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Semiactive Vibration Control of a Composite Beam using an Adaptive SSDV Approach

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Structures Journal of Intelligent Material Systems and
DOI: 10.1177/1045389X08099967 2009; 20; 939 srcinally published online Feb 9, 2009;
Journal of Intelligent Material Systems and Structures
Hongli Ji, Jinhao Qiu, Adrien Badel, Yuansheng Chen and Kongjun Zhu
Sources Based on LMS AlgorithmSemi-active Vibration Control of a Composite Beam by Adaptive Synchronized Switching on Voltage
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by Jinhao Qiu on June 2, 2009 http://jim.sagepub.comDownloaded from
Semi-active Vibration Control of a Composite Beamby Adaptive Synchronized Switching on VoltageSources Based on LMS Algorithm
H
ONGLI
J
I
,
1
J
INHAO
Q
IU
,
1,
* A
DRIEN
B
ADEL
,
2
Y
UANSHENG
C
HEN
1
AND
K
ONGJUN
Z
HU
1
1
Aeronautic Science Key Laboratory for Smart Materials and Structures, College of Aerospace EngineeringNanjing University of Aeronautics and Astronautics, #29 Yudao Street, Nanjing 210016, China
2
Laboratory of Systems and Materials for Mechatronics, Polytech’savoieSavoie, University, BP 80439, 74944 Annecy le Vieux Cedex, France
ABSTRACT:
In this article, an adaptive semi-active SSDV (Synchronized Switch Dampingon Voltage) method based on the LMS algorithm is proposed and applied to the vibrationcontrol of a composite beam. In the SSDV method, the value of voltage source in theswitching circuit is critical to its control performance. In the adaptive approach proposed inthis study, the voltage source is adjusted adaptively using the LMS algorithm. Two cases of theadjustment are considered. In the first case, as an improvement to the enhanced SSDV, thevoltage coefficient is adjusted by the LMS algorithm. In the second case, as an improvement tothe classical SSDV, the voltage value is adjusted directly. The new adaptive approach iscompared with the derivative-based adaptive SSDV proposed in the former study in thecontrol of the first mode of a composite beam. The control results show that adaptiveadjustment of voltage value and adaptive adjustment of voltage coefficient are equallyeffective in the vibration control of the composite beam and that LMS-based approachis slightly better than the derivative-based approach.
Key Words:
piezoelectric elements, synchronized switch damping, semi-active control,vibration damping, LMS algorithm.
INTRODUCTION
A
MONG
the many semi-active vibration controlmethods using piezoelectric actuators (Davis et al.,1997; Davis and Lesieutre, 1998; Richard et al., 1998,
2000; Clark, 1999, 2000), that based on a technique
of nonlinear synchronized switching, known asSynchronized Switch Damping (SSD), has attractedmuch attention due to many advantages compared withthe active and passive approaches: it is not sensitive tothe variation of the parameters of system; the semi-active control system is more compact and simpler thanactive control and it is more effective than passivecontrol. The SSD method consists in a nonlinearprocessing of the voltage on a piezoelectric patchembedded in the structure (Richard et al., 1998, 2000).
