Aircraft Engineering and Aerospace Technology
Emerald Article: Fractional controller design for suppressing smart beam
vibrations
Cem Onat, Melin Sahin, Yavuz Yaman
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Cem Onat, Melin Sahin, Yavuz Yaman, (2012),"Fractional controller design for suppressing smart beam vibrations", Aircraft
Engineering and Aerospace Technology, Vol. 84 Iss: 4 pp. 203  212
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Fractional controller design for suppressing
smart beam vibrations
Cem Onat
Department of Mechanical Engineering,Inonu University,Malatya,Turkey,and
Melin S¸ ahin and Yavuz Yaman
Department of Aerospace Engineering,Middle East Technical University,Cankaya,Turkey
Abstract
Purpose – The purpose of this paper is to detail the design of a fractional controller which was developed for the suppression of the ﬂexural vibrations
of the ﬁrst mode of a smart beam.
Design/methodology/approach – During the design of the fractional controller,in addition to the classical control parameters such as the controller
gain and the bandwidth;the order of the derivative effect was also included as another design parameter.The controller was then designed by
considering the closed loop frequency responses of different fractional orders of Continued Fraction Expansion (CFE) method.
Findings – The ﬁrst,second,third and fourth order approximations of CFE method were studied for the performance analysis of the controller.It was
determined that the increase in the order resulted in better vibration level suppression at the resonance.The robustness analysis of the developed
controllers was also conducted.
Practical implications – The experimentally obtained free and forced vibration results indicated that the increase in the order of the approximations
yielded better performance around the ﬁrst ﬂexural resonance region of the smart beam and proved to yield better performance than the classical
integer order controllers.
Originality/value – Evaluation of the performance of a developed fractional controller was realized by using different approach orders of the CFE
method for the suppression of the ﬂexural vibrations of a smart beam.
Keywords Controllers,Vibration,Smart beam,Lead zirconate titanate,Vibration control,Fractional control
Paper type Research paper
Introduction
Fractional order control systems have transfer functions with
fractional derivatives s
a
and fractional integrals s
2a
where ae
R.It is not an easy and straightforward task to compute the
frequency and time domain behaviours of such fractional order
transfer functions with available software packages.It is well
known that the commercially available simulation programs
have been prepared to deal with the integer power of derivatives
only.Also,the hardware required for the implementation of the
designed controllers use electronic components which are only
suitable for the integer order transfer functions.Although,there
are some recent works dealing with the implementation of a
controller using a fractance device (Nakagava and Sorimachi,
1992),this area deserves further studies.Therefore,the
problem of integer order approximations of fractional order
functions becomes a very important one to be solved.
A fractional transfer function can be replaced with an integer
order transfer function which has almost the same behaviour
with the real transfer function but much more easy to deal with.
There are several methods for obtaining rational
approximations of fractional order systems like Carlson’s
method,Matsuda’s method,Oustaloup’s method,the
Gru¨ nwaldLetnikoff approximation,Maclaurin series based
approximations,time response based approximations,etc.
(Podlubny et al.,2002).One of the most important
approximations for fractional order systems is the continued
fraction expansion (CFE) method.
In this study a fractional order controller,by using the CFE
method,was designed and implemented for the suppression of
the ﬂexural vibrations of a smart beam.The ﬁrst,second,third
and fourth order approximations of CFE method were studied
for the performance analysis of the controller.The robustness
analysis of the controllers was also conducted by attaching
various point masses to the free end of the smart beam.
Experimentally obtained results were presented for the
suppression of the free and forced vibrations of the smart beam.
Smart beam
The smart beam used in the study is shown in Figure 1(a).
It is a cantilever passive aluminiumbeamhavingthe dimensions
of 490 £ 51 £ 2 mm and with eight surfaces bonded
SensorTech – BM500 (25 £ 20 £ 0.5 mm) PZT (Lead
ZirconateTitanate) patches.A typical PZT patch is shown in
Figure 1(b) (Sensor Technologies Limited,2002).A thin
isolation layer is placed between the aluminiumbeamand each
PZTpatch,sothat eachPZTpatchmaybe employedas asensor
and an actuator independently.
