ScienceDirect International Journal of Machine Tools & Manufacture 48 (2008) 832–840 pptx

International Journal of Machine Tools & Manufacture 48 (2008) 832–840
Dynamic model of a centerless grinding machine based on
an updated FE model
I. Garitaonandia
a,Ã
, M.H. Fernandes
a
, J. Albizuri
b
a
Department of Mechanical Engineering, Faculty of Mining Engineering, University of the Basque Country, Colina de Beurko s/n 48902 Barakaldo, Spain
b
Department of Mechanical Engineering, Faculty of Engineering, University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain
Received 20 August 2007; received in revised form 29 November 2007; accepted 4 December 2007
Available online 15 December 2007
Abstract
Centerless grinding operations present some characteristic features that make the process especially susceptible to regenerative chatter
instabilities. Theoretical study of these vibrations present some difficulties due to the large amount of parameters involved in the process
and, in addition, such a study requires a precise determination of dynamic properties of the particular machine under study. Taking into
account the important role of the dynamic characteristics in the process, in this paper both analytic and experimental approaches are
used with the aim of studying precisely the vibration modes participating in the response. Using as reference results obtained from an
experimental modal analysis (EMA) performed in the machine, the finite element (FE) model was validated and improved using
correlation and updating techniques. This updated model was used to obtain a state space reduced order model with which several
simulations were carried out. The simulations were compared with results obtained in machining tests and it was demonstrated that the
model predicts accurately the dynamic behavior of the centerless grinding machine, especially concerning on chatter.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Centerless grinding; Finite element; Experimental modal analysis; Model reduction
1. Introduction
Centerless grinding is a chip removal process in which
the workpiece is not clamped, but it is just supported by the
regulating wheel, the grinding wheel and the workblade

(Fig. 1).
This configuration allows a simple and easy way to load/
unload workpieces with minimal interruption of the
process, providing high flexibility in the sense that high
productivities together with precis e dimensional tolerances
of the parts can be obtained.
On the other hand, problems associated with roundness
errors are very common in these machines because of the
floating center of the workpiece. As a consequence, surface
errors of the workpiece, after contacting the workblade and
the regulating wheel, produce a displacement of its center
that can lead to an error regeneration mechanism.
Several researchers have studied the origin and evolution
of roundness errors [1–4]. These researches have shown
that instabilities are produced due to geometric and
dynamic causes. Geometric instabilities are specific of
centerless grinding and they are produced as a consequence
of the geometric configuration of the machine, independent
of the structural characteristics and the workpiece angular
velocity. Dynamic instabilities have their origin in the
interaction between the regenerative process and the
dynamic of the structure. In this case, self-excited vibra-
tions appear limiting the surface quality of the workpieces.
The study of this last problem requires an adequate
knowledge of the dynamic properties of the machine, so it
is of great assistance to have numerical models including
these properties in order to predict the dynamic response of
the machine-process system both in its original design and
in a design with modifications. For the purpose of
obtaining an adequate model, in this paper finite element

(FE) models correlation-updating techniques are used by
means of experimental data.
ARTICLE IN PRESS
www.elsevier.com/locate/ijmactool
0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2007.12.001
Ã
Corresponding author. Tel.: +34 946 014967; fax: +34 946 017 800.
E-mail address: (I. Garitaonandia).
2. System modeling
Previously to the developm ent of this work, Albizuri
et al. [5] studied the vibratory response of the machine
under study using a lumped mass model, which character-
ized the moving components guided by the two ball screw s
in the feed direction (see Fig. 1). These are the components
with larger vibration amplitudes in chatter conditions.
This model, which has the advantage of its simplicity, is
somewhat limited because it supposes several infinitely
rigid components of the machine as the bed, the grinding
wheel and the grinding wheel head, so it does not model
adequately the force transmission path from the cutting
point between the workpiece and the grinding wheel to the
contact point between the workpiece and the regulating
wheel.
Due to the mentioned limitation, in this work special
attention has been paid to the development of an updated
FE model that will predict the dynamic response of the
machine under operation conditions, givin g an insight into
the real behavior of different components.
2.1. FE model

