Bipartitle Graphical
Integration of Grinding
Process Models
Satish Bukkapatnam, Ph.D.
Oklahoma State University, Stillwater, OK
Rajkumar Palanna, Ph.D.
Honeywell Aerospace, Torrance, CA
Abstract
In this paper the key grinding process models and relationships that were discovered by
previous research efforts have been unified in the form of a bipartite graph
representation, which is part of body of knowledge in constraint theory. The common
terminology resulting from the representation helps in consolidating grinding modeling
research works available in literature. The grinding bipartite graph has been used to
design the structures of real time process control models, as well as the process
instrumentation and experimentation strategies.
1. Introduction
The grinding process has historically been considered a complex domain by the
manufacturing researchers and practitioners because of the complicated characteristics of
the underlying dynamics (Bukkapatnam and Palanna, 2004). For a long time, industries
have been extensively using several configurations of these grinding processes for the
precision grinding of outer diameter (OD) of shafts and other radial symmetrical features.
Grinding of the outside diameter of shafts has numerous critical applications. The surface
finish of the shafts determines to a large extent the quality and durability of many
complex rotary products. Even though the shaft might cost only a small fraction of the
1
entire product cost, it is a major determinant of the product performance. Nowadays the
realization of the desired surface finish with minimum consumption of energy at
optimum cost is one of the main technological challenges. For example, Air Bearings is
one of the key technologies that give a competitive edge to aerospace companies like
Honeywell. Air bearings allow the rotating machines to reach speeds of around 100,000
RPM (more than 3 times faster than what conventional bearings allow) because there is
minimal friction between the rotating assembly and the stationary assembly. Higher
speeds are a definite technological advantage in high technology rotating assemblies like
aircraft ECS because they can lead to more powerful and lighter aircraft. An air bearing
assembly consists of mainly two component groups: the foil bearings that are made of
sheet metal and are part of the stationary portion of the system, and a shaft which forms
the rotating portion of the system. If the surface finish is too fine (Ra<5in*), the “takeoff” torque required for the air bearing is increased. If the surface finish is too rough
(Ra>10in*), then the load carrying capacity of the air bearing is reduced. Hence, a fine
balance has to be struck while specifying the surface tolerance. Usual surface tolerances
have significantly tightened over the last 60 years (see Figure 1). In the forties and fifties,
surface tolerances specified on prints were usually in 250 in Ra. Later, in the 1970s they
shrank to 64 in Ra. Today, 5 in Ra are not uncommon. Yet, for all the tightening of
design tolerances, machine tools have not changed drastically over the years [1].
Realizing consistent surface quality from grinding operations is among the major
industrial challenges today. The present efforts towards this end has met with limited
success; the major reason being the complexity of the process.
*
These values are only representative. Actual data is proprietary.
2
0.002500
0.002000
Inches
0.001500
0.001000
0.000500
?
0.000000
1940s
1950s
1960s
1970s
1980s
1990s
2000s
Year
Surface Roughness
Tolerance Discounting Surface Roughness
Figure 1: Shrinking of “Average” Tolerance Specifications during the Last 60 Years.
Broadly, the following four major phenomena, as summarized in Figure 2, influence
surface quality in grinding operations: (1) wheel friability and degradation, (2) plastic
deformation and other material removal sources that are distributed randomly in space
and time, (3) the resulting response of the machine structure that in turn is an assemblage
of many moving and rotary components, and (4) complicated thermomechanics (heat
transfer) pattern resulting from these distributed material removal sources. In order to
effectively control surface finish, one must first understand the vast set of relationships
and interactions connecting the process state variables and parameters that underpin these
four interacting phenomena. Over the past 80 years or so researchers have addressed
these various aspects of the grinding process and have provided several quantitative
models (both empirical and analytical) as well as qualitative relationships. In this paper,
the authors report a review of the previous grinding process modeling efforts, specifically
focusing on those research works relevant for surface finish control. The review has
captured the relevant multifarious relationships between grinding process parameters,
state variables and performance characteristics. Bipartite graph representation is
3
introduced as a means to graphically capture these relationships. A bipartite graph is a
concept borrowed from constraint theory body of knowledge. These graphs are generally
used to consolidate and graphically represent relationships and evaluate their suitability
for devising appropriate solutions (i.e. various aspects of solvability). A grinding bipartite
graph will provide guidelines for developing real time control strategies. One of the
applications of this consolidation using bipartite graph has been in the selection and
design of appropriate model structures in order to facilitate the implementation of “Model
Based Tampering” (MBT) of precision manufacturing operations [1,2].
Forces
Wh
Thermomechanics
eel
stat
e, s
our
ce d
For
ces
istri
buti
on
Wheel state, source distribution
Vibrations
Temperatures
Structural
Dynamics
Temperatures
Wheel state,
source distribution
Forces
Vibrations
Plastic Deformation
Wheel
Friability and
Degradation
Figure 2: The four major phenomena contributing to the complexity in overall grinding process mechanics.
