University of Huddersfield Repository

University of Huddersfield Repository
University of Huddersfield Repository
Rubio Rodriguez, Luis and De la Sen Parte, Manuel
An expert mill cutter selection system
Original Citation
Rubio Rodriguez, Luis and De la Sen Parte, Manuel (2005) An expert mill cutter selection system.
In: 10th IEEE International Conference on Emerging Technologies and Factory Automation, 19-22
September 2005, Catania, Italy.
This version is available at http://eprints.hud.ac.uk/16021/
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An expert mill cutter selection system
L. Rubio, M. De la Sen and A. Ibeas
Instituto de Investigación y Desarrollo de Procesos
Facultad de Ciencia y Tecnología, Campus Leioa
Universidad del País Vasco Apdo.644, Bilbao,Spain
{webrurol, webdepam, iebibhea}@lg.ehu.es
Abstract
This paper discusses the selection of tools in milling
operations. To carry out this research, it has been
developed an expert system hinged on numerical
methods. The knowledge base is given by limitations in
process variables, which let us to define the allowable
cutting parameter space. The mentioned process
variables are, instabilities due to tool-work-piece
interaction, knowing as chatter vibration, and the power
available in the spindle motor. Then, a tool cost model is
contrived. It is used to choose the suitable cutting tool,
among a known set of candidate available cutters, and
to obtain the appropriate cutting parameters, which are
the expert system outputs. An example is presented to
illustrate the method.
1. Introduction
Machining, in particular milling operations, is a broad
term used to define the process of removing material
from a work-piece. Furthermore, the milling operation
process planning is required, nowadays, to increase its
productivity, reducing cost and improving the final
product [1].
This paper brings forward the concept of selecting an
appropriate mill cutter, among a known set of candidate
cutters, and obtaining the adequate cutting parameters
for milling operations through an expert system.
There are several versatile approaches for tool and/or
cutting parameter selection based on expert systems on
manufacturing environments. Wong and Hamouda [2]
developed an on-line fuzzy expert system. The system
inputs, the tool type, the work-piece material hardness
and the depth of cut, and control the cutting parameters
at the machine, as output. Cemal Cakir et al. [3]
explained an expert system based on experience rules for
die and mold operations. In that paper, the geometry and
material of the work-piece, tool material and condition
and operation type are considered as inputs. Then, the
system provides recommendations about tool type, tool
specifications, work-holding method, type of milling
operation, direction of feed and offset values. Vidal et al.
[4] focused on the problem of choosing the
manufacturing route in metal removal process. They
select the cutting parameters by optimising the cost of
the operation taking into account various factors, such
as, material, geometry, roughness, machine and tool.
Carpenter and Maropoulus [5] designed a system, which
provides reliable tool selection and cutting data for a
range of milling operations. The method employs rule
based decision logic and multiple regression techniques
for a wide range of materials.
Here, the developed expert system consists of the
relative compliance between the tool and the work-piece,
and it is predicted with analytical methods. Moreover,
time and frequency domain milling process simulations
have been developed, which are, then, used in the expert
system definition.
Then, the knowledge base is explained. Basically, it
defines the allowable cutting parameters, which are
known as cutting parameter space, for a given tool-workpiece configuration. It is based on the chatter vibrations
avoidance, which limits the productivity of the process,
and on a spindle power limitation criterion.
On the other hand, a novel tool cost function is
designed. It depends on spindle power consumption,
material removing rate (MRR) and on a stability
criterion against possible perturbation in the spindle
speed variable.
The MRR is a parameter which measures the process
effectiveness. It is required to be as large as possible.
But, if the MRR increases beyond certain limits, chatter
vibrations are appreciated and the process becomes
unstable [6]. Other variable which limits the process
effectiveness is the power available in the spindle motor
[7]. The third parameter taking part into the cost function
is considered to ensure a well-posed behaviour of the
system if a perturbation in the spindle speed happened.
In conclusion, the proposed cost function is a measure
of how the milling process is being carried out at certain
operation conditions. The larger the cost function,
correspond to the worst operation condition. Thus, the
cutter and cutting conditions which minimise the
designed cost function are selected.
