Research Article Optimal Placement of Piezoelectric Plates ...downloads.hindawi.com/archive/2013/905160.pdf · the regions separated by the vibration nodal lines. In the paper by

Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013 Article ID 905160 8 pageshttpdxdoiorg1011552013905160

Research ArticleOptimal Placement of Piezoelectric Plates to ControlMultimode Vibrations of a Beam

Fabio Botta1 Daniele Dini2 Christoph Schwingshackl2 Luca di Mare2 and Giovanni Cerri1

1 Dipartimento di Ingegneria Universita degli Studi Roma Tre Via della Vasca Navale 79-00146 Roma Italy2 Department of Mechanical Engineering Imperial College London Exhibition Road London SW7 2AZ UK

Correspondence should be addressed to Fabio Botta fabiobottauniroma3it

Received 14 July 2013 Revised 11 October 2013 Accepted 21 October 2013

Academic Editor Marc Thomas

Copyright copy 2013 Fabio Botta et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Damping of vibrations is often required to improve both the performance and the integrity of engineering structures for examplegas turbine blades In this paper we explore the possibility of using piezoelectric plates to control the multimode vibrations of acantilever beam To develop an effective control strategy and optimize the placement of the active piezoelectric elements in termsof vibrations amplitude reduction a procedure has been developed and a new analytical solution has been proposed The resultsobtained have been corroborated by comparison with the results from a multiphysics finite elements package (COMSOL) resultsavailable in the literature and experimental investigations carried out by the authors

1 Introduction

The vibration control is a problem of great interest inmany engineering fields since it allows avoiding problemsconnected with the vibrations Blade vibrations in aircraftengines for example are often induced by interactionsbetween blades and fluid and the associated fatigue phe-nomena can give rise to catastrophic failures [1ndash3] Typicallypassive damping systems such as friction damping areused to increase the blade life These systems are veryeffective but in contrast to active damping elements theyare not able to change their characteristics depending onthe system response In the last two decades the adoptionof piezoelectric elements has received considerable attentionby many researchers for its potential applicability to differ-ent areas of mechanical aerospace aeronautical and civilengineering These elements have an interesting couplingbetween electrical and mechanical properties a deformationappears when an electric field is applied and vice versa [4 5]Their effectiveness to damp a particular excited mode or amultimode combination strongly depends on their positionin fact the study of their optimal position has receivedincreasing attention Typically the aim of these studies isto find the position that minimizes an objective functionor maximizes the degree of modal controllability (see [6 7]

for a review) The first study concerned with the optimalposition to damp a specified mode has been that of Crawleyand de Luis [4] They found that the actuators should bein regions of higher average strain Analogous results havebeen found by other researchers [8 9] For a cantilever beamSunar and Rao [10] Demetriou [11] and Bruant et al [12]found that the closer the piezoelectric actuators are to thefixed end the more efficient they are For a simply supportedbeamYang et al [13] found that to control one specificmodethe optimal position for the piezoelectric plates is withinthe regions separated by the vibration nodal lines In thepaper by Barboni et al [14] the possibility of exciting theflexural dynamics of an Euler-Bernoulli beam according toa single mode is examined The results show that to excitea desired mode the actuator must be placed between twoconsecutive points at which the curvature becomes zeroAldraihem et al [15] studied the optimal length and locationfor different boundary conditions They investigated beamswith one pair and two pairs of piezoelectric actuators Theiroptimization criterion was based on beam modal cost andcontrollability index moreover they added a penalty termto consider the actuator length (and the cost weight andspace factors associated to this) Baz and Poh [16] studied theeffects of varying the thickness and material of the bondinglayer as well as the position of the piezoelectric actuators

2 Advances in Acoustics and Vibration

They consider three beam elements to model a cantileverbeam and the results show that it is preferable to place theactuator in a region of large strain Subsequently Yang andLee presented an analyticalmodel for simultaneous optimiza-tion of noncollocated [17] and collocated [18] piezoelectricsensoractuator placement and feedback control gain Theresults show that this procedure can avoid the instability ofthe structural control system Q Wang and C M Wang [19]studied modal and multimodal vibrations They propose anew controllability index and illustrated various beam exam-ples with a pair of collocated piezoelectric actuators Morerecently studies about their use in blades of turbomachineryhave been carried out however only few of these concernactive damping [20ndash23] Unfortunately inmany real cases theloads applied to the structure excite more than one modewith different amplitudes and the excited modes can changeduring service Therefore the implementation of an activesystem that is capable of changing the work configurationof the piezoelectric plates can increase considerably theirefficiency in the damping of the vibrations In this paper anew function to find the optimal placement of piezoelectricplates to control the multimode vibrations of the cantileverbeam with different amplitudes of the single modes isdefined An analytical solution is proposed and the results arealso comparedwith the FEM simulations and results from theliterature with very good agreement

2 Governing Equations for PiezoelectricCoupled Beam

In Figure 1 an Euler-Bernoulli cantilever beam with attachedpiezoelectric patches is schematically shown the two PZTplates are applied in a symmetrical position with respectto the mid plane Indicating by 120575119871

119890 120575119871 in and 120575119871

119886the

virtual work of the elastic inertial and piezoelectric forcesrespectively the principle of the virtual work can bewritten as(in the following the PZT plates will be considered perfectlybonded to the structure their mass and inertia negligiblewith respect to the mass and inertia of the beam and theirthickness very lower than the thickness of the beam)