In this study,the piezoelectric patches are nominated with
respect to the positions on each surface of the aluminiumbeam
and are identiﬁed by number and surface names.As shown in
Figure 2,on surface A,piezoelectric patches are labelled
The current issue and full text archive of this journal is available at
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Aircraft Engineering and Aerospace Technology:An International Journal
84/4 (2012) 203–212
q Emerald Group Publishing Limited [ISSN 17488842]
[DOI 10.1108/00022661211237728]
203
from 1 to 4 in clockwise direction and on surface B,they are
labelled from1 to 4 in counter clockwise direction.Hence nA
and nB,where n ¼ 14,were symmetrically located on both
surfaces and were intended to provide bimorph type excitation
whenever necessary.
Experimental system identiﬁcation
The system identiﬁcation of the smart beam was conducted
experimentally.For this purpose,thesmart beamwas excitedby
using four piezoelectric actuator patches (1A1B and 4A4B)
and the response of the smart beam was obtained from the
piezoelectric sensor patch (2A).
Figure 3 shows the experimental setup used for the
determination of the experimental frequency response of the
smart beam.The excitation signal was a swept sine signal from
2Hz to18Hz with5Vpeaktopeak value andwas generatedby
HP33120A signal generator.This generated signal was
ampliﬁed 30 times by SensorTech SA10 high voltage ampliﬁer
which also uses SensorTech SA21 high voltage power supply.
The ampliﬁed excitation signal was then fed to piezoelectric
actuator patches 1A1B and 4A4B.Bru¨ el and Kjær PULSE
3560C platform was used for the determination of the
frequency response.
The mathematical model of the smart beam was obtained
by processing the measured frequency response data.By using
MATLAB’s “ﬁtsys” command located in m Analysis and
Synthesis Toolbox (Balas et al.,2001),the transfer functions
of the smart beamwas determined for second,sixth and tenth
order system models.The experimentally obtained transfer
function of the smart beamwas obtained within the frequency
range of 2Hz and 18Hz and that frequency range was known
to include the ﬁrst ﬂexural mode,approximately at 7Hz,of
the smart beam.
Figure 4 shows the magnitude and phase information of the
experimentally obtained system model together with the
second,sixth and tenth order developed system models of
the smart beam.It can be seen that the second order model is
insufﬁcient both in the representation of the magnitude and the
phase.Both the sixth and tenth order models can appropriately
represent the system characteristics.In order to reduce the
excessive computational efforts,in this study,a sixth order
model was selected.The transfer function of the smart beam
was accordingly obtained and is given in equation (1):
GðsÞ ¼
0:06449∙ s
6
þ13:42∙ s
5
þ288:7∙ s
4
þ54660∙ s
3
þ3:548∙ 10
5
∙ s
2
þ5:55∙ 10
7
∙ s þ7:102∙ 10
7
s
6
þ191:6∙ s
5
þ6085∙ s
4
þ741800∙ s
3
þ1:211∙ 10
7
∙ s
2
þ7:179∙ 10
8
∙ s þ7:89∙ 10
9
ð1Þ
Theory of the fractional systems and CFE method
The orders of the fractional calculus are the real numbers
(Jifeng and Yuankai,2005).Many different deﬁnitions for
general fractional integrodifferential operation can be found
in the literature.The commonly used for general fractional
integrodifferential expressions are those given by Cauchy,
RiemannLiouville,Gru¨ nwaldLetnikov and Caputo
(Vinagre et al.,2000).The Caputo expression for fractional
order differentiation is given as (Vinagre et al.,2000):
0
D
a
t
yðtÞ ¼
1
Gð1 2gÞ
Z
t
0
y
ðmþ1Þ
ðtÞ
ðt 2tÞ
g
dt ð2Þ
where a¼ m þy,m is an integer and 0,y,1.On the other
hand,the Caputo expression for fractional order integration is
deﬁned as (Xue et al.,2007):
0
D
g
t
¼
1
Gð2gÞ
Z
t
0
yðtÞ
ðt 2tÞ
1þg
dt;g,0 ð3Þ
Due to the simplicity in calculations,Laplace domain is
commonly used to express the fractional integrodifferential
operations.Thus,Laplace transform of the fractional order
differentiation can be given as (Xue et al.,2007):
Figure 1 (a) Smart beam used in the study;(b) a typical PZT patch
(a) (b)
Figure 2 Piezoelectric patches on the smart beam
Surface A
Piezoelectric Patch
Aluminium Beam
Surface B
2B 1B
3B 4B
2A1A
3A4A
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
204
L
0
D
a
t
f ðtÞ
¼ s
a
L½ f ðtÞ 2
X
n21
k¼1
s
k
½
0
D
a2k21
t
f ðtÞ
t¼0
ð4Þ
If the derivatives of the function f(t) are all equal to zero,the
following equation can be written (Xue et al.,2007):
L
0
D
a
t
f ðtÞ
¼ s
a
L½ f ðtÞ ð5Þ
A fractional differential equation for a fractional order control
system can be written as:
a
n
d
a
n
yðtÞ
dt
a
n
þa
n21
d
a
n21
yðtÞ
dt
a
n21
þ∙ ∙ ∙ þa
0
d
a
0
yðtÞ
dt
a
0
¼ b
m
d
b
m
xðtÞ
dt
b
m
þb
m21
d
b
m21
xðtÞ
dt
b
m21
þ∙ ∙ ∙ þb
0
d
b
0
xðtÞ
dt
b
0
ð6Þ
where y(t) is output and x(t) is the input of the system.
