Integrated and Consistent Active Control Formulation and Piezotransducer Position Optimization of...

Research ArticleIntegrated and Consistent Active Control Formulation andPiezotransducer Position Optimization of Plate Structuresconsidering Spillover Effects

Mojtaba Biglar and Hamid Reza Mirdamadi

Department of Mechanical Engineering Isfahan University of Technology Isfahan 84156-83111 Iran

Correspondence should be addressed to Hamid Reza Mirdamadi hrmirdamadicciutacir

Received 15 February 2014 Accepted 8 April 2014 Published 5 May 2014

Academic Editor Chao Tao

Copyright copy 2014 M Biglar and H R Mirdamadi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This study addresses new formulation for active vibration control of plates by optimal locations of attached piezotransducers Freevibrations are solved by Rayleigh-Ritz and transient by assumed modes methods Optimal orientations of patches are determinedby spatial controllabilityobservability as well as residual modes to reduce spillover These criteria are used to achieve optimalfitness function defined for genetic algorithm to find optimal locations To control vibrations negative velocity feedback controlis designed Results indicate that by locating piezopatches at optimal positions depreciation rate increases and amplitudes ofvibrations reduce effectively The effect of number of piezodevices is analyzed

1 Introduction

In recent years the active vibration control has been animportant challenge for space structures The importancehas been more remarkable and higher for larger and moreflexible space structures For vibration suppression smartmaterials such as piezoelectric transducersmay be used [1 2]Therefore determining smart actuator and sensor locations isa key subject for increasing the system efficiency

Many control algorithms have been used to eliminate thevibrations such as positive position feedback (PPF) controland direct velocity feedback control [3 4] Cupiał [5] pro-posed an optimization problem for finding the control voltageapplied to a single piezoelectric patch actuator or severalactuators for suppressing vibrations of a beam or rectangularplate Yiqi and Yiming [6] established an analytical modelfor the active vibration control of a piezoelectric FGM platebased on a higher-order shear deformation plate and elasticpiezoelectric theories He et al [7] presented a finite elementformulation for the vibration control of a functionally gradedplate (FGP) based on the classical lamination plate theory(CLPT) In the previous works of [6 7] the plates were fullycovered with integrated piezoelectric sensors and actuators

and they employed piezoelectric patches Thus in theseworks the optimization procedure was not required for anactive vibration control Orszulik and Shan [8] presenteda system identification and vibration control strategy fora flexible manipulator with a collocated piezoelectric sen-soractuator pair However they did not perform any opti-mization on the location of piezoelectric transducers Julaiand Tokhi [9] presented an active vibration control mecha-nism using GA and particle swarm optimization Howeverthe optimization procedure for sensoractuator locations wasnot performed in their work Kulkarni et al [10] presentednumerical and experimental studies on active control of acoupled bending-torsion vibration modes of a typical wingbox structure of an aircraft using piezoelectric ceramic stackactuators Zhao and Hu [11] developed a simplified methodfor analyzing and designing against alleviating the effect ofbuffeting in vertical tail system In [10 11] an optimizationprocedure was not performed and the formulation would bedifferent from that formulated in the present study

Kumar and Narayanan [12] determined the optimal loca-tions of the sensoractuator pair on a flexible beam In thiswork they used the linear quadratic regulator (LQR) strategyfor active vibration control Bruant et al [13] proposed a

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 276714 14 pageshttpdxdoiorg1011552014276714

2 Shock and Vibration

method based onminimizing the mechanical energy integralof the system for obtaining the optimal actuator locations andmaximizing the energy of the state outputs for obtaining theoptimal sensor locations on the flexible beam Hac and Liu[14] derived analytical modeling of a smart plate equippedwith piezoelectric sensors and actuators In their worktwo criteria for the optimal configuration of piezoelectricactuators were suggested using modal controllability and thecontrollability grammian

Bruant et al [15] developed a simple tool to simulatethe active control of laminated plates In a next step theyoptimized the geometry and the number of sensors andactuators Halim and Moheimani [16] suggested a criterionfor finding the optimal placement of collocated piezoelectricactuatorsensor pairs on a thin flexible plate However inthis work not only the equations of motion were not ofderived analytically but also the control algorithm had notbeen used for suppressing the plate vibration Yang and Lee[17] developed an integer-real-encoded GA to search forthe optimal placement and size of the piezoelectric patchesRader et al [18] presented an approach to optimizing theconfiguration of piezoelectric actuators for vibration controlof a flexible aircraft fin However not only they derivedthe equations of motion of smart structure explicitly butalso the residual modes were not considered in their workAmbrosio et al [19] presented an H

