Annals of the CIRP Vol. 56/1/2007 -347- doi:10.1016/j.cirp.2007.05.080
Intelligent Centerless Grinding: Global Solution for Process Instabilities and Optimal
Cycle Design
I. Gallego (3)
Manufacturing Department, Faculty of Engineering – Mondragon University, Mondragon, Spain
Submitted by R. Bueno (1), San Sebastian, Spain
Abstract
Centerless grinding productivity is largely limited by three types of instabilities: chatter, geometric lobing and
workpiece rotation problems. Regardless of its negative effect in manufacturing plants, no functional tool has
been developed to set up the process, because it involves the simultaneous resolution of several coupled
problems. In this paper, new simulation techniques are described to determine instability-free configurations,
making it possible to guarantee that the final workpiece profile is round. With this information and taking into
account other process restrictions, like system static stiffness and workpiece tolerance, the optimal grinding
cycle is designed. These results have been implemented into an intelligent tool to assist the application of this
research in industrial environments.
Keywords:
Centerless Grinding, Productivity, Simulation
1 INTRODUCTION
In centerless grinding, the workpiece is not clamped, but
simply supported between the grinding wheel, the blade
and the regulating wheel (figure 1), reducing the machine
idle time and avoiding the necessity of centring holes on
the workpiece. Due to the enormous manipulating time
and manufacturing cost saving that this implies,
centerless grinding is extensively used in the mass
production of components in automotive and bearing
industries for example.
Nevertheless, the process suffers from three kinds of
instabilities that may limit its precision and productivity.
1) Chatter, whose growing is much more pronounced than
in conventional grinding.

r
w
(t) be the radius and radius defect of the
workpiece in an infeed process (figure 1). The radius
defect varies due to strictly geometric reasons, because
the cutting forces cause changes in the deflection of the
machine, workpiece and wheels, and because vibrations
may arise. This way,
G
r
w
(t) can be expressed as:








ttttr
DKgw
HHHG
 (1)
where
H
g
(t) is the geometric displacement of the
workpiece due to roundness errors passing through the
contact points with the blade and regulating wheel.

rw
WG
tr the radius defect at the
contact points with the blade and regulating wheel.
g
b
and
g
r
are two geometrical parameters [1].
Geometric lobing stability has been studied in the
frequency domain [6,7]. Applying the Laplace transform to
equation (2), it is obtained:
T

J
r
h
M


M


J

J
r

J


)(
ewc
taktF (4)
The cutting force produces a deflection of the machine,
wheels and workpiece, which makes the radius reduction
to accumulate a delay in relation to the programmed feed.
From a stability point of view, it is not a matter of interest
to know the delay, but the variations in deflection
generated by the radius defect evolution. Defining
W
as
the rotation period of the workpiece, it is obtained:

trtrktF
wwwc
GWGG
(5)
The second term of equation (1) can be expressed as:


eq
c
K
k
tF
t
G
H
(6)

) and the
wheels/work contact stiffness (k
cs
, k
cr
). Determination of
equivalent stiffness and its dependencies on feed rate,
wheel type, etc. is essential to predict instabilities
accurately [8,9].
Regarding cutting stiffness, its value may be estimated
with analytical approximations, but a final experimental
calibration is recommended to have chatter prediction
maps and the optimal grinding cycle very close to reality.
Introducing expression (5) in equation (6):

trtrKtrtr
k
k
t
wwww
eq
w
K
GWGGWGH

(8)
K parameter represents the flexibility of the system and
relates the amount of deformation of the system to
different depths of cut.
Centerless grinding chatter was extensively studied by



Ư
á
á
ạ
ã
ă
ă
â
Đ


m
1
2
r
r
rr
22
r
r
wwD
2
1
N
r

rr
T
r
(10)
{X
r
}: vector containing the relative deformations at the
contact points of r mode.
{C}: vector quantifying the real displacement at the cutting
point due to a displacement of the contact points.
{P}: vector relating the forces at the contact points with
the normal force at the cutting point.
The last term in equation (9) is the referred correction to
H(s). We consider that this is an essential enhancement
of preceding models, as it has a significant influence on
the dynamic stability maps shown in section 4. In addition,
it should be pointed out that when vibrations are
introduced in the model, it is very important not to
disregard the static term
H
K
(t), because it may be proved
that, in that hypothetical case, geometric lobing could
appear with lobe numbers very far from integer, which is
not physically possible.
Rearranging all the terms deduced in equation (1):





ằ
ằ
ẳ

ô
ô
ơ
ê
Ư









W
W
W
W
ZZ[Z
s
N
r
s
s
s
e

<0. For that reason, it is necessary to find
the roots of the next function:



