Theoretical model of the surface roughness at the end milling with circular tips.
Slatineanu, Laurentiu ; Toca, Alexei ; Mazuru, Sergiu 等
1. INTRODUCTION
The milling is the cutting process which uses rotating tools having
many cutting teeth placed on the revolution or plane surface; there is a
feed motion achieved usually in a plan perpendicular on the tool axis.
There are milling techniques which use cylindrical mills, end mills,
conical mils etc. If in the case of the rough milling the interest is
focused on the material removal rate, on the tool wear, on the sizes of
the cutting forces, at the finish milling the main aspects are the
surface roughness and the machining accuracy. To obtain a diminished
surface roughness, there are used cutting teeth with secondary approach
angle k1=0 or teeth with circular inserts; the main advantage of the
last solution is the possibility to re-establish the cutting properties
of the worn tool by the rotation of the circular inserts.
[FIGURE 1 OMITTED]
At the plane surfaces milling by the using of the end mills with
circular inserts, there are (fig. 1) two situations: a) when the tool
axis is perpendicular on the plane of the feed motion, the asperities
generated by each tool tooth are intersected (fig. 1, a); b) if the tool
axis is a little inclined (practically, it is difficult to ensure the
absolute perpendicularity of the tool axis on the machined surface), the
asperities generated by each tooth do not intersect (fig. 1, b). In the
last case, the machined surface roughness is usually appreciated only
along the direction corresponding to the tool axis motion. The problem
is to establish if the surface roughness parameter [R.sub.a] is constant
or variable along other directions existing on the machined surface.
As above mentioned, different researchers measured the surface
roughness especially along the direction defined by the tool axis
motion.
Thus, Lu et al. (Lu et al., 1998) established a multiple regression
equation corresponding to the CNC end milling; they considered that the
model predicts the size of the surface roughness parameter [R.sub.a]
with about 90% accuracy. In the case of the theoretical profile of the
machined surface consisting of elliptical or circular arcs,
Jun Qu and Albert J. Shih developed (Qu and Shih, 2003) some
dimensionless form expressions for three surface roughness parameters
([R.sub.t], [R.sub.a], and [R.sub.q]). Amin et al. (Amin et al., 2007)
studied the influence exerted by the preheating on the surface
roughness, in the case of the end milling of carbon steel by the using
of the circular carbide inserts. There is not clear information
concerning the variation of the surface roughness parameters in
different directions included in the plan of the machined surface.
2. THEORETICAL MODELLING
A theoretical model could emphasize the eventual variation of
surface roughness parameters along the surface machined by the using of
the end mills with circular inserts. To establish such a model, the
information included in the figure 2 can be taken into consideration.
One can notice that if the rake angle has the value [gamma]=0[degrees]
and the face of the circular insert is placed in the axial plan, the
profile of the machined surface includes a concatenation of arcs of
circle along the direction corresponding to the tool axis motion (fig.
3). For any other direction, the surface profile includes arcs of curves
different of the arcs of circles.
[FIGURE 2 OMITTED]
The machining scheme is presented in the figure 1; the end mill
achieves the rotating motion and the feed motion. The radius of the
circular inserts is symbolized by R; the cutting edges of the circular
inserts are placed at the radius a.
If the circle corresponding to the insert active edge is placed in
the plan yOz, its defining relations are the following:
[(x - a).sup.2] + [z.sup.2] - [R.sup.2] = 0
y = 0 (1)
At the distance b from the plan zOy (this plan is defined by the
mill axis and by the direction of the feed motion), the profile of the
surface is given (Slatineanu, 2004) by the relation:
z = [[square root of [R.sup.2] - [square root of [y.sup.2] +
[b.sup.2]] - a).sup.2]] (2)
On the other hand, the surface roughness parameter [R.sub.a] is
generally given by the relation:
[R.sub.a] = 1/n [n.summation over (l)] [absolute value of
[y.sub.i]], (3)
where [y.sub.i] is the distance from a point of the profile to the
mean line and n is the number of the points on the segment of the
abscissa semiaxis where the roughness parameter [R.sub.a] must be
calculated. Because the surface roughness parameter is defined by taking
into consideration the position of the so-called mean line, defined by
the parameter m (the distance from the mean line to the axis Ox), in the
presented case the surface roughness parameter [R.sub.a] corresponds to
the relation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
A numerical method was used to determine the position of the mean
line. It is known (Surface, 2003) that the mean line must have such a
position that the sum of the squares of the deviation [z.sub.i] of the
profile from the mean line be minim:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Adequate software (Crefu, 1992) was used to determine the size of
the parameter m. This software is based on the following considerations
(Slatineanu et al., 2004); one can consider the initial position of the
mean line just near the axis O[y.sub.1]. By taking into consideration an
increment [DELTA]y, the first sum S1 of the squares of the profile
deviations [(z-m).sup.2] (for a pre established number n of the points
of the profile) can be determined.
