Theoretical Contributions Regarding the Modelling of the
Superfinishing Process
BADEA LEPADATESCU, FLAVIA FECHETE
Department of Manufacturing Engineering
Transylvania University of Brașov
Eroilor Bvd, nr.29, Brașov
ROMANIA
Abstract: - The mathematical modeling applied on the machining process by superfinishing includes the study of
the forces that appear during the action of the abrasive body on the surface of the parts subjected to processing,
as well as the study of the mathematical equations of the trajectories of the abrasive grains within the various
working methods used. This study is necessary to be able to obtain a quality of the processed surfaces in
accordance with the technical execution documentation. In this way, various movements of the abrasive tool and
the processed part can be combined in order to be sure of obtaining the quality of the part's surface finish in the
shortest possible time. The study allows obtaining mathematical formulas that facilitate the use of optimal
technological parameters for the processing of different shapes of pieces.
Key-Words: - Mathematical modeling, equations of crosshatch pattern, surface finish, optimal parameters.
Received: March 5, 2024. Revised: August 5, 2024. Accepted: September 9, 2024. Published: October 11, 2024.
1 Introduction
The theoretical foundation of the superfinishing
process is still not perfectly elucidated. That is why it
was necessary to study some aspects regarding the
shape and parameters of the trajectories of the
abrasive particles of the cutting tool, the effect of
these trajectories on the quality of the surface, the
determination of the cutting speed, the acceleration
of the particle, etc. Mastering these aspects will allow
ensuring a rigorous control of the cutting process and
at the same time will facilitate the recommendation
of optimal working parameters.
During superfinishing processing, a grain of abrasive
material travels a certain trajectory on the surface of
the part, which is generally a harmonic oscillatory
movement.
The study of the kinematics of a point of the abrasive
tool on the surface of the part arose from the need to
know the shape and length of the trajectory of a grain
during a rotation of the part. This is necessary to
determine the parameters of the chipping regime of
the length of the curve generated by a particle of the
abrasive tool on the surface of the part, to observe
which elements intervene in the equation of the
trajectory of the movement of an abrasive grain to
intervene on them and optimize the chipping regime.
Also, based on the relationships obtained to
determine the space traveled by an abrasive grain, the
respective velocities and accelerations can be
determined by differentiating with respect to time,
and it can be observed that during the entire
processing these parameters have variable values
over a period. This fact leads to a variation of the
parameters of the cutting regime and explains the fact
that the roughness obtained after processing is not
constant over the entire surface of the piece.
In the dynamics of the cutting process, the generation
of harmonic oscillatory motion can be achieved by
several methods such as: mechanical generation,
pneumatic, hydraulic, and electrical generation. The
methods of generating the oscillatory movement
correctly used in production are mainly the
mechanical and pneumatic ones, due to the simplicity
of the realization and safety in operation.
2 Development of a theoretical model of
the superfinishing process
The superfinishing process is a relatively new
processing process compared to other known
technological processes, being introduced in the parts
manufacturing lines in the period 1930-1935, after
this date superfinishing has become more and more
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widespread due to the advantages that has them in
comparison with other smoothing procedures.
From the desire to explain the phenomena that take
place in the superfinishing process, the specialists
created a theoretical model in which they believe that
from the point of view of the physical phenomenon
of the chipping process, the superfinishing process
with an abrasive bar is similar to that of broaching,
each abrasive grain being considered a pin tooth.
At the same time, a first limit of the theoretical model
is noted: that the distance between two abrasive
grains during superfinishing is much smaller than
that between two teeth of the broach, which leads to
the difficulty of evacuating the chips. It is considered
that the alternative movement of the abrasive bar is
the one that allows the evacuation of the chips and
avoids clogging of the pores. The authors of the
theoretical model explain the small size of the
detached chips by the relatively small length of the
chipping stroke of the abrasive grain. The theoretical
model explains the phenomenon of automatic
interruption of the chipping process during
superfinishing through the prism of the intervention
of the chipping liquid characterized by its viscosity.
The presented theoretical model explains the fine
roughness of the superfinished surface by "cutting"
the micro asperities that form the rough profile of the
part left over from the previous processing operation.
