# Geometrical analysis of the tool's rake angle at cutting.

1. INTRODUCTION

Nowadays, the functional rake angle [[gamma].sub.f] is defined as the angle between the plane normal to the instantaneous direction of the relative speed between tool and workpiece, known as pressure plane, and the plane tangent to the rake face of the tool. Based on this definition, the general equation of the functional rake angle, related to the geometrical parameters of the process, is (1), where: [v.sub.x], [v.sub.y], [v.sub.z] are the components of the relative speed tool/ workpiece, [[gamma].sub.N]--the orthogonal rake angle, [lambda]--the edge inclination angle and K--cutting edge angle. These angles, also named constructive angles, are considered on a certain point of the cutting edge (Belous & Cozminca, 1971).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The relation (1) has a great disadvantage, namely it represents a particularly value for [[gamma].sub.f], the one measured along the direction of maximum gradient. The chip should be removed after this direction only if it's perfectly fluid, and that's unreal. Using the equation (1) to estimate [[gamma].sub.f] for a possible practical situation, e.g. [[gamma].sub.N] = 0, [v.sub.x] = 0, [v.sub.y] = 0 we obtain cos [[gamma].sub.f] = cos [lambda], so [[gamma].sub.f] = [lambda]. It means that the chip, removing along the direction of maximum gradient, must flow along the cutting edge, and that's not possible (Trent & Wright, 2000).

As experimental tests have shown, the chip is removed along a direction at angle [eta], in the rake face plane, with the normal [N.sub.1] at the edge (Fig. 1). According to (Jawahir &von Lutterwelt, 1993) the angle [eta] has the same value as the inclination angle [lambda].

However, (Rapier & Wilkinson, 1959) showed that angle [eta] may reach larger values than [lambda], and (Russel & Brown, 1956) were concluded that the view [theta] of the chip's deviation angle n on a plane Q, normal to the cutting plane and parallel to the edge is practically equal to the inclination angle [lambda], [theta] = [lambda].

[FIGURE 1 OMITTED]

As Figure 1 shows, tan [eta] = tan [theta] cos [[gamma].sub.N1] so relation (2) result, where: [[gamma].sub.N1] is the rake angle measured in a plane orthogonal to the cutting edge.

[eta] = arctan(tan [theta] cos [[gamma].sub.N1]) = arctan(tan [lambda] cos [[gamma].sub.N1]) (2)

Therefore, relation (2) shows that always [eta] < [lambda]. The fact that angle [eta] is different than [lambda] was relieved also by (Zorev & Granovski, 1967) which have been concluded that most of the times [eta] exceeds [lambda], with maximum 5 ... 7 degrees and it's increasing with [lambda] and chip' width, at smaller values for tool rake angle [gamma].

On the other hand, the deviation angle [eta] decreases the cutting speed v is increasing, as the experimental relation (3) shows:

[eta] = [lambda] x [v.sup.-008] (3)

Equation (3) demonstrates that angle [eta] is always smaller than the inclination angle [lambda] , which is in accordance with the results of (Russel & Brown, 1956).

Based on the results presented above, we may conclude that for practical reasons we are allowed to approximate the deviation angle [eta] by the inclination angle [lambda], in case of free cutting. But the chip deviation phenomenon becomes more complicate in case of complex cutting. In this situation come in at chips formation processes not only the major cutting edge, but also the minor or secondary edges (Gunay et.al., 2004).

Therefore the chip removal direction D, inclined with the angle [eta] in the rake face plane regarding to the normal [N.sub.1] on the edge, cannot be determined only from geometrical reasons (as line with maximum gradient). We must consider also the aspects regarding the physics of chips formation.

That's why the real rake angle [[gamma].sub.r] has to be measured in the chip's removal plane, which contains the direction v of the relative speed tool/ workpiece, and the direction D of chip's removal. This angle is defined as the one between a plane normal to the direction of speed v (pressure plane) and the tangent plane to the tool's rake face, in the considered M point on the edge.

According to this definition, the real cutting angle [[delta].sub.r] = [pi]/2--[[gamma].sub.r] can be measured between the chip removal direction D and the speed direction v. See equation (4), where: [eta] [congruent to] [lambda], [[gamma].sub.N1]--the constructive cutting angle, measured into a plane orthogonal to the edge direction (Zorev & Granovski, 1967).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

But this relation its lack in generality, due to the fact that the cutting speed v was considered included into the edge plane and that's valid only in the particularly situations, when [v.sub.x] = 0, [v.sub.y] = 0 and = v.