W
W
W
ZZ[Z
s
N
r
s
s
e
V
ss
V
kK
egegs





á
á
ạ
ã
ă

w
rb
(12)
In the next section, the methodology to get all the
significant roots of f(s) is shown.
All these equations have been deduced for plunge
processes. The adaptation of the model for throughfeed
processes was explained by Meis [18] and Gallego et al.
[2].
Other remarkable approaches to improve productivity can
be found in the bibliography, like the one recently
proposed by Klocke et al. [19], which involves working
below center so that higher feed rates can be employed,
-349-
but using a new type of functional blade to avoid
geometric lobing. Other authors, like Harrison and Pearce
[20], have proposed changing machine configuration in-
process to allow a faster correction of the initial
roundness error of the workpiece.
Finally, it should be mentioned that the equations to
establish the limits for work rotation instabilities were
deduced by Hashimoto et al. [21].
2.2 Instabilities determination
In contrast to milling process, where it is just necessary to
know the chatter limits, in centerless grinding it is also
necessary to determine the absolute value of the stability
degree of the process. This is because at the optimal
configuration, where all the lobes are stable with the
maximum possible stability degree, initial roundness error
correction is faster. This way, the process is less sensitive

D
=0 axis. Those
configurations with positive real part roots (i.e.
[
<0), will
be unstable. Those configurations with negative real part
in all the roots, with the highest possible absolute value,
will be the optimal configurations.
The solution to this problem is to use an appropriate
mesh in the (
D
,
E
) plane and then, starting from each point
of the mesh, apply the best possible optimisation
algorithm to find the closest minimum as fast as possible.
With regard to the mesh, it is easy to demonstrate that
the characteristic function can not have two different
minima for the same value of
E
near
D
=0. This way, the
mesh in
D
can be avoided. The optimum mesh is a row of
points at
D
=0 from
E

it is necessary to determine the first and second
derivatives of the function analytically, leading to quite
complex expressions. Nevertheless, by rearranging terms
it is possible to include as many modes as desired in the
function without excessive complication of expressions.
The advantage of this technique is that it is possible to
determine whether a certain working configuration is
stable or not considering all instabilities in less than 0.1
seconds in an average computer. Repeating the calculus
for many configurations it is possible to plot stability maps
like the ones shown in the next sections.
Figure 2: |f(s)|
2
function.
3 GEOMETRIC LOBING SUPPRESSION
The two previous works on geometric lobing in infeed and
throughfeed [1,2] have led to the development of a
commercial set-up software, called Estarta SUA (Set Up
Assistant). As an example, in figure 3 a stability map of
the process is shown as represented in Estarta SUA.
Stability maps are 2D or 3D graphs that define stable and
unstable areas for different set-up parameters. In the
case of geometric lobing, stability maps are plotted as a
function of the blade angle (ș) and workpiece height
above centre (h), two variables that are easily controlled
by machine operators. Figure 3 has been obtained for the
next conditions: wheels and workpiece diameters
D
s
= 630 mm, D

obtain stability maps for any combination of blade angle
(ș), workpiece height (h) and regulating wheel rotating
frequency (
Z
r
), including at the same time geometric and
dynamic phenomena.
Figure 3: Geometric lobing stability map. In blue: stable
areas. In red: unstable configurations.
0 5 10 15 20 25
50
40
30
20
10
Height (mm)
Blade Angle (º)
5
7
36
32
28
24
20
16
31
35
22
18
33

36
34
9
36
26
10
15
10
5
0
0
0.5
-0.5
10
15
20
D
E
-350-
0 5 10 15
Height (mm)
Figure 4. Chatter and geometric lobing stability map. In
dark blue: stable areas. In red: unstable areas. Star size
is proportional to the experimental vibration amplitude,
round points represent tests without chatter.
In figure 4 a stability map is shown as a function of h and
Z
r
for a blade angle of 30º. Red areas represent
configurations susceptible to chatter, light blue zones

wheelhead opening frequency: 58.3 Hz;
T
= 30º; D
s
=
= 628 mm, D
r
= 340 mm; D
w
= 47 mm; L
w
= 368 mm;
K
eq
= 69.7 N/Pm; Q’ = 1.23 mm
2
s
-1
; feed: 1 mm min
-1
.
In the theoretical maps obtained for these cases, chatter
free and geometric lobing free areas can be observed
when using low heights and workpiece rotation speeds,
as well as some transient areas at higher rotation speeds.
There are also stable areas elsewhere, but they are too
small from a practical point of view to be used.
For machine 1, the map is checked with experimental
results in figure 4. The size of the stars is proportional to
the experimental vibration amplitude, while the round dots