Bipartite representation can also lay the groundwork for experimental studies,
particularly for sensor selection and design of experiments. We therefore anticipate that
the presented bipartite graph approach will be of immense value to many applications
involving grinding process sensing, control and experimentation. Although the presented
research focuses on MBT application to shaft OD grinding, the relevant modeling efforts
in surface grinding are also included. For future research and applications involving
grinding process sensing and control, the presented bipartite graph may, however, needs
to be slightly adjusted by including or excluding relevant models.
This paper is organized as follows: Section 2 provides an overview of the bipartite graph
technique, Section 3 describes the methodology for the bipartite graphical integration of
4
the various grinding process models. Section 4 shows the potential of this bipartite graph
representation for surface finish control.
2. A Concise Treatise on Bipartite Graphs
George J. Friedman has developed an analytic foundation called constraint theory for the
systematic determination of mathematical model consistency and computational
allowability [3]. Constraint theory’s primary application is in the construction and use of
heterogeneous, multidimensional mathematical models, especially when interdisciplinary
technical teams, systems analysts, and managers are involved.
This theory establishes a rigorous basis for the notion of a constraint. Constraint theory
separates a model from computations. It charts the flow of constraints throughout the
model and detects instances of over-constraint and under-constraint. Over-constraint is a
situation where more variables exists than that required for solving a group of equations
and under-constraint is a situation where fewer variables exists than that required for
solving a group of equations.
A bipartite graph is one of the representation tools given in the constraint theory. A
bipartite graph has two sets of vertexes: (1) nodes, representing the model’s relations and
(2) knots, denoting model variables. A knot will be connected to a node by an edge if and
only if the corresponding variable is present in the corresponding functional relationship
of a model. In this research nodes are represented using squares and knots are represented
using circles. A model graph can be thought of merely as the circuit diagram of an
analogue computer hood-up of the mathematical model where the nodes are the function
(or relation) generators and the knots are merely wired connections that permit the values
(voltages of variables) to pass from one function generator to another. The constraint
matrix method [3] can be used to develop elegant solution strategies using bipartite
graphs.
Based on the foregoing, one of the benefits of representing model equations group in
bipartite graph format is that it helps to develop a common language for unifying the
various previous models, likely developed independent of one another, and it also shows
5
the relationships between the various governing equations in the application domain. It
provides an insightful metamodel for both the mathematical model and all of its potential
computations.
The bipartite graph representation exhibits many computability characteristics far more
clearly than the original set of equations. Of course, we must return to the original
equations to solve the model and to perform computations, but a much better
visualization and more efficient control of the computations involving a large set of
equations can be accomplished by resorting to bipartite graphs.
6
3. Bipartite Representation of Grinding Process Models
The literature contains a vast body of efforts related to grinding process modeling. The
types of models ranged from analytical, physics-based models to empirical and
phenomenological relationships. The original terminology used in the reviewed papers
has been modified in order to establish a common language that integrates and
consolidates the various research works. The nomenclature used in this paper is provided
in Appendix 1.
The variables used in the grinding bipartite graph, shown in Figure 3, can be categorized
as output variables, state variables and process parameters. The output variables are
shown shaded in Figure 3. These include Temperature (), Normal and Tangential Force
vector ( F ), Acceleration, Acoustic Emission (Ae), and Grinding Power (P), which may
be measured in real-time using appropriate sensors. These variables comprise stateoutput models as well as on-line observers, which are necessary for real-time sensing and
control of processes. State variables such as those related to surface finish, wheel
condition, etc., could be tied together using common equation numbers and their
dependent variables. For example, a relationship to surface roughness (Ra ) can be
derived from knowing or measuring four different output variables as well as from
estimating different state variables as seen in Figure 3. This segment of the graph is of
significant relevance whenever the purpose is to determine the factors affecting
degradation of process that impact surface finish.
Since it is practically impossible for any research work to address the entire gamut of
issues in grinding process, many researchers tend to focus on a few specific aspects of the
process so that they can develop rigorous models. For convenience and easier
comprehension, this section categorizes the research works into four zones. Models in
each zone concentrate on variables within the zone or the equations that connect the
different zones. While the equations that tie the parameters together within the zones help
to provide a solid foundation for the integrated model, it is the equations that span
multiple zones that help to transgress between the zones and develop a holistic grinding
process model.
7
41
X3
12
X5
40
36
X5L
D
U
L
X4
38
P
x1
37
18
39
21
na
X1