120575119871119890+ 120575119871 in + 120575119871

119886= 0 (1)

Using the pin force model [4] the action of the PZT platescan be modeled by two flexural moments concentrated at theend of the plates (Figure 1) with

119872(119905) =120595

6 + 120595119864119886119888119879119886119879119887Λ (119905)

Λ (119905) =11988931

119879119886

119881 (119905)

120595 =119864119887119879119887

119864119886119879119886

(2)

M(t) M(t)

V(t)

h

a

xLb

Tb

Ta

Piezoelectricplates

Figure 1 Reference configuration and action of the PZT plates

So by indicating 119908 the vertical displacement the virtualwork of the PZTplates can bewritten as (the virtual quantitiesare oversigned by a tilde)

120575119871119886= 119872(

120597119908

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886+(ℎ2)minus

120597119908

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886minus(ℎ2)) (3)

The variables 119886 and ℎ can vary within the domain

0 le 119886 minusℎ

2le 119871119887

0 le 119886 +ℎ

2le 119871119887

0 le 119886 le 119871119887

0 le ℎ le 119871119887

(4)

and their value depends on the modes that must be damped[14]

The virtual work of the elastic and inertial forces can bewritten respectively as

120575119871119890= 119864119887119868119887int

119871119887

0

1205972

119908

1205971199092

1205972

119908

1205971199092119889119909

120575119871 in = minus120588119878int

119871119887

0

1205972

119908

1205971199052119908119889119909

(5)

Using the modal analysis technique indicating by 120601119894(119909)

the 119894th flexural modal displacement of the cantilever beamand by 119883

119894(119905) its amplitude the vertical displacement will be

approximated by

119908 (119909 119905) =

119873

sum

119894=1

119883119894(119905) 120601119894(119909) (6)

where 119873 is the number of the considered modesSubstituting the last in (3) (5) considering 119908 = 120601

119895(119909)

and successively in (1) the following governing equation isobtained

M

X (119905) + KX(119905) = B(119886 ℎ) 119881(119905) (7)

where X represents the vector of the amplitudes of themodes M and K are respectively the mass and the stiffness

Advances in Acoustics and Vibration 3

matrix and B(119886 ℎ) is the vector control (the prime denotesthe first spatial derivative)

B (119886 ℎ)

= [1206011015840

1(119886 +

ℎ

2) minus 1206011015840

1(119886 minus

ℎ

2)

1206011015840

2(119886 +

ℎ

2) minus 1206011015840

2(119886 minus

ℎ

2)

1206011015840

119873(119886 +

ℎ

2) minus 1206011015840

119873(119886 minus

ℎ

2)]

(8)

with = (120595(6 + 120595))119864119886119888119879119886119879119887(11988931119879119886)

If the Rayleigh damping is considered (7) becomes

M

X (119905) + CX(119905) + KX(119905) = B (119886 ℎ)119881 (119905) (9)

with

C = 120572M + 120573K (10)

Indicating by 120596 the diagonal matrix of the natural fre-quencies of the beam using the normal modes (M = I K =

1205962) and assuming 120573 = 0 (9) becomes

X (119905) + 120572X (119905) + 1205962X (119905) = B (119886 ℎ) 119881 (119905) (11)

If a single mode excitation is considered Barboni et al[14] found the optimal placement of PZTs However insome practical applications for example gas turbine bladesthe load spectrum can include various modes each ofwhich is characterized by different amplitudes and a newstrategy needs to be developed In this paper the multimodedamping will be obtained by applying a counterphase loadby PZTs plates to the external excitation The effectivenessof the piezoelectric elements will be measured by the ampli-tude of the vertical displacement of the free end so thatthe most effective (optimal) position identified in termsof 119886 and ℎ (Figure 1) will be the location which maximizesthis amplitudeTherefore assigning a general spectrum to thePZT load (which corresponds to the spectrum of 119881(119905) see(2))

119881 (119905) =

119873119904

sum

119895=1

119881119895cos (120596

119895119905) (12)

where 119873119904 is the number of excited modes The solution of(11) becomes

119883119894(119905)

=

119873119904

sum

119895=1

[[

[

minus

119890minus1199051205722

119861119894119881119895cosh((12)119905radic1205722minus 41205962

119894)radic1205722minus 41205962

119894(1205962

119894minus 1205962

119895)

radic1205722 minus 41205962119894(12057221205962119895+ (1205962119894minus 1205962119895)2

)

minus

119890minus1199051205722

119861119894119881119895(120572 sinh ((12) 119905radic1205722 minus 41205962

119894) (1205962

119894+ 1205962

119895))


)

+119861119894119881119895120572 sin (120596

119895119905) 120596119895

12057221205962119895+ (1205962119894minus 1205962119895)2+119861119894119881119895cos (119905120596

119895) (1205962

119894minus 1205962

119895)

12057221205962119895+ (1205962119894minus 1205962119895)2

]]

]

(13)

It is possible to obtain an approximate form of (13) con-sidering that the 119894th term is the dominant one (the 119894th term of119881(119905) is exciting119883

119894(119905) near its resonance) After some algebraic

manipulation (13) can be simplified to

119883119894(119905) cong

119861119894119881119894sin (120596

119894119905)

120572120596119894

minus

119890minus(12)119905(120572+radic120572

2minus41205962

119894)

