Download Jaanson, A. A New Approach to Mathematical Modelling of Metal

5th International DAAAM Baltic Conference
“INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOUR
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20–22 April 2006, Tallinn, Estonia
A NEW APPROACH TO MATHEMATICAL MODELLING OF
METAL CUTTING PROCESS CHARACTERISTICS
Jaanson, A.
least with a minimum number of passes. In
resolving such a task, coefficients of
stiffness of machine tool assemblies in
cutting process and mathematical models
of cutting process characteristics serve as
initial data. To get a solution of practical
importance the exactness of prediction
values of process characteristics must be
improved essentially. A way to tackle this
problem is dealt with in this paper. Up to
now no difference exists in achieving
working accuracy either on a conventional
machine tool or a modern precious CNC
machine tool. Possibilities to raise productivity, to lower the cost of production and
economize the energy consumed have not
been made use of.
Abstract: A review of mathematical
methods used for composing empirical
formulae and mathematical models of the
metal cutting process is brought. The
principle structure of process characteristics mathematical models in use and the
main differences of a perspective model of
a qualitatively higher level are given.
Keywords:
metal
cutting,
mathematical modelling.
turning,
1. INTRODUCTION
The development of metal machining may
be divided into two periods: up to the
introduction of modern computer controlled CNC machine tools, and afterwards.
It can be observed based on the papers
published till the end of 2005 that shifts
depending on stiffness coefficients of
machine tool assemblies have not been
taken into account by the cutting data
optimizing techniques. As machining
accuracy is to be guaranteed, multiple
passes (cuts) have been applied. There has
been no change in the dimensional
adjustment technique since modern CNC
machine tools were introduced. The cutting
tool is guided on a trajectory determined
by nominal dimensions of the work, just
like it has been done with conventional
machine tools. In principle, it is possible to
guide the cutting tool on a trajectory taking
into account its dimensional wear and
elastic shifts in the technological system.
As a result, a work with no axial profile
errors is achieved with a first pass or at
2. MATHEMATICAL METHODS
USED TO DATE
Metal cutting is probably the most widely
used process of manufacturing. Research
into this field of science began in the
middle of the 19th century. In principle,
the factors affecting cutting process, the
mechanism of chip forming and the phenomena accompanying the process are
known. The evergreen problem is how to
raise machining efficiency, i.e. how to
guarantee the required quality with higher
productivity and a lower cost of production. Mathematical description of process
characteristics is the first step to resolve
the problem. F. Taylor [1] was the first
investigator to compose an empirical tool
life (T) formula.
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number of well-known investigators in this
field of science, e.g [2, 3, 4, 5], lies in the
period from the 1940’s up to the 1960’s.
The 2k type design of experiments for
composing a tool life mathematical model
was first used by an American scientist
S. M. Wu in the year of 1964 [6]. (2k –
number of experiments for determining
estimations of coefficients of variables and
their interactions). When using 2k designs,
it is assumed that k factors are coded to the
standardised levels ±1. Wu’s model:
The principle form of Taylor’s polynomial:
(1)
T = Ca xp f y v n ,
where
C – coefficient of proportionality;
a p – depth of cut;
f
– feed rate;
v
– cutting speed;
x,y,n – exponents.
Taylor used his formula for optimizing
cutting data in turning and achieved a
multiple rise of productivity. The formula
(1) has been used by a number of investigators as a pattern in composing empirical
formulae for calculating metal cutting
process characteristics (cutting force components, surface roughness indicators,
cutting temperature). The values of the
characteristics do not depend on cutting
data only but also on the geometry of the
cutting tool, properties of a workpiece and
insert materials, and the environment. The
influence of factors that are not variables
of empirical formulae is taken into account
by correction coefficients:
C char = Ca xp f y v n K1 ...K n , (2)
ŷ = bo + b1x1 + b2x2 + b3x3 + b4x1x2 +
b5x1x3 + b6x2x3,
(3)
where
ŷ
– output of the model;
x1…x3 – coded inputs of the model;
bo
– value of output if x1=x2=x3=0;
b1…b6 – estimations of the model
coefficients.
Independent variables were the same as in
formula (1). The estimations of model (3)
coefficients were calculated by the method
of least squares. Response surface methodology is often used for visual illustration
of the dependence of output on input. Wu’s
model allowed to raise the exactness of
tool life prediction but did not resolve the
problem exhaustively – by including
essential factors into the model the number
of experiments grows unacceptably.
When using CNC machine tools chip
breaking must be guaranteed. From the
beginning of the use of CNC machine tools
investigations related to this problem have
been topical.