L
13

D
32
Fd
35
33
31
34
g0
Aw
n

w
19
14
20
vw
17
Q'w
X2
11
16
F
G
Sd
15
22
23


26
30
6
7
X2L
dw
8
tg
dc
5
Ae
Nr
vs
45
Ra
24
10
u
27
ds
25
r
dt
Nkin
29
42
Z
Ac
3
9
2
1
x/l
 max
44
t
28
dg
Cr
43
L
4
- Lag Operator
D
- Differentiator
#
- Equations
a
- Variables
L
Figure 3: Bipartite Graph Showing Relationships between Grinding Process Parameters
8
The bipartite graph and the corresponding model equations are partitioned into zones as
follows (also see Figure 4): The relationships ultimately connecting surface finish with
wheel degradation, material removal processes, forces and structural response and
temperature, respectively, are given in equation sets (1)-(10), (11)-(18), (19)-(33), (34)(37).
3.1. Zone 1 Models (Wheel degradation): The equations in this zone relate wheel
degradation to surface finish. Several researchers have developed empirical relationships
connecting surface finish with process parameters including material removal rate,
surface speed of the wheel, grinding depth, as well as parameters related to properties of
the wheel, including the dressing lead, dressing angle, and dressing depth. All models
available in literature are predominantly empirical in nature. For example, Malkin [4]
describes the following four equations for surface finish:
R a  K f  d1/ 3
 
Q'w x
vs
………………..(1)
Ra  K 2 S d1/ 2 a1d/ 4
 
Q 'w x
vs
……………….(2)
Ra ,  ( d gy )
……………….(3)
Ra  (
 w

C
 r




2
)
……………….(4)
9
41
X3
12
X5
40
36
X5L
D
U
L
X4
38
39
21
na
X1

L
13
x1
37
18
ZONE 2

D
P
32
Fd
35
33
31
34
g0
ZONE 3
Aw
n

w
19
14
20
vw
17
Q'w
X2
11
16
F
G
Sd
15
22
23


26
30
6
7
X2L
dw
8
tg
dc
5
Ae
Nr
vs
45
Ra
24
10
ZONE 1
27
r
ds
dt
Nkin
29
ZONE 4
42
Z
3
9
2
1
u
25
x/l
Ac
44
t
28
 max
dg
Cr
43
L
4
- Lag Operator
D
- Differentiator
#
- Equations
a
- Variables
L
Figure 4: Bipartite Graph for Grinding Process Showing The Various Zones
Eranki et al., related the material removal rate QW to surface finish Ra and wheel grade
g0 as [5];
Q ' w  35.0 g 00.78 Ra0.43 ,
………………………(5)
g0, with Ra and grinding ratio G as
g0  4.49G 0.46Ra Ra0.24
0.61
……………………….(6)
10
QW was related to Ra and G, as
Qw'  10.85G 0.36Ra Ra0.62
0.61
……………………….(7)
and dressing leads Sd was related to Ra as
S d  0.126 Ra1.6
………………………(8)
Fawcett and Dow provided the following relationship between Ra and feed rate fr [6];
Ra  kfr2
……………………….(9)
where k is an empirical constant. They also gave a relationship between Ra and nose
radius Nr as;
f r2
Ra 
12 5 N r
……………………….(10)
3.2. Zone 2 Model (Plastic deformation and material removal): The equations in this
zone largely pertain to material removal processes, and their relationship to the various
process parameters process state variables including those describing forces and energy.
11
We note that we have included a few elementary equations hereunder for the sake of
completeness of the bipartite graph.
The following two equations relate the specific grinding energy UE and the specific Q’w
[4]:
 Q' w
U E  K 4 
 vs