Then, the expert system takes tool characteristics,
related tool-work-piece material parameters and milling
operation as inputs and outputs the selected tool among
the candidates and robust programmed cutting
parameters.
Figure 1: End part of the milling system: tool and work-piece.
2. System description
A model, which represents the dynamic compliance
between the tool and work-piece in milling processes,
has been developed. In this case, it is predicted with
analytical methods. The model assumes the cutter to
have two orthogonal degrees of freedom and the workpiece to be rigid.
2.1. Dynamic model
The dynamic model of the milling cutter is assumed
to be a system with one mode of vibration in each
direction, x and y , while the feed direction is along the
x - axis. The milling system under consideration is
shown in figure 1. The milling cutter has n t teeth, which
are equally spaced. The dynamics of the system is given
by the differential equations [8],
mx ⋅ x + cx ⋅ x + k x ⋅ x = ∑ f xj ( t ) = f x ( t )
⋅⋅
⋅
nt
my ⋅ y + c y ⋅ y + k y ⋅ y = ∑ f yj ( t ) = f y ( t )
⋅⋅
⋅
j =0
nt
j =0
(1)
(2)
where mi , ci and k i are the mass, damping and
stiffness of the tool, f xj and f yj are the components of
the cutting force that is applied by the j th tooth, which
are obtained by projecting f into the two orthogonal
axis.
2.2. Cutting force model
A simple model of the cutting forces will be discussed
here which express the tangential cutting force to be
proportional with the instantaneous chip thickness.
Despite this simplicity, this model captures the essence
of the process. Hence,
f t = kt ⋅ b ⋅ h
(3)
where kt is the specific cutting force parameter, b is the
axial depth of cut and h is the instantaneous chip
thickness. In addition, the radial force may also be
expressed in terms of the tangential force as,
f r = kr ⋅ ft
(4)
where kr is a proportional constant. This cutting force
model has been used by several authors [6].
The most critical variable in (3) is the chip thickness
because it changes not only with the geometry of cutting
tool and cutting parameters, but also with the uneven
surface left by the previous passes of the cutting tool.
This process is known as regenerative mechanism [6].
The chip thickness is measured in the radial direction,
with the coordinate transformation,
ν j = − x ⋅ sin φ j − y ⋅ cos φ j
(5)
where φ j is the instantaneous angular immersion of
tooth j measured clockwise from the normal Y axis
(fig.1).
The resulting instantaneous chip thickness consists of
static part st ⋅ sin φ j , attributed to rigid body motion of
the cutter, and a dynamic component caused by the
dynamic displacements or vibrations of the tool at the
present, ν j , and previous tooth periods, ν oj . Then, the
total chip load can be expressed by,
h (φ j ) = ⎡⎣ st ⋅ sin φ j + (ν oj −ν j ) ⎤⎦ ⋅ g (φ j )
(6)
h j ( t ) = st ⋅ sin φ j + ⎡⎣ x ( t ) ⋅ sin φ j ( t ) + y ( t ) ⋅ cos φ j ( t ) ⎤⎦
(7)
− ⎡⎣ x ( t − T ) ⋅ sin φ j ( t ) + y ( t − T ) ⋅ cos φ j ( t ) ⎤⎦
in the tool rotate angle domain, or
in the time domain, where g (φ j ) is a unit step function
which determines whether the tooth is in or out of cut, st
is the feed rate per tooth, T is the tooth period and, if
the spindle rotates at N s ( rad ⋅ s −1 ) , the immersion angle
varies as φ j ( t ) = N s ⋅ t , and φ j ( t ) = 0 if the j -tooth is
not engaged with the part[6].
2.3. Time domain simulation
Since the system is excited by cutting forces that can
not be expressed by simple analytic functions, the
equations can not be integrated in a closed form. Hence,
the 4 th order Runge-Kutta method is employed to solve
the differential equations (1) and (2)[8]. A simulation
system, which reads the input data of cutting conditions,
machine tool characteristics, and other related
parameters, and outputs the forces and vibration
displacements of chatter in milling has been developed.