Modern science and technology constantly
require new materials of specific properties. As a rule, their machinability (easiness
with which a work material is machined
under a given set of cutting conditions) is
worse. According to Sandvik [7] the three
main parameters of machinability assessment are: (i) tool life; (ii) surface finish;
and (iii) cutting force. A number of new
investigation methods (FEM – finite
element method, neuron network method,
Taguchi method) are in use for resolving
where
Cchar – arbitrary characteristic of the
cutting process;
K1…Kn – correction coefficients.
To compose formulae like (1) and (2) the
one-variable-at-a-time approach has been
used.
As interactions of many factors are
essential, the correction coefficients are
valid only as to the factors, by which they
were determined. It is the reason, why the
type (2) formulae are of a very low (in
reality indeterminable) exactness. In workshops cutting data, geometry of the active
part of cutting tools and other quantities
needed are chosen as experience-based
estimates. (Through the age of metal cutting the skill of operators has been looked
at as an art in the sense of perfection). The
“golden age” of classic papers of a great
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number of earlier investigations beginning
from the 1920’s chip root micrographs do
not reflect the investigated process but a
changed image of chip formation. In
reality, chip formation process can be
investigated with the help of chip root
micrograhps gained by a quick stopping of
cutting process. A device enabling an
undisfigured chip root has been described
in [24]. The criteria that make it possible to
decide whether a micrograph reflects the
process under investigation or a transition
process are presented in [25]. An example
of a correct investigation of chip formation
is offered in [26].
Actually cutting process takes place within
a technological system components of
which are a machine tool, cutting tool,
workpiece and in many cases also a jig.
Astakhov’s theory of chip forming is not
based on the micrographs but on arbitrary
hand-drawn schemes and, therefore, leads
to deeply erroneous results. So does
Astakhov’s system concept in metal cutting [22, 23] as the cutting process cannot
take place without a machine tool. The
model inputs values are affected by the
parameters of the machine tool assemblies.
Reference systems are needed for mathematically describing the cutting tool and
cutting process. ISO 3002-1:1982 consists
of two reference systems: the tool-in-hand
system for describing cutting tool and toolin-use system for modelling the cutting
process. The standard was composed in a
period when cutting blades were bronzed
to the body of a cutting tool and the design
geometry of the tool was guaranteed by
grinding. The nose radius of such a tool
was almost equal to zero. To-day inserts
(cutting blades) are mechanically clamped
to the cutting tool body and the nose radius
varies in the limits of 0.4...10 mm. That is
why the ISO 3002-1:1982 is not fully
applicable.
The structure of a mathematical model of
cutting process characteristics in use is
depicted in Fig. 1.
problems of metal cutting. The opinion of
the users of the methods is that a lot of
problems can be resolved faster and
cheaper than when making costly and time
consuming experiments. Still the results
brought in many papers make us arrive at a
conclusion that the authors are not well
informed as to the achievements of metal
cutting theory. The results published in
papers [8, 9, 10, 11] prove that with the neuron
network and Taguchi methods trustworthy
results are not attained which makes the
methods unsuitable in solving metal cutting problems. In principle, FEM is suitable to resolve some problems of metal
machining [12]. Though, in some cases it
has also been used by researchers whose
main profession is obviously not metal
cutting. In such investigations, e.g. [13, 14, 15,
16
], incorrect calculation designs or formulae
can be found. As a rule, results of such investigations are useless for science and practice. Incorrect calculation designs can also
be found in investigations not related to
afore-mentioned calculation methods [17, 18,
19, 20
], making the results published useless.
Some investigators [21], presumably not
mechanical engineers, are dealing with the
problem of ensuring constant cutting force.
Such a set of problem is meaningless from
the position of resolving metal cutting
problems.
Astakhov et al. introduce in [22, 23] a “new
system concept” in metal cutting. It can be
read in [23], page 191, that cutting takes
place within a cutting system that is
defined as consisting of three components:
the workpiece, the cutting tool and the
chip. It is maintained in [22, 23]: “The
uniqueness of the cutting process, among
other closely related manufacturing
processes, is identified by chip formation,
which is considered to be caused by the
bending stress in the deformation zone”. A
similar mistake was made by E. Reuleaux
at the beginning of the last century.