f 1
……………….(11)
 P
UE  
 Q'
 w




…………………(12)
The specific grinding energy UE and the specific Q’w are related with the cutting force as
[7];
v F
UE  s
Q'w
……………………….(13)
G can be determined from knowing Sd and Q’w as given by Eranki et al., [5];
0.40
0.38
G  9.20 g00.52s d Sd0.49 (Qw' )0.56S d
………………………(14)
12
Where for equation (14) we have,
Q’w  (vw dc)
……………………….(15)
and
Q' w 
F  K 3 

 vs 
f
……………………..(16)
Mann et al. demonstrated the relationship between force F and Q’w [7];
F  ( Q'w )
……………………….(17)
The grinding power is a function of grinding width w, depth of cut d, Q’w and work speed
vw as given by Eranki et al., 1992 [5];
1
1
p  uwQw'  Bwd c 4vw4 (Qw' )
1
4
……………………….(18)
3.3. Zone 3 Models (Structural response and forces): The models contained in this
zone largely pertain to capturing the dynamics of the process, especially the
manifestation of wheel and work vibrations and the various forces under various process
conditions.
13
Kannappan and Malkin [8] have found that the main grinding forces are related to the
number of passes and g0 as;
F  ( n, g0 )
……………………….(19)
They have also developed a relationship between F and wear flat area [8];
……………………….(20)
F  ( Aw )
Thompson [9] in his seminal investigation of grinding dynamics provided a model of the
form;


m x1  c x1  kx1  Fd
……………………….(21)
where Fd is the normal grinding force.
The relationship between F and deflection of the loading tool is given by Nakayama et
al. [10];
  HF E
……………………….(22)
where, H and E are constant for a given grinding process setup.
Mann et al. [7] have characterized the relationship connecting d and F;
F (d )
……………………….(23)
The relationship between acoustic emission amplitude IAeI vs. truing depth dt and d is
given by Hundt et al. [11] as;
Ae  ( d , d t )
……………………….(24)
14
The relationship between maximum temperature and F is given by Mann et al., [7];
F
vs
v
bd 4 w
3
 max 
4
3 
1
1
1
1
( kC )w 2  ( kC )s 2 Ds 4
……………………….(25)
The set of equations that are related to grinding force is given below. The force is
determined using a series of variables according to the equation given by Hundt et al.,
[11].
F  f ( t g , Q'w ,Vs ,Vw , N kin ,  , d w , d s )
……………………….(26)
ds



 R(t ) sin(  wt )  2 sin((  s   w )t   kin1, 2 ) 
C1, 2 (t )  

d
R(t ) cos(  wt )  s cos(( s   w )t   kin1, 2 )
2


where
R (t ) 
dw  ds
 ut
2
and
 kin 
2
d s N kin
For Range A:
Chip thickness is 0.
For Range B:


h(t )  C1 (t  tkin (t ))  C2 (t )
15
For Range C:
h(t ) 

dw
 C2 (t )
2
For Range D:
Chip thickness is 0.
Now the cutting force F can be determined by Kinzle’s equation;
F  Ac K c where
Ac (t )  h(t )2 tan( )
2
kc  kc1.1h  mc
where
and
kc1.1  0.5kN / mm2 and
mc  3
The relationship between the grinding time versus infeed velocity ui and radial
displacement r is given by Malkin and Koren, [12];
tg 
vf
r
u
  [  (1  m ) ln 
ux
ux
ux
……………………….(27)
The relationship between chip thickness T and number of active cutting edges Nkin, vw, vs,
d and work diameter ds is given by Nakayama et al., [10];
T
4v
vN r
w
s
kin
r
d
d
c
……………………….(28)
s
The chip area Ac is a function of height dc and the w and is given by:
16
A  f (d , w)
c
c
……………………….(29)
P=
F (vs)
………………………(30)
As given by Kannappan and Malkin [8],
Aw  ( n , g 0 )
……………………….(31)
and the number of active grinding edges na is a function of number of passes and g0 as
given by;
na  ( n , g 0 )
……………………….(32)
The wear flat area Aw is a function of length L of contact and the w and is given by;
Aw  ( L , w )
……………………….(33)
The relationship between forcing on work piece Fd, Force F , forcing frequency  and
time t is given by Thompson, [9] as;
17
Fd  FSin(t )
……………………….(34)
According to Thompson [9], the relationship between Force F, force damping  and time
t is given by;
F  F0et
……………………….(35)
the dynamic wheel movement is governed by;
x2 (t )  x2 (t 
d s
vs
)
……………………….(36)
wheel wear x3 is a function of Force F given by;
x3  KG F
……………………….(37)
wheel wear x3 is a linear function of Area of wear Aw;
x3  K A AW
……………………….(38)
18
the work-piece wear (reduction in radius) x4 is given by Thompson, 1986 [9];
x4  KW F
……………………….(39)
the regenerative form for work displacement considering its abrasion is given by;
x5 (t )  x5 (t  60
vw
)  x4
……………………….(40)
and the relation between x1, x2, x5 and x3 was given as
x1  x2  x5  x3
……………………….(41)
3.4. Zone 4 Models (Thermomechanics): The model in this zone mainly focus on
temperature relationships.
The temperature is related to d [7] as;
19
 
