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

Strajescu, Eugen ; Sandu, Ion

1. INTRODUCTION

The complex surfaces are these geometric surfaces of that theoretical curves D or/and G has a such geometric form that for their generation are necessary compositions of sample movements cinematic correlated by a cinematic correlation ratio, RCCIN+ (Sandu & Strajescu, 2007). According to the cinematic principle of the generation of the real surfaces on machine tools, for the generation of complex surface in a general case, the generating element must effectuate a movement along the theoretic generatrix G and a movement along the theoretical directrix D (Sandu & Strajescu, 2006), (Strajescu & Sandu, 2006). At many type of complex surfaces, the movement along the curve D is a movement composed resulted from the composition of single movements, cinematic correlated. In that sense, at the generation of the helical cylindrical surfaces where the theoretic directrix curve D is a circular helix, it is used the composition of a rotation movement with the rectilinear translation movement of the generating element in order to generate a helical circular trajectory in conditions of cinematic correlation (Sandu & Strajescu, 2004), (Botez, 1967).

2. THE COMPOSITION OF A ROTATION MOVEMENT WITH A TRANSLATION RECTILINEAR MOVEMENT MADE IN AN AXIAL DIRECTION, NORMAL ON THE ROTATION PLANE

We consider a rotation right cylinder, with the radius R and a circular base in the plane YOZ of the coordinate system. To this cylinder we impart a rotation movement around the OX axe with the frequency n (rot/min). In the same time, an initial generating point [M.sub.o] of the cylinder makes a rectilinear translation movement on the axial direction of the OX axis, with the speed [[??].sub.A], and, after a certain moment, come up in M in its resulting movement, but relative to the cylinder. In order to obtain a relative movement, we consider the cylinder fixed by its rotation in an opposite sense with the frequency [n.sup.*] = - n, on the rotation angle [PHI] (fig. 1). By consequence, the relative speed's movement of the generating point M is the vector [[??].sub.M], given by the vectorial relationship:

[[??].sub.M] = [[??].sub.T] + [[??].sub.A] (1)

where: [[??].sup.*.sub.T] is the vector of the tangent speed from the rotation movement with the [[absolute value of n.sup.*.sub.T]] frequency, having the size:

[[absolute value of [v.sup.*.sub.T]] = 2[pi]Rn [rot/min] (2)

The right linear translation movement is an uniform movement made by the generating point M along the generatrix G of the cylinder, on cyclical cinematic distances with the size p.

A cinematic cycle is a rotation of the cylinder and is made in the time [T.sub.CIC] having the size:

[T.sub.CIC] = 1/n [min] (3)

In this case, the size of the axial speed [[??].sub.A] from the right linear uniform movement is given by the relation:

[v.sub.A] = [P.sub.E]/[T.sub.CIC] = [P.sub.E.sup.n] [m/min] (4)

In this way it results:

[R.sub.CCIN] = [v.sub.A]/n = [P.sub.E] [mm] (5) n

That imposes to the two corposants movements the cinematic coordination in the purpose that the generating point M cross cyclically along the generatrix G of the cylinder constant distances [p.sub.E].

If the vectorial relation (1) is projected on the coordinated system's axes OXYZ, we obtain the sizes of the components of the [[??].sub.M] speed on these axes, having the next expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The differential coordinates dX, dY, dZ of the generating point M are differential spaces crossed respectively with the speeds [v.sub.MX], [v.sub.MY] and [v.sub.MZ], in a differential time dT:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where:

dT = d[PHI]/2[pi] x n (8)

We replace now the relations (6) and (8) in the relation (7) and we obtain the expressions of the instantaneous components X, Y and Z of the generating point M.(9).

Effectuating the integrals we obtain the coordinates X, Y and Z of the point M that represent the parametric equations of the cinematic curve C, described by the generating point M as consequence of the composition of the two movements (10).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where integration constants [C.sub.X], [C.sub.Y] and [C.sub.Z] are determinate knowing that at [PHI] = 0, X = 0, Y = R and Z = 0, resulting [C.sub.X] = 0 [C.sub.Y] = 0 and [C.sub.Z] = 0.

[FIGURE 1 OMITTED]

By consequence, the parametric equations of the cinematic curve C are:

X = [p.sub.0][PHI];Y = Rcos[PHI];Z = R sin [PHI]. (11)

that represent the known parametric equations of the circular helix and in which

[p.sub.0] = [p.sub.E]/2[pi] (12)

is the parameter and pE is the step of the circular helix.

So, by composing a rotation movement with a right linear translation movement made on an axial direction, normal on the rotation plane, a generating point M generates as a cinematic curve C a circular helix, with the condition to respect [R.sub.CCIN] given by the relation (5) that impose to the two movements the cinematic correlation with the view to the generating point M generate the helical circular trajectory with the pitch [p.sub.E] and with the radius R given.

Practically, at the generation of the complex real surfaces the curve C as circular helix is generated as directrix D of the surface.

The most extended application of this case of simple movements composition is represented the generation of the directrix D at the generation of the main extended complex surfaces in techniques, the helical cylindrical surfaces with all kind of profiles.

3. CONCLUSION

The aspects shown in this paper represent original contribution concerning the cinematic generation of the circular helix used as directrix D at the generation of real complex surfaces of the type cylindrical helical surfaces.

The treatment of these aspects by original and unique mathematical demonstrations contributes to the improvement of the cinematic generation theory by simple movement composition of the plane and space curves used in practice as real complex curves D

This theory of the cinematic generation from composition of single movements cinematically correlated by ratios [R.sub.CCIN], for the directrix curves D and for the generatrix curves G of the geometrical surfaces, permits the cinematic correlation of the mobile parts of the machine-tools with NC at the generation of the complex surfaces on them, and, in the same time, the creation of the cinematic algorithms available for the generation of the most complex forms of the real surfaces on these machine tools.

We must precise that this area of researches is largely covered by the activities of the Department of Machines and Production Systems from the "Politehnica" University of Bucharest, in correlation with the problems of the machine-tools and their command (NC)

4. REFERENCES

Botez, E., (1967). Bazele generarii suprafetelor pe masiniunelte. (Basis of the surface generation on machine-tools). Editura Tehnica, Bucharest.

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.