It is implemented with a simple switch driven synchro-nously with the structural motion.In the srcinal SSD technique, called SSDS (synchro-nizedswitch dampingonshortage),theswitchisclosed todischarge thepiezoelectric elements. Due to the switchingprocess, a phase shift appears between the strain inpiezoelectric patch and the resulting voltage, thuscreating energy dissipation. To improve the controlperformance, an inductance can be connected to theshunt circuit to invert the voltage in the piezoelectricpatch. This method is called SSDI. The inversion processboosts the voltage, thus increasing energy dissipation.The objective of all switch control algorithms is tomaximize the energy dissipated in each cycle of vibration.To further improve the control performance, amethod called SSDV (synchronized switch damping onvoltage) has been proposed by Lefeuvre et al. (2006),Faiz et al. (2006), and Badel et al. (2006). In the classical
SSDV, a voltage source
V
cc
is connected to the shuntingbranch to further increase the voltage on the piezo-electric patch and accordingly to increase the dampingeffect. However, a stability problem arose if a constantvoltage source is used. Under a given excitation level,there is a value of the voltage source
V
cc
that totallysuppresses the vibration. This means that the forcegenerated by the voltage on the piezoelectric elementduring vibration theoretically cancels the effect of theexternal excitation force. Applying a voltage higher thanthe critical value leads to a stability problem because in
*Author to whom correspondence should be addressed.E-mail: qiu@nuaa.edu.cnFigures 3, 6 and 7 appear in color online: http://jim.sagepub.com
J
OURNAL OF
I
NTELLIGENT
M
ATERIAL
S
YSTEMS AND
S
TRUCTURES
, Vol. 20—May 2009 939
1045-389X/09/08 0939–9 $10.00/0 DOI: 10.1177/1045389X08099967
SAGE Publications 2009Los Angeles, London, New Delhi and Singapore
by Jinhao Qiu on June 2, 2009 http://jim.sagepub.comDownloaded from
this state the voltage has a driving effect instead of adamping effect (Badel et al., 2006). An enhanced SSDV
technique, in which the voltage source is proportionalto the vibration amplitude, has then been proposedby Badel et al. (2006) to improve the stability of SSDV.Theoretically, for a given value of the voltage coeffi-cient, the damping effect is not sensitive to the amplitudeof vibration. This is the main advantage of the enhancedSSDV. However, experimental results exhibit that theoptimal voltage coefficient depends on many factorssuch as the noise level of the measured signal, theproperty of the switch for example. Hence, in order toachieve optimal control performance, the voltage coeffi-cient should be adjusted adaptively according to thevibration amplitude and other experimental conditions.An adaptive enhanced SSDV technique, in which thevoltage coefficient is adjusted adaptively to achieveoptimal control performance, has already been proposedby the authors (Ji et al., 2008). The basic principle of theadaptive SSDV technique is that the voltage coefficient
is adjusted based on the sensitivity of the vibrationamplitude with respect to
. If the variation of amplitude is
D
u
Mi
due to an increment of the voltagecoefficient
D
i
, the sensitivity is defined as
D
u
Mi
/
D
i
.The voltage coefficient is then modified by
D
i
þ
1
¼
D
u
Mi
D
i
:
ð
1
Þ
Since
D
u
Mi
/
D
i
is an approximation of the derivative of amplitude
u
M
with respect to
, this approach is calledderivative-based adaptive SSDV. In the real system,
D
i
is not updated in each cycle of vibration because of the noise in the measured amplitude. Instead,
D
i
iskept constant for
n
cycles and the amplitudes
u
Mk
(
k
¼
1,
. . .
,
n
) are recorded. A parabolic curve is thenfitted from the points
u
Mk
and the slope at the final point
u
Mn
is defined as the sensitivity. In the former study(Ji et al., 2008), the number of points,
n
, was set to 20.In this study an adaptive SSDV method based onLMS algorithm to adjust the voltage source is proposed.Two cases of the adjustment are considered. In the firstcase, as an improvement to the enhanced SSDV, thevoltage coefficient is adjusted by the LMS algorithm.In the second case, as an improvement to the classicalSSDV, the voltage value is directly adjusted, using thesame algorithm. The new adaptive approaches areapplied to the control of the first mode of a compositebeam and the results are compared with the derivative-based adaptive SSDV proposed in the former study.
THE MECHANICAL MODEL OF THE BEAM
The structure used in the experiment is a cantileverGFRP (Glass Fiber Reinforced Plastics) compositebeam with an embedded piezoelectric patch, as shownin Figure 1. The composite beam was made from fourlayers of GFRP prepreg. The GFRP prepreg and thepiezoelectric patch are laminated in the following order:0
8
/90
8
/90
8
/PZT/0
8
, where 0
8
is the length direction of thebeam. The beam is 150mm long, 51mm wide, and0.8mm thick and its properties are given in Table 1.The piezoelectric patch is a 30mm
30mm square andits thickness is 0.2mm. It is polarized in the thicknessdirection and its properties are given in Table 2. Theequation of motion of the beam is a partial differentialequation. The equation of motion for each mode can beobtained from a modal analysis (Ji et al., 2008). In thisstudy, control of the resonant vibration at the firstnatural frequency is considered. Hence, the amplitude of the first mode is dominant and the high-order modes canbe neglected. That is, at the first resonant frequency, thestructure can be simplified to a single-degree-of-freedomsystem, represented in Figure 2. The equation of motioncan then be expressed in the following form:
M
€
u
þ
C
_
u
þ
K
E
u
¼
F
e
þ
F
p
,
ð
2
Þ
where
M
is the modal mass,
C
is the modal dampingcoefficient,
K
E
is the modal stiffness, and
F
e
and
F
p
aremodal force due to excitation and the piezoelectric patch,respectively. The modal mass
M
, modal stiffness
K
E
,
Table 1. Material properties of the composite beam.
