Vibration damping of turbomachinery components with
piezoelectric transducers: Theory and Experiment
B. Mokrani 1 , R. Bastaits 1 , R. Viguié 2 , A. Preumont 1 ,
1 Universit Libre de Bruxelles, Active Structures Laboratory,
Av. F. D. Roosevelt 50, B-1050, Brussels, Belgium
e-mail: bmokrani@ulb.ac.be
2
SAFRAN, Techspace Aero,
Route de Liers 121, B-4041, Lige, Belgium
Abstract
The first part of this paper discusses a design rule for the RL-shunts circuits used for vibration damping of
a bladed drum. These rules take advantage from the complexity and variability of the modal behavior of
the bladed drum to overcome the high level of uncertainties affecting the structure. In the second part, the
dynamic behavior of the bladed drum is described briefly. The performances granted by the proposed RL
shunt design are supported by experiments carried out on the prototype of a bladed drum. The proposed
solution proves very simple, efficient and robust with respect to all parameter change in the system.
1
Introduction
Air transport is facing two major conflicting requirements in the ever-increasing demand, both for individuals and goods, and the ever-increasing ecological standards and price of energy. This calls for innovative
solutions in the design and operation of aircrafts, one of the main axes of improvement being the reduction
of weight by increasing its functional efficiency. It can be performed through several aspects, such as the
use of new materials, of lightweight structural design and of smart structures (see [8]). These 3 aspects are
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coupled in our case study: the Bladed druM (or BluM⃝
). It consist of the low pressure stage rotor of a jet
engine developed by SAFRAN Techspace Aero (Fig.7) [3][4] .
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Compared to the classical design of the same engine part, the new monolithic design of the BluM⃝
provides
a mass reduction up to 25% at the expense of a very light natural damping of the blade modes (∼ 0.01%)
and inability of the use of conventional techniques of damping such as: adding viscous materials in the joints
between the blades and the drum, or using friction rings [6]. Under airflow loads during rotation, the blade
modes are subjected to high vibration levels which reduces the aerodynamics performance of the structure.
This leads to high cycle fatigue of blades eventually causing cracks and leading to major damage to the entire
engine. Therefore, a damping device should be integrated into the structure to reduce the response of the
most excited modes.
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In this paper, we propose to use shunted piezoelectric transducers as a damping system of the BluM⃝
[9]. The system consists of a set of piezoelectric patches glued on the support and connected to electrical
shunts. Only passive shunts are allowed because of the complexity to provide external power supply to the
rotating structure. Thus, the use of active circuits is no longer possible and the classical RL shunt [11] is
the best candidate comparing to other passive shunts such as pure resistive R-shunt wich has a very poor
performances or Synchronized Switch Damping SSD shunts known to be very complex to implement [1]
[10]. Indeed, the Linear RL shunt takes its advantage from its simplicity and its high performance.
The second section of this paper describes the classical linear RL shunt basics and shows the effect of the
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circuit mistuning on the obtained damping. A bladed rail vibration damping system is described in the third
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section and the design rules of the RL-shunt circuits are deduced. The BluM⃝
dynamics is highlighted in
section 4, and the experimental validation of the proposed damping technique is depicted in section 5.
2
2.1
Damping with RL-shunted piezoelectric transducers
Constitutive equations
Let us consider the uni-dimensional spring-mass system of Fig.1.a. The mass M is supported by a linear
piezoelectric actuator consisting of a stacking of n identical piezoelectric elements polarized through their
thickness.
F
x
f
M
n
Electrodes
+
V
à
Transducer
i=
2
1
dQ
dt
f
(a)
(b)
Figure 1: a) Uni-dimensional spring-mass system; b) piezoelectric linear transducer made of n identical
elements.
The constitutive equations of the piezoelectric transducer of Fig.1.b are [7]:
{
V
f
}
Ka
=
C(1 − k 2 )
[
1/Ka
−nd33
−nd33
C
]{
Q
x
}
(1)
where V is the electrical tension between its electrodes, Q the electrical charge, x is the elongation and f the
force applied at its tips; Ka is the stiffness of the transducer in short-circuit (i.e. V = 0), C is the electrical
capacitance when no forces are applied (i.e. f = 0), and d33 is the piezoelectric constant.
k is the electromechanical coupling factor. It measures the capability of the transducer for converting mechanical energy into electrical energy and vice versa. It is inherent to the material, and it can be expressed
as:
n2 d233 Ka
k2 =
(2)
C
For a multi degree of freedom structure, the effective electromechanical coupling factor Ki is used instead
of k. It is given by:
k2
νi
(3)
Ki2 =
1 − k2
where νi is the fraction of modal strain energy. It describes the amount of the strain energy in the location of
the piezoelectric transducer when the structure is vibrating according to mode i.