[FIGURE 3 OMITTED]
Within the next step, a new line placed at the distance from the
axis Oy increased with the increment [DELTA]m can be taken into
consideration and a new sum [S.sub.2] is calculated. The process is
retaken; initially, to series of the sum [S.sub.1], [S.sub.2], [S.sub.3]
etc. decreases; when the series begins to increase, the penultimate size
of m can be appreciated as offering information about the position of
the mean line. The limits of the interval used for the calculus (fig. 3)
are
[square root of [(a - f/2).sup.2] - [b.sup.2]], [[square root of (a
- f/2).sup.2] - [b.sup.2]].
Knowing the size of the parameter m, the size of the surface
roughness parameter [R.sub.a] can be determined, by the using of the
relation (4). This relation was used to calculate the size of the
parameter [R.sub.a], by taking into consideration two levels of the
radius a, the feed /, the distance b and the radius R of the circular
insert. The mathematical processing of the results thus obtained (by
means of software based essentially on the method of the smallest
squares) led to the relation:
[R.sub.a] = 20.51[R.sup.-1068] [a.sup.0.1214] [f.sup.1.977]
[b.sup.-0.0001579]. (6)
The analysis of this relation (based on a simplified geometrical
model) shown that the size of the surface roughness parameter [R.sub.a]
does not depend on the size of the distance b On the other hand, the
main influence on the size of the parameter [R.sub.a] is exerted by de
feed f, the absolute size of the exponent attached to this size having
the biggest value.
The established model offers an image concerning the influence
exerted by different factors on the surface roughness at the end
milling. In the future, one will try to use other methods to study the
surface roughness within a surface obtained by end milling with circular
inserts and practically verify the validity of such mathematical models.
3. CONCLUSION
A theoretical model able to emphasize the influence exerted by the
feed f, the radius R of the circular inserts, the radius where the
cutting edges are placed and by the position of the plane where the
surface roughness parameter [R.sub.a] is measured was proposed.
4. REFERENCES
Amin, K.M.N.; Abraham, I.; Khairusshima, N. & Mirghani I.
Ahmed, M.I. (2007). Influence of preheating on performance of circular
carbide inserts in end milling of carbon steel. Journal o/Materials
Processing Technology Vol. 185, No. 1-3, 2007-04-30, 97-105, ISSN 0924-0136
Cretu, G. (1992). Fundamentals of experimental research. Laboratory
handbook (in Romanian). Technical University "Gh. Asachi" of
Iasi, Romania
Lou, M.S.; Chen, J.C. & Li, C.M. (1998). Surface Roughness
Prediction Technique for CNC End-Milling. Journal o/ Industrial
Technology. Vol. 15, No. 1, November 1998-January 1999. Available /rom:
www.nait.org/jit/Articles/ lou1198.pdf. Accessed: 2008-06-23
Qu, J. & Shih, A.J. (2003) Analytical Surface Roughness
Parameters of a Theoretical Profile Consisting of Elliptical Arcs.
Machining science and technology, Vol. 7, No. 2, 281-294, ISSN 1091-0344
Slatineanu, L.; Dodun, O.; Coteata, M. & Santo, L. (2004).
Study of the surface roughness at the machining of the plane surfaces
with end mills. Progressivnye technologhii i sistemy machinostroenia.
Mejdunarodnyi sbornik nauchinyh trudov. ISBN 966-7907-16-3, Donetskii
Natsionalnyi Tekhnicheskii Universitet, Ukraine, 28, 265-270
*** (2003). Surfaces and Profiles (2003). Available from:
http://www.predev.com/smg/intro.htm. Accessed: 2005-03-02