max
4
RuV d
3  V l ( kC )
w
w c
w
……………………….(42)
The quasi steady state temperature distribution is given as a function of dimensionless
length as given by Kohli et al., [13];
  f (x l )
max
……………………….(43)
The max is related to depth of energy partition Z, as given by Kohli et al., [13];
  f (Z )
max
……………………….(44)
The average temperature is related to the material removal rate as given by Mann et al.,
[7];
  f (Q )
'
max
w
……………………….(45)
4. Application of Bipartite Representation to Modeling and Control
One may develop experimental as well as modeling strategies for grinding process
sensing and control using bipartite graph representation. One may note that every acyclic
path within a bipartite graph is equivalent to traversing a set of equations, and hence a
specified set of variables. One notes that while a few of these variables and parameters
are either set or measurable (eg: F, Ra, vw) the others are not (eg: Aw). Therefore, an
20
effective sensing strategy for estimating an unknown variable must involve traversing a
path connecting the variables of interest to one or more measurables. Evidently,
researchers need to explore multiple paths in order to integrate multi-sensor information
so as to circumvent practical experimental obstacles arising from measurement errors and
uncertainties and to increase the robustness of the estimates by determining the values in
multiple ways.
For example, as shown in Figure 5, Path 1 relates force signals ( F ) to surface finish.
Equation (25) relates force ( F ) to temperature (max ). Using equation (37), we are able to
relate temperature to material removal rate (QW ). With QW and knowing the wheel
grade (g0) we can use equation (5) to determine surface roughness (Ra).
Path 2 shown in Figure 6, is an alternate path between force signals ( F ) and surface
roughness (Ra). Equation (17) relates force ( F ) to material removal rate (QW ). Equation
(7) uses QW and grinding ratio (G) to give surface roughness (Ra).
21
41
X3
12
X5
40
36
X5L
D
U
L
X4
38
x1
37
18
39
21
na
X1

L
13

D
P
32
Fd
35
33
31
34
g0
Aw
n

w
19
14
20
vw
17
Q'w
X2
11
16
F
G
Sd
15
22
23


26
30
6
7
X2L
dw
8
tg
dc
5
Ae
Nr
vs
45
Ra
24
10
u
27
ds
25
r
dt
Nkin
29
42
Z
Ac
3
9
2
1
x/l
 max
44
t
28
dg
Cr
43
L
4
- Lag Operator
D
- Differentiator
#
- Equations
a
- Variables
L
Figure 5: Equations Solution Path 1
22
Path 3 shown in Figure 7 is the third path connecting forces ( F ) and surface roughness
(Ra). Force signals are related to material removal rate (QW) through equation (17). With
QW , grinding wheel grade (g0), and grinding ratio (G) along with the use of equation (14)
41
X3
12
X5
40
36
X5L
D
U
L
X4
38
x1
37
18
39
21
na
X1

