Download Active Vibration Control of Smart Cantilever Beam

Imperial Journal of Interdisciplinary Research (IJIR)
Vol-2, Issue-5, 2016
ISSN: 2454-1362, http://www.onlinejournal.in
Active Vibration Control of Smart Cantilever Beam
Shaikh Tauseef1 & R. K. Agrawal2
PG Student, Department of Mechanical Engineering, YTIET, Karjat 410201
Associate Professor, Department of Mechanical Engineering, YTIET, Karjat 410201
Abstract: In this paper vibration control of
cantilever beam which serve as basic model for
different type of structure used in various industry
fields is simulated using ANSYS 16 software. For
vibration control piezoelectric ceramic LEAD
ZIRCONATE TITANATE (PZT) patches are
attached near the fixed end at the top and bottom
surface serving as Actuator. The work considers
Active Vibration Control (AVC). Beam as well as
PZT patches are considered as Euler-Bernoulli
beam element. The cantilever beam along with
bonded PZT patch is called smart cantilever beam.
Different voltage is applied within limit to the PZT
patches and amplitude of vibration is observed.
2. Beam Theory
Beam under study is considered as Euler-Bernoulli
beam also PZT patches are considered as EulerBernoulli beam element. For a cantilever beam
deflection along length of beam at any instant
i.e.
is given by
Where,
1. Introduction:
Light weight, thin walled structures are applied in
many fields such as aircraft, turbines, space and
civil structures. Negative effects of these structures
includes low damping and high vibration
sensibility. One of the solution is to make
structures
smart
using
electrostrictives,
piezoelectric,
magnetostrictives
for
Active
Vibration Control. Technique of controlling the
vibration by applying force to the structure that is
equal in magnitude that of original force but
appropriately out of phase is known as “Active
Vibration Control”. Suresh Venna et al [1]
determined the electric potential that would be
generated in the piezoelectric vibration absorber
using FEA to determine optimal locations for
damping of the first mode. Zhang Shunqi [2] and
Saurabh Kumar [3] used PID algorithm and PID
controller respectively. Deepak Chhabra [4] used
Modified Control Matrix and Singular Value
Decomposition Approach to find optimal position
of Piezoelectric Actuators along length of beam.
Fabio Botta et al [8] developed a procedure and
proposed a new analytical solution for active
vibration control. From all above mentioned
research and experiments and many other as well it
is found that maximum damping or reduction of
vibration amplitude is obtained when PZT patch is
attached near the fixed end of cantilever beam. So
in this study, PZT patches are attached near the
fixed end each at top and bottom surface of the
beam.
Solving Equation (1) gives
………… (2)
Where,
Values of
is given in Table 1.
Table 1. Values of
n
1
2
3
4
5
L
1.875
4.694
7.854
10.995
14.137
3. FEA Modelling Of Smart Cantilever
Beam
An aluminium cantilever beam bonded with
collocated pair of Piezoelectric Actuator as shown
in Figure 1.
Figure 1.
Imperial Journal of Interdisciplinary Research (IJIR)
Page 1362
Imperial Journal of Interdisciplinary Research (IJIR)
Vol-2, Issue-5, 2016
ISSN: 2454-1362, http://www.onlinejournal.in
Using Lagrange’s equation, stiffness matrix and
mass matrix for regular and smart beam element is
given as,
D = Electric displacement i.e. charge per unit area
(Coulomb/m2)
 = Dielectric constant (Permittivity) under
constant stress
Stress developed in the Actuator is
Where,
= Young’s modulus of elasticity
The resultant bending moment produced by the
actuator is given by
= the applied actuator voltage
The force produced by the Actuator is given by
5. Smart Beam Simulation
Harmonic Excitation
Under
Simulation is done on ANSYS 16 using Ansys
Parametric Design Language (APDL) wherein
Three dimensional structural element (solid186) is
used for the metal part and piezoelectric patches are
modelled using three dimensional coupled elements
(SOLID5) which is shown in Figures 2(a),2(b) &
2(c).The dimension and material properties of
beam and PZT patch is described Table
2.
Where,
Moment of inertia of beam
Moment of inertia of piezoelectric patch
Length of beam
Length of piezoelectric patch
4. Modelling of PZT Actuator
Figure 2 (a).
Following Linear Piezoelectric Constitutive
equations are used for deriving the Actuator
equations.
Where,
 = Strain
 = Stress
SE = Compliance when electric field is constant
d31 = Piezoelectric Constant (Coulomb/N or m/V)
E = Electric field (Volt/m)
Imperial Journal of Interdisciplinary Research (IJIR)
Figure 2 (b).
