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Analysis of active vibration control in smart structures by ANSYS
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2004 Smart Mater. Struct. 13 661
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INSTITUTE OF PHYSICS PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 13 (2004) 661–667
PII: S0964-1726(04)77278-X
Analysis of active vibration control in
smart structures by ANSYS
H Karagülle1,3 , L Malgaca1 and H F Öktem2
1
Department of Mechanical Engineering, Dokuz Eylül University, 35100 Bornova,
Izmir, Turkey
2
Directorate of Construction and Technical Affairs, Ege University, 35100 Bornova,
Izmir, Turkey
E-mail: hira.karagulle@deu.edu.tr
Received 17 November 2003, in final form 24 February 2004
Published 18 May 2004
Online at stacks.iop.org/SMS/13/661
DOI: 10.1088/0964-1726/13/4/003
Abstract
It is possible to model smart structures with piezoelectric materials using the
product ANSYS/Multiphysics. In this study, the integration of control
actions into the ANSYS solution is realized. First, the procedure is tested on
the active vibration control problem with a two-degrees of freedom system.
The analytical results obtained by the Laplace transform method and by
ANSYS are compared. Then, the smart structures are studied by ANSYS.
The input reference value is taken as zero in the closed loop vibration
control. The instantaneous value of the strain at the sensor location at a time
step is subtracted from zero to find the error signal value. The error value is
multiplied by the control gain to calculate the voltage value which is used as
the input to the actuator nodes. The process is continued with the selected
time step until the steady-state value is approximately reached. The results
are obtained for the structures analyzed in other studies. The active vibration
control of a circular disc is also studied.
1. Introduction
It is desired to design lighter mechanical systems carrying out
higher workloads at higher speeds. However, the vibration
may become prominent factor in this case. Active control
methods can be used to eliminate the undesired vibration.
Using piezoelectric smart structures for the active vibration
control has been paid considerable attention over the last
decade. Some recent works are reported here.
The active vibration control of simple cantilever beams
is studied in [1–5]. Piezoelectric patches as actuators are
mounted on the beams. Another piezoelectric patch or a straingage can be used to sense the vibration level. The system
identification and pole placement control method is used in [1].
The beam and piezo-patches finite element model of the
structure is constructed and the closed loop control is applied
in [2, 3]. Singh [4] also used the beam and piezo-patches finite
element model, but applied modal control strategies.
Xu et al [5] reported results using the commercial finite
element package ANSYS. The control design is carried out
3 Author to whom any correspondence should be addressed.
0964-1726/04/040661+07$30.00 © 2004 IOP Publishing Ltd
in the state space form established in the finite element modal
analysis. The MatLAB Control System Toolbox is used in their
study for the control design. The influence of sensor/actuator
location is studied. It is observed that a location near to the
clamped end is better for vibration control.
Lim [6] studied the vibration control of several modes
of a clamped square plate by locating discrete sensor/actuator
devices at points of maximum strain. Quek et al [7] presented
an optimal placement strategy of piezoelectric sensor/actuator
pairs for the vibration control of laminated composite plates.
Xianmin et al [8] studied the active vibration control in
a four-bar linkage. The finite element model is used and the
reduced mode, standard H∞ , and robust H∞ control strategies
are analyzed.
Numerical simulations are reported in all the references
given. Experimental results are also reported in some
studies [1, 2, 5].
In this study, the product ANSYS/Multiphysics (version
5.4) is used to model smart structures [9]. The integration
of control actions into the ANSYS modeling and solution
is achieved. The success of the procedure is studied by
Printed in the UK
661
H Karagülle et al
Applying the Lagrange equation [11], the mathematical
model of the system in figure 1(a) can be found to be:
m1 0
ẍ1
c + c2 −c2
ẋ1
+ 1
ẍ2
−c2
c2
ẋ2
0 m2
k1 + k2 −k2
x1
f1
+
=
.
(2)
−k2
k2
x2
f2
(a)
(b)
Figure 1. (a) The two-degrees of freedom system (m 1 = 1.2 kg,
m 2 = 1 kg, k1 = 350 N m−1 , k2 = 300 N m−1 , c1 = 4 N s m−1 ,
c2 = 3 N s m−1 ) and (b) block diagram of the closed loop control
system.