L
13

D
P
32
Fd
35
33
31
34
g0
Aw
n

w
19
14
20
vw
17
Q'w
11
16
F
G
Sd
15
22
23


26
30
6
7
X2L
X2
dw
8
tg
dc
5
Ae
Nr
vs
45
Ra
24
10
u
27
ds
25
r
dt
Nkin
29
42
Z
Ac
3
9
2
1
x/l
44
t
28
 max
dg
Cr
43
L
4
- Lag Operator
D
- Differentiator
#
- Equations
a
- Variables
L
Figure 6: Equations Solution Path 2
we can get dressing leads (Sd ). With QW , Sd , and velocity of the work-piece (Vw) along
with equation two we can get surface roughness (Ra).
The bipartite representation is also valuable for determining experimentation strategies.
Among the main parameters vw, fr, d and the dwell , each of which can be a candidate
control parameter for experimental design, no three of these are directly involved in a
relationship (i.e., there does not exist a square node that directly connects any
combination of three or more of these parameters. Hence 3-level interaction effects may
23
be neglected, and as a result ResIV designs can be used whereby experimentation
overhead can be reduced by half..
Also the following qualitative understandings have also been inferred from the bipartite
representation:
1. Dynamics of the process including
-
How the system (structure) vibrates
-
How the wheel degrades
-
How the surface finish progresses
depends on the process condition, i.e., the dynamics will be different depending what
values of the process control parameters such as table speed, in-feed are chosen.
2. Also, at a particular setting of process parameters, dynamics of how the structure
vibrates and temperature and forces will vary with the level or severity of degradation.
41
X3
12
X5
40
36
X5L
D
U
L
X4
38
x1
37
18
39
21
na
X1

L
13

D
P
32
Fd
35
33
31
34
g0
Aw
n

w
19
14
20
vw
17
Q'w
X2
11
16
F
G
Sd
15
22
23


26
30
6
7
X2L
dw
8
tg
dc
5
Ae
Nr
vs
45
Ra
24
10
u
27
ds
25
r
dt
Nkin
29
42
Z
Ac
3
9
2
1
x/l
 max
44
t
28
dg
Cr
43
L
4
- Lag Operator
D
- Differentiator
#
- Equations
a
- Variables
L
Figure 7: Equations Solution Path 3
24
3. Degradation of wheel at a given small time interval depends on:
-
Wheel degradation at the beginning of the time, i.e., degradation at
previous time interval
-
Vibration, force, temperature dynamics
-
Current surface characteristics
-
Process parameters
4. Also, the effect of degradation on the performance including surface finish and
diameter at a given instance will vary depending on
-
How the vibrations, forces, temperature, etc., are behaving (i.e.,
dynamics at structural vibrations, forces, temperature)
-
And the value of degradation at the previous instance of time.
i.e., surface generated at a point on the wheel at a given time interval depends on:
1. Previous surface before that point is machined. i.e., surface from the
previous pass
2. Wheel condition at previous time interval
3. Vibrations, temperature, force dynamics at a specific time interval.
4. Process parameters
25
Level 1(t-1)
Level 1: Performance Variables
[2]
Level 2(t-1)
[4]
[5]
Level 2: Degradation Variables
Aw, tw
Lc, Rc, Cc
[1]
DOE Analysis
Level 3: Output Variables
[3]
Physics of what impacts
sensors only
Dynamics not
captured by
sensors
Models /
Signal Characteristics
Dynamics not
captured by
sensors
Dynamics that are not
impacted through
degradation variables
DOE Analysis
Level 4: Control Variables
Plus other conditions and noise
Figure 8: Analysis Relationship Map
Based on these domain understandings, a plan for analysis and modeling has been
developed for the grinding process parameters. The variables and parameters may be
mainly classified into four “levels”. Level 1 contains the performance variables. Level 2
will contain the degradation variables, Level 3 contain the output variables from the
sensors and Level 4 contains the control parameters that could be used for tamper. From
figure 8 it is evident that the following categories of models that have to be built in order
to develop an integrated model structures for grinding process control.
From Figure 8 the following model segments need to be built:
1) [Level 3 Matrix] = f([Level 4 Matrix], [Level1 {t-1} Matix], [Level 2{t-1} Matrix])
2) [Level 2 Matrix] = f([Level 4 Matrix], [Level 3 Matix], [Level 2{t-1} Matrix])
3) [Level 1 Matrix] = f([Level 4 Matrix], [Level 3 Matix], [Level 2 Matrix] [Level 1{t-1} Matrix])
We have successfully used these model structures to effectively control the surface finish
during grinding of shafts [2]. We further anticipate that our approach to consolidating
26
grinding model will leas to a better unification of the large body of work in grinding,
leading to a more comprehensive control of the process.
5. Summary
This paper provides a framework to integrate the various grinding process models using a
bipartite graph. A common language that integrates the various research works has been
developed. The equations have been categorized into different zones for ease of modular
application. Also, it is shown through examples how these bipartite graphs can serve as
good qualitative tools for developing thorough research and experimentation strategies.
27
REFERENCES
[1] Palanna R. and Bukkapatnam S.T.S., “The concept of model-based tampering for
improving process performance: An illustrative application to turning process,”
International Journal of Machining Sciences and Technology, 6(2):263-282, 2002.
[2] Palanna R., “Model based tampering for improved process performance: An
application to surface grinding of shafts”, PhD Thesis, University of Southern California
Thesis Archives, Aug. 2002.
[3] Friedman, G. J., “Constraint Theory: an Overview,” International Journal of Systems
Science, vol. 7, no. 10, pp. 1113-1151, 1976.
[4] Malkin S., Grinding Technology: Theory and Applications of Machining With
Abrasives, Society of Manufacturing Engineers Press, 1989
[5] Eranki J., Xiao G., and Malkin S., “Evaluating the performance of ‘seeded gel’
grinding wheels,” Journal of Materials Processing Technology, vol. 32, pp. 609-625,
1992.
[6] Fawcett S.C. and Dow T.A., “Development of a model for precision contour grinding
of brittle materials,” Precision Engineering, vol. 13, pp. 270-276, Oct 1991.
[7] Mann J.B., Farris T.N., and Chandrasekar S., “A Model for Grinding Burn,” Society
of Automotive Engineers, pp. 370-381, 1997.
[8] Kannappan, S. and Malkin, S., “Effects of Grain Size and Operating Parameters on
the Mechanics of Grinding,” Transactions of the ASME Journal of Engineering for Industry,
pp. 833-842, Aug 1972.
[9] Thompson R.A., “On the Doubly Regenarative Stability of a Grinder: The Theory of
Chatter Growth,” Transactions of the ASME Journal of Engineering for Industry, May
1986, Vol.108, pp. 75-82
[10] Nakayama, K., Becker, J., and Shaw M. C., “Grinding Wheel Elasticity,” Journal of
Engineering for Industry, vol. pp. 609-614, May, 1971
[11] Hundt W., Kuster F., and Rehsteiner F., “Model-Based AE Monitoring of the
Grinding Process,” Annals of the CIRP, vol. 46, pp. 243-247, January 1997.
[12] Malkin S. and Koren Y., “Optimal In-feed Control for Accelerated Spark-Out in
Plunge Grinding,” Transactions of the ASME, vol. 106, pp. 70-74, Feb 1984.
[13] Kohli S., Guo C., and Malkin S.,“Energy Partition to the Work-piece for Grinding
with Aluminum Oxide and CBN Abrasive Wheels,” Transactions of the ASME Journal of
Engineering for Industry, vol. 117, pp. 638-646, May 1995.
28
29
Appendix 1: Nomenclature