2.4. Stability lobes
Projecting f tj and f rj determined by equations (3)
and (4) into x and y axis, taking into account that the
static component of the chip thickness is dropped from
the expression (6), and summing for all teeth engaged
and rearranging the above expressions (3) and (4) in
matrix form, will yield to [2]:
⎧⎪ f x ⎫⎪ 1
⎡ axx axy ⎤ ⎧ ∆x ⎫
(8)
⎨ ⎬ = bkt ⎢
⎥⎨ ⎬
⎪⎩ f y ⎪⎭ 2
⎣ a yx a yy ⎦ ⎩∆y ⎭
where a xx , a xy , a yx , a yy can be easily obtained, and
they are angular position dependent.
Considering that the angular position of the
parameters changes with time and angular velocity,
equation (8) can be expressed in time domain in a matrix
form as:
⎪⎧ f x ⎪⎫ 1
(9)
⎨ ⎬ = bkt ⎣⎡ A ( t ) ⎦⎤ {∆r ( t )}
⎩⎪ f y ⎭⎪ 2
where {∆r ( t )} = ( x ( t ) − x ( t − T ) , y ( t ) − y ( t − T ) ) .
The time directional dynamic milling force
coefficients collected in A(t ) are periodic function of
the tooth passing period, T . Furthermore, A(t ) can be
expanded into a Fourier series. For the most simplistic
approximation, the average component of the Fourier
series expansion can be considered. The dynamic milling
expression for milling force will be reduced to:
mode, solving the eigenvalue equation(12), calculating
the critical depth of cut from (13), calculating the spindle
speed from (14) for each stability lobes, and repeating
the procedure by scanning the chatter frequencies around
all dominant modes of the structure [6].
1
b ⋅ kt [ Ao ]{∆r ( t )}
(10)
2
α
α
⎡ xx
xy ⎤
where [ Ao ] = ⎢
⎥ is the time-invariant but
α
α
yx
yy ⎦
⎣
immersion-dependent directional cutting coefficient
matrix [6].
Thus, being ⎣⎡φ ( iω ) ⎦⎤ the transfer function matrix at
the cutter contact zone, denoted by
f (t ) =
⎛ φxx ( iω ) φxy ( iω ) ⎞
⎡⎣φ ( iω ) ⎤⎦ = ⎜⎜
⎟⎟ . Furthermore,
⎝ φ yx ( iω ) φ yy ( iω ) ⎠
describing the vibrations at the chatter frequency, ωc , in
the frequency domain using harmonic functions,
{r ( iωc )} = ⎡⎣φ ( iω )⎤⎦ { f } eiωct , {ro ( iωc )} = e−iωcT {r ( iωc )} ,
{∆r ( iωc )} = {r ( iωc )} − {ro ( iωc )}
then, the equation (10) can be written as:
1
= bkt ⎡⎣1 − e− iωcT ⎤⎦ [ Ao ] ⎣⎡φ ( iωc ) ⎦⎤ { f } eiωc t (11)
Obtaining 2 the characteristic equation and its
eigenvalue, Λ :
{ f } eiω t
c
Λ=−
(
N
bkt 1 − e −iωc t
4π
)
(12)
For the case that the cross transfer functions of the
systems are neglected, the characteristic equation will be
reduced to a quadratic equation, and the eigenvalues Λ
can be obtained [6].
The critical axial depth of cut is calculated by
substituting the obtained eigenvalue into equation (12):
blim = −
2πΛ R
(1 + κ 2 )
Nkt
(13)
where κ = Λ I Λ R is the division between the
imaginary and real parts of the eigenvalue Λ .
Corresponding to the spindle speed N s = 60 N ⋅ T
(14) and the chatter frequency can be found [6] as:
ωcT = ε + 2kπ , where ε = π − 2ϕ , and ϕ = tan −1 κ .
T is the spindle period, and k is the integer number
of full vibration waves (i.e lobes), imprinted on the cut
arc. The lobes are calculated, selecting a chatter
frequency from transfer functions around a dominant
Figure 2: The milling system representation, stability chars,
force time response and force frequency response
Figure 2 shows, the lobes char, and the analytical time
and frequency domain response for a tool 2 system,
which characteristics can be seen in section 5. The
chatter stability lobes make up a spindle speed
(frequency) dependent dividing line between stable
(down part line) and unstable (up part line) depth of cut
for a certain width of cut. Stable state corresponded
figures present a delimit time response, and the tooth
passing frequency and its harmonics, frequency
response. Unstable state corresponded figures present a
not delimit time response, and the chatter frequency is
appreciated.
