Active vibration control of smart piezo cantilever beam using pid controller

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 01 | Jan-2014, Available @ http://www.ijret.org 392
ACTIVE VIBRATION CONTROL OF SMART PIEZO CANTILEVER
BEAM USING PID CONTROLLER
Saurabh Kumar1
, Rajeev Srivastava2
, R.K.Srivastava3
1
Research Scholar, Motillal Nehru National Institute of Technology, Allahabad - 211004, Uttar Pradesh, INDIA,
2
Associate Professor, Motillal Nehru National Institute of Technology, Allahabad - 211004, Uttar Pradesh, INDIA,
3
Professor, Motillal Nehru National Institute of Technology, Allahabad - 211004, Uttar Pradesh, INDIA,
Abstract
In this paper the modelling and Design of a Beam on which two Piezoelectric Ceramic Lead Zirconate Titanate ( PZT) patches
are bonded on the top and bottom surface as Sensor/Actuator collocated pair is presented. The work considers the Active
Vibration Control (AVC) using Proportional Integral Derivative (PID) Controller. The beam is assumed as Euler-Bernoulli beam.
The two PZT patches are also treated as Euler-Bernoulli beam elements. The contribution of mass and stiffness of two PZT
patches in the design of entire structure are also considered. The beam is modelled using three Finite Elements. The patches can
be bonded near the fixed end, at middle or near the free end of the beam as collocated pair. The design uses first two dominant
vibratory modes. The effect of PZT sensor/actuator pair is investigated at different locations of beam in vibration control. It can
be concluded from the work that best result is obtained when the PZT patches are bonded near the fixed end.
Keywords: Smart Beam, Active Vibration control, Piezoelectric, PID Controller, Finite Element
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1. INTRODUCTION
Active vibration control is a technique in which the
vibration of a structure is controlled by applying counter
force to the structure that is appropriately out of phase but
equal in amplitude to the original force. As a result two
opposite forces cancel each other and structure stops
vibrating. Piezoelectric and Piezoelectric Ceramic materials
can be used as sensors and actuators. These materials have
ability to transform mechanical energy to electrical energy
and vice-versa. A piezoelectric material is a crystal in which
electricity is produced by pressure (Direct Effect).
Conversely, a piezoelectric material deforms when it is
subjected to an electric field (Converse Effect). The
piezoelectric sensor senses the external disturbances and
generates voltage due to direct piezoelectric effect while
piezoelectric actuator produces force due to converse
piezoelectric effect which can be used as controlling force.
For generating the appropriate controlling force according to
the sensed signal controller is needed. Tran Ich Thinh , Le
Kim Ngoc [1] has developed a Finite Element (FE) model
based on the First-Order Shear Deformation Theory for the
static flexural shape and vibration control of a glass
fiber/polyester composite plate bonded with piezoelectric
actuator and sensor patches. The piezoelectric mass and
stiffness are taken into account in the model. The results
obtained were in good agreement with actual experimental
result. Aydin Azizi, Laaleh Durali, Farid Parvari Rad,
Shahin Zareie [2] used PZT elements as sensors and actuator
to control the vibration of a cantilever beam. They studied
the effect of different types of controller on vibration
control. Finite Element Analysis and generalized equation of
motion has been used in this paper. Premjyoti G.Patil [3]
provides a mathematical model for the deformation of
cantilever beam using Finite Element Method. Using the
mathematical model, the beam deformation is plotted using
MATLAB. Lucy Edery-Azulay, Haim Abramovich [4]
described that the active damping is obtained by using an
actuator and a sensor piezoceramic layer acting in closed-
loop. By transferring the accumulated voltage on the sensor
layer to the piezoelectric actuator layer, the beam can
actively damp-out its vibrations. An exact mathematical
model, based on a first order shear deformation theory
(FSDT) is developed and described. This model allows the
investigation of piezo-composite beams with two
actuation/sensing type mechanisms, extension and shear.
For obtaining the natural frequency and mode shapes
expressions were programmed in Maple 9. Effect of
different piezoelectric materials on damping was also
studied in this paper. Using the Euler–Bernoulli Beam
Theory R. Ly, M. Rguiti, S. D‟Astorg, A. Hajjaji, C.
Courtois, and A. Leriche [5] developed a model of
piezoelectric cantilever beam. The equations of motion for
the global system were established using Hamilton‟s
principle and solved using the modal decomposition method
which described dynamic behaviour of the beam for energy
harvesting. Then the model was implemented using
MATLAB software and will be able to integrate with the
circuit model for energy storage. The results obtained show
a good agreement with the experiments and other previous
works. Magdalene Marinaki, Yannis Marinakis, Georgios E.
Stavroulakis [6] focused on the design of a vibration control
mechanism for a beam embedded with piezoelectric sensors