Knowledge how the chip forms is essential
in modelling the metal cutting process. In a
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noise
Inputs of basic model:
v, a p , f , (M )
M
Outputs of model (fixed also by experiments):
Fx , Fy , Fz , T , Rz(DIN ) , Ra , θ
Correction coefficients K1…Kn
Figure 1. Inputs and outputs of cutting process characteristics mathematical models
(formulae) by using ISO 3002-1:1982 tool-in-hand system. v – cutting speed; ap – depth of
cut; f – feed rate; M – model; M – characteristic of workpiece material mechanical
properties; K1…Kn – correction coefficients; Fx, Fy, Fz – components of cutting force;
Rz(DIN), Ra – indicators of machined surface roughness; θ – cutting temperature.
describe cutting process characteristics a
polynomial model is used. Factors changing in the cutting process must be chosen
as model inputs. The characteristics of the
process are made up of two parts, differently depending upon the inputs. The intensity
of wear of various cutting wedge parts has
been substituted for tool life as one of the
model outputs. A different approach to the
problem to be resolved makes it possible to
compile a systematic mathematical model
of the cutting process, where interactions
of the model inputs and variation of the
process characteristic values in time are
considered. The possibility to raise the
prediction exactness of the cutting force
components values up to 20 times has been
proved experimentally. In the middle of the
20th century when intensive investigations
in metal cutting were done computer
engineering was not yet at a level to allow
solutions of today. To all appearance, the
users of the metal cutting theory of the
present day must be so accustomed to the
simplifications in use that no attempt has
been made to find better solutions, or
perhaps they have not been found.
Modern instrumentation allowing the
experimental results to be directly forwarded
to a computer for data processing makes it
possible to compile a model for any
material in a very short time (about 2 to 4
days). A structure of the new model is
shown in Fig. 2
3. A POSSIBILITY TO PREDICT
PROCESS CHARACTERISTICS
ON A QUALITATIVELY HIGHER
LEVEL
In the monograph “Fundamentals of
Mathematical Modelling of the Metal
Cutting Process” [27] (in Estonian) reasons
of inaccuracy of metal cutting process
mathematical models are analysed. It is
emphasized that most of the reasons are
subjective and therefore avoidable. The
metal cutting process and technological
system are treated as a unitary system.
In [27] the ISO 3002-1:1982 tool-in-hand
system has been adapted to modern cutting
tools. Two new reference systems have
been introduced. An insert reference
system is envisaged to describe the insert.
The tool-in-use system has been replaced
by the tool-in-process system making a
more precise determination of the face and
clearance angles in different points of the
cutting wedge possible. Coordinate
systems are made use of to calculate
parameters of the cutting process starting
from the design geometry of the insert,
insert bases orientation on the cutting tool
body and kinematics of the cutting tool.
The formal geometrical inputs of the
models can thus be replaced with these
harmonised with the physical essence of
the process, and at the same time the
number of inputs can be minimized. To
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Noise
Model outputs fixed on experiments
and got by data processing
Fγx , Fαx , Fγy , Fαy , Fγz , Fαz , dhα dt , dbγ /dt
dβ n2 dt , Rz(DIN ) , Rz (DIN )cal ,
Inputs of the model
Rz (DIN ) proc , Ra , Racal , Ra proc , θ
γ cp1 , γ cp 2 , bγ , auc , buc , v
M
Outputs of the model
Fγx , Fγy , Fγz , pn, pt, dhα dt , dbγ dt ,
dβ n2 dt , θ , Rz(DIN)proc, Raproc
Figure 2. γ cp1 , γ cp 2 – tool rake of the first and second face in cutting process; bγ – width of
the first face; auc – weighted average of uncut chip thickness; buc – width of uncut chip; v
– cutting speed; Fγx , Fγy , Fγz – components of the chip generating force; Fαx , Fαy , Fαz –
components of the force acting on the flank; dhα / dt – intensity of flank wear; dbγ / dt –
intensity of the first face land wear; dβ n 2 / dt – intensity of decreasing of the normal
wedge angle of the second face; Ra – arithmetical mean deviation of assessed profile;
Rz (DIN ) – ten-point height of irregularities; Ra proc , Rz ( DIN ) proc – components of
Ra and Rz (DIN ) depending on cutting process; Ra cal , Rz ( DIN ) cal – components of
Ra and Rz (DIN ) depending of the nose radius of the cutting edge and feed rate; θ –
cutting temperature; p n , pt – normal and tangential pressure on the flank of cutting tool.
4. CONCLUSION
sions) employed for turning. For taking
into use an applied research of the new
model is needed.
The new model is a real-time model with
the help of which a number of technical
problems unsolved so far with the conventional models can now be resolved.