3. Expert system
The main objective of the expert system is to obtain a
mill cutter, among the available ones, which have an
operating point or adequate cutting parameters, with
acceptable productivity (MRR), robustness stability
against spindle speed perturbations and less power
consumption than the spindle motor availability.
For this purpose, it is got the allowable cutting space
parameter, spindle speed, feed rate and axial depth of cut
for a constant radial depth of cut, taking into account the
regenerative chatter instability and the power available
in the spindle motor. Then, a novel cost function is
schemed. It is inversely proportional to MRR and a
parameter determinate as stability against spindle speed
perturbation, and proportional to power consumption.
Each term of the cost function have a proportionally
•
factor to have terms of the same magnitude. Also, there
is a weight factor which measures the importance of
each term. The weight factors are intended to be
programmed by the machine operator.
3.1. Milling process determination and preliminary
rules
In order to evaluate the system performance, it is
needed to select a suitable tool and performance indices.
Milling processes, basically, consists of two phases
roughing and finishing the surface. The main difference
between these operations is to decide the most
appropriate performance index for a given tool. The
quality and geometric profile of the cutting surface is of
paramount importance in milling finishing operation,
whereas roughing -milling consists on removing a large
amount of material from a blank.
This paper deals with roughing milling operation. The
rate at which the material is removed is called material
removing rate (MRR). This parameter measures the
productivity of machining processes. In milling
operations, MRR is defined as the multiplication
between axial and radial depth of cut, and feed per tooth.
MRR upper limit, is given by, chatter vibrations and
power deliver by the spindle motor. At certain
combinations of cutting parameters, such as spindle
speed, axial depth of cut and feed per tooth, either
chatter vibrations are appreciate, or the power available
by the spindle motor is insufficient. Then, these
parameters bound the roughing milling productivity.
For those reasons, at a first approximation, the input
cutting parameter space is given by the cutting
parameters, which are below the line at the stability
lobes char, and the power consumption is less than the
power available by the spindle motor.
But, due to the approximations in constructing
stability chars, the lobes are constructed, not by
replacing pure imaginary roots into the characteristic
equation, but adding a positive real number to them.
Furthermore, to have a robust system, it has been taken
into account a confine in a programmed maximum depth
of cut.
Then, the following algorithmic methodologies are
used, which are called preliminary rules:
• Rule1: Stability margin setting to ensure that the
system plays in a stable region, despite the system
model uncertainties.
• Rule 1.1: For calculating secure stability lobes
char, a small stability margin is selected, i.e, it
is supposed that the chatter vibrations happen at
δ + i ⋅ ω c instead of at i ⋅ ω c . The reason is that
the stability border line is calculated from a
linear approximation. Then, i ⋅ ω c is replaced
by δ + i ⋅ ω c , δ > 0 , when the stability border
line is calculated. This rule is applied to the
equation (13).
•
Rule 1.2: For improving the robustness of the
system, it has been taken into account a margin
at the final expression for chatter free axial
depth
of
cut,
equation
(18),
i.e,
b lim = α ⋅ b lim ,0 < α < 1 . This rule lets a better
control capacity in the spindle speed. On the
other hand, a better MRR selection is lost.
Rule 2: For searching the allowable input space
parameter, the set of spindle speed, Ns, axial depth
of cut, b and feed rate, st .
• Rule 2.1: Calculate the boundary points,
spindle speed and axial depth of cut pairs,
which compose the line between stable and
unstable zones, satisfying Rule 1. This rule is
obtained by plotting the stability lobes char,
which gives the line between stable and
unstable zones
• Rule 2.2: Calculate the admissible input space,
Q := ( N s , b, s t ) . The boundaries spindle speed
and axial depth of cut, gives the maximum
spindle speed and axial depth of cut pairs
without chatter vibrations (rule 2.1). The time
domain simulations output the system
dynamical force shape. As it will be seen in the
next section, the spindle power is force
dependent, which is spindle speed, axial depth
of cut and feed rate dependent. Then, for a
given spindle motor power available, the
admissible input cutting parameter space is
obtained.