Therefore we try to determine a general expression for the real functional rake angle related to the geometrical parameters, when all the possible components of the cutting speed [v.sub.x], [v.sub.y], [v.sub.z] occur.

2. THE GENERAL EXPRESSION OF THE REAL FUNCTIONAL RAKE ANGLE [[gamma].sub.r]

To determine a general relation indicating the dependency between yr and the geometrical parameters [lambda], [[gamma].sub.N1], [[gamma].sub.N] and [eta] we use the schematization illustrated in Fig.2.

It illustrates views in the constructive base plane, in the edge plane, in the rake face plane, where appears the chip's removal direction D, the direction orthogonal to the edge [N.sub.1] and the vector T tangent to the edge and in the chip's removal plane (Belous & Cozminca, 1971).

If we take apart the vector D after the tangent to the edge and the one orthogonal to the edge, the vectors T and [N.sub.1] result with sin[eta], respectively cos[eta] modules. Projecting these vectors to the x, y, z tool's axis we find the rectangular projections of the vector D.

The vector [N.sub.1] has the expression (5) and the vector T the expression (6), so for D the relation (7) results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

T = sin[lambda]cos[lambda]cosK x i + sin[eta]cos[lambda]sinK x j--sin[eta]sin[lambda] x k (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The functional real rake angle [[gamma].sub.r], measured in the removal plane and defined by v and D directions, is the angle between the pressure plane P, orthogonal to v, and the plane tangent to the rake face in

M point (see Fig. 2). It is the complementary angle of the real functional cutting angle, between v and D, so the equation (8) occurs.

Equation (8) represents the general expression of the functional real rake angle [[gamma].sub.r] and allow estimate it for any cutting process, according to the relative speed components [v.sub.x], [v.sub.y], [v.sub.z] and the geometrical parameters of the tool.

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

But for most of the tools types the main constructive rake angle is [[gamma].sub.N], measured in the plane orthogonal to the apparent edge, therefore the simplified equation (9) result, based on the connection relations, i.e. tan [[gamma].sub.N] = tan [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and neglecting the speed components [v.sub.x] and [v.sub.y]. For [eta] [congruent to] [lambda], the relation (10) results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

3. CONCLUSIONS

An analytical expression for estimating the real rake angle [[gamma].sub.r] was established; it can be used when the values for the speed components [v.sub.x], [v.sub.y], [v.sub.z], the edge constructive angles [[gamma].sub.N1], [lambda], K and the deviation angle [eta] are known.

If the speed components [v.sub.x] and [v.sub.y] may be neglected related to v, the simplified equations (9) and (10) are recommended. The influence of the normal constructive angle [[gamma].sub.N] on angle [[gamma].sub.r] is more pronounced as the absolute value of the inclination angle [lambda] is lower; the [lambda] sign also influences the [[gamma].sub.r] values. The influence of [lambda] on [[gamma].sub.r] is higher at lower values for [[gamma].sub.N].

Applying the above relations for the real functional rake angle allow us to consider the real cutting plane in order to evaluate the cutting forces values needed in tools design or in cutting process simulation.

4. REFERENCES

Belous, V. & Cozminca, M. (1971). Contributions to the real tool's rake angle study in cutting processes, Bulletin of the Polytechnic Institute of Iasi, XVII (XXI), Fasc. 3-4, pp. 145-154, ISSN 1011-2855.

Gunay, M.; Aslan, E.; Korkut, H. & Eker U. (2004). Investigation of the effect of rake angle on main cutting force, Int. Journal of Machine Tools and Manufacture, vol. 44, no. 9, pp. 953-959, ISSN 0890-6955.

Jawahir, J. S. & von Lutterwelt, C. A. (1993). Recent Developments in Chip Control Research and Application, CIRP Annals, vol. 42, no. 2, pp. 659-693, ISSN 1726-0604.

Rapier, A. C. & Wilkinson, P. T. (1959). Plasticity Report No. 156, Mechanical Engineers Research Laboratory, East Kilbride, Glasgow.

Russel, J. K. & Brown, R. H. (1956). The Measurement of Chip Flow Direction, Int. J. of Machine Tool and Manufacture, VI, Pergamon Press, London, pp. 129-138, ISSN 0890-6955.

Trent, E. M. & Wright, P. K. (2000). Metal Cutting, Fourth Edition, Publisher Butterworth--Heinemann, ISBN 0-7506-7069-X.

Zorev, N. N. & Granovski, G.I. (1967). Development of Metal Cutting Science. Publisher Mashinostroenia, Moskow.