Analytical approach to determining the optimal shape of the casting sprue for small-size metallic parts.

Deac, Cristian ; Tera, Melania ; Bibu, Marius 等

1. INTRODUCTION

The casting of small-sized metallic parts, such as those used in medicine for the realising of prostheses or implants, differs considerably from the conventional, industrial casting of large metallic parts, in terms of technology, equipment and auxiliary materials.

The procedure which is the most used for the casting of purpose is the so-called "lost wax" technique. It requires realising a wax model of the part which is to be cast (and of the corresponding gating system, the so-called sprue) and which is then embedded in a refractory mass, known as investment material. The resulted assembly is then heated to the melting temperature of the wax and later the cavity resulted after the evacuation of the molten wax, a metallic alloy is filled, which after cooling will form the desired part (Deac, 1995; Deac, 2003).

In order for the cast material to fill all the space in the casting mould, as fast as possible, procedures are used during which the transfer of the molten material is aided by a supplementary pressure, so that the filling occurs before the metal solidifies and so that the air is completely evacuated from the casting cavity. The best method which is currently used for this purpose is the centrifugal casting.

For the success of a centrifugal casting, the parameters of the employed casting machine are significant, but the dimensions and shape of the gate runners composing the sprue, are also very important.

Several authors (Al-Mesmar et al., 1999; Brauner, 1994; Chan, 1998) have indicated that a curved shape of the gate runners is the most desirable, but so far there were very few attempts to define mathematically and to optimise the shape of the casting sprue.

This paper presents a novel optimisation approach using a mathematical model based on the mechanics of the movement of the metal through the sprue.

2. THE MATHEMATICAL MODEL

In order to determine the optimal shape of the gate runners by mathematical modelling, it is necessary to analyse the kinematic elements (speeds and accelerations) and the dynamic elements (forces) which appear when a very small particle of molten material (considered as a material point) is sent from the melting crucible towards the casting cavity, for each type of movement which the particle is going through on its trajectory.

The functioning of a centrifugal casting machine presents two distinct phases. The first one, from the starting of the machine until the arm reaches the nominal rotation speed, is characterised by a constant angular acceleration ([epsilon] = constant) and a rapidly increasing angular speed. In the second phase, which lasts until the metal is completely solidified, the angular speed remains constant ([omega] = constant) and the movement is an uniformous rotation movement. The kinematic and dynamic elements have different expressions function of the movement type, so the two cases need to be addressed separately

The model presented here was defined based on a mobile cartesian coordinate system, in which the Ox axis is along the direction of the centrifugal arm and the Oz axis is superposed over the rotation axis of the centrifugal arm as presented in figure 1. Within this cordinate system, the particle has the coordinates (x(t), y(t), 0), the movement being free in the centrifugal arm's plane (horizontal plane). In the vertical plane, the force of gravity is much smaller than the centrifugal force and therefore has a negligeable effect).

In this paper only the case of uniformly accelerated rotation movement will be discussed, for which the speeds and accelerations have following expressions (fig. 2):

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

--the relative speed and acceleration:

[[bar.V].sub.r] = [v.sub.rx] x [bar.i] - [v.sub.ry] x [bar.j] = [??] x [bar.i] - [??] * [bar.j]; (1)

[[bar.a].sub.r] = [?? ]x [bar.i] - [??] x- [bar.j]; (2)

--the transport acceleration:

[[bar.a].sub.t] = [[bar.a].sup.[tau].sub.t] + [[bar.a].sup.[upsilon].sub.t] = ([epsilon] x [square root of [x.sup.2] + [y.sup.2]]) x [bar.[tau]] + ([[omega].sup.2] x [square root of [x.sup.2] + [y.sup.2]]) x [bar.[upsilon]] (3)

--the Coriolis acceleration:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The dynamic elements influencing the movement are (fig. 2):

--the transport force:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

--the Coriolis force:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

According to the second principle of mechanics and to the principle of effects superposition, the vectorial equation of the particle's relative movement is:

m x [[bar.a].sub.r] = [bar.G] + [[bar.F].sub.t] + [[bar.F].sub.c] + [[bar.N].sub.y] + [[bar.N].sub.z] (7)

If this equation is projected on the axes of the mobile coordinate system Oxyz, following equation system results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Replacing equations (5) and (6) in this equation system, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Or, after simplifying and replacing as adequate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This equation system can be solved with the help of numerical methods. For making the solving easier, the authors considered the case in which the terms containing the angular acceleration [epsilon] are negligeable and the angular speed varies linearly:

[omega] = [epsilon] x t (11)

The approximated solution which results for this system is:

x(t) = R + [a.sub.4][t.sup.4] - [a.sub.8][t.sup.8] (12)

y(t) = [b.sub.6][t.sup.6] - [b.sub.10] [t.sup.10] (13)

where the coefficients [a.sub.4], [a.sub.8], [b.sub.6], [b.sub.10] are defined by the expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

When plotting the resulting curve by means of a specialised software package, we obtain a graph as shown in figure 3.

[FIGURE 3 OMITTED]

3. CONCLUSIONS

The calculations and results presented above indicate that it is possible to determine the optimal shape of the casting gates starting from the mechanical conditions around a particle of the metal which is to run through the gates.

In the horizontal plane, this shape must be as close as possible to the optimal one, resulted from the calculations and shown in figure 3. For the best possible filling conditions, the main gate must be curved in the opposite sense to the rotation movement of the centrifugal arm.

In the vertical plane, the best results could be obtained if the gates would be inclined along the resultant between the gravity force and the resultant of the horizontal forces. However, the gravity force is much smaller than the centrifugal force, so the vertical angle can be neglected for the design of the sprue system.

Given the conditions imposed on it and the simplifications taken into account, the model described here is valid only for the case of small-size cast parts. Also, only the factors considered as most relevant for the given conditions have been considered.

In future, the authors intend to expand the model in order to take into account also other factors and to study elements such as the optimal size of the casting cone.

4. REFERENCES

Al-Mesmar, H.S., Morgano, S.M. & Mark, L.E. (1999). Investigation of the effect of three sprue designs on the porosity and completeness of titanium cast removable partial denture frameworks. J. of Prosthetic Dentistry, vol. 82, no.1, p. 15-21, ISSN 0022-3913

Brauner, H. (1994). 1st Titanguss mit konventionellen Hf-Schleudern moglich? Deutsche Zahnarztliche Zeitschrift, vol. 49, no. 8, p. 642-646, ISSN 0012-1029

Chan, D.C.N. et al. (1998). The effect of sprue design on the marginal accuracy of titanium castings. Journal of Oral Rehabilitation, vol. 25, no. 5, p. 424-429, ISSN 0305182X

Deac, C. (2003). Contributii la dezvoltarea tehnologica a topirii si turnarii unor materiale metalice utilizate in tehnica dentara (Contributions to the technological development of the melting and casting of metallic materials used in dental technics), Ph.D. thesis, "Lucian Blaga" University of Sibiu

Deac, V. et al. (1995). Turnarea titanului in protetica dentara (Casting of titanium in dental prosthetics), Editura Universitatii din Sibiu, ISBN 973-95604-4-9, Sibiu, Romania