3.2. Tool selection
In this section, an approach for tool selection is
suggested. For this purpose, a tool cost model function is
designed. The designed tool cost model is used to select
the appropriate tool between the candidates though the
optimisation Rules, explained below.
Then, the study requires a given set of candidates
milling cutters. Each one is characterised by the
following properties:
Ri = (ωnxi , ωnyi , ξ xi , ξ yi , k xi , k yi , nti , Di , β i )
where,
(ωxi , ω yi ) ∈ W is the tool natural frequency,
(ξ xi , ξ yi ) ∈ ξ is the tool damping ratio, ( kxi , k yi ) ∈ K is
the tool static stiffness, nti is the tool number of teeth,
Di is the tool diameter and βi is the tool helix angle.
Ri ∈ T , i = 1, 2,.., N , where N is the number of tools and
T is the set of tools available to the designer. W is the
set of tools’ natural frequencies, conformed by the pairs
(ωx , ω y ) for each tool, ξ is the set of tools’ damping
ratio, conformed by the pairs (ξ x , ξ y ) for each tool and
tools’ static stiffness is conformed by ( k x , k y ) for each
tool.
3.2.1. Tool cost model definition
To carry out the selection of a suitable tool, a novel
tool cost function has been conceived. The tool cost
model for a single milling process can be calculated
using the equation (20).
C ( Pt , MRR, ∆N s ; R, c1 , c2 ) = c1 ⋅ NF1 ⋅ Pt
(20)
NF
NF
+ c2 ⋅ 2
+c ⋅ 3
MRR 3
∆N s
nt
3
with ∑ ci = 1 , R ∈ T , where Pt = V ⋅ ∑ f tj (φ j ) ,
j =1 given below,
MRR i==1 a ⋅ b ⋅ st , ∆N s takes its definition
and q ≡ ( N s , b, st ) ∈ Q . Standardizing factors, NFi , are
defined as follow, NF1 = PtAv −1 , where PtAv is the power
available in the spindle motor, NF2 = MRRmax , where
MRRmax is the maximum MRR with the chatter vibration
and spindle power restrictions calculated among all the
candidate cutters and NF3 = ∆N max , where ∆N s max is the
maximum measured value of this variable among the
candidate cutters.
The tool cost function is designed to be MRR, power
consumption, and a range against possible perturbations
in tool rotational motion, dependent and inversely
proportional to MRR and a range against possible
perturbations and directly to power consumption.
These parameters have the following definitions:
Material or Metal Removing Rate ( MRR )
MRR = a ⋅ b ⋅ st , where, a is the radial depth of cut, b is
the axial depth of cut and s t is the linear feed rate. The
MRR is a parameter, which compares, the efficiency of
the milling process. A larger MRR improves the process
productivity.
Cutting power draw from the spindle motor ( Pt )
The cutting power, Pt , drawn from the spindle motor
is found from,
nt
Pt = V ⋅ ∑ f tj (φ j )
(21)
where V = πj =⋅1 D ⋅ N s is the cutting speed and N s is the
spindle speed. The tangential cutting force is given by:
f tj (φ j ) = K t ⋅ b ⋅ h (φ j )
(22)
where b is the axial depth of cut, K t is the cutting force
coefficient, which are material dependent and is
evaluated from experiments, and h (φi ) is the chip
thickness variation, which is feed rate st (mm/rev-tooth)
dependent.
Spindle speed security change ( ∆N s )
An additional term, spindle speed security change, is
added to the cost function model to be sure that chatter
vibrations are avoided. The spindle speed security
change, ∆N s , measure the nearest spindle speed at
which chatter vibrations happen to the supposed spindle
speed it will be operated. This fact allows to have an
error margin due to possible perturbations in this
variable.