2norm approach for

the actuator and sensor placement Sajizadeh and Darus[20] determined the optimal locations of sensors based onobservability concept Qian et al [21] formulated the optimalactuatorsensor location problem in a framework of a zero-one optimization problem which could be solved using GAHowever in this work residual modes were neglected Thusspillover effects could distort the system performance Bruantet al [22] formulated the optimal locations and number ofpiezoelectric sensors for active vibration control However in[20 22] they did not find the optimal locations of actuators

The spillover effect could be among one of the most chal-lenging issues that might spoil the active vibration control ofstructure Indeed when feedback controllers would be usedfor active vibration control of systems there would be noassurance that control algorithm could excite the residualmodes Thus for enhancing control algorithm operationa consideration of residual modes should be necessaryHowever in [14 23 24] residualmodes were neglectedThusspillover effects could distort the system operation Bruantet al [25] and Han and Lee [26] determined the sensorand actuator locations with the consideration of controlla-bility observability and spillover reduction Furthermorethey used GA to find efficient locations of piezoelectricsensorsactuators However in their papers the dynamiccharacteristics of both rectangular plate and piezoelectricsensorsactuators were not derived explicitly Bruant andProslier [27] presented an optimization problem based onoptimization criteria of ensuring either good observability orgood controllability of the structure as well as consideringresidual modes to limit the spillover effects However in [27]the dynamic characteristics of both rectangular plate andpiezoelectric sensorsactuators were not derived explicitly

u3 z

u1 x

u2 y

hst

lx

ly

120579

lys

lxs

Figure 1 Flexible plate and orientation of piezoelectric patches

In the present paper an integrated and consistent formu-lation is derived based on energy principles and variationalmethods for both the host plate structure and the patchtransducers together with a GA optimization algorithm foroptimal locations and orientations of these patches usingspatial controllability-observability-based optimality crite-ria In addition in this study the effects of residual modesare incorporated in the fitness function of optimization algo-rithm Based on the CPT and linear piezoelectric theory theequations ofmotion and sensor output equations of the smartplate are derived by using Hamiltonrsquos principle Rayleigh-Ritzapproximation procedure and the assumed modes methodFor increasing the system efficiency the optimal locationsand orientations of piezoelectric transducers are determinedbased on spatial controllability and observability and con-sidering residual modes to reduce the spillover effect Foractive vibration control a negative velocity feedback controlalgorithm is used

The remainder of the paper is organized as follows Wederive the basic equation in Section 2 In Section 3 we discussthe optimization of the location of sensors and actuatorsmounted on a rectangular plate The GA is used to findthe optimal location of sensors and actuators in Section 4In Section 5 designing controller for active vibration ofplate and several simulations for showing the influenceof optimization and active control algorithm are outlinedFinally in Section 6 we draw conclusions

2 Basic Equations21 Potential and Kinetic Energies and Virtual Work Con-sider a flexible plate (Figure 1) with119873

119886piezoelectric actuators

and 119873se piezoelectric sensors The following formulationcould be used for any plate geometry and any boundaryconditions The total potential energy of the structure andpiezoelectric patch is expressed as [28]

PE = int119881

st

1

2STcstSd119881

st

+

119873119886

sum

119895=1

int119881119886

119895

(1

2STcDS minus SThDj

+1

2DjT120573SD

j) d119881119886119895

+

119873se

sum

119895=1

int119881

se119895

(1

2STcDS minus SThDj

+1

2DjT120573SD

j) d119881se119895

(1)

Shock and Vibration 3

where S and Dj are strain and electric charge densityvector in local coordinates respectively cst cD h and 120573Sare the matrices of elastic constants of structure elasticconstants of piezoelectric patches under constant electriccharge density the piezoelectric constants and the inverse ofdielectric constant under constant strain ST and DjT are thetransposes of S and Dj 119881st 119881119886 and 119881se are the volumes ofstructure piezoelectric actuators and piezoelectric sensorsrespectively 119873