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and actuators. The modelling of the piezoelectric sensors
and actuators are based on the piezoelectric constitutive
equations and classical equations of motion using finite
element analysis. One nature- inspired intelligence method,
the Particle Swarm Optimization (PSO) which is a
population based swarm intelligence algorithm is used for
the vibration control of the beam. Three different variants of
the Particle Swarm Optimization were tested, namely, the
simple Particle Swarm Optimization, the inertia Particle
Swarm Optimization and the Constriction Particle Swarm
Optimization. The same problem has been solved with two
other optimization algorithms, namely a Genetic Algorithm
and a Differential Evolution and the results are compared.
Result shows that sufficient vibration suppression can be
achieved by means of PSO method. K Ramesh Kumar and S
Narayanan [7] used a finite element method based on Euler–
Bernoulli beam theory. Linear quadratic regulator (LQR)
controller is used for controlling. The LQR performance is
taken as the objective for finding the optimal location of
sensor–actuator pairs. The problem is formulated as a multi-
input multi-output (MIMO) model control problem. The
discrete optimal sensor and actuator location problem is
formulated in the framework of an optimization problem
which is solved using genetic algorithms (GAs). The study
of the optimal location of actuators and sensors is carried
out for different boundary conditions of beams like
cantilever, simply supported and clamped boundary
conditions. Jingjun Zhang, Lili He, Ercheng Wang, Ruizhen
Gao [8] takes a cantilever beam bonded with rectangular
shaped piezoelectric sensors and actuators. Two active
vibration control methods such as Linear Quadratic Gauss
(LQG) optimal control and robust H control are
investigated. The paper demonstrates that compared with the
LQG control method, H control has strong robustness to
modal parameters variation and has a good closed-loop
dynamic performance. C.M.A. Vasques, J. Dias Rodrigues
[9] used an analysis and comparison of the classical control
strategies, constant amplitude and constant gain velocity
feedback (CAVF and CGVF), and optimal control
strategies, linear quadratic regulator (LQR) and linear
quadratic Gaussian (LQG) controller in order to investigate
their effectiveness to suppress vibrations in beams with
piezoelectric patches acting as sensors or actuators. In the
paper a three-layered smart beam with two piezoelectric
surface layers were modelled. The transverse displacement
time history, at the free end is evaluated with the open- and
closed-loop classical and optimal control systems. The case
studies allow the comparison of their performances
demonstrating their advantages and disadvantages. T.C
Manjunath and B Bandyopadhyay. [10] used multirate
output feedback based discrete sliding mode control for
SISO systems in vibration control of Timoshenko smart
structure. The beam structure is divided in four Finite
Elements. The beam structure is modelled in the State Space
form using the concept of piezoelectric theory. The
performance of structure is investigated for first two
dominant vibratory modes as well as higher modes. The
effect of placing the piezoelectric collocated pair is
investigated at various locations on the beam. Michele Betti,
Georgios E. Stavroulakis and Charalambos C.
Baniotopoulos [11] used an active vibration control
technique for a smart beam. The structure is made of two
layers of piezoelectric material (PZT8) embedded on the
surface of an aluminium beam. Piezoelectric sensors and
actuators are perfectly bonded on the host elastic structure.
A Finite Element model for a composite smart beam was
developed. The integration of control actions is done within
the ANSYS. The ANSYS Parametric Design Language
(APDL) is used in order to develop a closed loop feedback
control law. The control law can be calculated from a
classical control theory, e.g. a linear feedback and LQR,
which is used in this paper. By taking into account the
modal shape of the beam, it is possible to suggest the
optimal position for piezoelectric patches.
Most of the present researchers have used
Finite element Analysis with different control laws to
suppress the vibration of a Piezoelectric Smart Structure.
The objective of this work is to design and analyse
Piezoelectric Smart Beam with commonly used control
method, The Proportional Integral Derivative (PID) when
Piezoelectric Sensor/Actuator pair is placed at different
locations on the beam. The paper is organized in three parts
i.e. FE formulation of Smart beam, Controller Design, and
Results.
2. MODELLING OF THE SMART CANTILEVER
BEAM
Consider an Aluminium Cantilever beam bonded with
collocated Piezoelectric Sensor/Actuator pair. Properties of
beam and PZT patches (Actuator and Sensor) are given in
Table-1. An external disturbing force FDist is acting at the
free end. FActu is the force generated by the Actuator. Fig- 1,
2 and 3 shows three positions of the PZT Sensor/Actuator
pair. The beam is divided in 3 Finite Elements. The beam
may be divided in more than 3 Finite Elements (say 4, 5, 6
etc.). The more Finite Elements we take the more accurate
results we get. The beam is considered as Euler-Bernoulli
Beam and PZT patches are also considered as Euler-
Bernoulli beam elements. We first start with the modelling
of regular beam element and then modelling of the Smart
beam element i.e. element having Piezoelectric
Sensor/Actuator pair also. Finally all the elements are
assembled using FE analysis. In modelling and analysis the
following assumptions are taken:-
 The beam, sensor and actuator are taken as Euler-
Bernoulli beam elements i.e. effect of transverse
shear forces is neglected.
 Sensor and actuator layers are thin compared with
the beam thickness.
 Cross- sections of beam, sensor and actuator
remain plane and normal to the deformed
longitudinal axis before and also after bending.
 Neutral axis of beam, sensor and actuator passes
through the centroid.