To calculate analytically, ∆N s , the following
algorithmic methodologies are carried out. They are
divided in two cases:
Case I: k = 0 , this case corresponds to pairs, spindle
speed, axial depth of cut, situated below the first lobe of
the stability chars. Then, there is no lobe in the right part
of the point as it can be shown in figure 3. Suppose that
( N sI , bI ) is the point which ∆N s has to be calculated:
a)
If bmin, cri > bI
∆N s = abs ( N s ,min,cri − N sI ) .
b)
If bmin, cri < bI
∆N s = abs ( N sI ,cri ( bI ) − N sI ) .
bmin,cri is the minimum value of the axial depth of cut
corresponding to the border line, N s ,min,cri is its
Figure 3: Spindle speed security change, ( ∆N s ) , case I .
corresponding spindle speed, N s ,cri ( bI ) is the leftprojection of the point ( N sI , bI ) into the nearest lobe.
Case II: k ≠ 0 , in this case, the point ,which ∆N s has
to be calculated, is situated between two lobes in the
stable region. Suppose that ( N sII , bII ) is the mentioned
∃k
point,
then
such
that N s ,min, cri ( k ) < N sII < N s ,min,cri ( k + 1) , where k is the
lobe number, k = 0,1..S − 1 , and S is the number of
printed lobes , and N s ,min, cri ( k ) is the spindle speed
corresponding to the axial depth of cut minimum value
on the border line, bmin,cri ( k ) , for the k-lobe. Then:
a)
If bmin,cri ( k ) > b < bmin,cri ( k + 1)
⎛ abs ( N s ,min,cri ( k ) − N s ) , ⎞
∆N s = min ⎜⎜
⎟⎟
⎝ abs ( N s ,min,cri ( k + 1) − N s ) ⎠
b)
If bmin,cri ( k ) < b > bmin,cri ( k + 1)
⎛ abs ( N s ,cri ( k ) − N s ) , ⎞
∆N s = min ⎜⎜
⎟⎟
⎝ abs ( N s ,cri ( k + 1) − N s ) ⎠
where N s ,cri ( k ) is the left-projection of the point
( N sII , bII ) into the k-lobe, and N s ,cri ( k + 1) is the rightprojection into the k+1-lobe.
The case under
consideration is graphically represented in figure 4.
Note that, standardization factors, NFi , are also added
to the cost function to have terms with the same
magnitude. Moreover, they make to have a relative term
between all the candidates cutters involved. On the other
hand, these terms ensure that the cost function will be
comparable among the different cutters.
The ci , i = 1,..,3 , values are the weights of the cost
function terms. They measure the importance of the cost
function terms. The below optimisation Rule 3 give a
pattern to program the ci .
The selection criterion is, mathematically, expressed
as:
Figure 4: Spindle speed security change, ( ∆N s ) , case II
3.2.2. Optimisation Rules
The above defined tool cost function is used to select
the appropriate tool and cutting parameters, through the
following optimisation rules.
Rule 3 : Weight factors selection
The weight factors are intended to be programmed by
the machine operator. An extended explanation of their
meaning and their adequate selection is given in this
section To select suitable values of ci , i = 1,..,3 , their
meaning has to be perceived. The c1 , measures the
importance of the spindle power consumption. The
larger c1 parameter is the more important to the spindle
power consumption in the cost model function. The
c2 measures the machine productivity, if the c2 is near to
one high productivity is required, and if it is near to zero
the productivity has no importance. The same reasoning
is applied to the c3 , which measures the stability against
possible perturbations in the spindle speed variable.
It has to take into account that the expert system,
ensures that the spindle power consumption is always
going to be smaller than the power available in the
spindle motor, through Rule 1. Also, that the cutting
parameter space has no problems due to chatter
vibrations through Rule 2.
Then, a possible criterion leading to a process with
acceptable productivity, which is the main objective of
the milling processes, c2 about 0.75, and the other two
constants will add 0.25 , suitable values are c1 = 0.1 and
c2 = 0.15 .
Rule 4 : Tool selection criterion
A simple tool selection criterion for cutter selection
has been developed. For a given values of c1,c2,c3, and
a given tool characteristics, the cost function value is
obtained for all the admissible input cutting parameter
space. The minimum value of the cost function is saved.
The procedure is repeated for all the available cutters.
Comparing the minimum value of the cost function for
all available or candidate cutters, the corresponding
cutter to the minimum value of the minimum value of
the cost function is the selected tool.
