MATLAB modeling of material surface roughness parameters at end milling process.

Dodun, Oana ; Ghenghea, Laurentiu ; Sarbu, Ionel 等

1. INTRODUCTION

It is known that the machined surfaces generated in ball-end or flat-end milling process have a cross-section consisting of elliptical or circular arcs, depending on the direction of the measurement line. This aspect of the machined material surface is due the perpendicularity deviation of the main shaft relative to the machined surface. In this case, the tool cutting edges, after a semi-rotation, did not reach the surface. If the deviation at perpendicularity is very small, the machined surface has curved tool paths on two directions.

In this paper we proposed a MATLAB function to establish, for a theoretical ideal profile, the following common surface roughness parameters: the arithmetic average roughness [R.sub.a], peak-to-valley roughness [R.sub.y], the average height of the asperities [R.sub.z] and root-mean-square roughness [R.sub.q]. The calculus establishes the coordinates of several points of the theoretic profile, determines the height of the least squares mean line and obtains the numeric values. To estimate the surface roughness parameters some approximation methods are commonly used. Since the 1960s, it is known the approximate solution to express the peak-to-valley roughness. The following mathematical relation was established assuming that [R.sub.y] is small and neglecting the higher order term [R.sub.y.sup.2] (Bohosievici, 1993, Boothroyd & Knight, 2006):

[R.sub.y] [congruent to] [f.sup.2]/8r (for f < 2r, r [not equal to] 0) (1)

where r is the tool radius and f is the feed. The feed f is the distance between two adjacent peaks of the surface profile. The approximation of the arithmetic average roughness [R.sub.a] can be expressed as (Bohosievici, 1993, Boothroyd & Knight, 2006):

[R.sub.a] [congruent to] 0.032 [f.sup.2]/r (2)

An empiric relation, legitimate for a cutting speed between 18 ... 44 m/min and feed of 14 ... 28 mm/min, to estimate the arithmetic average roughness [R.sub.a] at end milling process of a high-speed steel for bearings SH15 with a high-speed steel end mill was proposed in (Bohosievici, 1993):

[R.sub.a] = 4.83 [f.sup.1.69] x [t.sup.015]/[v.sup.123] x [r.sup.0.14] x [[gamma].sup.0.46] [micro]m, (3)

where t is the cutting depth and [gamma] is the front rake angle.

Other empiric relation is defined in (Slatineanu, 1980):

[R.sub.z] = [square root of [Cr.sup.u]/f x [t.sup.x] x [x.sup.z] x [x.sup.z.sub.l] [micro]m (4)

where C, u, x, z are constants that are experimentally established.

Other researchers (Qu & Shih, 2003) proposed a comparison of close-form section profile solution for calculating the three roughness parameters and the approximation methods using a parabolic curve to match the elliptical or circular arcs.

2. MODELING PROPOSAL AND DISCUSSION

Figure 1 illustrates the grooved surface machined by an end mill with a tool radius r that rotates all around a central axis. Due to the perpendicularity deviation of the main shaft relative to the machined surface, after a semi-rotation, the tool cutting edges did not reach the surface. We supposed that the cutting tool motion is plotting a torus with the following parameters: R -the radius of circle materialized by peak tool and r -the radius of the circle that rotates all around an axis to obtain the torus. The plane of the surface measurement profile is parallel and at a distance d relative to the plan X-O-Z. The equation of the torus generated by the tool corner radius can be expressed as a biquadratic equation in the unknown z (Coxeter, 1989):

[z.sup.4] + A x [z.sup.2] + B = 0 (5)

where the coefficients can be expressed as:

A = 2([x.sup.2] + [y.sup.2] + [R.sup.2 - [r.sup.2]]) (6)

B = [([x.sup.2] + [y.sup.2] + [R.sup.2 - [r.sup.2]).sup.2] - 4[R.sup.2]([x.sup.2] + [y.sup.2]) (7)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The first step to establish the surface roughness parameters is to resolve the equation (5). The coordinates of the points from theoretical ideal profile, x and y, shown in figure 2, can be represented by:

[x.sub.P1] [less than or equal to] x [less than or equal to] [x.sub.P2] (8)

where [x.sub.P1] = [square root of [(R - f / 2).sup.2] - [d.sup.2]] (9)

where [x.sub.P2] = [square root of [(R + f / 2).sup.2] - [d.sup.2]] (10)

and y = d (11)

Afterwards, we determined zeros of the polynomial represented by equation (5), we made a translation of the coordinate system, for making the calculus easier. Figure 3 shows the theoretical surface profile and the new coordinate system. After coordinate translation, the first point of the theoretical profile has null abscissa and the point with minimum z-coordinate has null elevation. In a particular case, when y = 0, the cross-section view of theoretical surface profile consists in circular arcs.