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 The polarization direction of the Sensor and
actuator layers is in the thickness direction ( z-
axis)
 The electric ﬁeld loading of the Sensor and actuator
layers is uniform uniaxial in the x-direction as
shown in Figures :-1,2 and 3
 The piezoelectric material is homogeneous,
transverse isotropic and elastic.
 Adhesive used in bonding the Sensor/Actuator does
not contribute in mass and stiffness of Smart beam
element.
In the present study suffix „b‟ is used for regular beam
element, suffix „p‟ is used for PZT patch element, suffix „a‟
is used for Actuator and suffix „s‟ is used for Sensor. Ab and
Ap are cross sectional areas of regular beam and PZT patch
elements respectively. Ib and Ip are moment of Inertias of
regular beam and PZT patch elements respectively.
PARAMETERS CANTILEVER
ALUMINUM
BEAM
PIEZOELECTRIC
SENSOR/
ACTUATOR
Length lb= 0.3 m lp=la=ls=0.1 m
Width bb=0.03 m bp= ba = bs = 0.03 m
Thickness tb= 0.003 m tp= ta = ts = 0.6×10-3
m
Young‟s Modulus
of Elasticity
Eb= 6.9×1010
N/m2
Ep=Ea=Es= 6.66×1010
N/m2
Density b= 2700 kg/m3
p=a =s= 7400
kg/m3
Piezoelectric
( PZT-5H ) Stress
Constant
g31=8.5×10-3
Vm/N
Piezoelectric
( PZT-5H ) Strain
Constant (d31)
d31= 265×10-12
C/N
Damping
Constants used
α= 0.001 and
β=0.0001
Table- 1: GEOMETRIC AND MATERIAL PROPERTIES
OF BEAM AND PZT PATCHES
Fig:-1 Sensor/Actuator placed near Fixed
End of the Beam
Fig:-3 Sensor/Actuator placed at Free End of
the Beam
Fig:-2 Sensor/Actuator placed at Middle of
the Beam

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A two nodes Finite element of a Smart Beam Element is
shown in Fig-4
2.1 FE formulation of regular beam element
A two nodes Finite element of a Regular Beam Element is
shown in Fig-5
The node undergoes both translational and rotational
displacements and they are u1, 1, u2 and 2. The linear
forces are f1 and f2 corresponding to linear displacements u1
and u2 and rotational joint forces i.e. Bending Moments are
M1 and M2 corresponding to the rotational joint
displacements 1 and 2 .The Transverse displacement with
in the element is assumed to be a cubic polynomial as
2 3
1 2 3 4( , )u x t a a x a x a x    (1)
Substituting the boundary conditions the shape functions of
beam elements can be obtained as
 1 2 3 4
2 3 2 3 2 3 2 3
2 3 2 2 3 2
{ ( )} ( ) ( ) ( ) ( )
3 2 2 3 2
1
T
b bb b b b b b
N x N x N x N x N x
x x x x x x x x
x
l ll l l l l l
 
 
      
 
(2)
The nodal displacement function can be written as
   1 1 2 2
T
q u u  (3)
The Lagrange‟s Equations gives the Kinetic Energy and
Potential energy of the system respectively as :
   
