Deviation of a machined surface in flank milling
A. Larue
a,∗
, B. Anselmetti
a,b
a
Laboratoire Universitaire de Recherche en Production Automatise
´
e, Ecole Normale Supe
´
rieure de Cachan, 61, av du pre
´
sident Wilson, 94235
Cachan Cedex, France
b
Institut Universitaire de Technologie de Cachan, 9, av de la division Leclerc 94234 Cachan Cedex, France
The flatness defects observed in flank milling with cutters of long series are mainly due
to the tool deflections during the machining
process. This article present the results of an identification procedure of the coefficients of a force model for a given tool workpiece
couple for the prediction of the defects of the tool during the cutting. The calibration method proposed meets a double aim: to define
an experimental protocol that takes the industrial constraints of time and cost into account and to work out a protocol which
minimizes uncertainties likely to alter the interpretation of the results (environmental, software or mechanical uncertainties). For that,
the procedure envisages the machining of a simple plane starting from a raw part formed by a tilted plane, allowing for the variation
of the tool engagement conditions. The tool deviation during the cutting process is indirectly identified by measuring the machined
surface. The observed straightness defect conditions can be explained by the evolution of the cutting pressures applied to the cutting
edges in catch during the cutter rotation. The precision was considerably improved by the taking into account of the cutter slope defect
in the calculation of the load applied to the tool. After identification of the tool-workpiece couple, the prediction model was applied to
some examples and allowed to determine the variations of form and position of the surface points with a margin of 5%.
Keywords: Cutter deformation; Force model; Flank milling; CAM integration
1. The milling
1.1. Tool paths generation in flank milling
‘Machining tool paths generation’ is to be carried out
by seeking the best possible geometrical quality on the
machined part while reducing times of manufacture. The
context of our study is that of the flank milling of free
forms parts with long tools. This very efficient process
from the point of view of productivity and of surface
quality is very much used in aeronautics and mould
manufacturing.
Flank milling of free surfaces traditionally implies the
generation of machining tool paths that will ensure an
optimal laying of the presumed rigid tool on the surface
∗
Corresponding author. Tel.: +33-1-47-40-27-65; fax: +33-1-47-
40-22-20.
E-mail address: larue@lurpa.ens-cachan.fr (A. Larue).
to machine [1–4]. In flank milling with long tools, the
geometrical errors generated by the tool deflection dur-
ing the cutting process can be very significant. Thus, to
obtain a surface of quality, it is necessary to take the
tool deformation under the cutting forces into account.
To foretell the tool deflections, this article presents a
method of identification for a tool workpiece couple in
flank milling. The experimental protocol is easily appli-
cable industrially. The data-processing exploitation of
the prediction model allows its integration in a
CAD/CAM environment.
Before detailing the identification procedure, let us
briefly come back on to the choice of force model.
1.2. Choice of force model
The numerous existing force models can be classified
into two modeling levels: ‘microscopic modelisation’
studies the interaction between the tool and the work-
piece material by using thermomechanical behavior laws
1
and ‘global modelisation’ considers the force resulting
from the contact between the tool and the part along the
cutting edges in catch.
The microscopic analysis of the chip formation makes
it possible to understand the physical phenomenon of
the material removal to apprehend the thermomechanical
phenomena generated during the cutting of the work-
piece material. The numerical simulation of machining
has to be very precise and requires a good knowledge
of the tool (sharpness of the cutting edge, forms of the
groove, ...). This type of approach is for the moment
essentially used in turning [5,6] and can lead to the
improvement of tool quality. The time simulation of
these approaches remains relatively significant, some-
times exceeding the hour of calculation. The integration
of a microscopic model in a CAD/CAM environment
of tool paths generation thus seems easier by using a
‘macroscopic’ force model.
The ‘macroscopic’ models describe the cutting
phenomenon by studying the evolution of the force
resulting from the chip formation along the cutting edges
in catch. These models are divided into two categories
according to whether they take the time in the machining
simulation into account or not. The approaches known
as ‘dynamic’ consider the instantaneous depth of cut
variation between the trajectory of the tooth which
machines and the preceding tooth. These models are
based on the regeneration mechanism of the machined
surface, considered as the main cause of the vibrations
appearance [7–10].