(minus1 + 119890119905radic1205722minus41205962

119894 )119861119894119881119894

120572radic1205722 minus 41205962119894

(14)

The second term of the right hand side of (14) representsthe transient part so that if this is neglected the amplitude ofthe displacement of the free end can be written as

100381610038161003816100381610038161003816

119908 (119886 ℎ)100381610038161003816100381610038161003816=

119873

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119861119894(119886 ℎ) 119881

119894120601119894(119871119887)

120572120596119894

100381610038161003816100381610038161003816100381610038161003816

(15)

In the following a bimodal excitation is considered indi-cating by 119894

1 1198942the indexes of the first and second considered

modes respectively and 119903 le 1 the relative contribution tothe excitation induced by the second mode with respect tothe first one (ie 119903 = 1 corresponds to the second modegoverning the excitation without any contribution from thefirst mode) From (12)

119881 (119905) = (1 minus 119903) cos (1205961198941119905) + 119903 cos (120596

1198942119905) (16)

Substituting (16) into (15) and introducing

1198921198941(119903 120585) =

120572

1 minus 119903

1205961198941

1206011198941(119871119887) 1206011015840

1198941(120585)

1198921198942(119903 120585) =

120572

119903

1205961198942

1206011198942(119871119887) 1206011015840

1198942(120585)

(17)

it is obtained that100381610038161003816100381610038161003816

119908 (119903 119886 ℎ)100381610038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161198921198941(119903 119886 +

ℎ

2) minus 1198921198941(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161198921198942(119903 119886 +

ℎ

2) minus 1198921198942(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

(18)


Indicating by (1205851= 119886+(ℎ2) 120585

2= 119886minus(ℎ2)) respectively

the position of the right and the left ends of the piezoelectricplates it will be

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(19)

If (1205851 1205852) are the coordinates of the absolute maximum

of |

119908 (119903 1205851 1205852)| it is possible to write

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(20)

Considering that 119892119896(119903 120585) has for all the considered val-

ues of 119896 its positive absolute maximum at 1205851= 119871119887 this value

will also be the abscissa of the positive absolute maximumfor |1198921198941(119903 1205851) minus 1198921198941(119903 1205852)| + |119892

1198942(119903 1205851) minus 1198921198942(119903 1205852)| This implies

that the PZT plates should always be placed with their rightedges terminating at the tip of the beam independently of theexcited modes and value of 119903 and hence

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161198921198941(119903 119871119887) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 119871119887) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(21)

Moreover since

0 lt 119892119895(119903 119871119887) gt 119892119895(119903 120585) forall120585 isin [0 119871

119887) forall119895 (22)

(21) becomes

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

= 1198921198941(119903 119871119887)

+ 1198921198942(119903 119871119887) minus 11989111989411198942

(119903 1205852)

(23)

where

11989111989411198942

(119903 1205852)

= 1198921198941(119903 1205852) + 1198921198942(119903 1205852)

=

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

1206011015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

1206011015840

1198942

(1205852)]

(24)

The ordinate 1205852of the absolute maximum of |

119908 (119903 1205851 1205852)|

will coincide with the absolute minimum of the 11989111989411198942

(119903 1205852)

This position changes with the considered modes and themodes ratio 119903 so that the optimal location of the left edge

of the PZT elements can be found by solving the followingsystem of equations

12059711989111989411198942

(119903 1205852)

1205971205852

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

12060110158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

12060110158401015840

1198942

(1205852)] = 0

1205972

11989111989411198942

(119903 1205852)

12059712058522

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

120601101584010158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

120601101584010158401015840

1198942

(1205852)] gt 0

(25)

The solution will provide all the local minima and theabsolute minimum will then be selected among these

3 Results and Discussions

This section reports analytical results for different bimodalexcitations and their comparison with FEM simulationsThese simulations have been performed by using the fre-quency module of the structural mechanics module ofCOMSOL Multiphysics software [24] Specifically a steelbeam of 30 cm of length has been taken into account withbimodal control in mind and focusing on combinations ofthe first five modes the wavelength of the highest modehas been divided into 50 subintervals of length 3mm sothat Δ119886 = Δℎ = 3mm and 5000 different combinationsfor 119886 and ℎ have been considered For each of these theamplitude of the response of the tip to a periodic load withthe frequency corresponding to one of the first five eigen-frequencies has been calculated Moreover the amplituderesponse to a linear combination of the previous loads hasbeen obtained by superposition of the response to two of theeigenfrequencies The optimal position has been chosen tobe the one which corresponds to the maximum amplitudeIn Figure 2 119891

12(119903 1205852) the position of its absolute minimum

maximum and the position of the center of the piezoelectricplates for different 119903 ratios are reported It is possible toobserve that for 119903 = 0 and 119903 = 1 (that correspond respto the excitation of the first and second modes) the optimalconfiguration of the PZT coincides with that obtained in [14]For 119903 = 0 119891

12(0 1205852) attains its absolute minimum at the

boundary points 1205852= 0 so that the optimal configuration

is with the piezoelectric plates distributed along the entirelength of the beam For increasing values of 119903 the position ofthe absolute minimum does not change until the derivativeof 11989111989411198942

(119903 1205852) becomes zero in 120585

2= 0 (251)

119903 =12060110158401015840

1(0) 1206011(119871119887)

120601101584010158401(0) 1206011(119871119887) minus 120601101584010158402(0) 1206012(119871119887) (12059611205962)cong 05 (26)