The Laplace transform of equation (7) can be obtained as
(Xue and Chen,2002):
GðsÞ ¼
YðsÞ
XðsÞ
¼
b
m
s
b
m
þb
m21
s
b
m21
þ∙ ∙ ∙ þb
0
s
b
0
a
n
s
a
n
þa
n21
s
a
n21
þ∙ ∙ ∙ þa
0
s
a
0
ð7Þ
where a
n
.a
n21
.∙ ∙ ∙.a
0
$ 0 and b
m
.b
m21
.∙ ∙ ∙.
b
0
$0 are satisﬁed,a
k
(k ¼ 0,1,2,...,n) and b
k
(k ¼ 0,1,
2,...,n) are constants.
The analysis of the Laplace transform and inverse Laplace
transform of fractional integrodifferential operation in time
domain are quite complicated and time consuming.
Hence alternative means were sought to simplify the
required algebra.A fractional transfer function can be
replaced with an integer order transfer function which has
almost the same behaviours with the real transfer function but
much easier to deal with.
There are several methods for obtaining rational
approximations of fractional order systems.For example,
Carlson’s method,Matsuda’s method,Oustaloup’s method,
the Gru
¨
nwaldLetnikoff approximation,Maclaurin series
based approximations,time response based approximations,
etc.(Podlubny et al.,2002).One of the most important
approximations for fractional order systems is the CFEmethod.
The CFE method is used for obtaining realization of s
a
(0,a,1) in this paper.This method can be expressed
in the form (Krishna and Reddy,2008;Krishna,2011;
Ozyetkin et al.,2010):
ð1þxÞ
a
¼
1
12
a∙x
1þ
ð1þaÞ∙ x
2þ
ð12aÞ∙ x
3þ
ð2þaÞ∙ x
2þ
ð22aÞ∙ x
5þ...
¼
1
12
a∙ x
1þ
ð1þaÞ∙ x
2þ
ð12aÞ∙ x
3þ
ð2þaÞ∙ x
2þ
ð22aÞ∙ x
5þ...
ð8Þ
In this formulation x ¼ s 21 used for the computation of s
m
.
By using equation (8),the ﬁrst,second,third and fourth order
integer approximations,which are dependent on m,can be
obtained as follows.