The objective of the ‘dynamic’ models is to envisage
the appearance of instabilities which cause chatter during
the machining process. The dynamic behavior of the tool
and/or part are thus described either by linearizing the
cutting pressures laws to study instabilities in the fre-
quential field [11–14], or by describing the geometry of
the machined surface step by step in the temporal field
[9,10,15].
The static approaches are based only on the concept
of cutting pressure applied to a theoretical chip section
and on the fixed beams theory to estimate the defor-
mations. In milling, the Kline and DeVor’s model [16]
has already been used by Seo [17] to treat flank milling
and the problems of machining tool paths compensation.
The context of the study proposed in this article is
that of flank milling at traditional cutting speed lower
than 200 m/min. The rotational frequency of the spindle
is low. We thus chose to adapt the Kline and DeVor’s
model for great axial engagements.
The article is organized as follows: section 2 describes
the model which is at the basis of the identification pro-
cedure. Section 3 details the experimental procedure and
the results obtained. Section 4 finally reconsiders speci-
ficities of the calculation of the angle of the tool real
engagement into the workpiece material, which is a sig-
nificant contribution of the method proposed, by show-
ing that the fact of taking the tool deflection into account
in this calculation improves the precision of the identifi-
cation.
2. Modelisation of the deformation
This article is only interested in flank milling of steels
at traditional speeds with solid plain milling cutters
whose cutting edges are helicoid. The defects of the sur-
faces machined with long series cutters are very signifi-
cant: localization defect of 0.8 mm, flatness defect of 0.2
mm. Taking the test conditions with a rigid part, a rigid
assembly and a rigid machine into account, we suppose
that the main cause of deformation is due to the tool
deflection.
2.1. Qualitative analysis of the cutting process
Modeling the cutting process requires the understand-
ing of the associated physical phenomenon. During a
flank milling process, the locations of the cutting edges
in catch simultaneously evolve in the workpiece
material. The machined surface is thus obtained by a
generating point P which moves along the cutting gener-
ator while the tool rotates.
For a given angle, the generating point is in P at a
distance z of the spindle nose considered as a housing.
After a small rotation of the plain milling cutter, the gen-
erating point is in P⬘ at a distance z⬘ (Fig. 1). This
explains the variation of the radial forces and of the
tool deformations.
An elementary cutting edge placed at point Q, which
is in the workpiece material, cuts an elementary chip
section by applying an elementary force dF. At each
point P, the resulting deformation is the sum of the
elementary deformations caused by the engagement of
all the elementary lengths constituting the cutting edges
simultaneously engaged in the workpiece material. Fig.
Fig. 1. Evolution of the generating point of a tool.
2
Fig. 2. Force along a cutting edge according to the cutter angular
position.
2 represents the evolution of radial force per unit of
length along a helicoid developed profile on the
machined surface. a is the total tool engagement angle
in the workpiece material. Each point Q is characterized
by the angular position b and by its distance z to the
spindle nose.
The variation of the radial forces and that of the dis-
tances from point P to the presumed housing along a
machined section makes it possible to qualitatively
understand that the resulting deformation of the section
is not linear.
Fig. 3. Tool deformation.
2.2. Definition of the deformation model
The calculation of the cutting pressures is approached
by making use again of the Kline and DeVor’s model
[16]. We hence propose a force model based on the
decomposition of the tool into elementary discs the
thickness dz of which is considered as constant. Each
elementary force applied in each point Q of the cutting
edge in catch considered is decomposed into a tangential
component dFt and a radial component dFr:
dFr ⫽ Kr.ep.dz ⫽ Kr
o
.(fz.sin(b)).dz.(fz.sin(b))
⫺0.3
dFt ⫽ Kt.ep.dz ⫽ Kt
o
.(fz.sin(b)).dz.(fz.sin(b))
⫺0.3
. (1)
Specific cutting pressures Kr and Kt are supposed to
answer the following model:
Kr ⫽ Kr
0
.ep
⫺0.3
and Kt ⫽ Kt
0
.ep
⫺0.3
(2)
where ep is the depth of cut given by ep ⫽ fz.sin(b).