   
1
[ ]
2
1
[ ]
2
T
T
T q m q
U q k q


 
(4)
Using Lagrange‟s Equation the Element Stiffness Matrix
and Mass Matrix of a beam element are computed that can
be found in Text Books of Vibration [14], [15] and written
as
2 2
3
2 2
12 6 12 6
6 4 6 2
[ ]
12 6 12 6
6 2 6 4
b b
b b b bb b
b
b bb
b b b b
l l
l l l lE I
k
l ll
l l l l
 
  
   
 
 
(5)
2 2
2 2
156 22 54 13
22 4 13 3
[ ]
54 13 156 22420
13 3 22 4
b b
b b b bb b b
b
b b
b b b b
l l
l l l lA l
m
l l
l l l l

 
  
 
 
   
(6)
The First, Second spatial derivatives of shape functions are
denoted as under that will be used in deriving the Sensor and
Actuator results.
   
   
'
1
2
''
2 2
( )
( ) ( ) and
( )
( ) ( )
dN x
n x N x
dx
d N x
n x N x
dx
 
  
 
 
  
 
(7)
2.2 FE formulation of smart beam element
When PZT patches are assumed as Euler-Bernoulli beam
elements the Elemental mass and stiffness matrices of PZT
beam element can be computed in similar fashion as:
2 2
3
2 2
12 6 12 6
6 4 6 2
[ ]
12 6 12 6
6 2 6 4
p p
p p p pp p
p
p pp
p p p p
l l
l l l lE I
k
l ll
l l l l
 
  
   
 
  
(8)
2 2
2 2
156 22 54 13
22 4 13 3
[ ]
54 13 156 22420
13 3 22 4
p p
p p p pp p p
p
p p
p p p p
l l
l l l lA l
m
l l
l l l l

 
  
 
 
    
(9)
The smart beam element is obtained by sandwiching the
regular beam element in between the two PZT patches as
shown in Fig.:-6
Fig-6 :A two nodes Finite element of a Smart Beam Element
Fig-4 : A two nodes Finite element of a Smart Beam
Element
Fig-5 :A two nodes Finite element of a Regular Beam Element

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In which 2b b p pEI E I E I  is the Flexural rigidity and
( 2 )b b b p pA b t t    is the mass per unit length of smart
beam element, tP is the thickness of PZT patches i.e.
thickness of Actuator and Sensor. So the Elemental mass
and stiffness matrices of Smart beam element are
2 2
3
2 2
12 6 12 6
6 4 6 2
[ ]
12 6 12 6
6 2 6 4
p p
p p p p
p pp
p p p p
l l
l l l lEI
k
l ll
l l l l
 
  
   
 
  
(10)
2 2
2 2
156 22 54 13
22 4 13 3
[ ]
54 13 156 22420
13 3 22 4
p p
p p p pp
p p
p p p p
l l
l l l lAl
m
l l
l l l l

 
  
 
 
    
(11)
3. SENSOR EQUATION
Following Linear Piezoelectric Constitutive equations
[6],[13] will be used for deriving the Sensor and Actuator
equations.
11 31
31 33D = +
E
x x z
z x z
S d E
d E
 
 
 
(12)
Where  is Strain,  is Stress, SE
is Compliance when
electric field is constant, d31 is Piezoelectric Constant
(Coulomb/N or m/V), E is Electric field (Volt/m), D is
Electric displacement i.e. charge per unit area
(Coulomb/m2
), 
is Dielectric constant ( Permittivity) under
constant stress. The direct piezoelectric effect is used to
calculate the output charge on the sensor layer created by
the strains in the beam. Since no electric ﬁeld is applied to
the sensor layer, we get
11 31z xD C d  (13)
Where C11 is the Young‟s modulus of elasticity (Inverse of
compliance).
The charge measured through the electrodes of the sensor is
given by
( ) z
S
q t D ds  (14)
The current on the surface of the sensor is given by
( )
( )
dq t
i t
dt
 (15)
From the Text books of Mechanics of Solids we know that
strain at a point in a beam is given as 2 2
xε =z d u/dx , where z
is a coordinate on the beam w.r.t. neutral axis. Width
bb=bs=ba. As such current generated can be written as [10]
 11 31 20
( ) ( ) { }
p
Tl
bi t zC d b n x q dx   (16)
Where z= tb/2+ts for maximum strain.
Voltage generated by the sensor is
( ) ( )S
SV t G i t (17)
Where GS is the gain of the signal conditioning device
11 31 1 1 2 2( ) [0 1 0 1]
T
s
s bV t G C d zb u u     
   (18)
This can be written as
( ) [0 1 0 1]{ }s
sV t C q   (19)
Where Cs =GsC11d31zbb is Sensor Constant. The above
equation can be written as
 ( ) { }
Ts
V t g q  (20)
Where {g} is a Constant Vector of size (4×1)
4. ACTUATOR EQUATION
From equation 12 the stress developed in the Actuator is
11 31x zC d E  (21)
Where Ez is the Electric Field.
The resultant bending moment produced by the actuator is
given by [10]
11 31 ( )
2
aa b
a
t t
M C d V t
 