GFRP
Elastic modulus
E
1
1.65
10
þ
10
(N/m
2
)Elastic modulus
E
2
3.52
10
þ
10
(N/m
2
)Poisson ratio 0.109Shear modulus 1.25
10
þ
10
(N/m
2
)Density 1800 (kg/m
3
)Thickness 0.8
10
3
(m)
Table 2. Material properties of the piezoelectric patch.
PZT
Elastic modulus 59
10
þ
9
(N/m
2
)Poisson ratio 0.345Density 7400 (kg/m
3
)Thickness 2
10
4
(m)Piezoelectric constant
d
31
260
10
12
(m/V)Capacitance
C
p
141
10
9
(F)
Excitation 150mm0.8mm
5 1 m m
PZT Beam
Figure 1.
Schematic representation of the system.
940
H. J
I ET AL
.
by Jinhao Qiu on June 2, 2009 http://jim.sagepub.comDownloaded from
and modal forces
F
e
and
F
p
can be derived from theequation of motion of the beam, but the dampingcoefficient
C
must be estimated experimentally.
F
p
canbe expressed in the following form:
F
p
¼
a
V, whereminus sign is to express the direction of force.The following energy equation is obtained by multi-plying both sides of Equation (2) by the velocity andintegrating over the time variable:
Z
F
e
_
u
d
t
¼
12
M
_
u
2
þ
12
K
E
u
2
þ
Z
C
_
u
2
d
t
þ
Z
V
_
u
d
t
:
ð
3
Þ
The provided energy is divided into kinetic energy,potential elastic energy, mechanical losses, and trans-ferred energy. The transferred energy
R
V
_
u
d
t
corre-sponds to the part of the mechanical energy which isconverted into electrical energy. For a certain vibrationlevel, maximization of this energy leads to minimizationof the mechanical energy in the structure. However, asthe vibration level is reduced, the absolute value of this energy may decrease. Hence the objective of all theSSD control approaches is to maximizing this energy ina given vibration.
CONTROL SYSTEMThe Principle of Classical and Enhanced SSDV
The schematic diagram of a SSDV control system isshown in Figure 3. In the classical SSDV, the output of the voltage source is fixed. As introduced above, theenhanced SSDV was proposed by Badel (2006) due tothe stability issue in the classical SSDV. In the enhancedSSDV, the output of the voltage source is proportionalto the amplitude of vibration:
V
cc
¼
u
M
ð
4
Þ
where
is the voltage coefficient. The enhancedSSDV is intrinsically stable if the coefficient
takes asuitable value.The principle of all the SSD approaches is tosuperpose a rectangular wave signal to the originalvoltage signal produced by strain. This rectangular waveis generated by the nonlinear processing of the voltage.In the SSDS approach, the amplitude of the rectangularsignal equals the voltage induced by the maximumstrain. The amplitude of the rectangular signal in theSSDI is boosted by the inversion process. In SSDV,the amplitude is further boosted by the voltage source inthe shunt circuit. As shown in Figure 4, the rectangularwave is a combination of two components: the voltagedue to inversion and the voltage from the voltage source,which is calculated from Equation (4) using the voltagecoefficient
. Hence the control performance is stronglyaffected by
. However, the optimal value of coefficient
depends on many factors such as the level of noise inthe measured vibration signal and the property of theswitch. Hence, adaptive approaches to optimize thevalue of
online are highly desirable. The authorsproposed a derivative-based adaptive SSDV in theformer study. In this study, a LMS-based adaptiveapproach is proposed.
LMS Algorithm
The schematic diagram of a general control systemusing LMS algorithm is shown in Figure 5, in which
h
is an FIR filter,
e
(
n
) is the error signal, D
m
is anoperator to create a vector of time series of length
m
,Z
1
is an operator which delays the signal by one timestep, and
y
(
n
) is the output of FIR filter (Hansen andSnyder, 1996). The function of D
m
is as follows,
D
m
f
e
ð
n
1
Þg ¼ f
e
ð
n
m
Þ
,
e
ð
n
m
þ
1
Þ
,
. . .