Therefore, to achieve a better damping performance, the location of the piezoelectric patches should be
selected to maximize Ki .
ACTIVE NOISE AND VIBRATION CONTROL
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Finally, the governing equations of the system are obtained by substituting f = F − M ẍ in the constitutive
equations of the transducer:
{
2.2
}
V
F − M ẍ
Ka
=
C(1 − k 2 )
[
1/Ka
−nd33
−nd33
C
]{
Q
x
}
(4)
Linear RL shunt
The piezoelectric element is now connected in series with a RL circuit, as shown in Fig.2. The governing
equation of the electrical charge Q becomes:
Q + RC(1 − k 2 )Q̇ + LC(1 − k 2 )Q̈ = nd33 Ka x
(5)
and, using the definitions of electrical frequency and damping, such that:
1
R
ωe = √
,
and
2ξ
ω
=
e
e
L
LC(1 − k 2 )
(6)
one gets:
2ξe
1
Q̇ + 2 Q̈ = nd33 Ka x
ωe
ωe
The dynamics of the mass M is governed by the second equation of (4):
Q+
(7)
Ka
nd33 Ka
Q+F
x=
2
1−k
C(1 − k 2 )
(8)
ẍ + ωn2 x =
nd33
1
ωn2 Q +
F
2
C(1 − k )
M
(9)
where ωn is the mechanical natural frequency:
ωn2 = Ka /(1 − k 2 )M
M ẍ +
which can be rewritten as:
F
x
M
R
L
Figure 2: Piezoelectric linear transducer connected to a RL-shunt.
Equations (7) and (9) describe two coupled resonant systems. The equivalent system behaves like a Den
Hartog Dynamic Vibration Absorber. The optimal value of the generated damping is obtained when the
electrical parameters are tuned as:
L=
1
2k
, and R =
ωn2 C(1 − k 2 )
ωn C(1 − k 2 )
(10)
Ki should be used instead of k for a multi degree of freedom system.
Note that the maximum damping obtained with an optimally tuned RL-shunt is ξRL = k/2, or, in the case
of a multi degree of freedom system ξRL = Ki /2.
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2.3 Effect of circuit tuning
Fig.3 shows the compliance X/F for different values of ωe /ωn when the piezoelectric transducer is connected to a RL circuit in series1 . Obviously, the damping is the highest when the electrical frequency ωe
matches the mechanical frequency ωn according to Equ.10.
X=F [dB]
20
!e
= 0:95
!n
0
!e
= 1:05
!n
!e
=1
!n
-20
!e=! n
0.6
0.8
1
1.2
1.4
Figure 3: Frequency response X/F for different values of the ratio ωe /ωn .
Fig.4 shows the influence of the electrical frequency tuning ωe on the obtained damping ξRL with respect to
the variation of ωe /ωn .
1
øRL
0.5
øRL=2
Attenuation (log scale)
0.5
0.1
øRL=10
+ 3%
+ 5%
0.8
0.9
1
1.1
1.2
!e=!n
1.3
1.4
Figure 4: Influence of the tuning of a RL-shunt on the generated modal damping. Effect of the mechanical
frequency tuning.
One can see that for 3%
√ error of the electrical frequency, the RL shunt circuit still provides damping close to
the optimal value (1/ 2 times less). This property will be used later to define the design rules of the circuit
tuning.
1
When the RL shunt is connected in parallel, the performances remains the same and only the parameters are tuned differently
(see de Marneffe [5] ).
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3 Bladed rail
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Aiming to assess the damping ability of the blade modes of the BluM⃝
, a preliminary study is performed on
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the bladed rail of Fig.5. The latter has a geometry similar to a portion of the first stage of the BluM⃝
. The
blades-to-support mass and stiffness ratios are representative of the real bladed drum.
Blade 1
2
3
4
Piezo patch 1
(excitation)
5
Piezo patch 2
(damping)
Figure 5: Bladed rail: the piezoelectric patches are glued under the support. Patch 1 is used for excitation
and patch 2 is used for damping. The velocity of each blade is measured at its tip.
Fig.6 shows the frequency response measured at the tip of blade 4 when patch 2 is connected to a RL shunt,
and patch 1 is used for excitation. The blade mode with frequency f = 601Hz is targeted, and the circuit
parameters are tuned according to Equ.10.
20
f ~ 589 Hz
~ 5 dB
~ 18 dB
~ 8 dB
Magnitude (dB)
0
f ~ 601 Hz
Open circuit
RLshunt
f ~ 605 Hz
-20
-40
-60
f [Hz]
570
580
590
600
610
Figure 6: Frequency Response Function measured on blade 4 with and without RL shunt. Patch 1 is used
for excitation and patch 2 for damping.