For values of 119903 above 119903 the optimal configuration grad-ually changes to reach the optimal one for the second mode


02 04 06 08 1

01

02

03

04

05

0 02 04 06 08 1

02

04

06

08

10

minus01

r

1205852 1205852

r = 10

r = 08r = 06

r = 04r = 02r = 00

f12(r120585 2)

Figure 211989112(119903 1205852) red line PZT platersquos left edge (analytical) green line PZT position of the centre (analytical) orange line PZT platersquos right

edge (analytical) times FEM simulations

02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f23(r120585 2)



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f34(r120585 2)

Figure 4 11989134(119903 1205852) red line PZT platersquos left edge (analytical) green line PZT position of the centre (analytical) orange line PZT platersquos

right edge (analytical) times FEM simulations


02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)



at 119903 = 1 The FEM results show that there is a very goodagreement between the numerical simulations and the analyt-ical results Different mode combinations are represented inFigures 3 4 5 6 and 7 the optimal configuration varies fromthe optimal value for the individual 119894

1-excited mode (119903 = 0)

to that for themode 1198942(119903 = 1) For all the considered coupled

modes the optimal position changes continuously except forthe couple ((1)ndash(3)) where a different type of behaviour isobserved (Figure 5) with a sharp transition at 119903 cong 08 Thisis due to the fact that when 119903 is low the absolute minimumof 11989113(119903 1205852) coincides with that obtained for the optimal

configuration of the first mode (119903 = 0) When the ratioreaches the value of 119903 cong 046 a relative minimum appearsbut this develops into an absolute minimum and thereforecoincides with the left end of the piezoelectric plates onlywhen 119903 gt 08 for such values the PZT elements should beactive only over about half of the length of the beam The

comparisons with the FEM are in agreement with all thecases analyzed The proposed analytical method has beenalso compared with the experimental investigations reportedin [25ndash27] To test different load spectra and configurationsan experimental apparatus has been designed and builtmoreover a MATLAB code has been developed to controlthe actuation and themodal contributions Four piezoelectricplates have been used in three different configurationsExperiments have been done with onemode andmultimodalvibrations A good agreementwas shown in all the consideredcases

4 Conclusions

In this work a new theoretical model for the optimalplacement of piezoelectric plates to control the multimode


02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)



vibrations of a cantilever beam is proposed After a detaileddescription of the theoretical model bi-modal excitationis considered The designs of the optimal configurationsfor different coupled modes and different relative contribu-tions are reported The comparison between numerical andtheoretical results is shown to confirm the validity of theoptimization technique This model can be very useful inmany real situations when the spectrum of the load excitesmore eigenmodes and has therefore implications for thedevelopment of strategies to control multimode vibrationsThis methodology has been extended to rotating systems [2829] Its application to real systems subjected to the centrifugalloads for example gas turbine blades [30] and impulse loads[31] will constitute the object of further contributions

Nomenclature

119886 Position of the center of the piezoelectric platesB Control vector119888 Beam widthC Damping matrix11988931 Piezoelectric coefficient

119864119886 Youngrsquos modulus of the piezoelectric material

119864119887 Youngrsquos modulus of the beam material

ℎ Piezoelectric plates length119868119887 Inertia moment of the beam

K Stiffness matrix119871119887 Beam length

119872 Piezoelectric bending momentM Mass matrix119903 Percentage coupling coefficient119878 Cross-section area of the beam119879119886 Piezoelectric thickness

119879119887 Beam thickness

119881 Voltage applied to the piezoelectric plates119908 Vertical displacement119908 Virtual vertical displacement

120572 Damping coefficient120601119894(119909) 119894th flexural mode of the cantilever beam

120588 Mass density119883119894(119905) Amplitude of the 119894th mode

120596119894 Natural frequency

References

[1] E Poursaeidi and M Salavatian ldquoFatigue crack growth simula-tion in a generator fan bladerdquo Engineering Failure Analysis vol16 no 3 pp 888ndash898 2009

[2] L Witek ldquoExperimental crack propagation and failure analysisof the first stage compressor blade subjected to vibrationrdquoEngineering Failure Analysis vol 16 no 7 pp 2163ndash2170 2009

[3] J Kubiak Sz G Urquiza B J Garcıa C and F Sierra E ldquoFailureanalysis of steam turbine last stage blade tenon and shroudrdquoEngineering Failure Analysis vol 14 no 8 pp 1476ndash1487 2007

[4] E F Crawley and J de Luis ldquoUse of piezoelectric actuators aselements of intelligent structuresrdquo AIAA Journal vol 25 no 10pp 1373ndash1385 1987

[5] F Botta and G Cerri ldquoWave propagation in Reissner-Mindlinpiezoelectric coupled cylinder with non-constant electric fieldthrough the thicknessrdquo International Journal of Solids andStructures vol 44 no 18-19 pp 6201ndash6219 2007

[6] M I Frecker ldquoRecent advances in optimization of smart struc-tures and actuatorsrdquo Journal of Intelligent Material Systems andStructures vol 14 no 4-5 pp 207ndash216 2003

[7] V Gupta M Sharma and N Thakur ldquoOptimization criteriafor optimal placement of piezoelectric sensors and actuatorson a smart structure a technical reviewrdquo Journal of IntelligentMaterial Systems and Structures vol 21 no 12 pp 1227ndash12432010