Dynamic characteristics of centerless grinding machine
were studied by means of a FE model using ANSYS
software. This model, which consists of 53,200 nodes and
37,807 elements, is depicted in Fig. 2.
This figure shows the global coordinate system used in
the model, where z-axis was defined as the longitudinal axis
of the machine, x-axis as the transversal one and y-axis as
the vertical one.
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Nomenclature
f
FEM
mode shape obtained from FE model
f
EMA
mode shape obtained from EMA
U matrix of mode sh apes
X diagonal matrix of natural frequencies
n diagonal matrix of modal dampings
A, B, C, D matrices defining state space model
L
u
input forces influence matrix
q modal coordinate vector
u state space output vector
y state space input vector
x state space state vector
j
1
, j

2
, g, y, h centerless grinding geometric parameters
(see Fig. 8)

0
¼
sin j
2
sinðj
2
Àj
1
Þ
depth of cut increment due to unitary
waviness error of the workpiece at workblade
contact point
1 À  ¼
sin j
1
sinðj
2
Àj
1
Þ
depth of cut increment due to unitary
waviness error of the workpiece at regulating
wheel contact point
s Laplace operator
o
t

, o
p,
o
a
angular velocity of grinding wheel, work-
piece and regulating wheel, respectively (Hz)
K cutting stiffness (N/mm)
b workpiece width (mm)
k
0
cr
contact stiffness per unit width between grind-
ing wheel and workpiece (N/mm/mm)
k
0
cs
contact stiffness per unit width between reg-
ulating wheel and workpiece (N/mm/mm)
G(s) machine transfer function
Fig. 1. Scheme of the centerless grinding machine.
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 833
A total of 15 mode shapes and natural frequencies were
extracted in the 0–160 Hz frequency range. This range of
interest was defined taking into account that for the
grinding machine under study, in chatter conditions, the
most severe vibrations were observed in the neighborhood
of 60 Hz, while other less severe vibrations appeared at
about 130 Hz.
This model predicts adequately the elastic-inertial
properties of different components of the machine. A

major prob lem arises when introducing the stiffness and
damping properties of joint elements into the model
because when studying joints, there are a lot of uncertain-
ties that make not possible to model them precisely [6].In
order to overcome this obstacle, the development of a good
FE model requires the use of experimental data.
2.2. Experimental analysis
An impact testing experimental modal analysis (EMA)
was performed in the machine. Responses were measured
in 69 points using triaxial accelero meters, so aceleration/
force frequency response functions (FRF) corresponding
to 207 degrees of freedom were obtained. Fig. 3 illustrates
the geometry used in the analysis, in which the arrow shows
the excitation point and direction. This excitation direction
was selected in order to excite modes with high modal
displacement components in z direction.
The FRFs obtained in the analysis were studied in LMS
Cada-X software, obtaining 10 natural frequencies, mode
shapes and damping factors in the frequency range of
interest.
2.3. Theoretical–experimental correlation
Numerically obtained mode shapes were correlated with
the experimental ones in FEMtools software using the
modal assurance criterion (MAC) [7]:
MAC FEM; EMAðÞ¼
ðf
T
FEM
f
EMA

Þ
2
ðf
T
FEM
f
FEM
Þðf
T
EMA
f
EMA
Þ
. (1)
MAC values obtained between the first 15 numerical
mode shapes and the first 10 experimental ones are shown
in Table 1. In this table, four values of MAC corresponding
to as many paired mode shapes have been highlighted due
to their importance in the development of this work. The
first three pairs show MAC values above 85%, while the
last pair shows a lower value. Although these MAC values
point out that the correlation between the corresponding
numerical and experimental mode shapes is adequate, it
can be seen that there are significant differences in the
natural frequencies of these mode shapes, so it was
necessary to impr ove the FE model by means of an
updating process.
In the updating process, the three numerical natural
frequencies with higher MAC values were selected as
responses to be improved. To select adequate parameters

to be updated, a sensitivity analysis was performed and it
was concluded that the parameters most influencing the
estimation of the mentioned natural frequencies were the
stiffness values of the joint elements connecting the bed to
the foundation and the axial stiffness of the lower slide
ball screw.
These stiffness values were improved iteratively in order
to match the numerical natural frequencies to the experi-
mental ones. A Bayesian parameter estimation technique
was used [7] for this purpo se. Fig. 4 shows the MAC matrix
obtained after the updating process.
In this figure, it can be seen that an adequate corre-
lation remains between the previously paired mode shapes.
Moreover, Table 2 shows that the difference between the
updated natural frequencies and the experimental ones are
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Fig. 2. FE model of the machine.
Z
Y
X
Fig. 3. Geometry of the EMA.
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840834
small. Numerical frequency 13 also was improved in the
updating process.
3. Updated FE model characteristics
Once an updated FE model has been obtained, in the
development of a dynamic analysis it is important to study
the model in order to identify the modes with higher
contribution to the response in operation conditions. In
this analysis, it is necessary to take in mind that in