(
)
• Compute,
C Ptj ( q j ) , MRR j ( q j ) , ∆N sj ( q j ) ; Ri , c1 , c2 ; (23)
for each Ri ∈ Ti , i ∈ N , and N is the set of
candidate tools and ∀q j ≡ ( N sj , b j , stj ) ∈ Q where
j ∈ N p = {1,.., N p } is a discrete sub-space of the
cutting parameters space where the cost function
(20) is calculated.
• For obtaining the selected tool, ST, compute
(
)
⎪⎧C Pt ( q j ) , MRR ( q j ) , ∆N s ( q j ) ; Ri , c1 , c2 , ⎪⎫
ST = arg min ⎨
⎬
i∈N
⎩⎪q j ∈ Q
⎭⎪
with ST ∈ T , obtaining the appropriate tool according
to the criterion.
Following the rules, the expert system provides an
appropriate cutter among the candidates.
Rule 5 : Cutting parameter selection
• Rule 5.1 : General case
To select the cutting parameters, there are two
possibilities. First of all, directly, calculate the cutting
parameters, which correspond to the selected tool, which
gives the minimum value of the cost function. It can be
expressed mathematically as,
• Compute the following equation (24)
q * ≡ ( N s* , b * , st* ) =
{ (
arg min C Pt ( q j ) , MRR ( q j ) , ∆ N s ( q j ) , ST , c1 , c2
q j ∈Q
)}
obtaining an input cutting parameter for the selected
tool.
The cutting parameter space is obtained by checking
all possible values of spindle speed and axial depth of
cut which are below the stability line in the stability char
according to Rule 1. These values join to the allowable
feed rates, which do not make consume more spindle
power than the available, Rule 2, form the cutting
parameter space. For the selected tool, the trio of cutting
parameters which minimize the cost function are, then,
selected.
• Rule 5.2 : Refinement case
In order to have a more accurate possibility, it has
been taken into consideration that the cutting parameters
can be searched with a more fine integration step around
the point where the cost function gives its minimum
value. Now, the cutting parameter space is given by a 3tuple Q* = ( N s( k )* , b( k )* , st( k )* ) around q* , for k = 1,.., p , where
p is the number of points to be considered, according to
Rules 1 and 2. The procedure for obtaining the required
cutting parameter is the same as used in Rule 5.1 trough
equation (24) for the above defined new cutting
parameters.
Mathematically expressed :
• Compute
q** = arg min
C Pt q*( k ) , MRR q*( k ) , ∆N s q*( k ) , ST , c1 , c2
*( k )
q
∈Q*
{( ( )
(
)
Obtaining the refined cutting parameter.
(
)
)}
Rule 6 : Process malfunctions : tuning c1 , c2 , c3 values
Nevertheless, in programming the selected tool and
cutting parameters, malfunctions of the process may lead
to a poor behaviour of the process. The most important
are tool wear and burr formation. These phenomena,
which are common in the manufacturing processes,
make that the analytical and experimental testes are not
always in concordance. If it is happened, the follow
algorithmic methodology could be applied:
While
Achatter > Atoothpast
c2 ← 0.99 ⋅ c2
c3 ← 0.01 ⋅ c1 + c3
end
where Achatter is the chatter frequency vibration amplitude,
and Atoothpast is the highest amplitude among the tooth
passing frequency and its harmonics. So, a more stable
state is obtained.
Finally, figure 5 shows a scheme of the expert system.
The developed expert system takes the α and δ
constants, the tools´ modal parameters such as its natural
frequency, damping ratio, tool static stiffness, number of
teeth, the radius of the tool, the helix angle, and the
cutting constants for the work material and cutter (tools´
characteristics), the spindle power available and the cost
function weight factors, as inputs and outputs the
appropriate tool among the candidates and robust
programmed cutting parameters.
4. Example
For the validation of this method, the above study has
been applied for two practical straight cutters and a fullimmersion up-milling operation. The example considers
the tools to have the following characteristics, according
with
the
section
3.2
notation,
R1 = ( 603, 666,3.9,3.5,5.59,5.715,3,30, 0 ) ,
and
R2 = ( 900.03,911.65,1.39,1.38, 0.879, 0.971, 2,12.7, 0 ) .