Then, the transfer functions can be written as the following,
after substituting the values of the masses, damping and spring
constants:
3(s + 100)
H21 (s) =
(3)
D(s)
(1.2s 2 + 7s + 650)
D(s)
comparing the results with the analytical results obtained
for the active-vibration control of a two-degrees of freedom
system. The smart structures studied in references are then
analyzed by ANSYS. The results for the active vibration
control of a circular plate are also presented.
where
2. Two-degrees of freedom system
Substituting X r (s) = 0 and F2 (s) = 1, the transfer function
of the closed loop system in figure 1(b) is found to be
Analytical solution
The system considered and a block diagram of the closed
loop control system are shown in figure 1. f 2 is the vibration
generating force, and f 1 is the controlling force. X r , X 2 , F1
and F2 are the Laplace transforms of the reference input (xr ),
output displacement (x2 ), the forces f 1 and f2 , respectively.
G 1 is the transfer function of the control action, and it is taken as
KI
G 1 (s) = K P +
+ KDs
s
(1)
for the PID (proportional-integral-derivative) control. K P ,
K I , K D are the proportional, integral, derivative constants,
respectively [10]. H21 (s) is the transfer function from F1 to X 2 ,
and H22 is the transfer function from F2 to X 2 . The reference
input, xr , is taken as zero for the vibration control. xe (t) is
defined as the error signal, where xe (t) = xr (t) − x2 (t) [10].
The vibration generating force is taken as a unit impulse in this
study, and thus F2 (s) = 1.
H22 (s) =
D(s) = 1.2s 4 + 10.6s 3 + 1022s 2 + 2250s + 105 000.
X 2 (s) = {s(1.2s 2 + 7s + 650)}{s D(s) + 3[K D s 3
+ (K P + 100K D )s 2 + (100K P + K I )s + 100K I ]}−1 .
For the solution by ANSYS, MASS21 and COMBIN14
elements are used. The system in figure 1(a) is modeled.
Modal analysis is performed and two undamped natural
frequencies are found. The time step (t) can be taken as
1/(20 fh ), where fh is the highest frequency. However, it
is taken as 1/(60 fh ) because the differential control action
requires smaller time steps for higher accuracy. ts is the time
at which the steady-state response is approximately reached.
The undamped natural frequencies for the open loop system
are found as 1.75 and 4.27 Hz. So, t = 0.0039 s.
ANSYS Solution
0.04
0.02
0.02
X (m2)
X (m2)
Analytical Solution
0
-0.02
-0.02
-0.04
-0.04
Control Off
Control On
(a)
0.5
1
1.5
Time (s)
2
2.5
Control Off
Control On
(b)
-0.06
0
0.5
1
1.5
Time (s)
2
Figure 2. (a) Analytical and (b) ANSYS solutions (K P = 100, K I = 40 and K D = 10 for control on).
662
(6)
Solution by ANSYS
0.04
0
(5)
x2 (t) can be found by taking the inverse Laplace transform of
X 2 (s). The time histories of x2 (t) are given in figure 2(a) for
the uncontrolled and controlled cases.
0.06
0.06
-0.06
0
(4)
2.5
Analysis of active vibration control in smart structures by ANSYS
Figure 3. Block diagram of the analysis.
The value of f 2 is 1/t at t = t, and it is zero otherwise.
The value of f 1 is zero at t = t. The part of the macro which
enables the calculations for the closed loop analysis for t > t
is given below:
sum=0
errp=0
*do,t,2*dt,ts,dt
*get,e1,node,2,u,x
err=0-e1
sum=sum+err*dt
dif=(err-errp)/dt
f1=kp*err+ki*sum+kd*dif
f,1,fx,f1
errp=err
time,t
solve
*enddo
The variables dt, ts, kp, ki and kd are defined in the previous
part of the macro, and they correspond to t, ts , K P , K I and
K D , respectively. The variable f1 corresponds to the actuation
force f 1 .
The time histories of x2 (t) obtained by the ANSYS
solution are given in figure 2(b) for the uncontrolled and
controlled cases. It is observed from figures 2(a) and (b) that
the analytical and ANSYS solutions are in agreement.
After testing the success of the ANSYS solution by
comparing it with the analytical solution for the two-degrees
of freedom system, the ANSYS solution is used for the smart
structures below.