119886and 119873se are the number of actuator and

sensor patches By using the following relation the localcoordinate of system for each piezoelectric element can betransformed to global coordinates

S = RjSS Dj

= RjDD

j (2)

where RjS and R

jD are strain and electric charge density trans-

formation matrices respectively The total kinetic energy ofthe structure and piezoelectric patches are obtained as [28]

KE = 1

2int119881

st120588st(2

1+ 2

2+ 2

3) d119881st

+1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(2

1+ 2

2+ 2

3) d119881119886119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(2

1+ 2

2+ 2

3) d119881se119895

=1

2int119881

st120588st(uTu) d119881st +

1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(uTu) d119881119886

119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(uTu) d119881se

119895

(3)

where 1 2 and

3are the velocity components in the 119909 119910

and 119911 directions and 120588st 120588119886 and 120588se are the mass densities ofstructure piezoelectric actuators and piezoelectric sensorsrespectively The virtual work of external forces is written as[28]

120575119882ext=

119873119886

sum

119895=1

V119886119895120575119902119886

119895+

119873se

sum

119895=1

Vse119895120575119902

se119895+

119873119891

sum

119895=1

120575u(119909119895 119910119895)119879

119891119895 (4)

where V and119891 are voltage that applied to piezoelectric patchesand concentrated force and 120575119902 and 120575u are variation of electriccharge and mechanical displacement

22 Discretized Displacement Field In the Rayleigh-Ritzmethod the spatial displacement field of midplane for avibrating rectangular plate is written as

1199061(119909 119910 119905) = Nu

1

(119909 119910)U1(119905)

1199062(119909 119910 119905) = Nu

2

(119909 119910)U2(119905)

1199063(119909 119910 119905) = Nu

3

(119909 119910)U3(119905)

(5)

where 1199061 1199062 and 119906

3are the midplane displacements of the

plate along the 119909 119910 and 119911 directions respectively U1 U2

and U3are the generalized coordinates of plate response

and Nu1

Nu2

and Nu3

are the shape function that need tosatisfy geometric boundary conditions Equation (5) can beexpressed as a matrix relationship

u =

1199061

1199062

1199063

= [

[

Nu1 0 00 Nu

2

00 0 Nu

3

]

]

U1U2U3

= NuU (6)

23 Discretized Strain-Displacement Relationship The rela-tionship between strain and displacement based on theclassical plate theory is expressed as [28]

119878119909119909

119878119910119910

119878119909119910

=

1205971199061

120597119909

1205971199062

120597119910

1205971199061

120597119910+1205971199062

120597119909

+ 119911

minus12059721199063

1205971199092

minus1205971199063

120597119909

minus212059721199063

120597119909120597119910

(7)

In terms of generalized coordinates (7) becomes

119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[

[

120597Nu1

1205971199090 0

0120597Nu

2

1205971199100

120597Nu1

120597119910

120597Nu2

1205971199090

]]]]]]]]

]

U1U2U3

+

[[[[[[[[[[

[

0 0 minus119911

1205972Nu3

1205971199092

0 0 minus119911

1205972Nu3

1205971199102

0 0 minus119911

1205972Nu3

120597119909120597119910

]]]]]]]]]]

]

U1U2U3

= BmUU + Bb

UU

(8)

where BmU and Bb

U are respectively strain-displacementtransformation matrices for membrane and flexural actions

24 Discretized Electric Charge Density In this subsectionwe attempt to discrete the electric charge density Forachieving this purpose we assume that the electric chargedensity is constant within a layer of a patch Additionally weassume that electric charge density acts only in the thicknessdirection of piezoelectric patches and the electric charge oneach patch is chosen as a generalized coordinate With thisassumption a mathematical relationship between electric


charge density and the corresponding generalized coordinatefor each piezoelectric patch 119895 is written as [28]

D119895 = B119895qq119901997904rArr B1q =

[[[[[

[

0 0 0

0 0 sdot sdot sdot 0

1

1198601119901

0 0

]]]]]

]

B2q =[[[[[

[

0 0 0


01

1198602119901

0

]]]]]

]

B119873119901

q =

[[[[[

[

0 0 0


0 01

119860119873119901

119901

]]]]]

]

