Simulation of an active vibration control system in a centerless grinding
machine using a reduced updated FE model
M.H. Fernandes
a
, I. Garitaonandia
a,
Ã
, J. Albizuri
b
, J.M. Herna
´
ndez
a
, D. Barrenetxea
c
a
Faculty of Mining Engineering, Department of Mechanical Engineering, University of the Basque Country, Colina de Beurko s/n, 48902 Barakaldo, Spain
b
Faculty of Engineering, Department of Mechanical Engineering, University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain
c
IDEKO Pol. Industrial de Arriaga, 2. 20870 Elgoibar, Spain
article info
Article history:
Received 22 August 2008
Received in revised form
28 October 2008
Accepted 13 November 2008
Available online 27 November 2008
Keywords:
Centerless grinding
Active control

conditions can differ from the chatter-free configurations.
Other procedures have been based on structural modifications
to stiffen the most flexible components in the force transmission
loop [6,7]. This stiffening produces an increase in the first
resonant frequency, widening the low workpiece rotational speed
stability zone. Nevertheless, such an alternative involves a
redesign of the machine structure, giving rise to solutions that
can be economically unfeasible.
Taking into account the above-mentioned limitations, Albizuri
et al. [8] proposed a novel approach based on the application of
active vibration control. Using commercial piezoelectric actuators
(A) in a feedback loop, they reduced the structural vibration level
and, consequently, the roundness errors of the workpieces.
Although the results they obtained were promising, they were
presented without a previous mathematical development, which
would allow them to optimise the active vibration system by
means of simulations, concerning both the programmed con-
trollers and the used sensors and actuators.
Later, Garitaonandia et al. [9] characterized the dynamic
properties of the machine combining finite element (FE) model
updating and model order reduction techniques. The proposed
approach was effective for the estimation of machine dynamic
behaviour under usual cutting conditions, but the study was not
extended to the simulation of any active vibration control scheme.
The availability of a theoretical model describing both the
structural and the controller characteristics is of major interest, as
it permits one to predict the effectiveness of different control
alternatives, giving insight into their behaviour before their
practical implementation and providing valuable information to
select the most suitable one. Therefore, the high costs (both

plate was presented in the longitudinal Z direction (see Fig. 1).
Other less-severe vibrations were detected at about 130 Hz when
the secondary chatter mode was excited, dominated by a local
bending deformation of the workblade.
3. Integration of the actuators in the FE model
To simulate the active control system, it is necessary to modify
the FE model in order to incorporate the piezoelectric actuators.
With this purpose, the design presented in [8] was followed,
where two PI-247.30 piezoelectric actuators had been placed in
the upper spindle support (D) area to obtain a strong control
authority on the main chatter mode. This design is depicted in
Fig. 2, where it can be seen that the integration of the actuators
requires a design modification of the upper spindle support in
such a way that two holes are made in it.
Each actuator, which is located collinear to a force transducer
(B), exerts the control force on its left side (according to Fig. 2)
over support D and on its right side over an auxiliary component
(E) that connects the two actuator-force transducer groups. This
component transfers the two control forces to the lower spindle
(C). Therefore, both actuators are placed in a parallel configuration
in the loading path from the lower slide to the swivel plate, and
they transmit the force to the lower spindle in a series
configuration.
Piezoelectric actuators like the ones used in this application
consist of n-stacked ceramic layers of PZT material (lead–zirco-
nate–titanate) that change in length when electrically charged. An
important aspect to consider is the modelling of the force they
exert. The relation among the externally applied voltage (V
pzt
), the

where b is the influence vector of the axial end displacements of
the actuator.
Substituting Eq. (2) into Eq. (1) and considering the centerless
grinding machine structure formed by its mass matrix M, its
damping matrix C and its stiffness matrix K, the incorporation of
the forces exerted by the actuators in the equation of motion
governing the dynamic equilibrium leads to [11],
ARTICLE IN PRESS
Fig. 1. Updated FE model of the centerless grinding machine.
Fig. 2. Detail of piezoelectric actuators location area.
M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245240
M
€
z þ C
_
z þðK þ K
pzt
b
1
b
T
1
þ K
pzt
b
2
b
T
2
Þz ¼ L

can be properly modelled through an axial stiffness K
pzt
between
its ends (modelling the passive behaviour) and a pair of opposite
forces of value K
pzt
d
33
nV
pzt
applied axially in the same ends
(modelling the active behaviour).
Following the design illustrated in Fig. 2, the lower spindle area
of the FE model (Fig. 3a) was modified in order to incorporate the
actuators, as shown in Fig. 3b. Each actuator was modelled with
an element having the same stiffness as the piezoelectric stack
and two lumped masses of half the total mass of the actuators
were placed at their ends. The actuators were connected with
the upper spindle support and with two elements modelling
the auxiliary component. These elements were idealized as
undeformable because it was considered that the displacement
of the lower slide relative to the swivel plate was much bigger
than the deformation of the auxiliary element. Finally, the rigid
elements were connected to the lower spindle.
It was verified that these structural modifications did not alter
appreciably the natural frequencies corresponding to the main
and the secondary chatter modes; hence, the passive behaviour of
the structure remained basically unchanged.
4. Definition of the control strategy
The active damping control strategy, which had been im-