3. Smart structures
In this section, the active vibration control in smart structures
is simulated by ANSYS. The block diagram of the analysis is
shown in figure 3. ANSYS/Multiphysics [9] can be used to
model piezoelectric and structural fields. Fe is the vibration
generating force. The instantaneous value of the vibration
generating force Fe can be defined at each time step. It
is taken as F0 at t = t and zero for other time steps
in the analyses below. The strain at a sensor location, ε,
is calculated. The reference input is zero for the vibration
cancellation. K s , K c and K v are the sensor, control and power
amplification factors, respectively. K s and K v are taken as
1000 by inspection, and K c is changed in the analyses below.
Only the proportional control is applied. The multiplication of
K s K c K v is the proportional constant for the actuator voltage,
Va . So, changing the values of K s , K c and K v , keeping
their multiplication the same, does not affect the results. The
calculated deflection at a location, dt , is observed to evaluate
the performance of the vibration control.
Figure 4. Configuration of the beam type structure.
Beam or plate type structures
First, beam or plate type structures are considered. The
configuration of the structure is shown in figure 4. The strain
value at the sensor location is taken as the feedback. The
dimensions and the distances for the cases studied are given in
table 1. This type of structure is studied in many studies [1–
5]. The corresponding reference where the same structure is
studied is indicated in table 1.
A macro has been written for ANSYS. The macro starts
with the definition of the variables for the dimensions of the
structure. Then the three dimensional material properties are
assigned. The part of the macro where the material properties are assigned is given below. Material 1 is the metal, and
material 2 is the actuator material.
mp,ex,1,68e9
! Elasticity modulus for metal
mp,dens,1,2800
! Density
mp,nuxy,1,0.32
! Poisson’s ratio
mp,dens,2,7500
! Density for piez. material
mp,perx,2,15.03E-9
! Permittivity in x direction
mp,pery,2,15.03E-9
! Permittivity in y direction
mp,perz,2,13E-9
! Permittivity in z direction
tb,piez,2
! Define piez. table
tbdata,16,17
! E16 piezoelectric constant
tbdata,14,17
! E25
tbdata,3,-6.5
! E31
tbdata,6,-6.5
! E32
tbdata,9,23.3
! E33
tb,anel,2
! Define structural table
tbdata,1,126E9,79.5E9,84.1E9! C11,C12,C13
tbdata,7,126E9,84.1E9
! C22,C23
tbdata,12,117E9
! C33
tbdata,16,23.3E9
! C44
tbdata,19,23E9
! C55
tbdata,21,23E9
! C66
The macro is continued to create nodes and finite elements.
SOLID45 elements are used for the metal part, and SOLID5
elements are used for the piezoelectric part of the structure.
The finite element model for case 1 is shown in figure 5.
Fixed boundary conditions are defined for the nodes at x = 0.
The degrees of freedom, VOLT, are coupled for the nodes at
the top and bottom surfaces of the actuator by the ANSYS
command cp.
Modal analysis is performed to determine the time step.
Only the reduced method (Householder method) can be used
for the structures which have coupled-field solids in ANSYS.
The time step is chosen as t = 1/(20 fh ), where f h is the
highest natural frequency to be considered. The three natural
frequencies for the undamped system are given in table 2.
663
H Karagülle et al
Table 1. Dimensions and distances for the cases.
Case
Reference
Dimensions of
the structurea (mm)
1
2
3
4
[5]
[5]
[2]
[1]
504 × 25.4 × 0.8
224.25 × 25 × 0.965
160 × 25.4 × 2
348 × 24 × 1
a
b
Dimensions of
the actuatorb (mm)
Actuator
distance, da (mm)
Sensor
distance, ds (mm)
72 × 25.4 × 0.61
39 × 25 × 0.75
46 × 20.6 × 0.254
72 × 24 × 0.5
12
9.8
5.7
12
48
29.3
28.5
48
Aluminum.
PZT-5H.
Table 2. Natural frequencies for the undamped system (control off).
Natural frequencies (Hz)
Figure 5. Finite element model for case 1.
The first mode is considered to calculate the time step and
t is 0.0159, 0.0025, 0.0007 and 0.0055 for cases 1, 2, 3 and
4, respectively.