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Imperial Journal of Interdisciplinary Research (IJIR)
Vol-2, Issue-5, 2016
ISSN: 2454-1362, http://www.onlinejournal.in
Figure 2 (c).
Table 2. Material Properties
Properties
Aluminium
PZT-5H
Length
400 mm
20 mm
Figure 3.
6. Result
Width
40 mm
40 mm
Thickness
2 mm
0.15 mm
68e9 (N/m2)
2800 (kg/m3)
----7500
(kg/m3)
Young Modulus
Density
Poison’s ratio
0.32
Compliance Matrix
S11
16.5
S12
-----
-4.78
S13
-8.45
S22
16.5
S23
-8.45
S33
20.7
S44
2.33
S55
43.5
S66
42.6
Piezoelectric strain matrix
d13
-274
d23
-274
d33
593
d52
720
d61
720
Relative permittivity
1977.4
x
1977.4
y
2395.4
z
x 10 -12
(m2/N)
x 10-12
(m/V)
By applying voltage of 25V and 50V in thickness
direction of PZT patch i.e. polarization direction,
amplitude of vibration is reduced as shown in Table
3.
Table 3. Amplitude of Vibration
Frequency
Amplitude of vibration (cm)
(Hz)
Without
With
With
Patch
Patch
Patch
(25V)
(50V)
1.9
1.31
1.26
1.22
2.8
1.37
1.31
1.27
3.7
1.46
1.39
1.34
1.39
1.59
1.51
1.45
5.5
1.79
1.69
1.60
6.4
2.10
1.96
1.83
7.3
2.64
2.41
2.20
8.2
3.72
3.26
2.86
9.1
6.87
5.4
4.30
10.0
124.73
20.09
9.76
(F/m)
7. Conclusion
Beam is excited harmonically by applying external
force of magnitude 1N in frequency range of 1 to
10 Hz as for the dimension of the beam first
fundamental natural frequency is 10Hz.It is found
that maximum amplitude of vibration occurs at first
natural frequency as shown in Figure 3, So in this
study amplitude is reduced for first mode of
vibration of beam.
Imperial Journal of Interdisciplinary Research (IJIR)
Vibration of beam can be controlled by using PZT
as done in this work. Different voltage is applied
across PZT patch to obtain different result.it is
observed that when voltage is high vibration
reduction is more but it should be noted that
maximum voltage applied must be within the
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Imperial Journal of Interdisciplinary Research (IJIR)
Vol-2, Issue-5, 2016
ISSN: 2454-1362, http://www.onlinejournal.in
capacity of PZT patch which depends on Type,
Material and dimension of PZT patch.
8. References
[10] Ruo Lin Wang, H. Gu, and G. Song, “Adaptive
Robust Sliding Mode Vibration Control of a
Flexible Beam Using Piezoceramic Sensor and
Actuator,”Mathematical
Problems
in
Engineering Volume 2014, Article ID 606817, 9
pages
Research Papers:
[1] Suresh Venna, Yueh-Jaw Lin,An Effective
Approach for Optimal PZT Vibration Absorber
Placement on Composite Structures, Modern
Mechanical Engineering, Vol.3 No.1, February
27, 2013
Website:
[1] www.nptel.ac.in
[2] http://emweb.unl.edu/Mechanicspages/scottWhitney/325hweb/beams.htm
[2] Shunqi Zhang, Rüdiger Schmidt, Xiansheng
Qin, Active vibration control of piezoelectric
bonded smart structures using PID algorithm,
Chinese Journal of Aeronautics, Volume 28,
Issue 1, February 2015, Pages 305-313, ISSN
1000-9361
[3] Saurabh
Kumar,
Rajeev
Srivastava,
R.K.Srivastava, Active Vibration Control Of
Smart Piezo Cantilever Beam Using PID
Controller, International Journal of Research in
Engineering and Technology eISSN: 23191163 | pISSN: 2321-7308
[4] Deepak Chhabra, Gian Bhushan, Pankaj
Chandna, Optimal Placement of Piezoelectric
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Matrix and Singular Value Decomposition
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Aerospace, Industrial and Mechatronics
Engineering Vol:7 No:3, 2013
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Botta,
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905160, 8 pages
[6] M. Kerboua, A. Megnounif, M. Benguediab,
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[7] Zhi-chengQiu, Hong-xin Wu, Chun-de Ye,
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[8] Ting Zhang, Hong Guang Li, Guo Ping Cai,
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[9] Riessom W,Prasad Krishna,K.V. Gangadharan,
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