Figure 3 Experimental setup for the system identiﬁcation of the smart beam
SENSORTECH SA21 HIGH
VOLTAGE POWER SUPPLY
SENSORTECH SA
10 HIGH VOLTAGE
AMPLIFIER
Piezoelectric Patch
1A1B4A4B
Actuator Input
Piezoelectric Patch
2A Sensor Output
1A 2A
4A
Signal Conditioner
Unit
Signal Conditioner
Unit Output
PULSE HARDWARE
3,560C
SIGNAL
GENERATOR
HP33120A
Signal Generator
Voltage Output
Pin Connector
BNC Connector
Crocodile Chips
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
205
First order approximation:
s
m
ø
ð1 þmÞ ∙ s þð1 2mÞ
ð1 2mÞ ∙ s þð1 þmÞ
ð9Þ
Second order approximation:
s
m
ø
ðm
2
þ3∙ mþ2Þ ∙ s
2
þð22∙ m
2
þ8Þ ∙ s þðm
2
23∙ mþ2Þ
ðm
2
23∙ mþ2Þ ∙ s
2
þð22∙ m
2
þ8Þ ∙ s þðm
2
þ3∙ mþ2Þ
ð10Þ
Third order approximation:
s
m
ø
ðm
3
þ6∙m
2
þ11∙mþ6Þ ∙ s
3
þð23∙m
3
26∙m
2
þ27∙mþ54Þ ∙ s
2
þð3∙m
3
26∙m
2
227∙mþ54Þ ∙ s
þð2m
3
þ6∙m
2
211∙mþ6Þ
ð2m
3
þ6∙m
2
211∙mþ6Þ ∙ s
3
þ 3∙m
3
26∙m
2
227∙mþ54ð Þ ∙ s
2
þð23∙m
3
26∙m
2
þ27∙mþ54Þ ∙ s
þðm
3
þ6∙m
2
þ11∙mþ6Þ
ð11Þ
Fourth order approximation:
s
m
ø
ðm
4
þ10∙ m
3
þ35∙ m
2
þ50∙ mþ24Þ ∙ s
4
þð24∙ m
4
220∙ m
3
þ40∙ m
2
þ320∙ mþ384Þ ∙ s
3
þð6∙ m
4
2150∙ m
2
þ864Þ ∙ s
2
þð24∙ m
4
þ20∙ m
3
þ40∙ m
2
2320∙ mþ384Þ ∙ s
þðm
4
210∙ m
3
þ35∙ m
2
250∙ mþ24Þ
ðm
4
210∙ m
3
þ35∙ m
2
250∙ mþ24Þ ∙ s
4
þ 24∙ m
4
þ20∙ m
3
þ40∙ m
2
2320∙ mþ384ð Þ ∙ s
3
þ 6∙ m
4
2150∙ m
2
þ864ð Þ ∙ s
2
þð24∙ m
4
220∙ m
3
þ40∙ m
2
þ320∙ mþ384Þ ∙ s
þðm
4
þ10∙ m
3
þ35∙ m
2
þ50∙ mþ24Þ
ð12Þ
These expressions will be used in the modelling and the
realization of the fractional effect during the design and
implementation of the controllers (Onat et al.,2010,2011a,b).
Fractional controller design
Development of an active controller for a dynamic system is
analogous to the determination of a suitable viscous damping
ratio for the same system.Since the viscous damping force
Experimental
2
nd
order analytical model
6
th
order analytical model
10
th
order analytical model
Experimental
2
nd
order analytical model
6
th
order analytical model
10
th
order analytical model
10
0
10
–1
10
–2
6 7 8
Frequency (Hz)
6 7 8
Frequency (Hz)
Magnitude (db)
Phase (Degree)
–50
0
30
–100
–150
–180
(a)
(b)
Notes: (a) Magnitude; (b) phase
Figure 4 Frequency responses of the experimentally obtained and analytically estimated smart beam models
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
206
is proportional with the velocity and the velocity is the time rate
of change of the displacement,the knowledge of the differential
effect becomes important for the controller design.Inthis study
the differential effect was included as the fractional one and the
active vibration controller was synthesized in two steps.First,
the fractional differential effect of the smart beamwas derived
fromthe experimentally measured response signal by using the
fractional derivative effect s
m
.In this study various
approximations for s
m
was considered by using ﬁrst,second,
third and fourth degree approach of CFE method.A relevant
ﬁlter was then designed in order to incorporate the dynamical
characteristics of the ﬁrst ﬂexural mode of the smart beam.The
designed ﬁlter,H(s),is given in equation (13):
HðsÞ ¼
1
s
2
þ8:554∙ s þ1829
ð13Þ
The block diagramof the studied closed loop systemis shown
Figure 5.In addition to H(s) and s
m
which are already deﬁned;
W(s) and Y(s) stand for the systeminput and the systemoutput
inLaplace domain,respectively,andKrepresents the controller
gain.
Simulations conducted for the smart beam
The frequency responses of the open loop system and closed
loop systems are shown in Figure 6 for different values of the
fractional order m.The gain of the controller was kept
constant as K¼ 100.Figure 6 indicate that the increase in the
fractional order mprovides more effective suppression in the
resonance region of the open loop frequency response.