centerless grinding operations the normal force generated
in the cutting point between the workpiece and the grinding
wheel is produced mainly in z direction, so the relative
excitability of the different modes in that direction was
evaluated calculating their modal participation factors
(MPF). The result is shown in Fig. 5, where the MPF
have been normalized so that the largest value has unit
magnitude.
From this figure it can be concluded that there are three
modes with highest contribution to the dynamic response.
In the mode shape at 33.48 Hz all the components of the
machine move in phase in the longitudinal direction in a
suspension movement with respect to the supports of the
bed. The mode shape corresponding to natural frequency
of 58.59 Hz is the one which is excited normally when
chatter vibrations appear in the centerless grinding
machine, so it is called the main chatter mode. Fig. 6
shows the animation for this mode shape. It is seen that
this mode corresponds to an out of phase movement
between the heads of the two wheels.
The mode shape at 127.41 Hz corresponds to a mode,
which has been excited only in some tests performed in
the machine, always leading less vibration amplitudes than
the previous one.
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Table 1
MAC values before updating, in %
FE mode
shapes (Hz)
EMA mode shapes (Hz)

12345678910
(33.41) (48.43) (58.91) (76.96) (90.52) (108.44) (122.03) (128.48) (129.86) (144.02)
1 (30.16) 0.618 0.155 0.182 0.922 3.69 0.776 0.0768 0.18 0.0025 1.15e-4
2 (31.84) 95.7 5.45 18 0.0242 0.713 11.8 1.84 0.204 0.736 0.937
3 (45.9) 0.391 88 0.289 10.9 2.03 1.32eÀ3 0.0741 7.04 0.529 0.158
4 (54.85) 2.78 0.307 94.5 17.2 1.2 0.0131 5.53 5.8eÀ3 3.99 1.29
5 (67.38) 0.388 7.26eÀ5 0.753 7.88 6.07 10.9 8.6 2.99eÀ3 2.55 1.32
6 (73.5) 0.176 0.00821 0.0938 17.9 1.02 8.79 0.0144 0.217 0.669 8.12eÀ3
7 (82.27) 2.56 0.836 0.163 36.2 22.2 4.56eÀ4 0.0427 0.793 1.21 2.35
8 (93.03) 0.463 0.333 0.0339 0.242 19.3 7.9 10.9 0.49 0.916 4.92
9 (96.84) 0.472 0.212 0.423 5.5 25.2 42.8 42.1 8.44 4.76 8.03
10 (102.02) 0.901 0.0206 0.114 0.829 7.89 50.9 14 0.467 36.8 0.116
11 (113.28) 1.08eÀ3 2.4 0.0251 0.125 2.23 0.587 0.0959 0.451 0.322 0.316
12 (122.13) 0.565 0.234 3.76 1.49 0.987 1.08 10.9 0.734 17.7 8.2
13 (126.82) 0.0334 0.0806 2.43 0.747 0.332 0.0645 3.88 64.2 20.5 8.87
14 (149.88) 4.32 0.688 6.23 0.0856 0.229 3.93 1.35 3.23eÀ3 38.4 2.24
15 (159.44) 1.26 0.2 3.67 9.15 19.1 3.12 10.4 4.84 8.41 1.6
Bold numbers correspond to important mode shape pairs.
Fig. 4. MAC matrix after the updating process, in %.
Table 2
Comparison between updated numerical values and experimental results
Pair FE
mode
Hz EMA
mode
Hz Diff.
(%)
MAC
(%)
1 2 33.48 1 33.41 0.20 96.2