[8] K D Dhuri and P Seshu ldquoPiezo actuator placement and sizingfor good control effectiveness and minimal change in originalsystem dynamicsrdquo Smart Materials and Structures vol 15 no 6pp 1661ndash1672 2006

[9] K R Kumar and S Narayanan ldquoActive vibration control ofbeams with optimal placement of piezoelectric sensoractuator


pairsrdquo Smart Materials and Structures vol 17 no 5 Article ID055008 2008

[10] M Sunar and S S Rao ldquoDistributed modeling and actuatorlocation for piezoelectric control systemsrdquo AIAA Journal vol34 no 10 pp 2209ndash2211 1996

[11] M A Demetriou ldquoNumerical algorithm for the optimalplacement of actuators and sensors for flexible structuresrdquo inProceedings of the American Control Conference pp 2290ndash2294Chicago Ill USA June 2000

[12] I Bruant G Coffignal F Lene and M Verge ldquoA methodologyfor determination of piezoelectric actuator and sensor locationon beam structuresrdquo Journal of Sound and Vibration vol 243no 5 pp 861ndash882 2001

[13] Y Yang Z Jin and C K Soh ldquoIntegrated optimal design ofvibration control system for smart beams using genetic algo-rithmsrdquo Journal of Sound and Vibration vol 282 no 3ndash5 pp1293ndash1307 2005

[14] R Barboni A Mannini E Fantini and P Gaudenzi ldquoOptimalplacement of PZT actuators for the control of beam dynamicsrdquoSmart Materials and Structures vol 9 no 1 pp 110ndash120 2000

[15] O J Aldraihem T Singh and R C Wetherhold ldquoOptimalsize and location of piezoelectric actuatorsensors practicalconsiderationsrdquo Journal of Guidance Control and Dynamicsvol 23 no 3 pp 509ndash515 2000

[16] A Baz and S Poh ldquoPerformance of an active control systemwith piezoelectric actuatorsrdquo Journal of Sound and Vibrationvol 126 no 2 pp 327ndash343 1988

[17] S M Yang and Y J Lee ldquoOptimization of non-collocatedsensoractuator location and feedback gain in control systemsrdquoSmart Materials and Structures vol 2 no 2 pp 96ndash102 1993

[18] S M Yang and Y J Lee ldquoVibration suppression with optimalsensoractuator location and feedback gainrdquo Smart Materialsand Structures vol 2 no 4 pp 232ndash239 1993

[19] Q Wang and C M Wang ldquoOptimal placement and sizeof piezoelectric patches on beams from the controllabilityperspectiverdquo Smart Materials and Structures vol 9 no 4 pp558ndash567 2000

[20] A Hohl M Neubauer S M Schwarzendahl L Panning andJ Wallaschek ldquoActive and semiactive vibration damping ofturbine blades with piezoceramicsrdquo in Active and Passive SmartStructures and Integrated Systems vol 7288 of Proceedings ofSPIE San Diego Calif USA March 2009

[21] I Goltz H Bhmer R Nollau J Belz B Grueber and J RSeume ldquoPiezo-electric actuation of rotor blades in an axialcompressor with piezoceramicsrdquo in Proceedings of 8th EuropeanConference on Turbomachinery (ETC rsquo09) Graz Austria March2009

[22] A J Provenza and C R Morrison ldquoControl of fan bladevibrations using piezo-electric and bi-directional telemetryrdquo inProceedings of the ASME Turbo Expo Vancouver Canada June2011

[23] S M Lin I CMao and J H Lin ldquoVibration of a rotating smartbeamrdquo AIAA Journal vol 45 no 2 pp 382ndash389 2007

[24] Comsol Multiphysics Userrsquos Guide Version 43[25] F Botta N Marx S Gentili et al ldquoOptimal placement of

piezoelectric plates for active vibration control of gas turbineblades experimental resultsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol8345 of Proceedings of SPIE San Diego Calif USA March 2012

[26] F Botta N Marx D Dini R de Lieto Vollaro and G BattistaldquoExperimental results for optimal placement of piezoelectric

plates for active vibration control of a cantilever beamrdquo Interna-tional Journal of Engineering and Technology vol 5 no 5 2013

[27] N Marx Counter vibration on gas turbine compressor blades viapiezoelectric plates [MS thesis] Imperial College London UK2012

[28] F Botta N Marx C W Schwingshackl G Cerri and D DinildquoA wireless vibration control technique for gas turbine bladesusing piezoelectric plates and contactless energy transferrdquo inProceedings of the ASME Turbo Expo San Antonio Tex USAJune 2013

[29] F Botta D Dini and R de Lieto Vollaro ldquoA new functionfor the optimal placement of piezoelectric plates to controlmultimode vibrations of a rotating beamrdquo International Journalof Engineering and Technology vol 5 no 5 2013

[30] G Cerri M Gazzino F Botta and C Salvini ldquoProductionplanning with hot section life prediction for optimum gasturbine managementrdquo International Journal of Gas TurbinePropulsion and Power Systems vol 2 no 1 pp 9ndash16 2008

[31] F Botta andG Cerri ldquoShock response spectrum in plates underimpulse loadsrdquo Journal of Sound andVibration vol 308 no 3ndash5pp 563ndash578 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