However,further increase causes a shift at the resonance
frequency towards lower values and a performance loss at low
frequency region accompanies this shift.This effect is more
prominent for higher values of the fractional order m.This can
better be explained with the help of Figure 7 which shows the
polezero map of the closed loop dominant poles.As it can be
seen from Figure 7 that as mincreases the right half plain
poles make the system as unstable and approximately after
m¼ 0.83 system becomes unstable.Due to this and as shown
in Figure 8,the response level at the resonance increases until
m¼ 0.83 and drops afterwards.
For the simulations a fractional order value of m¼ 0.2 was
selected.This value was determined to yield comparatively
good performances both at the resonance and also at the off
resonant regions and furthermore not found to shift the open
loop resonance value.
Figure 7 Polezero map of the closed loop mvalues
PoleZero Map
Real Axis
Imaginary Axis
1.5
–41.6
–41.7
–41.8
–41.9
–42
–42.1
–42.2
–42.3
–42.4
–1 –0.5 0 0.5 1
µ = 0.87
µ = 0.84
µ = 0.81
µ = 0.78
µ = 0.75
µ = 0.93
µ = 0.90
µ = 0.96
µ = 0.99
µ = 0.72
Figure 5 Block diagram of the developed fractional controller
Figure 6 Frequency response of the smart beam for different values of
the fractional order m
6 7
Frequency (Hz)
8
Open loop
µ = 0.1
µ = 0.2
µ = 0.3
µ = 0.4
µ = 0.5
µ = 0.6
µ = 0.7
µ = 0.8
µ = 0.9
–50
–40
–30
–20
Magnitude (db)
–10
0
10
Figure 8 Frequency response of the smart beam for different mvalues
in stability border
30
20
10
–10
–20
–30
–40
–50
6 7
Frequency (Hz)
Magnitude (dB)
8
µ = 0.99
µ = 0.96
µ = 0.90
µ = 0.87
µ = 0.84
µ = 0.83
µ = 0.82
µ = 0.81
Open loop
µ = 0.93
0
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
207
The designed fractional controller of K ¼ 100 andm¼ 0.2 was
studied by considering four different CFE approximations as
the ﬁrst,second,third and fourth order approximations.The
controller which was developed by using the ﬁrst order
approximation of the CFE method was named as CFE1.The
others were named accordingly in ascending order.
Figure 9 shows the simulated closed loop frequency
responses which are around the resonance value of
approximately 7Hz for different approximation orders of the
CFE method together with the open loop frequency response
of the smart beam.
Experiments conducted on the smart beam
Thefreeandforcedvibrationexperimentswereconductedonthe
smart beam.For the free vibration experiments the smart beam
was given an initial 8mmtip deﬂection and the ensuing motion
was recorded for open and closed loop time responses.In closed
loop experiments the developed controllers were utilized.The
experimental time domain results are shown in Figure 10.The
settling times were recordedtobe nearly 19.5,12.7,4.2and8.1s
for CFE1,CFE2,CFE3 and CFE4 cases,respectively.
Then the forced vibrations were considered.The smart beam
was excited at its ﬁrst resonance frequency (approximately at
7Hz) by the help of PZTpatches.The effects of controllers on
the suppression of the forced vibrations are shown in Figure 11.
For these cases,the suppression rate at the ﬁrst resonance
frequency,which is given in equation (14),was calculated
approximately as 34.4,64.5,87.5 and 78.5 per cent,for CFE1,
CFE2,CFE3 and CFE4 cases,respectively:
SuppressionRate
¼
ðOpenLoopMagnitudeÞ
max
2ðClosedLoopMagnitudeÞ
max
ðOpenLoopMagnitudeÞ
max
£100
ð14Þ
Figures 10 and 11 yield that the controller developed by using
CFE3 approach shows better performance among the ones
considered.
The experimentally obtained open and closed loop
frequency response curves of the smart beam are shown in
Figure 12.It has been determined that the controllers with
CFE1,CFE2 and CFE3 cases had been shifted to 6.875Hz
whereas the CFE4 case did not undergo any resonance shift
and presented an effective vibration suppression at the open
loop resonance region.