The natural frequency is measured in hertz, the tool
damping is in %, the tool stiffness is in KN ⋅ mm −1 and
the diameter of the tool is in mm . The work-piece is a
Figure 5 : Schematic expert system representation.
rigid aluminium block whose specific cutting energy is
chosen to be k^ t1,2 = 600kN ⋅ mm −2 and the proportionally
factor is taken to be kr1 = 0.3 , for the tool one, and
kr 2 = 0.07 for the other one. Other expert system
parameters are, the stability margin factor, δ = 0.05 and
the stability margin factor for the axial depth of cut,
α = 0.95 .
The analytical test for mill cutter selection was
conducted using spindle speeds with increments of
1000rpm , axial cutting depth started with its minimum
value in the stability border line divided by ten, and it is
increased in steps of this same size, for a given spindle
speed. The operation constrain on the maximum feed per
tooth is 0.55mm and the step integration is taken to be
0.05 . The spindle power availability is 745.3W .
Figure 6: Minimum-C function vs.
c1 varies, with c 3 = 0.075 .
The resultant tool is that leading to the minimum tool
cost function value. In figure 6, it is shown the values of
tool cost function as c1 parameter varies, the c3 has been
taken as a constant c3 = 0.075 and the c2 follow the rule
c2 = 1 − c1 − c2 . This study has been done to illustrate the
influence of the ci parameters in the tool cost function. It
is observed the tool R1 has a better behaviour respect to
the tool R2 for all possible value of c1 and c2 , with
c3 = 0.075 . Analysis with other values of c1 , c2 and c3 ,
have been carried out and the results are similar, and the
tool R1 has a better behaviour. Then, a more general
analysis shows in figure 7, in which the minimum value
of the tool cost function for all possible combinations of
c1 , c2 , c3 , with the restriction c1 + c2 + c3 = 1 is displayed.
The analysis has revealed that the first tool has a better
behaviour than the second one for all combinations of
the ci parameters. Thus the output of the expert system is
the first tool.
of work-pieces to be manufactured for each tool in order
to minimise the changes of tools.
5. Conclusion
Figure 7: Minimum-C function versus c1, c2 , c3 varies.
For the cutting parameters selection, two steps have
been done. First, the cutting parameter corresponding to
the minimum of the tool cost function for the selected
tool for values of c1 = 0.2, c2 = 0.725, c3 = 0.075 is
obtained. These values are q* = (5800, 0.4924, 0.2722) .
It can be a well-done first approximation. For a more
accurate solution, the tool cost function is evaluate
around the above mentioned cutting parameters. Then,
another integration is taken into account around q* . The
test for cutting parameters selection is conducted using
spindle speeds between 5700 and 5900 rpm, and a step
of 20, axial depth of cut between 0.48 and 0.52 and a
step of 0.01, and the feed per tooth between 0.25 and
0.35 and a step of 0.025. The resulted programmed
cutting parameters are q ** = (5780,0.494,0.28) .
An efficient approach for mill cutter selection has
been developed through an expert system. The expert
system is instructed with the characteristics of the
candidates tools, as well as with the stability margin and
constrains of operations, such as, power availability and
robust. Furthermore, a tool cost model function, built
from the expert systems preliminary rules, is proposed to
evaluate the possible performance of each candidate tool
in milling process. This performance index is then used
to select an appropriate tool and cutting parameters for
the operation which lead to the maximum productivity,
while respecting stability and power consumptions
margins though optimisation rules. A simulation
example which shows the behaviour of the system is
presented.
Acknowledgements
The Authors are very grateful to MCYT by its partial support through
grant 2003-00164 and to the UPV/EHU through Project 9/UPV
00I06.I06-15263/2003.
References
[1]
[2]
[3]
Figure 8: Situation of the point q** in the stability diagram and
tool displacement and power consumption time domain
responses for the selected tool.
Figure 8 shows the situation in the stability lobes of
the programmed point q ** , the tool displacement and
the power consumption. It is observed that the point is
robustly stable and the power consumption is less than
the power availability in the spindle motor, while
acceptable MRR.
This method can be applied to any number of selected
tools generating in a automatic task the best one to be
used in the system. Moreover, the method can be used to
schedule the relative compliance between the available
tools and the used work-pieces materials. Finally, the
expert system can be used to optimise the manufacturing
process, in the sense of planning the adequate sequence
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