This model was many times validated in the context
of turning, coefficient ⫺0.3 appearing as a rather
stable constant.
The geometrical variations of the machined surface
are mainly due to the plain milling cutter deformation
in the normal plane to the surface. Only the component
dN of this cutting force interests us.
dN(b) ⫽ dFr(b).cos(b) ⫹ dFt(b).sin(b). (3)
We consider that the tool is a fixed beam of constant
moment of inertia I
gz
, that is the case when the tool used
has a teeth number multiple of 4. The resulting defor-
mation dy at point P from coordinate z, due to the radial
force dN applied to point Q is given by:
dy ⫽
1
E.I
gz
.dN(b).
冋冉
(z⫺L(b))
3
3
冊
.
冉
(z⫺L(b))
2
.L(b)
2
冊册
(4)
where L(b) ⫽ (h × b)/(2p)ifQ belongs to the same
tooth as P, h being the helicoı¨dal pitch and E being the
Young’s modulus of the tool. The moment of inertia is
given by I
gz
⫽ p·d
4
eq
/64 with d
eq
⫽ m·2R, R being the
plain milling cutter radius and m an equivalence ratio
identified during a static test similar to the test presented
in Fig. 5. An error in the determination of I
gz
has no
influence on the identification precision as this error is
made up for at the time of the matrix system resolution
leading to the specific cutting coefficients values Kr
0
and Kt
0.
The total theoretical deformation yth
i
at point P
i
is the
sum of the elementary deformations dy generated by all
the points Q of the cutting edges engaged simultaneously
in the workpiece material.
yth
i
⫽
冘
N
t
k ⫽ 1
冕
(b ⫽ a)
(b ⫽ 0)
dy
ik
⫽
冘
N
t
k ⫽ 1
冉
1
E.I
gz
冕
(b ⫽ a)
(b ⫽ 0)
dmf
ik
冊
(5)
where the bending moment dmf
ik
is given by:
3
dmf
ik
⫽ dN(b).
冉冉
(z
i
⫺L(b)
ik
)
3
3
冊
(6)
⫹
冉
(z
i
⫺L(b)
ik
)
2
.L(b)
ik
2
冊冊
where i is the index of point P
i
along the studied line,
k is the number of the considered cutting edges and N
t
is the number of the teeth of the plain milling cutter.
The engagement angle a, useful for the integral calcu-
lation Eq. (5) depends on the cutting edge and on the
tool orientation. During this study, a basic model was
used by considering that the angle a is constant and cal-
culable for a radial engagement er given by a ⫽
acos((R⫺er)/R). We then noted rather significant dif-
ferences between measurements and the estimates of
variations of the surface machined. We observed that
under effect of the the tool deformations, the maximum
engagement angle a considerably decreases. It was thus
necessary to set up a process to take the deformation in
the engagement angles calculation into account. Section
4 will specifically detail this point.
Ultimately, the approach proposed in this article
extends the Kline and DeVor’s model [16] to the taking
into account of great axial engagements. The integral
relates to the deformation due to each elementary cutting
edge length whereas, in the original model, the integral
relates to the forces, the resultant forces being used to
calculate the deformation.
3. Identification procedure
3.1. Introduction
The determination of the force model coefficients is
a significant stage for the prediction of the machining
defects. The estimate of these parameters classically
requires the realization of a significant campaign of tests
during which the cutting pressures are generally meas-
ured with dynamometers [18–24]. These experimental
data are then treated by an optimization method and lead
to the determination of the coefficients. The results
obtained always depend on the conditions of realization
of the tests and on the hypotheses of calculation set up.
The most current identification strategies concern the
direct measurement of characteristic components of the
cutting process for a tool-workpiece couple (Kr
0
, Kt
0
characterized) for given conditions. The measurement of
the force is difficult in an industrial cycle, because it
requires the maintainance of a cutting force turntable, its
central processing unit and a very qualified technician.
The examination is also rather delicate because there are
generally several teeth in catch. After that, it is necessary
to associate a tool deflection model to the phenomenon
resulting from the presumed cutting forces.