It also provides an explanation of the self-
interruption of the chipping process. It is considered
that the cutting process is interrupted automatically,
when following the "recutting" of the tips of the
initial geometric micro asperities (the roughness of
the semi-finished product), the bearing surface of the
processed material increases, the contact pressure
decreases, and the cutting fluid forms a bearing film
that no longer allows contact between abrasive grains
and metal.
Unlike the theoretical models developed for other
cutting processes, the theoretical model of the
superfinishing process considers the cutting fluid as
an active element in the cutting tool-workpiece
contact area. According to the model, this liquid - the
superfinishing oil, non-emulsifiable oil characterized
by viscosity, has the primary role in interrupting the
cutting process.
The proposed theoretical model [6] starts from the
following specifications:
- the active part of the abrasive grain (which has a
diameter Dmax = 9 µm) not embedded in the binder,
has dimensions comparable to the maximum height
of the micro asperities (about 4 µm) that form the
profile of the processed part (Rt= 4-5 µm).
- the chipping liquid present in the processing area
can be assimilated to a continuous environment of
spherical particles that try to form an adsorption layer
on the surface of solid bodies but, due to the
dimensions close to those of the microrelief of the
part, they fail to penetrate the micro asperities profile.
In the first seconds of the superfinishing process, the
abrasive granule "squeezes" the oil drops from the
surface of the piece, making direct contact between
the granule and the metal ridges, simultaneously
achieving the "cutting" of the metal tips and the
blunting, breaking, or tearing of the abrasive granule
from the binder.
Fig.1
Schematic representation of superfinishing process.
Superfinishing is a chip removing machining process
used to refine the surface of a metal component to an
extremely fine surface with low roughness.
All the superfinishing grains in contact with the
workpiece create a machining pattern by means of
overlaying individual sinusoidal lines that cross each
other at a particular angle (Fig.1). This generates a
specific, defined pattern of grooves and plateaus,
which, in turn, results in the advantages of the
superfinishing process. The grooves act as channels
to aid uniform distribution of the lubricant, while the
plateau guarantees a high percentage contact area.
Part of the metal tips cut are fragmented and pulled
from the machining area by the cutting liquid, and
others remain and penetrate into the "depths" of the
valeys. The alternative movement of the tool favors
the "stuffing" of the chip pulled into the microrelief
of the part.
In the next 10 seconds of machining, the abrasive tool
removes the entire layer of material that constituted
the initial microrelief and begin to "dig" into the
compact material. The traces left by the abrasive
grains in the machined part are proportional to the
size of the active part (approximately 4 µm) and
therefore much smaller than those left by the abrasive
grain in the body to be rectified (250-300 µm), so the
roughness of the machined surface is also much
smaller. The peaks of the superfinished surface are
much more frequent and finer, so the bearing surface
is larger and the pressure lower, the oil particles are
no longer pierced and the superfinishing process is
much more difficult (Fig.3).
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Fig. 2 The first phase of machining.
At the same time, the impurities settled in the
microstructure’s valeys contribute to the hardening of
the surface layer (max.1µm). Thus, the abrasive
grains can no longer detach chips and the machining
process self-stops after approximately 150-200
seconds.
Fig.3 The second phase of machining.
3 Kinematics of a current point of the
tool in relation to the part
The kinematics of a current point of the tool in
relation to the part will be studied for three processing
methods: (a) superfinishing with plunge feed; b)
superfinishing with through feed; c) superfinishing
with two oscillatory movements.
a) Superfinishing with plunge feed
During the processing of the parts by superfinishing,
the abrasive grains describe trajectories of different
shapes and lengths on the surface of the part. The
length of these trajectories is considered to represent
the "degree of coverage" of these surfaces with traces
of processing. There is a direct link between the
degree of coverage and the quality of the surface of
the piece, in the sense that with the increase of the
Fig.
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degree of coverage, the roughness of the surface, the
bearing coefficient, the state of the surface layer
improve.
Fig.4 shows the kinematic diagram of superfinishing
with plunge feed.
Fig.4 Superfinishing with plunge infeed.
where:
B, L, M = dimensions of the abrasive stone.
P = pressing force of the abrasive stone on the part
surface.
a = the amplitude of the oscillation movement.
Vp= peripheral speed of the piece.
α = the angle at the center covered by the abrasive
tool.