They consider three beam elements to model a cantileverbeam and the results show that it is preferable to place theactuator in a region of large strain Subsequently Yang andLee presented an analyticalmodel for simultaneous optimiza-tion of noncollocated [17] and collocated [18] piezoelectricsensoractuator placement and feedback control gain Theresults show that this procedure can avoid the instability ofthe structural control system Q Wang and C M Wang [19]studied modal and multimodal vibrations They propose anew controllability index and illustrated various beam exam-ples with a pair of collocated piezoelectric actuators Morerecently studies about their use in blades of turbomachineryhave been carried out however only few of these concernactive damping [20ndash23] Unfortunately inmany real cases theloads applied to the structure excite more than one modewith different amplitudes and the excited modes can changeduring service Therefore the implementation of an activesystem that is capable of changing the work configurationof the piezoelectric plates can increase considerably theirefficiency in the damping of the vibrations In this paper anew function to find the optimal placement of piezoelectricplates to control the multimode vibrations of the cantileverbeam with different amplitudes of the single modes isdefined An analytical solution is proposed and the results arealso comparedwith the FEM simulations and results from theliterature with very good agreement

2 Governing Equations for PiezoelectricCoupled Beam

In Figure 1 an Euler-Bernoulli cantilever beam with attachedpiezoelectric patches is schematically shown the two PZTplates are applied in a symmetrical position with respectto the mid plane Indicating by 120575119871

119890 120575119871 in and 120575119871

119886the

virtual work of the elastic inertial and piezoelectric forcesrespectively the principle of the virtual work can bewritten as(in the following the PZT plates will be considered perfectlybonded to the structure their mass and inertia negligiblewith respect to the mass and inertia of the beam and theirthickness very lower than the thickness of the beam)

120575119871119890+ 120575119871 in + 120575119871

119886= 0 (1)

Using the pin force model [4] the action of the PZT platescan be modeled by two flexural moments concentrated at theend of the plates (Figure 1) with

119872(119905) =120595

6 + 120595119864119886119888119879119886119879119887Λ (119905)

Λ (119905) =11988931

119879119886

119881 (119905)

120595 =119864119887119879119887

119864119886119879119886

(2)

M(t) M(t)

V(t)

h

a

xLb

Tb

Ta

Piezoelectricplates

Figure 1 Reference configuration and action of the PZT plates

So by indicating 119908 the vertical displacement the virtualwork of the PZTplates can bewritten as (the virtual quantitiesare oversigned by a tilde)

120575119871119886= 119872(

120597119908

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886+(ℎ2)minus

120597119908

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886minus(ℎ2)) (3)

The variables 119886 and ℎ can vary within the domain

0 le 119886 minusℎ

2le 119871119887

0 le 119886 +ℎ

2le 119871119887

0 le 119886 le 119871119887

0 le ℎ le 119871119887

(4)

and their value depends on the modes that must be damped[14]

The virtual work of the elastic and inertial forces can bewritten respectively as

120575119871119890= 119864119887119868119887int

119871119887

0

1205972

119908

1205971199092

1205972

119908

1205971199092119889119909

120575119871 in = minus120588119878int

119871119887

0

1205972

119908

1205971199052119908119889119909

(5)

Using the modal analysis technique indicating by 120601119894(119909)

the 119894th flexural modal displacement of the cantilever beamand by 119883

119894(119905) its amplitude the vertical displacement will be

approximated by

119908 (119909 119905) =

119873

sum

119894=1

119883119894(119905) 120601119894(119909) (6)

where 119873 is the number of the considered modesSubstituting the last in (3) (5) considering 119908 = 120601

119895(119909)

and successively in (1) the following governing equation isobtained

M

X (119905) + KX(119905) = B(119886 ℎ) 119881(119905) (7)

where X represents the vector of the amplitudes of themodes M and K are respectively the mass and the stiffness



B (119886 ℎ)

= [1206011015840

1(119886 +

ℎ

2) minus 1206011015840

1(119886 minus

ℎ

2)

1206011015840

2(119886 +

ℎ

2) minus 1206011015840

2(119886 minus

ℎ

2)

1206011015840

119873(119886 +

ℎ

2) minus 1206011015840

119873(119886 minus

ℎ

2)]

(8)

with = (120595(6 + 120595))119864119886119888119879119886119879119887(11988931119879119886)


M

X (119905) + CX(119905) + KX(119905) = B (119886 ℎ)119881 (119905) (9)

with

C = 120572M + 120573K (10)



X (119905) + 120572X (119905) + 1205962X (119905) = B (119886 ℎ) 119881 (119905) (11)


119881 (119905) =

119873119904

sum

119895=1

119881119895cos (120596

119895119905) (12)


119883119894(119905)

=

119873119904

sum

119895=1

[[

[

minus

119890minus1199051205722

119861119894119881119895cosh((12)119905radic1205722minus 41205962

119894)radic1205722minus 41205962

119894(1205962

119894minus 1205962

119895)


)

minus

119890minus1199051205722

119861119894119881119895(120572 sinh ((12) 119905radic1205722 minus 41205962

119894) (1205962

119894+ 1205962

119895))


)

+119861119894119881119895120572 sin (120596

119895119905) 120596119895

12057221205962119895+ (1205962119894minus 1205962119895)2+119861119894119881119895cos (119905120596

119895) (1205962

119894minus 1205962

119895)

12057221205962119895+ (1205962119894minus 1205962119895)2

]]

]

(13)




119883119894(119905) cong

119861119894119881119894sin (120596

119894119905)

