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LOGO
ByMandipalli Naresh
12MC10F
Multi Input Multi Output
Active Vibration Control ofSmart Structures using PZT
GuideDr. K V Gangadharan
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Introduction
Vibration control
Passive Semi active Active
m
k c
m
k c
m
k c
Fd Fd Fd Fc
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Summary of literature review
Majority of literatures are on single input andsignal output systems but in reality many systems
are multi input and multi output (MIMO).
Majority of literatures are based on simulation
studies which require to be validatedexperimentally.
Basic controllers can result even the system to be
unstable, with large changes in system parameters.
This problem can be avoided using robust control
and adaptive control design techniques like
LQR/LQG controller.
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Summary of literature review (cont..)
Pole placement technique is the basic state feedbackcontroller.
Controller performance had been tested for
impulse i.e., for free vibration which is required to
be improved for forced vibration.
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Problem Statement
To analyze the nature of vibration in structures and
suppress the vibration effectively using pole
placement state feedback technique for MIMO
system
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Objectives
Comparative study of conventional controllers andstate feedback controller (Pole placement controller)
for active vibration control.
Vibration control for few dominant modes ofvibration and impulse vibration.
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Modeling and Simulation
System Configuration
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Modeling and Simulation (Cont)
Beam is modelled as two node element which has2DOF at each nodes ( Transversal displacement
and slope )
Assuming cubic polynomial equation for transverse
vibration of beam
2 3
0 1 2 3( )u x a a x a x a x
1u 2u
2
1
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Modeling and Simulation (Cont)
The interpolation can be written in shape functionform after applying boundary conditions,
1 1 2 1 3 2 4 2( ) (x)u ( ) ( )u ( )u x w w x w x w x
2 3
2 3
2 31
22
2 33
2 3
4
2 3
2
3 21
2
3 2
x x
l l
w x xx
w l l
w x x
w l l
x x
l l
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Modeling and Simulation (Cont)
Comparing finite element equation with mechanicaltranslational motion equation,
2 2
2 2
156 22 54 13
22 4 13 3[M]
54 13 156 22420
13 3 22 4
l l
l l l l Al
l l
l l l l
2 2
3
2 2
12 6 12 6
6 4 6 2
[ ] 12 6 12 6
6 2 6 4
l l
l l l l EI
K l ll
l l l l
[ ] [M] [K] ; 0.01, 0.001C
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Modeling and Simulation (Cont)
Piezoelectric as sensor and actuator Direct and converse piezoelectric equations
Voltage produced by sensor
Equivalent force is given by piezoelectric
31
t
fd E s
31z fD d e E
c 31 c 310
(t) G e zb ' G e zb[0 1 0 1] '
pl
T
sV B d dx d
31 [ 1 0 1 0]V (t)
c p aF E d bz
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Modeling and Simulation (Cont)
Model order reduction
System model equation is
Reduced model order equation can be found by
considering required modes
d cM d C d K d F F
d T q
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Modeling and Simulation (Cont)
System matrices
Experimental FEM ANSYS
1stmodeFrequency(Hz)
29.604 32.33 32.782
2ndmodeFrequency(Hz)
169.5 172.76 183.71
3 0.103327733887813 -0.0000000000000134[m] 10-0.0000000000000134 0.0151176776852316
4.264094632032168 -0.0000000000814512[k]
-0.0000000000814014 18.639762290157726
0.000426512790937 -0.000000000000008[c]
-0.000000000000008 0.001863991346693
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Modeling and Simulation (Cont)
State space model
0 0 1 0
0 0 0 1A
-4.1268e+04 8.7175e-07 -4.1278 8.7175e-11
-5.0048e-06 -1.2330e+06 -5.0048e-10 -123.2988
0 0 0
0 0 0
0.0112 0.0242 0.0356
0.1522 0.2523 0.2404
B
0 0 1.9515 05 3.2191 05
0 0 1.6118 05 8.5565 06
e eC
e e
0 0 0
0 0 0D
x Ax Bu
y Cx Du
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Modeling and Simulation (Cont)
Particle system response
Voltage
Time
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Modeling and Simulation (Cont)
Mode shapes in ANSYS
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Experimental Setup
ChargeAmplifier
Voltage
Amplifier
Beam with Sensor and
Actuator
PXI RTSystem
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Pole placement technique
Consider a system
State feedback control law is:
K is feedback gain vector
x Ax Bu
y Cx Du
1
2
1 2 3
( )
( )
.. . .
.
.
( )n
x t
x t
K K K
x t
( )u Kx t
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Pole placement technique (Cont..)
Closed-loop control system with u=Kx
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Pole placement technique (Cont..)
Substituting u in open loop system
Eigenvalues of A BK gives the closed loop poles of
the system
Thus the controller gain K will be determined by
placing closed loop poles at fixed desired position
Necessary and sufficient condition for arbitrary
pole placement is that the system be completely
state controllable and observable.
x( ) ( ) x(t)t A BK
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Pole placement technique (Cont..)
Condition for controllability
Condition for observability
Where n = number of states
m = number of outputs
1. . . nrank B AB A B n
2 1. . .
nrank CB CAB CA B CA B D m
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Pole placement technique (Cont..)
State feedback with controller
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Proportional controller result
Percentage reduction : 93.95
Time
Voltag
e
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Conclusion
The detail system dynamics is analysed successfully;
Mathematical model of the system is developed using
FEM technique for controller design and simulation;
It has been attempted to implement a classical controller
namely proportional controller for first mode of
vibration;
The Percentage strain (in voltage) reduction is observed
to be 93.95% with additional harmonic frequency;
The conventional controller implemented is not still
effective for the 2nd mode of vibration, thus it requiresfurther study.
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Future work
In the future it is planned to design state feedbackcontroller for MIMO active vibration system based on
pole placement technique.
Controller
Full state observer Experimentally validating the effectiveness of the
designed controller
Comparing the state feedback controller with classical
controller.
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