β = the angle of inclination of the trajectory of the
abrasive grain relative to the axis of the part.
An abrasive particle has a periodic sinusoidal
movement during processing, whose equation is:
x(t) = asin ωt (1)
where ω is angular velocity of the part being
machined.
An Oxyz system related to the part will be considered
(Fig.5).
Fig.5 Coordinate system in the case of processing
with transverse advance.
The radius vector OM will be projected on the three
coordinate axes, resulting in:
(2)
where: θ = ωτ; (3)
ω = πn/30 (4)
The projection on the Oz axis is zero because both
the part and the tool have no longitudinal
displacements when processing with transverse feed.
Substituting (3) in (2) we get:
(5)
Differentiating with respect to time, we obtain
velocity:
(6)
To find out the length of the trace of the trajectory of
an abrasive grain on the surface of the part, unfold the
surface of the part and work in the Ouz coordinate
system, using the variable u = Rθ, (Fig.6).
Fig.6 Trajectory of an abrasive grain.
In the new coordinate system using the notation
(7)
relations (2) will become:
(8)
And relations (6) have the form:
(9)
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According to Fig.6, the equation of the trajectory of
an abrasive grain in the Ouz coordinate system is:
z = asin( (10)
where a is the amplitude of the motion.
The length of the elementary arc ds will be:
ds = du
(11)
Differentiating eqution (10) with respect to u we get:
(12)
Taking into account relations (11) and (12), the arc of
the curve included between the abscissas u1 and u2
will have the length:
S =
(13)
If it is considered that only the arc length of the
trajectory of the abrasive grain will be taken into
account at one rotation of the part, then the limits of
the parameter u will be 0 and 2πR, it will result:
S =
(14)
From relation (14) it can be seen which factors lead
to the increase in the length of the trajectory of an
abrasive grain on the surface of the part, because the
quality obtained after processing is directly
proportional to this length. The amplitude of the
oscillatory movement also has an important
influence, because as the amplitude increases, so
does the quality of the processed surface.
The value of the amplitude of the oscillatory
movement cannot be too high, because it is limited
by technological and functional considerations. The
technological considerations are given by the need
for a small grain stroke because the space between
the grains does not allow the accumulation of a large
volume of chips resulting from chipping. Functional
and constructive considerations are related to the
complexity of realizing a greater amplitude of the
work system and to the large moments of inertia that
may occur. That is why the optimal amplitude for this
method of processing is 1.5 - 3 mm.
b) Superfinishing with through feed
If the length of the processed parts is greater than the
length of the abrasive bar, then in addition to the
harmonic oscillatory and plunge feed, the tool or part
must also perform a longitudinal feed. For example,
let's assume that the workpiece has a rotation
movement n and a through feed with an axial speed
v0 (Fig.7).
Fig.7 Machining with through feed.
The abrasive bar has a harmonic oscillatory
movement with an oscillating stroke and is pressed
on the surface of the workpiece with a force P. In this
case, the elements of the technological system are
related to the Cartesian coordinate axes Oxyz, where
we determined the equation of the trajectory of a
certain point M of the abrasive grain (Fig. 8).
Fig.8 The trajectory of an abrasive grain in the Oxyz
coordinate system.
Thus, from Fig.8, the radius vector OM of a certain
point M of the abrasive tool is broken down into
components on the 3 coordinate axes given by the
relations:
(15)
where (16)
Using (16), z will become:
z =
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where, k =
The system of relations (15) becomes:
(16)
Differentiating with respect to time, we obtain
velocity:
(17)
Unfolding the surface of the part in Fig.7, in the Ouz
coordinate system, one can observe the shape of the
trajectory of an abrasive grain during one rotation of
the part.
Fig.9 The trajectory of an abrasive grain in the Ouz
coordinate system.