In the transient analysis, the coefficients of Rayleigh
damping (α and β) are defined. α = β in this study. Fe = F0
for t = t and Fe = 0 at the subsequent time steps. Va = 0
at t = t. The strain is calculated at the selected sensor
location and it is multiplied by K s , and then it is subtracted
from zero. The zero value is the reference input value to control
the vibration. The difference between the input reference and
the sensor signal is called the error signal [10]. The error value
is multiplied by K c and K v to determine Va at a time step. The
part of the macro which enables the calculations for the closed
loop analysis for t > t is given below.
*do,t,2*dt,ts,dt
*get,u1,node,nr,u,x
*get,u2,node,nr1,u,x
err=0-ks*(u2-u1)/dx
va=kc*kv*err
d,nv,volt,va
time,t
solve
*enddo
The variables ks, kc and kv correspond to K s , K c and
K v , respectively. nr and nr1 are the node numbers used to
calculate the strain. These nodes are adjacent in the x direction,
and dx is the distance between them.
The values of Rayleigh damping coefficients, α and β,
the impulsive force, F0 , and the control gain, K c , are taken
differently for each case. The values of F0 and K c are limited
by the maximum value of the actuator voltage, which can
664
Case
First
Second
Third
1
2
3
4
3.15
19.85
70.83
9.13
18.12
104.78
400.28
45.31
45.97
259.22
805.83
106.92
be applied to the actuator safely without breaking it. The
maximum voltage per thickness of the piezoelectric material
is taken as 235 V mm−1 . So, the actuator voltages are kept
below 143.4, 176.3, 59.7 and 117.5 V for cases 1, 2, 3 and 4,
respectively.
The tip deflections and actuator voltages for different
values of the control gain for case 1 are shown in figure 6.
It is observed that as K c increases the vibration settling time
decreases and the actuator voltage increases. The case K c = 5
cannot be applied because the absolute value of the actuator
voltage exceeds the limit value of 143.4 V.
The tip displacement and the actuator voltages for cases
2, 3 and 4 are shown in figures 7 and 8. Similar results are
obtained in the references given in table 1. Different control
strategies are applied in the references and some of the results
are verified by experiments [1, 2, 5].
Axially symmetric structures
The configuration of the axially symmetric structure and its
finite element model are shown in figure 9. The y axis is the
axis of symmetry. PLANE42 elements are used for the metal
part, and PLANE13 elements are used for the piezoelectric
part of the axially symmetric structure. The part of the macro
where the piezoelectric material properties are defined for the
axially symmetric structure is given below:
mp,dens,2,7730
mp,perx,2,1.503e-8
mp,pery,2,1.300e-8
tb,piez,2
tbdata,2,-6.5
tbdata,5,23.3
tbdata,8,-6.5
tbdata,10,17
tb,anel,2
tbdata,1,126e9,79.5e9,84.1e9
tbdata,7,117e9,84.1e9
tbdata,12,126e9
tbdata,16,23e9
Analysis of active vibration control in smart structures by ANSYS
1
0.015
Kc=0
Kc=2
2
3
Kc=3.3
4
Kc=5
1
200
Kc=0
2
Kc=2
3
Kc=3.3
Kc=5
4
Case 1
2
0.005
100
Actuator Voltage (V)
0.01
Tip Deflection (m)
Case 1
1
3
4
0
-0.005
0
1
4
2
3
-100
-200
-0.01
(a)
(b)
-0.015
0
0.5
1
-300
0
1.5
0.5
1
1.5
Time (s)
Time (s)
Figure 6. (a) Tip deflections and (b) actuator voltages for different values of control gain (F0 = 0.1, α = 0.001).
1.5
x 10
-3
100
Case 2
Case 2
50
Actuator Voltage (V)
Tip Deflection (m)
1
0.5
0
-0.5
-1
-1.5
0
0.1
0.2
0.3
Time (s)
0.4
-50
-100
-150
K c=0
K c=5
(a)
0
K c=0
K c=5
(b)
0.5
-200
0
0.1
0.2
0.3
Time (s)
0.4
0.5
Figure 7. (a) Tip deflections and (b) actuator voltages for case 2 (F0 = 0.2, α = 0.0003).