2 3 48.43 2 48.43 0.00 87.5
3 4 58.59 3 58.91 À0.55 93.2
4 13 127.41 8 128.48 À0.83 64.6
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 835
4. FE model order reduction
The large number of degrees of freedom of the updated
FE model implies computationally expensive calculations
in order to simulate the vibration behavior of the machine.
This restriction makes necessary the reduction of the size in
such a way that the reduced model and the original model
will have the same frequen cy response charact eristics in the
frequency range of interest.
The big size of the FE model was reduced using the
modal coordinate reduction [8], which is based on the
fact that the dynamic response of the centerless grinding
machine in the frequency range of interest is dominated by
the first 15 structural modes, so it is possible to simulate
its dynamic behavior using these modes and neglecting
the rest.
Several representative degrees of freedom were selected
defining the points in which application of forces or
acquisition of responses was required. Using as reference
the different lists obtained from the updated FE model,
truncated U matrix was created retaining the modal
contributions of the mentioned degrees of freedom for
the first 15 mode shapes. The first 15 updated natural
frequencies were used to create X matrix and the damping
properties obtained experimentally were used to construct
n matrix.
These matrices were used in MATLAB environment

to obtain a modal model of the structure in state space
defined by
_
x ¼ Ax þ Bu;
y ¼ Cx þ Du:
(2)
The state vector was selected as follows [9]:
x ¼
Xq
_
q
"#
. (3)
In Eq. (3), the size of the state vector (and thus the order
of the modal model) is twice the modes included in the
model, being this size much less than the order of the
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Fig. 5. Modal participation factors.
Fig. 6. Main chatter mode animation.
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840836
updated FE model used as reference. The resulting state
space model A and B matrices are
A ¼
0 X
ÀX À2nX
"#
; B ¼
0
U
T

L
u
"#
. (4)
C and D matrices of system (2) depend on the required
outputs, so the described model can be used to simulate
displacements, velocities and accelerations of selected
degrees of freedom.
FRFs acceleration/force obtained both experimentally
and using the state space modal model between input degree
of freedom j and output degree of freedom k (see Fig. 2)
were compared. The result is displayed in Fig. 7 .
This figure shows that the state space modal model
reflects adequately the system dynamic below approxi-
mately 70 Hz, whi le above this frequency the discrepancies
with the experi mental results are higher. Likewise, it can be
seen that both the FRFs show three important resonance
peaks corresponding to the mode shapes with higher MPF
obtained in Fig. 5.
5. Experimental verification
In order to evaluate the effectiveness of the state space
reduced model obtained in previous section, it was used to
perform a theoretical study of chatter instabilities in the
centerless grinding machine under study. Fig. 8 shows a
detail of the configuration of the machine and Fig. 9 shows
the block diagram used to study its stability, which is based
on similar diagrams presented in previous works [3,4].
The term G(s) takes into account the dynamic flexibility
of the machine and it was obtained considering the three
modes with major dynamic contribution in feed direction

(see Fig. 7). This idea was carried out using controllability
and observability criteria [10]. The state space modal model
defined by Eq. (2) was transformed in the balanced
realization, in which the controllability and observability
matrices are equal and strictly diagonal, being the terms of
the diagonal a quantitative measure of the relative
importance of the different states in the input–output
behavior of the system. This realization was divided into a
dominant subsystem formed by the six more controllable
and observable states, and a weak one, formed by the rest
of the states. This last subsystem was eliminated residualiz-
ing the least controllable and observable states in order to
include their static contribution in the response [11].
5.1. Analysis of chatter frequencies
The characteristic equation of the block diagram shown
in Fig. 9 is
1 À e
À2pðs=o
p
Þ

K
ð1 À Þ
bk
0
cr
þ
1
bk
0

cs
þ GðsÞ
!
þ 1 À 
0
e
Àj
1
ðs=o
p
Þ
þð1 À Þe
Àj
2
ðs=o
p
Þ
¼ 0. ð5Þ
To guarantee stabl e cutting conditions, all the roots of
this equation must be in the left side of the complex plane.
If one of the roots is located in the right side of the complex
plane, the system is unstable and during the grinding
process the response grows in time causing the regeneration
of a roundness error in the workpiece.
The complete resolution of the roots of the characteristic
equation is not an easy task due to the transcendental
nature of the equation to be solved, formed by three time
delays, so there are infinitely many solutions satisfying it.
In this application, these roots were solved using the
root locus technique [12], obtaining the solutions of the

characteristic equation (Eq. (5)) for increasing values of
cutting stiffness in the 0-
N
range. This technique plots
the evolution of the different roots, so it can be determined
which one becomes unstable.
The application of this method requires the resolution of
the characteristic equation for a zero cutting stiffness value.
These solutions are:
 the poles of the transfer function G(s),
 an infinite number of poles of ð1 À e
À2pðs=o
p
Þ
Þ located at
minus infinity,
 the roots of the geometric characteristic equation
1 À 
0
e
Àj
1
ðs=o
p
Þ
þð1 À Þe
Àj
2
ðs=o
p

