Jigs based on hypocycloid curves for universal turning machine.

Rosca, Adrian Sorin ; Dumitru, Nicolae ; Craciunoiu, Nicolae 等

1. INTRODUCTION

There are situations when is more economic to manufacture a surface on a machine tool which was not designed to generate that kind of surface. A typical case can appear at a small workshop, equipped with universal turning machine tools, which must to produce parts from shaft class, with side plane surfaces, in large volumes. In figure 1 we can see such a part, (command shaft from a hydraulic valve), with the plane surfaces at the right end, disposed in section as a square shape.

The literature doesn't offer too much information for this kind of problems, but (Rosculet, 1983) presents the solution from figure 2, as a separate jig that can be attached at a turning machine. The work piece is fixed and centered in universal fixing device and longitudinal slider. The gear train [Z.sub.1]-[Z.sub.2] receives the spinning movement from main shaft and passes next to [Z.sub.3]-[Z.sub.4] gear train, which also ensures an opposite spinning for rotating head. This one is mounted on tool slider, which can execute in the same time the feed and rotation movement, due to universal joint and involute spline joints. With this cinematic solution the tip of the tool describes a hypocycloid curve.

If we intend to realize this solution for a SNB 400 universal turning machine, we can emphasis the following problems:

* dimensions at gear train z3-z4 should be around 800 mm;

* is mandatory to close all the gears inside of a large case;

* at the given work piece, the angles which appears at the universal joint can reach 70[degrees], more over the acceptable limit;

* the gear train [Z.sub.1]-[Z.sub.2] must to be open, which is a serious treat for safety;

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

2. MATHEMATICAL BACKGROUND

2.1 Hypocycloid curves

The aspects related to the hypocycloid curves are wide presented in literature such as Mathematics at a glance (1980) and can be summarized as:

* the curve is generated by a fixed point P, related to a circle, which is rolling slidingless inside of another fixed circle, as in figure 3.a;

* his parametric equations are in presented in relation (1):

where: r--radius of mobile circle, R--radius of fixed circle, [phi]--angle described by P point, a--distance from the centre of circle to the P point.

x = (R--r) x cos [phi] + a x cos R--r/r [phi]

y = (R--r) x sin [phi]--a x sin R--r/r [phi] (1)

As we can see in fig. 3.b, to obtain a hypocycloid curve with a square aspect, is necessary to ensure the following two conditions:

* respect a ratio R : r = 4 : 1;

* use an adequate value for the term a--smaller than mobile circle radius--r (see eq. 1).

The main disadvantage for this solution is the switching position of the tool, relative to the work piece: in some areas, the tool (presented as a short straight line in fig. 3.b), is situated inside the work piece and in another areas is situated outside of the work piece.

2.2 Hypocycloids curves with straight segments

In this situation we will use at our solution a larger value for term a (a > r), that can generate a 4 leaves hypocycloid as in figure 4.a.

For a real application we propose a rotating head with two tools disposed on a diameter, as in figure 5. In this case we will use a ratio R : r = 2 : 1, each one of the tool will generate a separate curve, oriented at 90[degrees], as in figure 4.b. This solution gives a better balance of the rotating parts, and also reduces to half the rotation speed of the same subassembly. Each one of the tools will cut 2 opposite sides of the square, approximated with the elongated hypocycloids, (Popescu, 1998).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

2.3 Theoretical error for twin tool system

To determine the maximum error [[epsilon].sub.max], between the hypocycloid and the desired square, we are doing some consideration related at figure 6, where the curves are deformated to emphasis the error.

As we can see, the biggest error is at the corner, at the intersection of the two curves. In this real point both coordinates x and y are equal. To derive the maximum error we must to calculate the angle 9 corresponding at the hypocycloid point, where the two coordinates are equal. Replacing in relation (1) and taking in consideration that the relation between the radius of the fix and mobile circle must to be R=2r, we are obtaining the relation (2):

r cos [phi] + a cos [phi] = r sin [phi]--a sin [phi] (2)

Separating the terms with the unknown value (angle [phi]), from the constructive parameters, and applying some elementary transformation we are deriving the relation (3):

[phi] = 1/2 arcsin ([r.sup.2]--[a.sup.2])/([r.sup.2] + [a.sup.2]) (3)

In the above relation the constructive parameters are: r = 155 mm (adopted to have a decent diameter for the other circle) and a = 148 mm (position for the tool tip corresponding at the tool Kennametal MDJNR-2525K mounted in the twin system).

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

In this situation the maximum error is generated for the angle [phi]=1.3234295[degrees], where the coordinates of the real point are: X=6.99813, Y=6.99813 (instead of 7). If the side of the desired square is 14 mm (with an error of 0.0026407 mm on x and y axis), along the hypotenuse of the triangle from figure 6, we get the maximum error with the value [epsilon]max = 0.003734 mm. If we are looking in the settlement SREN 20286-1/1997 for class IT12 r, for free dimensions (Margine 2001, 2007, 2009), we are getting the maximum tolerance field with the value 0.18 mm, which is over the theoretic error calculated.

3. FINAL SOLUTION OF THE DEVICE

Considering the estimations before mentioned, we intend to realise a modified version as follows:

* the spinning movement will be taken from the back of the main shaft, as results from (Cernaianu, 2002);

* the gear trains will be replaced with chain transmissions: one on the back which doubles the speed, and one on the front which ensures a proper angle for the universal joints used at the rotating head (Dumitru, 2008);

In figure 7 we can see the entire device, above the existing turning machine SNB 400. The two chain transmissions are disposed in separate casings, and are coupled with a shaft.

4. CONCLUSIONS

The theoretical errors produced by the hypocycloid devices can be maintained well under the required tolerances with proper constructive values for circle diameters and distance to the generating point. The proposed solution for this device can be realistically applied on the SNB 400 turning machine, enhancing the basic capabilities of this universal machine. The calculated precision at this work piece is bellow the usual tolerances. The casing used at the chain transmissions ensures a good protection and a precise positioning for all the parts.

5. REFERENCES

Cernaianu, A. (2002). Machine tools, elements of structural and cinematic design, Ed. Universitaria, Craiova

Dumitru, N. (2008). Machine parts--shafts and bearings, Ed. Tehnica, ISBN 978-973-31-2332-3, Bucuresti

Margine, Al. (2001, 2007, 2009). Machine parts, Vol.1, 2, 3, Ed. Universitaria, ISBN 973-8043-39-4, Craiova

Bachmann, K.H., & others (1980). Mathematics at a Glance--Romanian version, Ed. Tehnica Bucuresti

Popescu, I. (1998). Algorithms and new methods in mechanisms design, Ed. Universitaria, Craiova

Rosculet, S., V. (1983). Designing of jigs and fixtures, Ed. Tehnica, Bucuresti