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)