Compiling programs for CNC machine
tools makes it possible to achieve the
required quality with a minimum number
of cuts, and actually optimise the cutting
data. So far the relation of the feed and
speed alone has been optimised. The
monograph also provides principles for
building up a CNC machine tool control
program. In [27] turning of outer surfaces
has only been treated. Still, the cutting
process mathematical model can be usable
for the majority of machining methods by
correspondingly adapting the mathematical
apparatus (about 700 mathematical expres
5. REFERENCES
1. Taylor, F. M. On the Art of Cutting
Metals. Trans. ASME, 1907, 28.
2. Merchant, M. E. Some observations on
the past and present of research on
machining and grinding. ASME, 1993, 3.
3. Shaw, M. C. Some observations concerning the mechanics of cutting and
grinding. ASME, 1993, 3.
4. Zorev. N. N. Metal Cutting Mechanics.
Pergamon, Oxford, 1965.
137
17. Grezik, W. Experimental investigations of the influence of adhesion on the
5. Reznikov, A. N. Thermophysics of
Cutting. Mashinostroyenie, Moscow,
1969 (in Russian).
6. Wu, S. M. Tool life testing by response
surface methodology. Part i and part ii.
Trans. ASME, 1964, B 86, 105–116.
7. Modern Metal Cutting. Sandvik
Coromant, Sweden, 1994.
8. Lin, W. S. et al. Modeling the surface
roughness and cutting force in turning.
J. Mater. Process. Technol., 2001, 108,
286–293.
9. Bhaskara Reddy, S.V. et al. Optimal
sub-division of the depth of cut to
achieve minimum production cost in
multipass turning using a genetic algorithm. J. Mater. Process. Technol.,
1998, 79, 101–108.
10. Szecsi, T. Cutting force modeling using
artificial neural networks. J. Mater. Process. Technol., 1999, 92–93, 344–349.
11. Yang, W. H., Tarng, Y. S. Design
optimization of cutting parameters for
turning operations based on the Taguchi
method. J. Mater. Process. Technol.,
1998, 84, 122–129.
12. Phan, A.-V. et al. Finite element and
experimental studies of diametral errors
in cantilever bar turning. Applied Mathematical Modelling, 2003, 27, 221–232.
13. Silva, M. B. da, Wallbank, J. Cutting
temperature: prediction and measurement methods – a review. J. Mater.
Process. Technol., 1999, 88, 195–202.
14. E.-G. Ng et al. Modelling of temperature and forces when orthogonally
machining hardened steel. Int. J. Mach.
Tools Manufact., 1999, 39, 885–903.
15. El-Gallab, M., Sklad, M. Machining of
AL/SiC particulate metal matrix composites. Part III: Comprehensive tool
wear models. J. Mater. Process. Technol., 2000, 101, 10–20.
16. Grezik, W. et al. Finite element modelling of temperature distribution in the
cutting zone in turning process with
differently coated tools. J. Mater. Process. Technol., 2005, 164–165, 1204–
1211.
frictional conditions in the cutting process. Tribology International, 1999, 32,
15–23.
18. Fang, N. An improved model for
oblique cutting and its application to
chip-control research. J. Mater. Process.
Technol., 1998, 79, 79–85.
19. Chang, C.-S. A force model for nose
radius worn tools with a chamfered
main cutting edge. Int. J. of Machine
Tools & Manufacture, 1998, 38, 1467–
1498.
20. Choudhury, S. K., Appa Rao, I.V.K.
Optimization of cutting parameters for
maximizing tool life. Int. J. of Machine
Tools & Manufacture, 1999, 39, 343–
353.
21. Lian, R.-J. et al. A grey prediction
fuzzy controller for constant cutting
force in turning. Int. J. of Machine Tools
& Manufacture, 2005, 45, 1047–1056.
22. Astakhov,
V.P.
Metal
Cutting
Mechanics. CRC Press, 1998.
23. Astakhov, V.P., Shvets, S.V. A system
concept in metal cutting. J. Mater.
Process. Technol., 1998, 79, 189–199.
24. Hastings, W.F. A new quick-stop
device and grid techniques for metal
cutting research. Ann. CIRP, 1967, XV,
109–116.
25. Jaanson, A. About quick-stop devices
for metal cutting research. Proc. TPI,
1971, 306, 3–13 (in Russian).
26. Komanduri, R., Brown R.H. On the
mechanics of chip segmentation in
machining. J. of Engineering for Industry, 1981, 1, 33–51.
27. Jaanson, A. Fundamentals of Mathematical Modelling of the Metal Cutting
Process. TUT Press, Tallinn, 2005 (in
Estonian).
6. CORRESPONDING ADDRESS
Arvo Jaanson
Tallinn University of Technology
Department of Machinery
138
Ehitajate tee 5, 19086 Tallinn, Estonia
Arvo.Jaanson @ ttu.ee
139