The kinematics of generation of the helical surfaces of the cylindrical symmetric ruled worms.

Sandu, Ioan-Gheorghe ; Strajescu, Eugen

1. INTRODUCTION

The worms are machine parts that have in the main cases as a conjugate element a worm gear. The so formed gearing is named worm gearing. The axis of the component of the ale worm gearing are not parallel and not concurrent, usually perpendicular one to another one. The most used worm gearing in engineering are the worm gearing cinematic cylindrical at which the worm is cylindrical ruled symmetrical (Sandu 2008).

The worm gearings with axis not parallel and not concurrent are in the main cases reducing gears that mean that the worm is the motor element.

The cylindrical worm gearing has at the basis of it geometric and dimensional construction and at the base of it generation a theoretic worm named reference worm (figure 1).

[FIGURE 1 OMITTED]

It reference cylinder has the reference diameter [D.sub.0] and it is part of the exterior cylinder having the diameter [D.sub.e] and contains the interior cylinder with the diameter [D.sub.i].

Dimensional parameters are calculated with the relationships (Maros 1966):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

in which: q is named diametral coefficient, tabulated in function of [m.sub.a];

[m.sub.a] is the axial modulus of the worm gearing;

[a.sub.0], [b.sub.0]--the height of the head, respectively of the root of the tooth of the reference worm.

The circular helix characteristic to the reference worm is the helix corresponding to the reference cylinder and is named the mean circular reference helix of the worm; it has the pitch [p.sub.E] and the inclination [[theta].sub.0] having the size:

ctan [[theta].sub.0] = q/k (2)

in which k is the number of the beginning of the reference worm.

2. THE THEORY OF THE KINEMATIC GENERATION OF THE CYLINDRICAL SYMMETRIC RULED WORMS

Considering their advantages at the surface generation, the most used cylindrical worms are the cylindrical symmetric ruled worms. The flank form of these worms is given by the curve's C form (figure 2) that is generated in a frontal section, normal on the worm's axis. As it was that the types of the cylindrical symmetric ruled worms are different between them by the curve's C form.

In the study of the generation mode of the curve C it is considered a generating right line d, tangent at a circle (cylinder) named base cylinder with the radius [R.sub.b] from that is unitive connected the generating element M of the generation tool. The right line d is inclined posed towards the basic cylinder with the angle [[theta].sub.b] that is the inclination angle of the circular helix described by the right line d on the base cylinder.

To the right line d it is impressed the rotation movement around the axis of the base cylinder with the frequency [n.sub.d] (rot/min) and the axial displacement movement with the speed [[??].sub.DA], as a complementary generation movement. (Sandu & Strajescu 2007).

[FIGURE 2 OMITTED]

These movements are cinematically coordinated by RCCNN given by the relationship (3), so that the right line d therefore the generating element M generates the helical circular trajectory with the given pitchpE (Strajescu & Sandu 2006 and 2008).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[R.sub.CCIN] = [v.sub.DA]/[n.sub.d] = [p.sub.E] (3)

In the same time, the line d roll up with the angular speed cod without sliding on the base cylinder, so that the generating element M describes a curled involute C, tangent in the exterior of a circle (cylinder) with the radius [R.sub.dr] named directory circle (cylinder).

The so generated helical surface is named convolute helicoids of the first specie, and it is obtained only by the shown kinematics, if the sense of the inclination of the line d and the sense of the helix coincide. The cylindrical worm having the flanks helical surfaces similar to helicoids of the first specie is named convolute worm of the first specie.

The main inconvenience of this kind of worm is that in a normal section on the average reference helix of the tooth (of the spire or of the tooth space (the space between two teeth) it is not possible to obtain a rectilinear profile.

By consequence, the flanks of this kind of worm can not be generated simultaneously with a generating cutting tool having rectilinear generating cutting edges. Because these difficulties, generally, the importance of this worm disapierred, and, for this reasons it is established that it is named convolute worm zero.

The second form of the curve C is a curled involutes tangent in the interior to the director circle (cylinder) (figure 3).

The generated helical surface is named convolute helicoids of the second specie. This surface is generated by the right line d with the same kinematics as at the convolute worm zero, but the senses of inclination of the line d and of the circular helix describe

The cylindrical worm having the flanks as helical surfaces of a convolute helicoid of the second specie has the property that in the normal section on the reference medium helix of the tooth or of the tooth space. It is obtained a right linear profile. As consequence, the flanks of that worm can be simultaneously generated by generating cutting tools with right linear edges. In the same time, that kind of worm presents a right linear profile in a section made with a plane parallel with the axial plane of the worm and tangent at the director cylinder [R.sub.dr], bed by it are opposite.

The cylindrical worm having the flanks helical surfaces of the type convolute helicoids of the second specie is named convolute worm without other specifications.

The convolute worm is named too N worm, with the variants ND worm and NG worm.

At the worm of the type ND, the right linear profile is obtained in a normal section on the average reference helix of the tooth, and at the worm of the type NG, the right linear profile is obtained in a normal section on the average reference helix of the tooth space.

The third form of the C curve is a normal involute (figure 4,a), at which the director cylinder is confounded with the base cylinder ([R.sub.dr] = [R.sub.b]).

The helical surface is generated in the same kinematical mode as at the zero convolute worms, but the generation tool is posed with the rectilinear generating edges in a parallel plane with the axial plane of the worm and tangent to the base cylinder, above the first. That surface is named involute helicoid.

The cylindrical worm having the flanks as helical surfaces of the type involute helicoid is named involute worm or E worm.

The flanks of this kind of worm cannot be simultaneously generated with a generation cutting tool having right linear cutting edges, but only separately. The separate generation of its flanks is possible because this worm have the property that in a section made with a plane parallel with the axial plane of the worm and tangent to the base cylinder, a right linear profile is obtained.

The fourth form of the curve C is an Archimedes spiral surface (figure 4,b). In that case do not exists a director cylinder ([R.sub.d] = 0).

The helical surface so generated is named Archimedes helicoid, and the generation kinematics is the same at the convolute worm. In that case the generation tool is put on with the generating edges in the axial plane of the worm. The cylindrical worm having the flanks helical surfaces of the type of Archimedes helicoid is named Archimedes worm or A worm.

That worm presents in the section made with an axial plane a right linear profile. As consequence, this type of worm can be too generated simultaneously on the both flanks with cutting tools having right linear generating cutting edges.

3. CONCLUSIONS

The presented study put in evidence the kinematics generation, specific for the four kinds of cylindrical ruled symmetric worms utilized in techniques--ND, NG, E and A. This kinematics stay at the base of the real helical surfaces' generation of these worms on machine tools by specific proceedings detailed anteriorly and mathematically justified, at which the cutting tools are profiled tools and with rectilinear generating edges, corresponding to every kind if worm.

4. REFERENCES

Maros, D., (1966). Worm gearings. Editura Tehnica, Bucuresti

Sandu, I. Gh., (2008). Surface Generation. Treatise. Editura Academiei Romane, Bucharest

Sandu, I., & Gh., Strajescu E., (2007). New Theoretical Aspects Concerning the Generation of the Complex Surfaces. ICMaS 2007 Bucharest, Editura Academiei Romane

Strajescu, E., Sandu I., Gh., (2006) Theoretical Studies about the Generation of the complex Surfaces ICMaS 2006, Bucharest, Editura Academiei Romane

Strajescu E., & Sandu, I., Gh., (2008). The Kinematical Generation of the Directrix Curves of the Geometrical Surfaces. 19th International DAAAM Symposium. Tarnava, Slovakia