The attenuation levels of the control cases considered are
deﬁned in equation (15):
Attenuation Level ¼ ðOpen Loop Vibration LevelÞ
2ðClosed Loop Vibration LevelÞ
ð15Þ
The attenuation levels at 7Hz were determined as 10.34dBfor
CFE1,11.55dB for CFE2,15.95dB for CFE3 and 14.90dB
for CFE4.When one considers these at the shifted resonance
frequency of 6.875Hz the values become 21.71,21.71,1.72
and 4.92 dB,respectively,for the ascending order
representation.Hence considering Figure 12 and the
attenuation levels CFE4 type controller was determined to
give the best performance.
Experimental robustness tests of the developed
controllers
The robustness tests are usually conducted by additional point
masses (Onat et al.,2007,2009).Since those types of
experiments were easy to conduct and at the same time
resonance shifts can easily be achieved,this approach is quite
favourable by the researchers.In this study two different point
masses (i.e.a single axis accelerometer of 5.23g anda threeaxis
accelerometer of 17.54g) were attached to the free end of the
smart beam.
Figure 13 shows the time domain responses of 5.23g
attached mass case for 8mm initial tip displacement and for
four different CFE values.It can be seen that the increase in
CFE order results in better settlement time.
Figure 9 Response of the smart beam for different order
approximations of the CFE method
5
5
0
0
–5
–5
–10
–10
Open loop
CFE1
CFE2
CFE3
CFE4
Open loop
CFE1
CFE2
CFE3
CFE4
–15
–15
6.8 7 7.1
–20
–25
–30
–35
–40
–45
–50
6 7
Frequency (Hz)
(a)
(b)
Magnitude (dB)Magnitude (dB)
Frequency (Hz)
8
Notes: (a) Response between 68 Hz; (b) zoomed response
between 6.87.1 Hz
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
208
Figure 10 Experimental free vibration responses of the smart beam
0
20
35
–8
–6
–4
–2
0
2
4
6
8
Open loop
Magnitude (mm)
0
20
35
–8
–6
–4
–2
0
2
4
6
8
CFE1 CFE 2 CFE 3 CFE 4
0
20
35
–8
–6
–4
–2
0
2
4
6
8
Time (s)
0
20
35
–8
–6
–4
–2
0
2
4
6
8
0
20
35
–8
–6
–4
–2
0
2
4
6
8
Figure 11 Experimental forced vibration responses of the smart beam at its ﬁrst resonance frequency
0
0.5 1.5 2.5 3.5 4.5
1
2
3
4
5
0
–10
10
Magnitude (mm)
Time (s)
0 0.5 1.5 2.5 3.5 4.51 2 3 4 5
0
–10
10
0 0.5 1.5 2.5 3.5 4.51 2 3 4 5
0
–10
10
0 0.5 1.5 2.5 3.5 4.51 2 3 4 5
0
–10
10
0 0.5 1.5 2.5 3.5 4.51 2 3 4 5
0
–10
10
Open loop
CFE 1
CFE 2
CFE 3
CFE 4
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
209
The series of experiments were repeated for the second mass
of 17.54g and the experimentally obtained time domain
responses are shown in Figure 14.
Figure 14 indicates that,like Figure 13,the increase in CFE
order results inbetter settlement time.However,comparisonof
Figures 13and14alsoreveals that the increase inattachedmass
also increases the settling time for corresponding CFE cases.
The frequency domain responses of both cases are shown in
Figure 15.Figure 15(a) shows them for 5.23g mass and
Figure 15(b) shows the responses for 17.54g mass.
As it can be seen in Figure 15(a),the frequency domain
responses indicate that the addition of 5.23g mass reduces the
open loop resonance value to 6.547Hz and CFE1,CFE2 and
CFE3 controllers reduce the resonance values to 6.531,6.500
and 6.469Hz values,respectively.CFE4 application did not
affect the openloopresonance value.The highest vibratory level
of the open loop was determined as 4.35dB.The controllers,
with ascending order,suppressed that to 0.03 21.36,26.10
and 214.50dB values.
Figure 12 Open and closed loop experimental frequency responses of the smart beam
10
–10
–20
–30
–40
–50
–60
6 7
Frequency (Hz)
Magnitude (dB)
Open loop
CFE 1
CFE 2
CFE 3
CFE 4
8
0
Figure 13 Experimental time domain responses of the smart beam having 5.23 g tip mass
0
20
35
–8
–6
–4
–2
20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0
2
4
6
8
Open loop CFE 1
CFE 2 CFE 3 CFE 4
Magnitude (mm)
0
0
2
4
6
8
0
2
4
6
8
Time (s)
0
2
4
6
8
0
2
4
6
8
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
210
Figure 15(b) gives a 5.750Hz open loop resonance value.