example when high feed velocities or dull wheels are
employed, the workpiece dragging becomes unstable,
generating shakes, irregular velocities, jumps and
accelerations, with risks for machine operators.
In the absence of other kind of instabilities, these
phenomena are the limiting factor to productivity, because
the process feed defines the cutting force exerted by the
grinding wheel and the required brake moment to control
the movement of the workpiece. On the other hand,
another function of the regulating wheel is to rotate the
workpiece before beginning the grinding process. If the
workpiece does not rotate, a flat band may be generated
in the periphery of the workpiece.
As mentioned before, the model and equations describing
these phenomena were fully developed by Hashimoto et
al. [21]. The model is conditioned by how precisely the
values of the friction coefficients in the contacts (
P
b
,
P
r
)
are introduced. With this objective two methodologies
have been developed: one in laboratory and another in
situ in the process. In laboratory, tribometer
measurements have been performed with disc-on-disc
geometry, due to its similarity with the real process, using
pressures at the contact area identical to real processes.
To obtain the values in situ, two force sensors in the

8
10
12
14
16
18
20
22
21
19
9
11
13
15
17
21
19
9
11
13
15
17
7
Regulating wheel speed (min
-
1
)
-5
-351-
Rough dressing Fine dressing

1,6 45 0,25 1,6 52 0,20
3,2 59 0,25 3,2 73 0,20
5,7 72 0,25 5,7 89 0,20
8 86 0,24 - - -
Table 2: Relationship between tangential and normal
forces at regulating wheel-steel contact as obtained in a
sensorised grinder for different dressing conditions and
pressures.
In a future paper, a complete study of friction coefficient
values for different wheels and conditions will be shown.
6 OPTIMAL CYCLE DESIGN
The main reference on optimal cycle calculation in
grinding processes is the work developed by Malkin [25].
Based on this work, the author has successfully
developed the GrindSim software to set-up and optimise
cylindrical grinding processes.
Two possible criteria can be used to design a centerless
grinding cycle: 1) Minimise the process cycle time or 2)
Adjust the cycle to a previously established process time,
minimising wheel wear. This last option is very common in
production lines.
The process parameters to be optimised include the feed
for each stage of the infeed cycle, the stock removal in
each stage and the spark-out time. The restrictions to
apply are given by the maximum power to employ in the
roughing process free from burning problems [25] and the
required tolerance and roughness of the final workpiece.
To define the optimal cycle, the feeds to use are fixed
first. The first feed will depend on the criteria chosen to
optimise the cycle. In the first case, it is deduced from the

programmed final position of the wheel exceeds slightly
the desired dimension of the workpiece (figure 6b), in the
same quantity as the average accumulated radius defect
at the end of the process.
CONCLUSIONS
The main conclusions of this work can be summarised as
follows:
1. Both an enhanced model and computation algorithm
for centerless grinding have been developed.
2. All the process instabilities can be predicted together.
At the optimal configuration, roundness correction is
faster, making the process less sensitive to changes
in the quality of entering workpieces.
3. Grinding cycles have been designed to obtain the
required workpiece tolerance at minimum production
time or, alternatively, minimum wheel wear.
Practical application of these results may increase
significantly precision and productivity of many industrial
processes, reducing set-up time and decreasing
production stops motivated by chatter or out-of-roundness
issues.
0
2 4
6
8
0
2 4
6
8
(a)

Transition Zone
Spinning Zone
Blade angle (º)
-5 0 5 10 15
10
15
20
25
30
35
40
45
50
Height (mm)
-352-
To assist the implementation of grinding simulation in
industry, an intelligent software tool has been developed
(Estarta SUA), which can be incorporated into the CNC
control of centerless grinders.
7 ACKNOWLEDGMENTS
This work has been carried out with the financial support
of the Basque Country Government (projects UE 2005-4
and IT-2005/043) and the Spanish Government (projects
FIT-020200-2003-72 and DPI2003-09676-C02-01).
The author wishes to acknowledge his colleagues from
Ideko Tecnological Centre (R. Lizarralde, D. Barrenetxea
and G. Aguirre), Mondragon University (J. I. Marquínez, J.
Madariaga and R. Fernández), Estarta (I. Muguerza) and
Manhattan Abrasives (P. Cárdenas) for their contribution
to this work.

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the Relationship between Setup Parameters &
Lobing behavior in Centerless Grinding, Annals of
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[11] Miyashita M., Hashimoto F., Kanai A., 1982,
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223.
[12] Rowe, W.B., Bell, W.F., Brough, D., 1986,
Optimization studies in high removal rate centreless
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