Contact model of ellipsoidal worm-gears.

Cioban, Horia ; Butnar, Lucian ; Pop, Nicolae 等

1. INTRODUCTION

Double-enveloping worm gearing comprises enveloping worms mated with fully enveloping worm gears. It is also known as globoidal worm-gearing, globoid or hour glass gear and has advantage versus a traditional worm gear because of increased driving efficiency with 6-10% and increased loading capacity (about 30%). The disadvantage of a globoidal gear is higher manufacturing cost. On the other hand, the technology is more or less a secret of the manufacturers.

Comparing globoidal with ellipsoidal worm-gears there are similarities regarding the geometry and the generation method of the flanks. The mathematical model of the ellipsoid worm gears consists in a set of equations of the flanks surfaces. Based on these equations, the virtual model can be obtained using different CAD systems (www.wikipedia.org, 2009). Assembling the worm and the wheel together, first information about the surfaces in contact can be obtained studying the collisions of the 3D models. Importing the models in a FEA environment, the contact can be studied taking into consideration the material properties, loads and supports. A gear drive containing ellipsoidal worm-gearing was never produced before; it was only the subject of theoretical studies.

This paper presents an original method to obtain the virtual model of the gear that can be also applied to the globoids.

2. THE MATHEMATICAL MODEL

Figure 1 presents a fixed coordinate system ([X.sub.f][Y.sub.f][Z.sub.f]) and two coordinate systems linked the worm (1) and the wheel (2).

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

A point [P.sub.1] ([x.sub.1], [y.sub.1], [z.sub.1]) that belongs to the surface of the worm flank has the equation

[W.sub.1] = [T.sub.12] x [W.sub.2] (1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

is the transfer matrix from [X.sub.1][Y.sub.1][Z.sub.1] to [X.sub.2][Y.sub.2][Z.sub.2] coordinate system (Litvin & Fuentes, 2004). Having the rotation angle [phi]1 of the worm and the distance u to the point [P.sub.1] as parameters, like in figure 2, the flank surface equations of the worm are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

A similar procedure is used to obtain the flank surface of the wheel (Dudas, 2000).

3. THE VIRTUAL MODEL

Equations (3) can be used to calculate the significant points from worm surface, having [i.sub.21], [r.sub.0], a and a as input data, keeping u as a fixed parameter and giving values to [[phi].sub.1] parameter. A programming language like C or LISP used in a CAD environment can be the support to create the 3D model of the helix. Increasing the value of the u parameter a new helix can be obtained that belongs to the surface of the same flank. In this way, the 3D model of the flank surface can be built, like in figure 2.

The virtual model of the surface has only geometrical relevance, but cannot be used for simulation purposes. A surface model cannot take the properties of a given material and to obtain in this way the behaviour of a real model. Not all CAD applications can apply Boolean operations like union or intersection between two surface bodies. For this reasons, the surface model has some disadvantages in the simulation process that can be avoided by a solid model.

[FIGURE 3 OMITTED]

Parametric design is recommended to generate the solid model. Equations (3) give the points of the middle helix, drawn as a spline curve. The middle helix is used as path and guide rail to sweep the flanks profile. From the revolved ellipsoid, the sweep is subtracted through Boolean operations. A 3D solid like in figure 3 is created and saved as a new part.

The wheel is generated in a similar way. A parametric CAD application gives the possibility to assembly the worm and the wheel together. At the beginning, when the components are placed in the assembly environment, all the parts have all the 6 degrees of freedom (DOF). Some of DOF are removed applying constrains with respect to the movement conditions. So, the worm is placed in [X.sub.1][Y.sub.1][Z.sub.1] coordinate system and has the possibility to be rotated with ([[phi].sub.1] degrees around the [Y.sub.1] axis (figure 1). The relation between [[phi].sub.1] and [[phi].sub.2] is given by the transfer ratio

[i.sub.21] = ([[phi].sub.2] / [[phi].sub.1] (4)

Having all the constrains applied, the assembly looks like in figure 4.a. Parametric applications give the possibility to analyze the interference between the two solids. Figure 4.b shows collisions between the worm and the wheel symmetric arranged from the middle to the end. The reason if these collisions can be found in the difference of the curvature of the worm and the wheels flanks and can be avoided by decreasing the section of the flanks. The interference tool gives also the value of the interfered volume and this information is very useful in the manufacturing process.

4. CONTACT ANALYSIS USING FEM

Finite Element Analysis is a computer simulation approach used in engineering analysis that uses a numerical technique called the finite element method. This analysis gives information regarding the behavior of the gearing parts under constrains like moments, forces or pressure (Sharif et all, 2001). First step is to import the virtual model into the FEA environment and to choose the material properties of the parts. Meshing is the process in which the geometry is spatially divided into elements (tetrahedrons) and nodes, like in figure 5a. This mesh along with material properties is used to mathematically represent the stiffness and mass distribution of the assembly (Chira, F., Banica, M., Lobontiu, M., 2008).

The solver of the FEA application needs information regarding the connections between the parts, the type of the contact between the worm and the wheel flanks. It is also necessary to define the supports and loads.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In the case of static structural analysis, the solution determines the displacements, stresses, strains, and forces in structures or components caused by loads that do not induce significant inertia and damping effects. Figure 5b is a colored chart that shows the distribution on the flanks of the equivalent Von Misses stress. A similar map can be obtained for the deformations.

The results show the high values displayed in red for the stress in the root fillet area and for deformations at the top of the teeth. Low values, displayed in blue, are obtained on base of the teeth for deformations and on top of the teeth for the stress. The highest values have to be compared with the admissible stress and it have to make corrections in geometry or to change the material in case of low safety factor.

The outputs of the solver can be images, animations or reports in documents and html format.

After the solution is solved, the results offer information regarding the field of interest and the designer have to decide what type of changes have to apply: depending of the case, the material or the shape of the parts can be optimized.

5. CONCLUSION

The ellipsoidal worm gearing can be treated as a special case of globoidal worm gearings. The study of the ellipsoidal worm gears modelling offers useful information for the manufacturing process and part of the results can be applied also for the globoidal worm gears. Even an ellipsoidal worm gearbox was not produced yet, relevant conclusions can be obtained from researches of the virtual model.

Considering the similarities with the globoidal worm gear, the studies will continue in the frictional contact domain, in a range of the friction factor, comparing values for Von Misses stress and displacements. The mathematical model can be also improved, to include all types of worm gearings in a same set of equations.

6. REFERENCES

Chira, F., Banica, M., Lobontiu, M., 2008, On the variation of the functional parameters of the asymmetric gears on relation with the designing variables, Annals of DAAAM for 2008 & Proceedings of The 19th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions", 22-25th October 2008, Trnava, Slovakia, ISSN 1726-9679, pg. 0237-0238

Dudas, I. (2000). The Theory and Practice of Worm Gear Drives, Penton Press, ISBN 9781 9039 9661 4, London.

Litvin, F. & Fuentes, A. (2004). Gear Geometry and Applied Geometry, Cambridge University Press, ISBN 0 521 81517 7, USA

Sharif, K.; Kong, S., Evans, H. P. & Snidle, R. W. (2001), Elastohydrodynamic analysis of worm gear contacts, Proceedings of the International Tribology Conference, Japanese Society of Tribologists, Nagasaki, pp. 1867-1872 ISBN 4-9900139-6-4

*** (2009) http://en.wikipedia.org--Finite Element Method, Accessed on: 2009-04-15