(9)

where 119860 is the area of piezoelectric patches By using (9) theelectrical generalized coordinate vector is defined as

q119901 =

[[[[[[[[[[

[

119902119901

1

119902119901

2

119902119901

3

119902119901

119873119901

]]]]]]]]]]

]

(10)

where 119901 = 119886 or 119901 = se In this content a means actuatorand se means sensor Inserting (8) and (9) into the equationof potential energy that is (1) yields the following expressionthat is a function of generalized coordinates [28]

PE = 1

2UTKstU + 1

2UTKse

DU +1

2UTKa

DU

minus UTKseUqq

seminus UTKa

Uqqa

+1

2qseTKse

qqqse+1

2qaTKa

qqqa

(11)

whereKstKseD andK

aD are respectively the stiffness matrices

of the structure piezoelectric sensors and piezoelectricactuators Kse

Uq and KaUq are coupling matrices of sensors

and actuators and Kseqq and Ka

qq are capacitance matrices of

sensors and actuators respectively The matrices are calcu-lated as

Kst= int119881

st(Bm

UTcstB

mU + Bb

UTcstB

bU + Bm

UTcstB

bU

+BbUTcstB

mU) d119881

st

KpD =

119873119901

sum

119895=1

int119881119901

119895

(BmUTRj

STcDRj

SBmU + Bb

UTRjSTcDRj

SBbU

+BmUTRj

STcDRj

SBbU + Bb

UTRjSTcDRj

SBmU)

times d119881119901119895

K119901Uq =119873119901

sum

119895=1

(int119881119901

119895

BmUTRj

SThRj

DBjqd119881119901

119895

+int119881119901

119895

BbUTRjSThRj

DBjqd119881119901

119895)

Kpqq =

119873119901

sum

119895=1

int119881119901

119895

BjqTRjDT120573sR

jDB

jqd119881119901

119895

(12)

where 119901 = 119886 or 119901 = se If we insert (6) into (3) kinetic energycan be rewritten as

KE = 12UMstU + 1

2UMseU + 1

2UMaU (13)

whereMstMse andMa are mass matrices of the plate piezo-electric sensors and piezoelectric actuators The matrices arecalculated as

Mst= int119881

st120588stNT

uNud119881st

Mp=

119873119901

sum

119895=1

int119881119901

119895

120588119901

119895NT

uNud119881119901

119895

(14)

where 119901 = 119886 or 119901 = seBy inserting (6) into (4) the external work can be written

as

120575119882ext= 120575qaTBa

VVa+ 120575qseTBse

VVse+ 120575UTFc (15)

where Fc is

Fc =119873119891

sum

119895=1

NTu (119909119895 119910119895) 119891119895 (16)

In addition BaV and Bse

V are defined as

Bak = INa

Bsek = INse

(17)

where I is an identity matrix


25 Governing Equations of Motion Utilizing Hamiltonrsquosprinciple the governing equations for the plate vibrationequipped with piezoelectric sensor and actuator patches canbe derived

MU + KUUU minus KseUqq

seminus Ka

Uqqa= F (18)

minusKseqUU + Kse

qqqse= Bse

k k(119905)se (19)

minusKaqUU + Ka

qqqa= Ba

kk(119905)a (20)

where KUq = KqUT M = Mst

+ Mse+ Ma and KUU =

Kst+ Kse

+ Ka Further k(119905)a and k(119905)se are the externalvoltages applied to the actuators and sensorsM andKUU arerespectively the total mass and passive stiffness matrices ofthe system and (19) can be rewritten as

qse = Kseqqminus1Bse

k kse+ Kse

qqminus1Kse

qUU (21)

It is supposed that there is no external applied voltage acrossthe sensor patches thus kse = 0 Doing the above operationfor (20) results in actuator electric charge

qa = Kaqqminus1Ba

kka+ Ka

qqminus1Ka

qUU (22)

Substituting (21) and (22) into (18) yields the followingexpression for the equations of motion of the plate

MU + KU = Υaka+ Fc (23)

where

Υa = KaUqK

aqqminus1Ba

k (24)

is the influencematrix of input voltage applied across actuatorpatchesThe total active and passivematrix of system stiffnessis expressed as follows

K = KUU minus KseUqK

seqqminus1Kse

qU minus KaUqK

aqqminus1Ka

qU (25)

