed vectors [[??].sub.TM], as a tangent speed in the rotation movement of the rolling curve algebraic composed with the rotation movement of the center [O.sub.r] and [[??].sub.TOr], as a tangent speed in the center [O.sub.r] in its rotation movement in face to the center O. By conseqruence the resulting speed in the point M will be the vector [[??].sub.M], given by the vectorial relationship:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The size of the component speeds are similar. So:

[V.sub.TM] = [R.sub.R] ([[omega].sub.1] - [[omega].sub.2]) = [R.sub.R] [[omega].sub.1] - [R.sub.R] [[omega].sub.2]. (8)

Considering the equality (5), it results:

[V.sub.TM] = ([R.sub.B] - [R.sub.R]) [[omega].sub.2] (9)

On another side, it is to observe that the size of the speed vector is given too by the relation (9).

If we project the vectorial relation (7) on the co-ordinate axes we obtain the sizes of the speed components on these axes, that have the next relations:

[v.sub.MX] = -([R.sub.B] - [R.sub.R]) [[omega].sub.2] sin [epsilon] - ([R.sub.B] - [R.sub.R]) [[omega].sub.2] sin [PHI]

[v.sub.MX] = -([R.sub.B] - [R.sub.R]) [[omega].sub.2] cos [epsilon] - ([R.sub.B] - [R.sub.R]) [[omega].sub.2] cos [PHI] (10)

From the fig. 1 we can observe that

[epsilon] = [phi] - [PHI]. (11)

Replacing the relation (3) in the relation (11), we obtain:

[epsilon] = ([R.sub.B] - [R.sub.R]/[R.sub.R]) [PHI] (12)

If we consider the (12) relationship in the relations (4) and ( 10), we obtain the final expressions of the sizes of the speed [[??].sub.M] components on the coordinates axes:

[v.sub.MX] = -([R.sub.B] - [R.sub.R]) [[omega].sub.2] sin ([R.sub.B] - [R.sub.R]/[R.sub.R]) [PHI] -

- ([R.sub.B] - [R.sub.R]) [[omega].sub.2] sin [PHI]

[v.sub.MX] = -([R.sub.B] - [R.sub.R]) [[omega].sub.2] cos ([R.sub.B] - [R.sub.R]/[R.sub.R]) [PHI]-

-([R.sub.B] - [R.sub.R]) [[omega].sub.2] cos [PHI] (13)

The differential coordinated dX and dF of the generating point M are differential space traversed respectively with the speeds [v.sub.MX] and [v.sub.MY], in a differential time dT:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where:

dT = d[PHI]/dT (15)

If the relationships (13) and (15) are replaced in the relationship (14), we obtain the expressions of the instant coordinates X and F of the generating point.

[FIGURE 1 OMITTED]

By the accomplishing of the integrals, we obtain the coordinates X and F) of the point M, that represent the parametric equations of the cinematic curve C, described by the generating point M as a consequence of the composition of the two rotation movements.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

where the integration constants [C.sub.x] and [C.sub.y] are determinate knowing that at [PHI] = 0, X = [R.sub.B] and F = 0, resulting [C.sub.X] = 0 and CF = 0.

So, the parametric equations of the cinematic curve C are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

that represent the parametric equations of the hypocycloid.

In conclusion, by composing of two movements of plane rotation (in the same plane), a generating point M of a circular rolling curve described as a cinematic trajectory C a hypocycloid with the condition to respect the [R.sub.CCIN] given by the relationship (6) that impose to the two movements the cinematic coordination of the rolling without sliding of the rolling curve inside the circular base.

Practically, at the generation of the real surfaces, the cinematic curve C as hypocycloids generated as a generating curve G of the complex surface.

An example of the applicability of this case of composition of simple movements is the realization of the generatrix G at the generation of the cylindrical wheel gears' flanks with cycloid profile.

3. CONCLUSION

The aspects shown in this paper represent original contributions concerning the cinematic generation of the hypocycloid curves used as generatrix G at the generation of the type cycloid denture from the watches industry.

4. REFERENCES

Botez, E., (1967). Bazele generarii suprafetelor pe masiniunelte (Basis of the surfaces' generation on machine tools). Editura Tehnica, Bucuresti

Sandu, I., Gh., Strajescu, E., (2007) New Theoretical Aspects Concerning the Generation of the Complex Surfaces. ICMaS 2007, Bucharest, Editura Academiei Romane, ISSN 1482-3183, pag. 181-184

Sandu, I., Gh., Strajescu, E., (2006). Contributions at the theory of the Generation of the Directrix and Generatrix Curves of the Geometrical Surfaces. ICMaS 2006, Bucharest, Editura Academiei Romane, ISSN 1482-3183, pag. 327-330

Sandu, I., Gh., Strajescu, E., (2004). The Cinematic Generation of the Directrix and Generatrix Curves of the Geometrical Surfaces. ICMaS 2004, Bucharest, Editura Academiei Romane, ISSN 1421-2943, pag. 171-173

Strajescu, E., Sandu I., Gh., (2006) Theoretical Studies about the Generation of the complex Surfaces ICMaS 2006, Bucharest, Editura Academiei Romane, ISSN 1482-3183, pag. 327-330