The proposed MATLAB function follows these steps to obtain two vectors named: xR -vector containing the abscises and zR -vector containing the height of the points from theoretical profile. With the calculated points, we can draw the theoretical measurement profile. In program we can control n--the number of points of the profile by modifying the increment q between the abscissa of point P1 and point P2. The implicit value of q is 5 [micro]m. We may obtain an increase of the calculus accuracy by growth the number of points that must diminish the value of q. The xR and zR vectors have the length equal with n.

According to STAS 5730/2-85, the reference mean line in surface roughness is the least squares mean line, a straight line parallel to the X-axis. Based on the definition of the least squares mean line, we calculate a parameter M defined as:

[n.summation over (1)] [z.sup.2.sub.i] = min (12)

Some results obtain with the MATLAB functions are presented in the figure 3 and below:

-for r = 1200 [micro]m, R = 60000 [micro]m, f = 400 [micro]m /rev, d = 0 [micro]m, we obtain n = 401 points and Ra = 4.2885 [micro]m; Rz = 18.3371 [micro]m; Ry = 16.7840 [micro]m; Rq = 85.8762;

-for r = 1200 [micro]m, R = 60000 [micro]m, f = 400 [micro]m /rev, d = 50000 [micro]m, we obtain n = 724 points and Ra = 4.2713 [micro]m;

Rz = 18.2662 [micro]m; Ry = 16.7840 [micro]m; Rq = 114.9302 [micro]m.

The proposed MATLAB function verifies the result by comparison the value of the first point form the theoretical profile with a parameter l that corresponds the geometrically establish equation:

[(f/2).sup.2] = l x (2R - l) (13)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

3. CONCLUSION

The proposed MATLAB function offers a solution to obtain theoretically four surface roughness parameters, [R.sub.a], [R.sub.z], [R.sub.y]., [R.sub.q]. We supposed that the trace for surface finish measurement is perpendicular relative to the velocity vector of the end mill. The mathematical model is based purely on geometric considerations. The real surface generated in machining processes will deviate from this theoretical profile owing to the tool wear, vibration of the tool during machining, elastic deformation and recovery of the tool and workpiece, build-up edge at tool tip, resolution of the measurement machine etc.

From the particular calculus presented above and using many times the proposed function one may observe an interesting fact that the roughness parameters are not influenced by the distance between the feed vector and the surface measurement profile d. This theoretical model can be used also as prediction of the surface roughness. By comparison with measured surface roughness value it is possible to investigate the effects and interdependences influences of the workpiece material, tool behavior, cutting parameters and other factors on the machined surface.

4. REFERENCES

Bohosievici, C. (1993). Modern technologies for mechanical manufacturing (in Romanian), Polytechnic Institute, Iasi

Boothroyd, G. & Knight, W. (2006). Fundamentals of Machining and Machine Tools, CRC Press, ISBN: 1574446592, Boca Raton

Coxeter, H. (1989). Introduction to Geometry, Wiley, ISBN: 978-0-471-50458-0, New York

Cretu, G. (1992). Fundamentals of experimental research. Laboratory handbook (in Romanian). Technical University "Gh. Asachi" of Iasi, Romania

Ou, J. & Shih, A.J. (2003). Analytical Surface Roughness Parameters of a Theoretical Profile Consisting of Elliptical Arcs, In: Machining Science and Technology, Vol. 7, Issue 2, January, 281-294, ISSN 1532-2483

Slatineanu, L. (1980) Contributions to the study of the some Romanian steels mahinability (in Romanian). PhD thesis, Technical University of Iasi, Romania, (in Romanian)