The current through the thickness of each sensor patch isequal to the time derivative of the electric charge accumulatedon the surface of each sensor as given in (21)

ise (119905) = dqse (119905)d119905

(26)

Whenever a piezoelectric sensor is used for sensing a strainrate the current can be converted into the output open circuitsensor voltage [3]

Φse(119905) = Gci (119905) (27)

where Gc is the matrix of constant gains of the currentamplifier and converts the sensor current vector to an outputsensor voltage vector Using (21) and (26) the output sensorvoltage can be expressed as

Φse(119905) = GcK

seqqminus1Kse

qUU = CU (28)

where (28) is the sensor equation which transforms the strainrate to a voltage

26 Modal Equations The eigenvalue problem of (23) can besolved for determining the eigenvalues and eigenvectorsTheorthonormality property must be satisfied as

HTMH = I HTKH = Λ2n (29)

where 119899 is the number of the modes Λ2n = diag([120596211205962

2

1205962

3sdot sdot sdot 1205962

119899]) is the eigenvaluematrix andH is the eigenvector

matrix The modal coordinates are introduced as

U = HR (30)

R is generalized displacement vector By using modal coordi-nates modal equations of motion andmodal sensor equationcan be obtained as

R + Λ2nR = Υ

1015840

aka+HTFc Φ

se(t) = C1015840R (31)

where Υ1015840a = HTΥa and C1015840 = CH The addition of a structural

damping ratio matrix into the modal equations of motionresults in the following expressions [29]

R + 2ZΛ nR + Λ2nR = Υ

1015840

aka+HTFc (32)

where Z is a diagonal matrix of modal damping ratios Λ n isa diagonal matrix of natural frequencies andΛ2n is a diagonalmatrixwhich is equal to the square of natural frequenciesThematrices of Z Λ n and Λ

2

n are written as

Z =[[[[[[

[

12057710 sdot sdot sdot 0 0

0 1205772

0 0

d

0 0 120577119899minus1

0

0 0 sdot sdot sdot 0 120577119899

]]]]]]

]

Λ n =

[[[[[[

[


0 1205962

0 0

d

0 0 120596119899minus1

0

0 0 sdot sdot sdot 0 120596119899

]]]]]]

]

(33)

3 Optimal Locations of the PiezoelectricSensors and Actuators

To determine the optimal piezoelectric sensor and actuatorlocations for a simply supported plate a modified optimiza-tion criterion based on the concept of spatial H

2norm is

used [30] In this method two criteria for determining theoptimal placement of piezoelectric actuators and sensors areproposed using spatial controllabilityobservability and con-sidering the effect of residual modes to reduce the spillovereffect for the simply supported plate Here we consider astate of no in-plane extension and assume that the plate isonly in a condition of pure bending With this assumptionthe spatially discredited displacement field of the midplane iswritten as

1199061= 0 119906

2= 0 119906

3= Nu3U3 (34)


31 Optimal Placement of Actuators We assume the voltagesapplied to the actuator patches are as follows

ka = [V1198861 V1198862sdot sdot sdot V119886119873119886

]119879

(35)

By the assumption that the initial conditions and externalmechanical forces are zero the transfer function of the plate

resulting from Laplace transforming of (32) can be writtenas

119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)

Equation (36) is the transfer function from the voltagesapplied to the actuator patches k(119905)a to the plate purebending 119906

3 Υ119886

119894 is defined as

Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063

is the ith component of the vectorNu3

Nowwe are ina position to define the spatialH

2normof a transfer function

as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)

where trace is defined to be the sum of the elements on themain diagonal of a matrix and

119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)

We find the optimal location of the jth piezoelectric actuatorpatch by defining the spatial controllability function 120595

119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)

where (119909119886(119895) 119910119886(119895)) and 120579119886(119895) are the corner coordinates ofthe jth actuator patch and its orientation respectively Theactuator orientation may change in the interval (0 le 120579 lt

120587) In this study the variables of the actuator locations thatshould be optimized are (119909119886(119895) 119910119886(119895) 120579119886(119895)) 119895 = 1 2 119873