x
f
), are located in the complex plane to produce an adequate
migration of closed loop poles as the feedback gain g is increased.
The minus sign in Eq. (4) is included to produce a negative
feedback of the acceleration.
As is shown in Fig. 4, the accelerometer is located very close to
the actuators. The control configuration, where the measurement
is done in the same degree of freedom (dof) as the excitation, is
called the collocated control, and it is very demanding in practical
applications because it enjoys the very attractive property of
unconditional stability [12,13]. Physical constraints avoid exact
collocation in this application.
5. Reduction procedure
Once the FE model has been modified incorporating the active
elements and the control strategy has been described, it is
necessary to apply a reduction procedure to complete the required
simulations at a reasonable computer cost. For this purpose, the
modal truncation method was used [14], which is based on
characterizing machine dynamics by the dominating vibration
modes within the frequency range of interest. To cover the main
and the secondary chatter modes, this frequency range was
established between 0 and 160 Hz, where 15 vibration modes
were calculated. For these modes, both natural frequencies and
modal displacements corresponding to dof’s, where application of
forces or acquisition of responses was required, were selected.
Following the procedure described in [9], the extracted modal
parameters, together with modal damping factors obtained
experimentally, were used to obtain a state space model of order
30 designed for the purpose of predicting displacements,

space representation was obtained, described by the three most
controllable and observable modes (6 states), defined by
_
x ¼ A
r
x þ B
r
F
n
K
pzt
d
33
nV
pzt
()
, (5)
y
m
a

¼ C
r
x þ D
r
F
n
K
pzt
d

one responsible for the appearance and evolution of the most
important roundness errors in workpieces, so
o
f
was adjusted to
55 Hz. Filter damping was fixed in
x
f
¼ 0.5 [12].
6.1. Evolution of structural roots
A very important feature to consider in the simulation process
is the choice of the feedback gain, as it modifies the structural
dynamic behaviour, changing closed loop roots location. The
evolution of these roots for increasing values of feedback gain is
shown in Fig. 6, where the trajectories of compensator poles and
structural roots selected in the balanced truncation process can be
distinguished. The structural roots correspond to the following
vibration modes:

a suspension mode at 209 rad/s (33.3 Hz),

the main chatter mode at 363 rad/s (57.8 Hz),
ARTICLE IN PRESS
Fig. 5. Feedback loop.
Fig. 6. Root locus for increasing feedback gain.
Fig. 7. (a) Evolution of the root corresponding to the main chatter mode and (b) comparison of receptance FRFs.
M.H. Fernandes et al. / International Journal of Machine Tools & Manufacture 49 (2009) 239–245242

the secondary chatter mode at 797 rad/s (126.8 Hz).
None of the trajectories shown in Fig. 6 crosses the imaginary

system for different feedback gain values to assure that admissible
limits are not exceeded. For this purpose, time domain simulation
of the process was programmed, which is well suited to obtain
such quantitative values. This procedure implies integrating the
control loop shown in Fig. 5 in the chatter loop of the centerless
grinding process, previously presented by several authors [6,9].
Fig. 8 shows the integrated model, where
e
0
,(1À
e
),
j
1
and
j
2
are variables depending on the geometric configuration of the
machine, s is the Laplace operator,
o
p
is the angular velocity of the
workpiece, K is the cutting stiffness and k
eq
is the equivalent
contact stiffness. The control system influence over the structure
ARTICLE IN PRESS
Fig. 8. Centerless grinding process chatter loop with control algorithm integrated.
Fig. 9. Theoretical evolution of accelerations: (a) at the grinding wheel head, (b) at the regulating wheel head and (c) at the workblade.
Fig. 10. Experimental evolution of accelerations: (a) at the grinding wheel head, (b) at the regulating wheel head and (c) at the workblade.

This study was undertaken using the normal grinding force
evolution as input in the developed state space model to obtain
some acceleration predictions as outputs. Figs. 9a–c show the
theoretical evolution of accelerations in three dof’s located at the
grinding wheel head, the regulating wheel head and the work-
blade, respectively, both before and after the application of control
law. Experimental measurements of the same variables are shown
in Fig. 10 .
Theoretical results agree with experimental measurements
quite well, as they predict adequately the quantitative values of
accelerations in different components of the machine. Further-
more, the maximum reachable vibration reductions in these
components are predicted correctly. Additionally, it can be seen
that no vibration reduction is obtained in the workblade, as it
could be expected taking into account that vibration of this
component is dominated by the secondary chatter mode, which
cannot be actively damped (Fig. 7b). In this case, simulation
results show even an increase in acceleration amplitudes when
control is applied.
6.4. Comparison of final workpiece roundness errors
The final profile of the workpiece gives a quantitative
measurement of the maximum error reduction that can be
achieved with the SOF controller. Fig. 11a shows the theoretical
shape simulated before the application of the control law whereas
Fig. 11b illustrates the profile after its application. Figs. 12a and b
show the profiles obtained experimentally.
The theoretical model predicted a roundness error reduction of
41.7%, whereas the experimental error reduction had been of 32%.
This result shows that computer calculations were highly realistic,
which is a statement that is reinforced comparing theoretical and

field of design of controllers integrated in machine tool structures.
The availability of the developed model is an essential require-
ment to tackle an optimisation process of the active vibration
control system in the centerless grinding machine, as it permits
one to evaluate in the design stage how different control
algorithms contribute to improve the dynamic behaviour of the
machine.
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