A determination of the displacements of the contact surface.

Dinu, Gabriela ; Hadar, Anton ; Vlasceanu, Daniel 等

1. INTRODUCTION

The displacements and the contact deformations influence the accuracy of the machine-tools. The role of the machine-tool's foundation is to achieve a highly uniform distribution of static and dynamic loads on ground, ensuring a proper rigidity for the assembly and, therefore, a better static and dynamic stability for the relative positions of the machine's elements.

For illustrating a calculus methodology using a self program, regarding the vertical displacements of the contact surfaces points of the bearing elastic small plates for a machine-tool stand, there are considered a parallelepiped shaped foundation and three bearing small plates, according to the Fig. 1.

The computation program, which is based on the analytical expressions of the flat surface contact between a linear elastic plate and an elastic foundation (Vlasov & Leontiev, 1966), the flat surface plate being loaded with a concentrated force applied eccentrically and with a loads uniformly distributed, leads to the displacements of the points from any position within the contact area, by resolving the differential equations which describe the phenomenon of the two bodies' deformation (Dinu,1998).

[FIGURE 1 OMITTED]

The original elements of the method are: the introduction of the initial and "ad limit" conditions, the development of the terms that depend on the type of the loads and on the acquirement of the expressions of the plate and foundation displacements.

The computation program is easy to be used, being a useful instrument in studying the tackled problem (Dinu et al., 2008).

For being able to apply the principle of the effects' superposition, the small plates are situated so that the influence areas of the maximum loads on these ones should not interfere.

In this way, the problem could be dissociated and the study could be made for each small plate individually (Hadar et al., 2008).

For estimating the area around the small plate, in which significant vertical displacements are registered (longer than 0,002 mm), the specialized COSMOS/M computation program was applied.

2. THE AREA WITH SIGNIFICANT DISPLACEMENTS AROUND THE BEARING SMALL PLATE

There is considered the example of a foundation, with a big enough surface, on which the stand of a machine-tool bears, by means of three small plates.

If there are used the usual elastic characteristics for the small plates and the foundation--the small plate of steel, the foundation of concrete--and it is presumed that the potential maximum loads is P = 50 kN, corresponding to a uniform distributed loads of p = 2,778MPa, then, knowing the value of the vertical displacements which results from the program with finite elements, the influence area around a single small plate could be determined.

The foundation is considered to be embedded at the base, having the dimensions of 600mm * 750mm * 500mm, with the Young modulus [E.sub.f] = 4 x [10.sup.4] MPa and the Poisson ratio [v.sub.f] = 0,22. The bearing small plate is made of steel, having the dimensions of 120mm x 150mm x 40mm, with [E.sub.p] =21 x [10.sup.4]MPa and [v.sub.p] = 0,3.

The foundation was meshed in 3000 elements of SOLID type and the small plate was meshed in 12 elements of the same type (Fig.2).

In Fig. 3, the influence area of the displacements can be delimitated to a strip with a width of 70mm, beyond which the vertical displacements are lower than 0,002mm.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

3. THE DISPERSION OF LOADS IN THE CASE OF A BEARING ON THREE SMALL PLATES

There is analyzed the case plotted in Fig. 1 and Fig. 4, for which the projection on the bearing plane of the position of the mass centre is known, so, the application point of the concentrated force, P, is also known.

The force P is the vertical component of the resultant of the forces which reacts on the foundation (Dinu, Baciu et al., 2008).

The dispersion of loads on the three bearing small plates results from the equations of equilibrium of the forces system which reacts on the foundation,

where [P.sub.2] = 0,2292P, [P.sub.3] = 0,6042P and [P.sub.4] = 0,1666P,

represent the forces concentrated in the small plate's centers of gravity, as reactions of the bearing surfaces (Fig.4).

If P=100kN, the uniform distributed load which was taken into consideration when settling the influence area of the displacements is highly enough, the resultant of the force uniform distributed on a small plate being 0,5P.

The setting of the three small plates (Fig.4) was chosen so that the spaces from the small plates to the edges and the ones between the small plates to be longer than 70mm.

4. THE APPLICATION OF THE PROPOSED CALCULUS METHODOLOGY

There has been considered P=[10.sup.5]N for which the forces distributed on the small plates are: [P.sub.2]=22920N, [P.sub.3]=60420N and [P.sub.4]=16660N and the uniformly distributed loads are: [p.sub.2]=1,2734MPa, [p.sub.3]=3,3566MPa and [p.sub.4]=0,92555MPa.

The foundation with the three small plates is loaded simultaneously with the distributed loads [p.sub.2], [p.sub.3], and [p.sub.4].

The values obtained by means of program are compared with the values obtained by COSMOS/M program. The digitization of the structure is made in 3036 elements of SOLID type and 3756 nodes.

In conclusion the results are the same and the values are terms of microns.

The displacements obtained by COSMOS/M program, using the SOLID finite element type, also include the vertical components of the displacements, corresponding to the bending of the foundation's plate.

For making a comparison between the results of the proper program and the COSMOS/M program, the bending values, readied in the fringe area, obtained was subtracted, and are showed in the following figure (Fig.3).

[FIGURE 4 OMITTED]

5. CONCLUSION

The information aforementioned could represent a first work stage for the specialist in designing machine-tools when estimating the accuracy.

That could be obtained during the treatment treating as far as the effects of the bearing of a machine- tool on the foundation are concerned.

The layout of the small plates on the foundation was made so that they are not supposed to influence each other, as much as possible, letting between them and from the small plates to the edges large enough spaces.

If the torsor of the forces which react on the whole assembly, calculated in relation to its centre of gravity or centre of mass, is composed of a resultant and a resulting moment.

It can be considered- pursuant to their distribution on the bearing small plates- as being a case of superimposing the effects of a distributed load and a concentrated force, applied eccentrically.

The example taken into consideration in this chapter was the one in which the torsor consists only of the vertical component of the resultant, situation that is equivalent to the plates' loading with a uniformly distributed load of different values.

6. REFERENCES

Dinu, G., (1998), Researches regarding the influence of the contact deformations-strains on the accuracy of the machine-tools, Doctoral dissertation, Politehnica University of Bucharest

Dinu, G., Baciu, Fl., Vlasceanu, D., Draghici, S., (2008), Study the strength and stiffness of a boring and milling machine bed, Proceedings of the 19th international DAAM Symposium, Trnava, Slovakia, p. 381-382;

Dinu, G., Atanasiu, C., Baciu, Fl., Nutu, E., (2008), Study of deformations in ram-boring spindle, Proceedings of the 19th international DAAM Symposium, Trnava, Slovakia, p. 379-380

Hadar, A. Motomancea, A.; Szabo, A.; Gheorghiu, H.; Marin, D.; Ocnarascu, C. (2008) Consideration about One-way Hydrocylon with Ring Shaped, The 19th International DAAAM Symposium, ISSN 1726-9679, Trnava, Slovakia, p.1035

Vlasov, V.Z.; Leontiev, N.N. (1966.) Beams, plates, and shells on elastic foundation, Jerusalem, Israel Program for Scientific Translations (translated from Russian)