To define a protocol of identification answering the
industrial constraints, we use an indirect measurement
by quantifying the tool deflection with the measurement
of the machined surface. The opposite problem is then
solved by comparing the theoretical deformation of the
tool with the deformation of the machined surface.
In this context, Landon and al. propose the machining
and the measurement of a master part gathering charac-
teristic machinings from which we can infer a law gov-
erning the machining defects [25]. The approach pro-
posed can be applied to all the types of machining and
is not based on a theoritical force model.
3.2. Principles of the procedure
To make the identification procedure easily reproduc-
ible industrially while reducing uncertainties related to
its implementation and its interpretation, a part should
be defined that allows to characterize a tool workpiece
couple on a given machine in stabilized mode.
3.2.1. Principles of the test selected
Taking the constraints previously evoked into account,
we define a test protocol based on the flank milling in
concordance of a simple plane which is realizable on
any milling machine. To vary the depth of cut of 0.5 up
to 3 mm, the raw part has a specific form mainly consti-
tuted of a tilted plane (Fig. 4) and of surfaces dedicated
to the realization of references to facilitate measure-
ments.
The raw part is divided into four zones:
앫 In the identification zone, the raw surface is a tilted
plane giving a depth of cut variation which is included
between 0.5 and 3 mm
앫 The reference zones which require two narrow bands
Fig. 4. Specifications of the raw part.
4
with a small depth of cut of 0.3 mm in order to limit
the cutting forces and the tool deflections
앫 A lateral slot that lets the tool end out of the work-
piece material.
In order to isolate the influence of the tool deflection
on the part defect, the tool end is left free to avoid the
disturbances caused by a parasitic friction of the tool
end.
3.2.2. Experimental validation of the assumptions
To obtain an experimental model representing the cut-
ting process as precisely as possible, the following
hypotheses are posed:
1. The part is very rigid and firmly taken in on the
part holder.
2. The part holder is very rigid.
3. The tool is considered as a fixed beam.
4. The spindle of the machine unit is considered as a
rigid body.
To quantify the spindle deformation, a force of the
same order of magnitude (maximum 150 N) as the cur-
rent cutting force is applied on a very rigid test holder,
using a dynamometric ring. The induced deflection is
obtained using two comparators for one point of load
(Fig. 5). The deviation of the spindle axis on the level
of the tool end is about 6 µm, which is negligible com-
pared to the observed variations on the part. In static,
the spindle can thus be regarded as a rigid bogy. The
point A, located near the spindle nose, is retained as the
presumed housing point.
To analyze the rigidity of the machine, a comparator
is fixed on the spindle casing during the preceding
Fig. 5. Validation of the static rigidity of the machine.
identification test to measure the displacements of the
machine table compared to the spindle. The maximum
deformation noted is lower than 15 µm and will be neg-
lected too.
We finally consider that the dynamics of the machine
spindles does not disturb machining, because the
machined surface is a simple plane which is milled at
constant feedrate with a long stabilization and acceler-
ation distance before machining the central part
(section 3).
3.2.3. Principles of measurement
From an operational point of view, the identification
consists in machining the test part and measuring the
defects of the machined surface. These measurements
make it possible to establish a surface statement of the
deformations. This statement gives the actual value of
the deformation at each point of a measurement grid
posed on the machined surface. The step of the grid is
of 1 mm.
To measure the defects of the test part correctly, a
reference mark was conceived on the basis of reference
surfaces (Fig. 6). This reference mark is built by
bypassing the part with the tested tool by taking a weak
depth of cut of 0.3 mm on an axial engagement of 5
mm. It thus allows for an effective registration between
the reference mark part and the reference mark of the
CMM machine. This type of machining allows the con-
struction of reference surfaces while being freed from
the gauge and tool deflection defects. The two zones
with low depth of cut located at the level of the part
ends are used to validate the registration of the measured
points cloud.
Within the framework of this article, the part test was
machined under the following conditions:
앫 HSS tool whose diameter is 20 mm and whose active
length is 88 mm
Fig. 6. Reference mark useful for measurement.
5