Positive Rake Angle

Rake Angle

Deflection Of Beam
d
Dressing Angle
 max
Quasi Steady State Temperature

Mean Density

Time Constant
Ac
Chip Area
Aw
Wear Flat Area
ad
Dressing Depth
B
Wheel Width
C
Specific Heat
d
Depth Of Cut, In-Feed
dg
Average Grinding Dimension
ds
Grinding Wheel Diameter
dt
Truing Depth
dE
Embedding Dimension
F
Magnitude Of The Force Vector (A 2 Dimensional Vector)
f
Constant Between 0.4-0.9
fr
Feed Rate
G
Grinding Ratio
g
Grades Of Wheel
g0
Wheel Grade
30
lc
Length Of Wheel
L
Length Of Contact
N
Number Of Passes
Nkin
Kinematic Number Of Cutting Edges
Nr
Nose Radius
na
No. Of Active Edges
P
Power For Grinding
QW
Material Removal Rate
R
Radial Displacement
Ra
Surface Roughness (in)
rr
Mean Ratio Of Chip Width To Chip Thickness
r
Radial Advancement
Sd
Dressing Leads
t
Time
T
Chip Thickness
Tg
Grinding Time
UE
Specific Grinding Energy
ui
In-Feed Velocity
vs
Peripheral Grinding Wheel Speed
vw
RPM Of Work-Piece
vf
Peripheral Velocity
w
Grinding Width
x/l
Dimensionless Length
Z
Depth For Energy Partition
m
Work-Piece Mass
31
c
Damping
k
Work-Piece Stiffness

Acceleration Of Work-Piece
x1

x1
Velocity Of Work-Piece
Fd
Forcing On Work Piece

Forcing Frequency

Force Damping
x2(t)
Wheel Displacement At Time t
x3
Wheel Wear
AW
Area Of Wear On Grinding Wheel
x5(t)
Overall Wheel Displacement At Time t
32