Using relations:
,
The system of relations (16) becomes:
(18)
The Cartesian components of the velocity of point M
will be:
(19)
In the Ouz coordinate system, point M has the
coordinates:
z = asin ( (20)
(21)
The length of the elementary arc ds is calculated with
the relation:
ds = du
(22)
Taking into account (21) and (22), the arc of the curve
described between the abscissas u1 and u2 has the
length:
S =
(23)
Calculating the length of the trajectory curve of the
abrasive grain at a single rotation of the part, then the
integration limits will be 0 and 2π, and relation (23)
becomes:
S =
(24)
From relation (24) it can be seen that the length of the
curve traced by an abrasive grain on the surface of
the part will be greater if the amplitude of the
oscillatory movement and the axial speed v0
impressed on the tool or the part are increased. The
amplitude of the oscillation cannot be affected too
much, because a value higher than 3 mm would lead
to a worsening of the evacuation of the chips and
therefore the machining capacity of the abrasive
grains. You can increase the advance speed v0, or you
can intervene on the angular speed ω of the part,
which must not have values that lead to cutting
speeds higher than 30-35 m/min.
The greater the length of the curve described by the
abrasive grain, the higher the surface finish of the
machined surface.
c) Superfinishing with two oscillatory movements
To increase the intensity of the machning process and
implicitly the productivity of the process, it is
recommended that the abrasive tool perform two
types of oscillatory movements with different
amplitudes and frequencies. Thus, for example, an
oscillatory movement produced pneumatically
with a smaller amplitude (1-3 mm) and a high
frequency,1500-2000 cpm (cycles per minute)
and another mechanically generated oscillatory
movement with a higher amplitude, 5-20 mm,
with a lower frequency, 60-150 cpm. The
composition of the two oscillatory movements
increases the degree of coverage of the machined
surface with cross hatch patterns of abrasive grains,
resulting in a better surface finish in a shorter time.
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We will study the case where the oscillatory motion
of the abrasive tool is composed of an oscillatory
movement produced by a mechanical generator and
another oscillatory movement produced
pneumatically. The mechanical generation of
oscillations is produced by a crank-slider plane
mechanism as seen in Fig.10.
Fig.10 Generation oscillations by a crank-slider
mechanism.
For the mechanism in Fig.10, the explicit form of the
displacement motion has the next form:
(25)
where:
The principle kinematic diagram of the mechanism
with two oscillatory movements is shown in Fig.11.
Fig.11 Kinematic diagram of the mechanism with
two oscillatory movements.
From Fig.11 it can be seen that the path left by an
abrasive grain on the surface of the part is actually a
composition of two harmonic oscillatory movements.
The part has a rotational movement with angular
velocity ω, and the abrasive tool is under the
influence of two oscillatory movements, one with the
amplitude a1 mechanically generated by the
mechanism in Fig.10, and the other with the
amplitude a2 generated pneumatically by a
pneumatic vibration generator. In order to increase
the machining capacity, a feed movement with axial
speed v0 is also applied to the part.
In order to obtain the equation of the displacement of
an abrasive grain on the surface of the part,
everything is related to an Oxyz coordinate system
(Fig.12).
Fig.12 The path left on part surface in the case of
two oscillating movements.
In Fig.12, the radius vector OM of a point of the tool
is broken down into the following coordinates on the
three coordinate axes:
(26)
where s is given by (25).
Considering that:
So,
(26)
where
and
(27)
where =
Using (26) and (27) system (26) becomes:
(28)
If the cylindrical surface of the part is unfolded
(Fig.13), in the Ouz coordinate system, we have:
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Fig.13 Path left by an abrasive grain in Ouz
coordinate system.
where
But
where
and
where
The system (28) becomes:
(29)
The length of the elementary arc is:
(30)
Using (29) will have:
(31)
Taking into account relations (30) and (31), the arc of
the path included between the abscissas u1 and u2 will
have the length:
S =
(32)
To find out the length of the trajectory of an abrasive
grain at one rotation of the part, 0 and 2πR are taken
as integration limits.
5 Conclusion
From relation (32) we can draw the conclusion
regarding the factors that influence the length S of the
trajectory of an abrasive grain on the surface of the
part. Thus, it can be seen that to increase the length S
of the trajectory of the abrasive grain, the angular
speed ω1 of the radius r and the length l of the
mechanical oscillation generation mechanism must
be increased. Also, to increase the length S, the
amplitudes a1 and a2 of the two oscillatory
movements can be increased.
Also from relation (32) it can be seen that in order to
have as long as possible the length of the curve S, we
must reduce the factor k, and therefore reduce the
eccentricity e of the origin of the rotation movement
from the mechanical generator of oscillations with
respect to the axis of translation of the tool which
support abrasives stones.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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