120572120596119894

minus

119890minus(12)119905(120572+radic120572

2minus41205962

119894)


119894 )119861119894119881119894

120572radic1205722 minus 41205962119894

(14)


100381610038161003816100381610038161003816

119908 (119886 ℎ)100381610038161003816100381610038161003816=

119873

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119861119894(119886 ℎ) 119881

119894120601119894(119871119887)

120572120596119894

100381610038161003816100381610038161003816100381610038161003816

(15)




119881 (119905) = (1 minus 119903) cos (1205961198941119905) + 119903 cos (120596

1198942119905) (16)


1198921198941(119903 120585) =

120572

1 minus 119903

1205961198941

1206011198941(119871119887) 1206011015840

1198941(120585)

1198921198942(119903 120585) =

120572

119903

1205961198942

1206011198942(119871119887) 1206011015840

1198942(120585)

(17)


119908 (119903 119886 ℎ)100381610038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161198921198941(119903 119886 +

ℎ

2) minus 1198921198941(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161198921198942(119903 119886 +

ℎ

2) minus 1198921198942(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

(18)


Indicating by (1205851= 119886+(ℎ2) 120585



100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(19)


of |


100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(20)






100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161198921198941(119903 119871119887) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 119871119887) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(21)

Moreover since

0 lt 119892119895(119903 119871119887) gt 119892119895(119903 120585) forall120585 isin [0 119871

119887) forall119895 (22)

(21) becomes

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

= 1198921198941(119903 119871119887)

+ 1198921198942(119903 119871119887) minus 11989111989411198942

(119903 1205852)

(23)

where

11989111989411198942

(119903 1205852)

= 1198921198941(119903 1205852) + 1198921198942(119903 1205852)

=

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

1206011015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

1206011015840

1198942

(1205852)]

(24)


119908 (119903 1205851 1205852)|


(119903 1205852)



12059711989111989411198942

(119903 1205852)

1205971205852

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

12060110158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

12060110158401015840

1198942

(1205852)] = 0

1205972

11989111989411198942

(119903 1205852)

12059712058522

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

120601101584010158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

120601101584010158401015840

1198942

(1205852)] gt 0

(25)









(119903 1205852) becomes zero in 120585

2= 0 (251)

119903 =12060110158401015840

1(0) 1206011(119871119887)

120601101584010158401(0) 1206011(119871119887) minus 120601101584010158402(0) 1206012(119871119887) (12059611205962)cong 05 (26)



02 04 06 08 1

01

02

03

04

05

0 02 04 06 08 1

02

04

06

08

10

minus01

r

1205852 1205852

r = 10

r = 08r = 06

r = 04r = 02r = 00

f12(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f23(r120585 2)



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f34(r120585 2)




02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)









4 Conclusions



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











B (119886 ℎ)

= [1206011015840

1(119886 +

ℎ

2) minus 1206011015840

1(119886 minus

ℎ

2)

1206011015840

2(119886 +

ℎ

2) minus 1206011015840

2(119886 minus

ℎ

2)

1206011015840

119873(119886 +

ℎ

2) minus 1206011015840

119873(119886 minus

ℎ

2)]

(8)

with = (120595(6 + 120595))119864119886119888119879119886119879119887(11988931119879119886)


M

X (119905) + CX(119905) + KX(119905) = B (119886 ℎ)119881 (119905) (9)

with

C = 120572M + 120573K (10)



X (119905) + 120572X (119905) + 1205962X (119905) = B (119886 ℎ) 119881 (119905) (11)


119881 (119905) =

119873119904

sum

119895=1

119881119895cos (120596

119895119905) (12)


119883119894(119905)

=

119873119904

sum

119895=1

[[

[

minus

119890minus1199051205722

119861119894119881119895cosh((12)119905radic1205722minus 41205962

119894)radic1205722minus 41205962

119894(1205962

119894minus 1205962

119895)


)

minus

119890minus1199051205722

119861119894119881119895(120572 sinh ((12) 119905radic1205722 minus 41205962

119894) (1205962

119894+ 1205962

119895))


)

+119861119894119881119895120572 sin (120596

119895119905) 120596119895

12057221205962119895+ (1205962119894minus 1205962119895)2+119861119894119881119895cos (119905120596

119895) (1205962

119894minus 1205962

119895)

12057221205962119895+ (1205962119894minus 1205962119895)2

]]

]

(13)




119883119894(119905) cong

119861119894119881119894sin (120596

119894119905)

120572120596119894

minus

119890minus(12)119905(120572+radic120572

2minus41205962

119894)


119894 )119861119894119881119894

120572radic1205722 minus 41205962119894

(14)


100381610038161003816100381610038161003816

119908 (119886 ℎ)100381610038161003816100381610038161003816=

119873

sum

119894=1

100381610038161003816100381610038161003816100381610038161003816

119861119894(119886 ℎ) 119881

119894120601119894(119871119887)

120572120596119894

100381610038161003816100381610038161003816100381610038161003816

(15)




119881 (119905) = (1 minus 119903) cos (1205961198941119905) + 119903 cos (120596

1198942119905) (16)


1198921198941(119903 120585) =

120572

1 minus 119903

1205961198941

1206011198941(119871119887) 1206011015840

1198941(120585)

1198921198942(119903 120585) =

120572

119903

1205961198942

1206011198942(119871119887) 1206011015840

1198942(120585)