6
x 10
-4
4
x 10
-3
Case 3
Case 4
3
Tip Deflection (m)
Tip Deflection (m)
4
2
0
-2
2
1
0
-1
-2
-4
-6
0
K c=0
K c=1
(a)
0.02
0.04
0.06
0.08
Time (s)
0.1
0.12
K c=0
K c=4
-3
0.14
-4
0
(b)
0.2
0.4
0.6
Time (s)
0.8
1
Figure 8. Tip deflections for (a) case 3 (F0 = 2, α = 0.0001) and (b) case 4 (F0 = 0.2, α = 0.0006).
The three natural frequencies for the undamped system
are given in table 3.
The first mode is considered to calculate the time step, and
t is 0.000 49 and 0.000 53 for cases 5 and 6, respectively.
The center displacements for different actuator radii are
shown in figure 10. The radius of the structure, Rc =
101.6 mm, and the sensor distance from the center is taken
as 2/16 times Rc . The extreme actuator voltages are −124.6
665
H Karagülle et al
Table 4. Characteristics of vibration signals.
Control off
Cases ξ
Figure 9. Configuration of the axially symmetric structure and its
finite element model.
Table 3. Natural frequencies for the undamped system (control off).
Natural frequencies (Hz)
Case
First
Second
Third
5
6
102.94
94.50
422.96
445.46
1012.8
1120.3
and −124.3 V for cases 5 and 6, respectively. It is observed that
the vibration cancellation is faster for the increasing actuator
size for the same level of the maximum actuator voltage.
Characteristics of vibration signals
The vibration signals given in figures 6, 7, 8 and 10 can be
approximately modeled by the signal d(t) = Ae−ξ ωn t sin(ωt),
where ω = ωn 1 − ξ 2 . A is the amplitude, ξ is the damping
ratio, ωn is the undamped frequency and ω is the damped
frequency. The maximum and minimum values of the actuator
voltages (Vmax and Vmin ), and the values of ξ and fd are listed
for different cases in table 4, where f d = ω/(2π). It is
observed from table 4 that the closed loop control increases
the damping ratio and decreases the damped-frequency.
1
2
3
4
6
Control on
f d (Hz) K c
0.011 3.11
0.019 19.61
0.023 69.10
0.017 9.13
0.029 93.34
3.3
5
1
4
25
ξ
f d (Hz) Vmax (V)
0.061 2.83
0.153 16.54
0.056 62.96
0.044 8.59
0.145 85.91
115.4
97.2
49.7
82.7
124.3
Vmin (V)
−139.0
−155.0
−58.1
−91.8
−69.9
known. The calculated instantaneous value of the sensor
quantity is subtracted from the reference quantity, which is zero
for the vibration cancellation, to calculate the error signal value
at a time step. The error value is multiplied by the control gain
to determine the instantaneous value of the actuator voltage.
The process is continued in a loop until the steady-state value
of the observation quantity is approximately reached. The
control gain is determined by increasing its value. It is limited
by the maximum actuator voltage which can be applied to
the piezoelectric actuator without breaking it. The validity of
the control loop in ANSYS is first tested by comparing with
the analytical results for a two-degrees of freedom system.
Then the procedure is applied to the active vibration control
problem of the smart structures for which other simulation
and experimental results are available in the references. It is
observed that the procedure can successfully be applied for the
smart structures.
Modeling of smart structures, locating the actuators and
sensors, determining the feedback gain and evaluating the
performance of the design are the main steps in active
vibration control. This can be achieved by the computer aided
engineering software ANSYS. Structures with complicated
geometries can be analyzed.
4. Conclusions
Acknowledgment
ANSYS/Multiphysics can successfully be used for the
simulation of the active vibration control of smart structures.
The instantaneous value of the vibration generating force is
6
x 10
This research was supported by the Dokuz Eylul University
Research Fund, Project Number: 03.KB.FEN.053.
-5
6
x 10
-5
Case 5 (Ra/Rc=2/16)
Case 6 (Ra/Rc=3/16)
Center Deflection (m)
Center Deflection (m)
4
2
0
-2
2
0
-2
-4
K c=0
K c=15
(a)
-6
0
4
0.02
0.04
0.06
Time (s)
0.08
K c=0
K c=25
(b)
0.1
-4
0
0.02
0.04
0.06
Time (s)
0.08
0.1
Figure 10. Center deflections for different actuator sizes (F0 = 1, α = 0.0001): (a) case 5 (Ra /Rc = 2/16), (b) case 6 (Ra /Rc = 3/16).
666
Analysis of active vibration control in smart structures by ANSYS
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