119886

The patches orientation is shown in Figure 1For enhancing the performance of feedback control

we should reduce the spillover effects For achieving thispurpose it is necessary that with due consideration of theresidual modes we minimize the authority of the actua-tors over the high-frequency modes Furthermore we canenhance the performance of control with considering onlythe first 119873

119862modes and place the actuators at the locations

that they provide a high-authority over the first 119873119862modes

The actuators optimal locations can be found by maximizingthe following criterion

119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)

where 120582 is a weighting constant 119873119862

and 119873119877

are number of controlled and residual modesThe terms radicsum

119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 are respectively the spatialcontrollability of the first 119873

119862modes and the remaining

residual modes Regarding that the spatial controllabilityis the same as spatial H

2norm for obtaining it a limited

number of modes are considered The optimization criterionfor locating actuators can be normalized as

119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)

In (42) since the components of the numerator do not havethe same numerical range normalizing all of them to thesame value that is the maximum value is performed If thisnormalization procedure would not have been taken intoaccount the resulting locations and orientations could nothave been optimized correctly

32 Optimal Placement of Sensors For optimizing piezoelec-tric sensor locations we rewrite (32) as

Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)

where 119862119894is defined as


Table 1 Terminology for GA

Terminology DescriptionFitness function The function that should be optimized (119869 in (42) and (47))

IndividualAny point that is utilized in the fitness function is an individual The number of variables for findingpiezoelectric device positions and orientations is either 3 times 119873se or 3 times 119873119886 that is equal to the number ofindividuals

Population An array of individuals forms the population If the size of population is119873119894and the number of variables or

individuals is 3 times 119873se or 3 times 119873119886 then the population is shown by119873119894by 3 times 119873se or119873119894 by 3 times 119873119886 matrices

Beginning Creating a casual population is the first step of GAEvaluation The value of 119869 in (42) and (47) to be evaluated for each populationSelection Select two parents having the largest value of the fitness functionCrossover Incorporate two parents that produce new children

Mutation In order that this operator produces the children it applies a random change in parents New children areplaced in new population

119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902

)119894sdot sdot sdot 119866

119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)

We find the optimal location of the jth piezoelectric sensor bydefining the spatial observability function Θ

119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)

where (119909se(119895) 119910se(119895)) and 120579se(119895) are the corner coordinatesof the 119895th sensor and its orientation respectively The ori-entation of sensors changes in the interval (0 le 120579 lt 120587)In this work the variables of sensor locations that should beoptimized are as (119909se(119895) 119910se(119895) 120579se(119895)) 119895 = 1 2 119873se

The optimal placement of sensors can be found bymaximizing the following criterion

119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)

which can be normalized as

119869se =sum119873se119895=1radicsum119873119862

119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

maxsum119873se119895=1radicsum119873119862

119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

maxsum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)

By normalizing each Rayleigh-Ritz mode with respect tocorresponding maximal energy that could be achieved fromits corresponding mode the criterion would search for the

optimal configuration In this criterion the residual modesare considered and their influence on the optimizationcriterion is controlled byweighting constant By comparing(42) and (47) it is understood that for optimizing bothsensor and actuator locations the same procedure should beimplemented

4 Application of GA for Optimal Location ofPiezoelectric Devices

In this study the GA is utilized for optimizing the piezoelec-tric device locations on the flexible plate GA is a means bywhich the machine can simulate the mechanism of naturalselection This operation is done by searching in the designspace for finding the best solution The GA starts a searchfrom a series of points and for performing the searchprocedure it does not require the Jacobian of functions [33]Many researchers for optimizing their work have used theGA[34ndash37] The terminology used for GA is brought in Table 1

Usually with the progress in the next generations thealgorithm converges to an asymptote point and the amount offitness function improves When the convergence is reachedor the stopping criteria are satisfied the search process stopsFor optimization a GA with the following configuration isconsidered in the work population size 200 crossover rate08 and number of generations 50

Numerical simulations of the optimization process areanalyzed in this section We consider a simply supportedrectangular plate for which the piezoelectric patches areattached to both upper and lower surfaces The mechanicalproperties of aluminum that is used in this paper are 120588 =

2770 kgm3 V = 03119864 = 70GPa and 120577119894= 00002We suppose

that the piezoelectric sensors and actuators are perfectlybonded to the surfaces and they are made from the same


Table 2 Geometrical properties of the plate and the piezoelectricsensors and actuators