,
e
ð
n
1
Þg
:
In most cases the LMS algorithm is used in feed-forwardsystems, in which the input to the FIR filter is the sameas the system input. In the case of vibration control, thesystem input is the force of excitation. Hence, a feedback-type system is preferable in a vibration control system.
InductorPZTVoltagesourceDiodeSwitchSwitchcontrolExtremumdetection
V
Figure 3.
Schematic diagram of a SSDV control system.
K
E
C F
(
t
)
u
(
t
)
FP
PZT
M
Figure 2.
Diagram of the electromechanical model.
Semi-active Vibration Control of a Composite Beam
941
by Jinhao Qiu on June 2, 2009 http://jim.sagepub.comDownloaded from
In a feedback-type system, in which the error signal of the system is used as the input of the FIR filter, theoutput of the filter is calculated from the followingequation:
y
ð
n
Þ ¼
h
ð
n
Þ
e
ð
n
1
Þ ¼
h
ð
1
Þ
e
ð
n
1
Þþ
h
ð
2
Þ
e
ð
n
2
Þ þ þ
h
ð
m
Þ
e
ð
n
m
Þ ð
5
Þ
where
n
denotes the discrete time,
m
is the numberof filter coefficients and
e
ð
n
1
Þ ¼
D
m
f
e
ð
n
1
Þg
. Sincethe response always lags the input, the displacementamplitude of the former step is used to compute thecontrol input of the present step. Hence, a delayoperator is necessary in the diagram.The output of the system,
e
(
n
), is the sum of two parts,one induced by the disturbance from the exciter and theother induced by the controller, as shown in Figure 5.If the output due to disturbance is denoted by
d
, theoutput
e
(
n
) can be written in the following form:
e
ð
n
Þ ¼
d
ð
n
Þ þ
y
ð
n
Þ ¼
d
ð
n
Þ þ
h
ð
n
Þ
e
ð
n
1
Þ ð
6
Þ
From Equations (5) and (6), the equation to update
h
forminimization of the square of the error signal
e
(
n
) is
h
ð
n
þ
1
Þ ¼
h
ð
n
Þ
2
e
ð
n
Þ
e
ð
n
1
Þ
;
ð
7
Þ
where
is a parameter for step size. Large value of
speeds up the convergence of the iteration, but itcan become unstable if
is too large. In this study
isset to 0.05.
Application of LMS Algorithm to SSDV
In SSDV, the parameter to be optimized is the voltage
V
cc
or the voltage coefficient
. Their values are definedat each switching point, not the discrete time
n
.Hence the detected displacement amplitude
u
M
,instead of the displacement
u
itself, is used as the error
e
to the FIR filter. The output
y
of the FIR filter is thevoltage or the voltage coefficient
at the switchingtimes, instead of the voltage value at each discrete time,and the calculated voltage is held constant until the nextswitching time so that a rectangular wave is generatedautomatically by the switching circuit. Hence the LMS-based system is a sub-system, which is not executed ateach discrete time, but triggered and executed at eachdetected extrema as shown in Figure 6.In the case of the optimization of
, the value of
iscalculated from:
ð
n
0
Þ ¼
h
ð
n
0
Þ
u
m
ð
n
0
1
Þ ¼
h
ð
1
Þ
u
m
ð
n
0
1
Þþ
h
ð
2
Þ
u
m
ð
n
0
2
Þ þ þ
h
ð
m
Þ
u
m
ð
n
0
m
Þð
8
Þ
u u u u
g
V
m
V
m
V
m
(a) (b) (c)
V
m
V
s
Figure 4.
Decomposition of the voltage signal in SSDV: (a) The voltage V on the PZT and the structural deflection; (b) The voltage V
M
due to inversion and the voltage source; (c) The voltage V
S
due to strain.
PZTLasersenseorBeamDSPSwitchcontrolSwitchVoltagesourceVInductanceExtremumdetector
u
M
LMS-basedSSDVVoltage or
b
Figure 6.
Diagram of switch control system based on LMS-algorithm.
DisturbanceBeamControl systemD
m
Z
−
1
ControllerFIR filterLMSalgorithm
y
(
n
)Erroroutput
e
(
n
)
e
(
n
)
e
(
n
−
1)
Figure 5.
Control system using LMS algorithm.
942
H. J
I ET AL
.
by Jinhao Qiu on June 2, 2009 http://jim.sagepub.comDownloaded from

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