Þ
¼ 0
ÀÁ
. The initial esti-
mations of these roots were obtained using an iter-
ative graphical procedure, which consisted in modifying
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Fig. 7. FRFs between j–k degrees of freedom.
Fig. 8. Geometry of centerless grinding.
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840 837
sequentially the values of the possible roots until the real
and imaginary parts of the geometric characteristic
equations were annulated. These initial estimations
were used to obtain the final solutions using Newton–
Raphson method [13].
The first set of roots obtained for K ¼ 0 were used as
initial estimations for the next increment of the cutting
stiffness using Newton–Raphson method, and so on until
reaching the cutting stiffness value obtained experimentally
for the particular geometric configuration of the machine
under study.
In order to compare the results obtained theoretically
with those obtained experimen tally, several simulations
were performed programming geometric conditions with
which previously cutting tests had been done in the
machine. These conditions are shown in Table 3.
Contact stiffness values assumed in the simulations
corresponded to typical values for centerless grinding of
steel components using a vitrified grinding wheel together
with a rubber-bonded regulating wheel [13].

For illustration purposes, Fig. 10 shows the evolution of
the root loci for a workpiece angular velocity of 6.2 Hz. In
this figure, structural pole on 58.59 Hz migrates towards
the imaginary axis for increasing values of cutting stiffness
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Fig. 9. Block diagram of centerless grinding.
Table 3
Cutting conditions
Workblade angle, y 301
Center height angle, g 81
Regulating wheel diameter 340 mm
Grinding wheel diameter 628 mm
Workpiece diameter 46.2 mm
Workpiece width 368 mm
Fig. 10. Root locus for increasing cutting stiffness. o
p
¼ 6.2 Hz.
Fig. 11. Comparison between theoretical and experimental chatter
frequencies.
I. Garitaonandia et al. / International Journal of Machine Tools & Manufacture 48 (2008) 832–840838
until the pole crosses it and thus, it is instabilized. Chatter
frequency is determined by the imaginary part of the
unstable root after concluding the loci.
This procedure was repeated for different values of
workpiece angular velocity in the 0–20 Hz range. Chatter
frequencies obtained theoretically were compared with the
experimentally measured ones, as it is shown in Fig. 11.
This figure shows that theoretical predictions are in
agreement with experimental results.
5.2. Time domain simulations

The reduced order model validation was completed with
various time domain simulations in order to qua ntify the
number and the amplitudes of the undulations produced in
the workpiece in chatter conditions. The workpiece was
discretized in 360 equal segments and its rotation was
simulated segment by segment. Taking as reference the
process block diagram shown in Fig. 9, time evolution of
the roundness errors of the workpiece was obtained and in
each segment rotation, integrating numerically this evolu-
tion using Runge–Kutta algorithm [14], the errors were
calculated as a function of the dynamic response of the
machine and the errors in the previous pass and in the
contact points with the workblade and the regulating
wheel. Nonlinear effects as contact loss between the
workpiece and the grinding wheel and spark-out process
were taken into account [15]. The simulations were done
programming a regulating wheel infeed rate of 1 mm/min, a
total feed of 0.2 mm and a spark-out time of 2 s. Fig. 12a
shows the final theoretical profile obtained for a workpiece
angular velocity of 6.2 Hz, while Fig. 12b shows the real
profile obtained under the same conditions programmed in
the simulations.
It is shown that workpiece profiles obtained both theore-
tically and experimentally are quite similar. Moreover,
theoretically predicted roundness error is within the same
order of magni tude of the experimentally measured one.
6. Conclusions
In this work, a dynamic model of a centerless grinding
machine has been developed performing a detailed study of
mode shapes that are excited in chatter conditions.

The combined use of numerical FE model updating
techniques via experimental modal data and model reduc-
tion techniques resulted in a state space model representing
adequately the modes with major modal contribution in
machine vibrations. The presented methodology advances
the state-of-the-art in modeling procedures of centerless
grinding machines.
Simulations demonstrated that the model predicts
accurately both the appearance of chatter vibrations for
different machine configurations and the time evolution of
workpiece roundness errors under unstable operation
conditions. Thus, this model represents a powerful tool
to define optimal set up conditions in order to increase the
productivity in centerless grinding practice.
Acknowledgments
The authors are grateful to IDEKO Technological
Center for the provision of numerical and experimental
facilities to conduct this work.
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Fig. 12. Final workpiece profile: (a) theoretical and (b) experimental.
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