(17)


119908 (119903 119886 ℎ)100381610038161003816100381610038161003816=

100381610038161003816100381610038161003816100381610038161198921198941(119903 119886 +

ℎ

2) minus 1198921198941(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161198921198942(119903 119886 +

ℎ

2) minus 1198921198942(119903 119886 minus

ℎ

2)

10038161003816100381610038161003816100381610038161003816

(18)


Indicating by (1205851= 119886+(ℎ2) 120585



100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(19)


of |


100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(20)






100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161198921198941(119903 119871119887) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 119871119887) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(21)

Moreover since

0 lt 119892119895(119903 119871119887) gt 119892119895(119903 120585) forall120585 isin [0 119871

119887) forall119895 (22)

(21) becomes

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

= 1198921198941(119903 119871119887)

+ 1198921198942(119903 119871119887) minus 11989111989411198942

(119903 1205852)

(23)

where

11989111989411198942

(119903 1205852)

= 1198921198941(119903 1205852) + 1198921198942(119903 1205852)

=

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

1206011015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

1206011015840

1198942

(1205852)]

(24)


119908 (119903 1205851 1205852)|


(119903 1205852)



12059711989111989411198942

(119903 1205852)

1205971205852

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

12060110158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

12060110158401015840

1198942

(1205852)] = 0

1205972

11989111989411198942

(119903 1205852)

12059712058522

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

120601101584010158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

120601101584010158401015840

1198942

(1205852)] gt 0

(25)









(119903 1205852) becomes zero in 120585

2= 0 (251)

119903 =12060110158401015840

1(0) 1206011(119871119887)

120601101584010158401(0) 1206011(119871119887) minus 120601101584010158402(0) 1206012(119871119887) (12059611205962)cong 05 (26)



02 04 06 08 1

01

02

03

04

05

0 02 04 06 08 1

02

04

06

08

10

minus01

r

1205852 1205852

r = 10

r = 08r = 06

r = 04r = 02r = 00

f12(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f23(r120585 2)



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f34(r120585 2)




02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)









4 Conclusions



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Indicating by (1205851= 119886+(ℎ2) 120585



100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(19)


of |


100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816=100381610038161003816100381610038161198921198941(119903 1205851) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 1205851) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(20)






100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161198921198941(119903 119871119887) minus 1198921198941(119903 1205852)10038161003816100381610038161003816

+100381610038161003816100381610038161198921198942(119903 119871119887) minus 1198921198942(119903 1205852)10038161003816100381610038161003816

(21)

Moreover since

0 lt 119892119895(119903 119871119887) gt 119892119895(119903 120585) forall120585 isin [0 119871

119887) forall119895 (22)

(21) becomes

100381610038161003816100381610038161003816

119908 (119903 1205851 1205852)100381610038161003816100381610038161003816max

=100381610038161003816100381610038161003816

119908 (119903 119871119887 1205852)100381610038161003816100381610038161003816max

= 1198921198941(119903 119871119887)

+ 1198921198942(119903 119871119887) minus 11989111989411198942

(119903 1205852)

(23)

where

11989111989411198942

(119903 1205852)

= 1198921198941(119903 1205852) + 1198921198942(119903 1205852)

=

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

1206011015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

1206011015840

1198942

(1205852)]

(24)


119908 (119903 1205851 1205852)|


(119903 1205852)



12059711989111989411198942

(119903 1205852)

1205971205852

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

12060110158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

12060110158401015840

1198942

(1205852)] = 0

1205972

11989111989411198942

(119903 1205852)

12059712058522

=

119872

120572[(1 minus 119903)

1206011198941(119871119887)

1205961198941

120601101584010158401015840

1198941

(1205852) + 119903

1206011198942(119871119887)

1205961198942

120601101584010158401015840

1198942

(1205852)] gt 0

(25)









(119903 1205852) becomes zero in 120585

2= 0 (251)

119903 =12060110158401015840

1(0) 1206011(119871119887)

120601101584010158401(0) 1206011(119871119887) minus 120601101584010158402(0) 1206012(119871119887) (12059611205962)cong 05 (26)



02 04 06 08 1

01

02

03

04

05

0 02 04 06 08 1

02

04

06

08

10

minus01

r

1205852 1205852

r = 10

r = 08r = 06

r = 04r = 02r = 00

f12(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f23(r120585 2)



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f34(r120585 2)




02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)









4 Conclusions



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 04 06 08 1

01

02

03

04

05

0 02 04 06 08 1

02

04

06

08

10

minus01

r

1205852 1205852

r = 10

r = 08r = 06

r = 04r = 02r = 00

f12(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f23(r120585 2)



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f34(r120585 2)




02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)









4 Conclusions



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 04 06 08 1

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

01

02

03

04

05

minus01

1205852 1205852

f13(r120585 2)



02 04 06 08 1

01

02

0 02 04 06 08 1

02

04

06

08

10

minus01

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

1205852 1205852

f25(r120585 2)









4 Conclusions



02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










02 04 06 08 1

005

010

015

0 02 04 06 08 1

02

04

06

08

10

r

r = 10

r = 08r = 06

r = 04r = 02r = 00

minus005

minus010

12058521205852

f35(r120585 2)




Nomenclature












References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Optimal Placement of Piezoelectric Plates ...downloads.hindawi.com/archive/2013/905160.pdf · the regions separated by the vibration nodal lines. In the paper by