CFE1 and CFE2 controllers shift the closed loop responses
to 5.719Hz.CFE3 and CFE4 controllers,on the other
hand,shift to 5.688Hz and 5.656Hz,respectively.The
highest vibratory level of the open loop was 4.04dB.The
controllers,with ascending order,suppressed that to 1.62,
0.36,20.78 and 26.04dB values.
Comparison of frequency responses also shows that the
increase in the added mass,as expected,reduces the
resonance frequencies of respective cases.
Conclusion
A fractional controller was developed for suppressing the ﬁrst
ﬂexural resonance level of a smart beam.In addition to the
controller gain and the bandwidth;the order of the derivative
effect was also taken into the consideration as another design
parameter.The realization of the designed fractional controllers
was then conducted by using the ﬁrst,second,third and fourth
order approaches of the CFE method.The time domain and
frequency domain experimental results showed that increase in
Figure 14 Experimental time domain responses of the smart beam having 17.54 g tip mass
0
20
35
–8
–6
–4
–2
20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0 20 35
–8
–6
–4
–2
0
2
4
6
8
Open loop CFE 1 CFE 2 CFE 3 CFE 4
Magnitude (mm)
0
0
2
4
6
8
0
2
4
6
8
Time (s)
0
2
4
6
8
0
2
4
6
8
Figure 15 Open and closed loop experimental frequency responses of the smart beam having tip mass of (a) 5.23 g;(b) 17.54 g
10
–10
–20
–30
–40
–50
–60
5 5.5 6 6.5
Frequency (Hz)
(a)
(b)
Open loop
CFE 1
CFE 2
CFE 3
CFE 4
Magnitude (dB)
7 7.5 8
0
10
–10
–20
–30
–40
–50
–60
5 5.5 6 6.5
Frequency (Hz)
Open loop
CFE 1
CFE 2
CFE 3
CFE 4
Magnitude (dB)
7 7.5 8
0
Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
211
the order of the controllers also increased the performance and
robustness of the controller.
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About the authors
CemOnat is a graduate of Mersin University,
Turkey (BSc,1999,Mechanical Engineering),
Inonu University,Turkey (MSc,2001,
Mechanical Engineering),Yildiz Technical
University,Turkey (PhD,2006,Mechanical
Engineering) and Middle East Technical
University,Turkey (Post Doc.,2011,
Aeronautical Engineering).His specializations are structural
dynamics and experimental analysis of vibrating structures and
its control,smart structure applications,activevibrationcontrol
and wind turbine blade design.He has been an Assistant
Professor at InonuUniversity inthe Department of Mechanical
Engineering since 2009 and is the author of seven international
scientiﬁc papers and conference proceedings.Cem Onat is
the corresponding author and can be contacted at:
cem.onat@inonu.edu.tr
Melin S¸ ahin is a graduate of Middle East
Technical University,Turkey (BSc,1996,
Aeronautical Engineering;MSc,1999,
Aeronautical Engineering) and University of
Southampton,UK (PhD,2004,School of
Engineering Sciences,Ship Science).His
specializations are structural dynamics and
experimental analysis of vibrating structures,smart structure
applications,active vibration control,lightweight structures
and composite applications,structural health monitoring and
damage identiﬁcation.He has been an Assistant Professor
at Middle East Technical University in the Department of
Aerospace Engineering since 2005 and is the author of
25 international scientiﬁc papers and conference proceedings.
Yavuz Yaman is a graduate of Middle East
Technical University,Turkey (BSc,1981,
Mechanical Engineering;MSc,1984,
Mechanical Engineering) and University of
Southampton,UK(PhD,1989,Department of
Aeronautics and Astronautics).His research
interests are structural dynamics,smart
structures,active vibration control and aeroelasticity.He has
been a Professor at Middle East Technical University in the
Department of Aerospace Engineering since 2001 and has
published 39 papers in international scientiﬁc journals and
conference proceedings.
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Fractional controller design for suppressing smart beam vibrations
Cem Onat,Melin S¸ ahin and Yavuz Yaman
Aircraft Engineering and Aerospace Technology:An International Journal
Volume 84 ∙ Number 4 ∙ 2012 ∙ 203–212
212
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