  
 
(22)
Where Va
(t) is the voltage applied on the actuator which is
given by
( ) ( )a S
V t Controller gain V t  (23)
The force produced by the Actuator is given by
   11 31 1( ) ( )
2 a
aa b
Actu b
l
t t
F C d b V t n x dx
 
  
 
 (24)
This can also be expressed as
    ( )a
ActuF H V t (25)
Where {H} is a constant vector of size (4×1) and is given as
   
   
11 31 1 0 1 0
2
or
1 0 1 0
T a b
b
T
a
t t
H C d b
H C
 
  
 
 
(26)
Where Ca is the Actuator Constant and is given by
11 31
2
a b
a b
t t
C C d b
 
  
 
(27)
5. CONTROL LAW USING PID CONTROLLER
Next goal is to achieve an appropriate controlled voltage
that can be fed to the Actuator, for that a PID controller is

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used in the present study. A typical PID control law that can
be used for Active Vibration Control is:
( ) ( ) ( )p i dy t K K e t dt K e t    (28)
Where y (t) is control signal, Kp, Ki and Kd are Proportional,
Integral and Derivatives gains respectively and e(t) and e(t)
are error signal and its derivative respectively. These three
gains can be tuned in order to provide fine control for the
application. For fine tuning of the controller the following
gain values are taken into consideration KP=100, Kd =10 &
Ki=50.
A typical PID controller is shown in Fig.- 7
6. DYNAMIC EQUATION OF SMART STRUCTURE
Next step is to formulate and solve the Equation of motion
of entire structure that is given by
         Dist ActuM q K q F F    (29)
Consider a generalized coordinate using a transformation
    q x for the first two dominant vibratory modes
then equation of motion becomes
       r r r r
Dist ActuM x K x F F         (30)
If damping of the structure is also considered then assuming
proportional damping as
     C M K   (31)
The generalized dynamic equation of motion is given as
         r r r r r
Dist ActuM x C x K x F F               (32)
7. STATE SPACE FORMULATION FOR THE FIRST
TWO DOMINANT VIBRATION MODES
Let the {x}={y} as
   1 1
2 2
x y
x y
x y
   
     
   
(33)
And
       1 3 3
2 4 4
and
y y y
x y x y
y y y
     
         
     
 
   
 
(34)
Equation of motion now can be written as
   
3 3 1
4 4 2
r r r
r r
Dist Actu
y y y
M C K
y y y
F F
     
                
     
 

 (35)
This can be simplified as
   
1 13 1 3
4 2 4
1 1
r r r r
r r r r
Dist Actu
y y y
M K M C
y y y
M F M F
 
 
     
                     
     
       

 (36)
The above equation can be written in State form as
   
 
   
 
   
1 1
2 2
1 1
3 3
4 4
1 1
0
0 0
( ) ( )
r r r r
a
T Tr r
y y
Iy y
y yM K M C
y y
V t u t
M H M f 
 
 
   
    
    
                       
   
   
    
            




(37)
Where u (t) is the magnitude of external force and {f} is the
unit force vector. The sensor voltage is taken as output of
the structure which can be written as
       1 2 3 4( ) 0
TTs
V t g y y y y 
 
(38)
So the State Space Model of smart structure for the first two
dominant vibratory modes is given by
        
     
( ) ( ) ( )
and
( ) ( ) ( )
a
Ts a
y A y t B V t D u t
V t E y t F V t
  
 

(39)
Where
 
   
 
 
   
 
 
   
 
 
   
 
1 1
1 1
0
,
0 0
, ,
0
, and
r r r r
T Tr r
T
I
A
M K M C
B D
M H M f
E F Null Matrix
g
 