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From this result, the following conclusions were obtained:
• Blade modes can be damped using piezoelectric patches glued on the support, although their modal
fractions of strain energy νi are higher in the blades than in the support. This is due to the very low
natural damping of the blade modes: ξ < 0.01%.
• A RL shunt tuned on one blade mode can add damping for the other blades modes since their generalized electromechanical coupling factors Ki are not null, and their frequencies ωi are quite close to the
electrical frequency ωe . As presented in Fig.4, this can be seen as a mistuning of the circuit frequency
with regard to the modes.
Therefore, from these two conclusions, a new tuning approach of RL shunt circuit parameters is defined
when several patches are used for damping. The proposed approach, referred to as the mean shunt is the
following:
• Each patch has its own RL-shunt.
• All the RL-shunt circuits are built with the same components (R and L).
• The shunts are all tuned on the expected mean resonance frequency of the targeted family of blade
modes
1 ∑
ωe = ω̄ =
ωi
N
the tightness of the interval around ω̄ ensures a good authority of the patches over the whole frequency
bandwidth.
• R, is chosen using Equ.10 as well; such that Ki∗ is used instead of Ki : Ki∗ = max max {Ki,j }.
patch j mode i
This mean shunt approach tackles the high level of complexity and uncertainties of the structure by taking
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average values. It is very simple and robust [2], and it will be used for tuning the RL shunts of the BluM⃝
.
ACTIVE NOISE AND VIBRATION CONTROL
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4 Bladed druM : BluM⃝
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The current study concerns the damping of the monolithic bladed drum (BluM⃝
) of Fig.7. It represents the
rotor of a jet engine low pressure stage and consists of three blade wheels welded by friction to the drum.
(a)
(b)
(c)
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Figure 7: Prototype of the BluM⃝
: a) CAD model of the whole structure; b) finite element model of the one
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R
⃝
stage BluM ; c) experimental prototype of the one stage 76 blade BluM⃝
.
Mainly, blade modes having their frequencies and shapes close to the engine excitation order will be excited
under air flow loads. Thus, these modes will be targeted in order to increase their damping.
In this section, we describe the vibration damping system based on RL-shunted piezoelectric patches. The
direct application of the mean shunt approach shown in the previous section is applied to the one stage
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BluM⃝
. For simplicity and availability purposes, the whole experiments and model validation are performed
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only on the one stage 76 blade BluM⃝
(Fig.5.b and Fig.5.c).
4.1
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BluM⃝
dynamics
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Besides the axis symmetry feature of the one stage BluM⃝
, its particular geometry consisting of two substructures (blades and drum) makes its dynamic behavior more complex and different from usual structures
such as beams and plates. In fact, its mode shapes are dominated either by the deformation of the blades, or
by the deformation of the drum or by the deformation of both. They can be classified into three families:
• Drum dominant modes: the strain energy is mostly concentrated in the drum, Fig.8.a ;
• Blades dominant modes: the strain energy is mostly concentrated in the blades, Fig.8.b ;
• Coupled modes: unlike the previous cases, the strain energy is evenly distributed in the whole structure,
Fig.8.c.
In addition, owing to the axisymmetry of the system, the mode shapes occur in degenerate orthogonal pairs.
Both modes of the pair have the same shape, but are rotated with respect to each other by 90o around the axis
of symmetry (sines and cosines modes). This is depicted in Fig.8.b and Fig.8.d.
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a)
c)
b)
d)
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Figure 8: Different mode shapes of BluM⃝
: a) Drum dominant mode with 4 nodal diameters; b) Blade
dominant mode, first blade bending family with 9 nodal diameters (sine mode shape); c) Coupled blade
and drum dominant mode with 5 nodal diameters; d) Blade dominant mode with 9 nodal diameters (cosine
mode).
Obviously, drum-dominant and coupled modes have high νi in a patch located on the drum, and they will be
damped much better than the blade dominant modes. Although the strain energy of the latter is concentrated
in the blades, they too can be damped by the same patch due to the drum flexibility. This flexibility allows
the transfer of energy between blades through the drum, and thus, allows energy to be extracted by the patch.
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Indeed, if one assume that the drum is infinitely stiff, the BluM⃝
can be considered as a set of 76 independent
cantilevered blades. Then, for each mode shape family, e.g. 1st blade bending, one gets 76 decoupled modes
with equal resonance frequencies such that the blades have the same modal amplitudes but only one blade
vibrates for each shape (the global mode shape matrix is identity), thus, νi = 0 in the drum.
Nevertheless, since the drum is flexible, the blades modes occur with different frequencies and shapes, and
then νi in the support is nonzero. Therefore, for a given family of the blades modes, the global mode shapes
fall into three classes depending on the relationship between their phases and their amplitudes [12]:
• Each blade has the same mode shape amplitude and phase as its neighbors, and this is the 0 nodal
diameter mode.
