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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To cite this document:

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¨ nwald-Letnikoff 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-

Zirconate-Titanate) 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

www.emeraldinsight.com/1748-8842.htm

Aircraft Engineering and Aerospace Technology:An International Journal

84/4 (2012) 203–212

q Emerald Group Publishing Limited [ISSN 1748-8842]

[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 1-4,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 (1A-1B and 4A-4B)

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 with5Vpeak-to-peak 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 1A-1B and 4A-4B.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 integro-differential operation can be found

in the literature.The commonly used for general fractional

integro-differential expressions are those given by Cauchy,

Riemann-Liouville,Gru¨ nwald-Letnikov 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 integro-differential

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 integro-differential 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

¨

nwald-Letnikoff 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

1A-1B-4A-4B

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

pole-zero 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 Pole-zero map of the closed loop mvalues

Pole-Zero 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 three-axis

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 6-8 Hz; (b) zoomed response

between 6.8-7.1 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,light-weight 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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