 
 
 
 
                 
   
    
            
 
  
  
(40)
Fig-7 :A Typical PID Controller

_______________________________________________________________________________________
8. RESULTS
The beam is divided in the three Finite elements. Time
response of the structure is studied after bonding the
Sensor/actuator pair at different locations on the beam say
near the fixed end, at the middle and at the free end. The
results are shown in Figures- 8,9 and 10 respectively
CONCLUSIONS
Present study is useful in controlling the vibration of modern
day machines, Engineering structures, Automobiles,
Gadgets Spacecrafts, Bridges, Marine equipments, Machine
Tools, Off shore structures, High rise buildings etc. Present
work deals with Active vibration control of a Cantilever
Beam bonded with two Piezoelectric patches as collocated
pair. It is observed that without control, response is
paramount but after applying control force sufficient
vibration suppression has been achieved. Results are taken
after placing the Piezoelectric Actuator/Sensor Pair near
fixed end, at middle and near free end of the beam. It can be
concluded from the work that best result is obtained when
the Piezoelectric patches are bonded near the fixed end.
REFERENCES
[1]. Tran Ich Thinh , Le Kim Ngoc , “Static behavior
and vibration control of piezoelectric cantilever
composite plates and comparison with
experiments”, Computational Materials Science
49 (2010) S276–S280
[2]. Aydin Azizi, Laaleh Durali, Farid Parvari Rad,
Shahin Zareie , “Control of vibration suppression
of a smart beam by piezoelectric elements”, 2009
Second International Conference on
Environmental and Computer Science, IEEE
Computer Society
[3]. Premjyoti G.Patil, “An Efficient Model for
Vibration Control by Piezoelectric Smart Structure
Using Finite Element Method”, European
Journal of Scientific Research ISSN 1450-216X
Vol.34 No.4 (2009), pp.485-494
[4]. Lucy Edery-Azulay , Haim Abramovich , “Active
damping of piezo-composite beams”, Composite
Structures 74 (2006 458–466
[5]. R. Ly, M. Rguiti, S. D‟Astorg, A. Hajjaji, C.
Courtois, A. Leriche, “Modeling and
characterization of piezoelectric cantilever
bending sensor for energy harvesting”, Sensors
and Actuators A 168 (2011) 95–100
[6]. Magdalene Marinaki , Yannis Marinakis, Georgios
E. Stavroulakis, “Vibration control of beams with
piezoelectric sensors and actuators using particle
swarm optimization”, Expert Systems with
Applications 38 (2011) 6872–6883
[7]. K Ramesh Kumar and S Narayanan, “Active
vibration control of beams with optimal placement
of piezoelectric sensor/actuator pairs”, Smart
Mater. Struct. 17 (2008) 055008
[8]. Jingjun Zhang, Lili He, Ercheng Wang, Ruizhen
Gao, “Active Vibration Control of Flexible
Structures Using Piezoelectric Materials”,2008
International Conference on Advanced Computer
Control, IEEE Computer Society
[9]. C.M.A. Vasques, J. Dias Rodrigues , “Active
vibration control of smart piezoelectric beams:
Fig- 10 : Tip Displacement of Beam when Sensor/Actuator
pair is placed at Free End of the Beam
pair is placed at Middle of the Beam
pair is placed near Fixed End of the Beam

_______________________________________________________________________________________
Comparison of classical and optimal feedback
control strategies”, Computers and Structures 84
(2006) 1402–1414
[10]. T.C. Manjunath, B. Bandyopadhyay, “Vibration
control of Timoshenko smart structures using
multirate output feedback based discrete sliding
mode control for SISO systems” , Journal of
Sound and Vibration 326 (2009) 50-74
[11]. Michele Betti, Georgios E. Stavroulakis and
Charalambos C. Baniotopoulos, “Active vibration
suppression of smart beams”, PAMM · Proc. Appl.
Math. Mech. 6, 799–800 (2006)
[12]. Sinan Korkmaz, “A review of active structural
control: challenges for engineering informatics”,
Computers and Structures 89 (2011) 2113-2132
[13]. A. Preumont, “Vibration Control of Active
Structures-An Introduction”, 3rd Edition ,
Springer
[14]. William T. Thomson, "Theory of vibration with
Applications” CBS Publishers & Distributors
[15]. S.S. Rao, “Mechanical Vibrations” Addison
Wesley Publishing Co.

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