• Each blade has the same mode shape amplitude as its neighbors, but it is vibrating in antiphase with
them, and this is the 39 (N/2) nodal diameters mode.
• Each blade has different mode shape amplitudes from its neighbors, and this is the 1 to 38 nodal
diameters modes.
ACTIVE NOISE AND VIBRATION CONTROL
353
Note that the first and the second class modes does not occur in pairs because rotating their shapes by 90o
leaves them unchanged. Only the third class occurs in pairs of degenerate orthogonal modes (sine and
cosine), which added to the first and the second classes constitute 76 global blade modes. Therewith, all that
modes has different resonance frequencies packed in a very narrow frequency range with a nonzero νi in the
drum.
4000
3000
Frequencie [Hz]
|
{z
}
1st blades torsion
(1T)
2000
1000
|
{z
}
1st blades bending
(1F)
Index [/]
0
40
80
120
160
200
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Figure 9: Natural frequencies distribution of the one stage BluM⃝
model.
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The first 200 natural frequencies of the one stage BluM⃝
are depicted in Fig.9. One can observe two
distinct families: the first family consisting of the first blade bending modes and packed between 1180Hz
and 1250Hz, and the second family of the first blade torsion modes packed between 2810Hz and 2930Hz.
The other frequencies correspond to drum and coupled modes.
Note that the larger the frequency range of the blade modes family, the stronger the coupling is between the
blades and the drum.
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5 Experimental validation
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Experiments are conducted on the one stage BluM⃝
prototype equipped with 28 piezoelectric patches glued
on the inner side of the drum, Fig.10.
Gyrator
circuits
Blade support
drum
Blades
Piezoelectric
patches
Blades
Support (Inner side)
R
.
Figure 10: Experimental set-up of the one stage BluM⃝
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The BluM⃝
is excited through a voltage applied to 6 different patches and damped by shunting the other
22 patches. Each patch is connected to a tunable Antoniou gyrator[13] simulating the behavior of a pure
inductor.
The electrical shunts are tuned using the mean shunt approach. The first blade bending modes frequencies
are packed between 1180Hz and 1250Hz. Thus, the electrical frequency is chosen approximately in the
middle at ωe = 1215Hz.
Because it is very difficult to identify the generalized coupling factors Ki experimentally, the circuits are
tuned manually according to model prediction of the Ki s, and the resistors R are chosen accordingly.
Fig.11 shows the frequency response function FRF between the applied voltage and the measured velocity
at a one blade tip with and without shunts.
20
Mag [dB]
Open circuit
0
RL shunt
-20
-40
1180
1200
Freq [Hz]
1220
1240
R
Figure 11: FRF of the BluM⃝
between the applied voltage and the velocity of one blade tip.
Mostly, the whole blade modes are damped with different values of damping depending on their coupling
factors Ki . Indeed, the well damped modes are those having strong coupling with the drum.
ACTIVE NOISE AND VIBRATION CONTROL
355
The identified natural damping of the blade modes without shunt is ξ < 0.02%. However, when the structure
is damped, one can see on Fig.11 that the peak amplitudes are not only reduced, but shifted as well. Thus, it
is difficult to assess the obtained damping for each mode.
Therefore, the cumulative root mean square (RMS) of the frequency response is used in order to assess the
shunt effect. For a white noise excitation in the frequency range [f1 − f2 ], the cumulative mean square of the
frequency response T is given by:
2
σ (f2 ) =
∫
f1
f2
|T (jf )|2 df
Fig.12 shows the integrated root mean square of the frequency response of Fig.11.
26.2
25
RMS [/]
20
15
10
Open circuit
1180
6.5
RL shunt
5
1200
1220
1240
Freq [Hz]
Figure 12: Integrated RMS of the blade frequency response.
The RMS value of the blades in the frequency bandwidth of their first bending mode is reduced by 4, it has
σoc = 26 in open circuit, and σRL = 6.5 with RL shunt.
6
Conclusion
The dynamic behavior of the bladed drum has been briefly highlighted, and its damping system using RLshunted piezoelectric patches is described. The proposed design rules of RL shunts is validated experiR
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mentally on the one stage BluM⃝
. Experiments conducted on the BluM⃝
shows a reduction of the blade
response by a factor of 4 in the blade modes family bandwidth. This damping enhancement takes its advantage from the drum flexibility which provides a good coupling with blades sufficient to increase the very low
inherent damping ratio.
Importance should be given to the dynamic analysis of the structure in order to identify specific modes
subjected to high vibration during rotation. In addition, effort will be focused on the integration procedure
of the damping system in a rotating engine under real operating conditions.
Acknowledgment
This research was performed in the ”Health-Monitoring +” project supported by Skywin - Rgion Wallone,
involving SAFRAN Techspace Aero as an industrial partner.
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