Plate Piezoelectric patch119909-length 119897

119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004

Thickness ℎ119904119905and ℎ

119901(m) 0002 00001

piezoelectric materials and symmetrically attached to theplate Tables 2 and 3 give the geometrical and mechanicalproperties of materials that are used in this study

By using the RayleighndashRitz method and expanding thetransverse displacement of the plate in terms of a time depen-dent modal shape series satisfying the geometric boundaryconditions we have

1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)

where 119896 and 119897 are the total number of mode shapes in thelongitudinal119909 and lateral119910directions respectively 120581 and 120580 arethe number of half-waves in the longitudinal 119909 and lateral 119910directions this series can be expressed as a matrix expansion

1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)

The boundary conditions of plate are simply supported

1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)

where 119897119909119904and 119897119910119904are the dimensions of rectangular plate

For a simply supported plate the modal shape functionsthat are used in this study are expanded in terms of doubleharmonic functions

119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904

are the length and width of rectangularplate For the purpose of vibration control only the firstfive modes are considered that their natural frequencies arepresented in Table 4 We have taken into account the influ-ences of piezoelectric patches into our general formulationHowever we have neglected these effects in our solution fora simply supported rectangular plate since the dimensionsof piezoelectric patches are small in comparison with thehost plate We neglected their influences in calculating thenatural frequencies and eigenmodes Thus in (18) M = Mst

and KUU = Kst There are several examples considered herefor showing the influence of GA that is used for locatingthe piezoelectric devices attached to a thin plate At firstwe discuss the values of 120582 in the optimization procedure

The different values of 120582 that are substituted in (42) for thecase of a single actuator and sensor are listed in Tables 5ndash7In Table 5 the first mode is taken into account that wouldhave the highest controllability and else the next four modesare considered as the residual modes As can be seen whenthe values of 120582 increase the values of (40) decrease and thecontrollability decreases It means that for large values of120582 GA could not find the optimal position and orientationof the actuator In Table 6 the second mode is taken intoaccount that would have the highest controllability and elsethe other modes are considered as the residual modes As canbe understood similar to previous case when the values of120582 increase the values which appeared in (40) decrease andthe controllability decreases meaning that for large valuesof 120582 GA cannot find the optimal position and orientation ofthe actuator In Table 7 the first two modes are unresidualmodes and else the other three modes are considered asthe residual modes For this case several optimal positionsare obtained that some of them are the nearly close for thedifferent values of 120582 Thus the selection of 120582 from disjointranges as presented in Table 7 does not have any observableinfluence on finding the optimal positions and orientationsThus we would choose the value of 120582 equal to two

In the first two simulations the first two lowest modeswould seem to have the highest controllability and observ-ability and else the remaining three modes are considered asthe residual modes Because the fitness functions for findingsensor and actuator (transducer) locations and orientationsare the same herein we only discuss how to find the actuatorlocations and orientations In Figure 2(a) the best and meanvalues of the fitness function for finding the optimal locationof one transducer attached to the plate are shown As can beseen the best value for the objective function is equal to 168The optimal corner point coordinates for a single-transducerare (0657m 0206m) while the transducer orientation inthis point is 09 rad The best and mean fitness functionvalues for finding the optimal locations of two transducersare indicated in Figure 2(b) In the first generations the bestvalue of fitness function is improved rapidly that is becauseof being too far away from the optimal point In the othersimulations of this section the first three lower modes arecontrolled and the remaining two modes are considered asresidual modes The evolution of the best and mean valuesof the fineness function is presented in Figure 3 The optimalpositions of these simulations are listed in Tables 8 and 9

5 Results and Discussion

51 Controller Design For active vibration control a negativevelocity feedback control algorithm is used This feedbackstrategy increases the depreciation rate thus it is an effectiveway for reducing the oscillatory amplitude A simple algo-rithm of this type is used in such a way that the actuatorvoltage can be obtained with an amplifier gain and a changeof polarity on the sensor voltage as follows

k(119905)a = minusGCoΦse(119905) (53)

where GCo is a matrix of gains of the amplifier that is used inthe feedback control


0 5 10 15 20 25 30 35 40 45 50Number of generations

2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue

Best valueMean value

(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50


Number of generations

(b)

Figure 2 Evolution of the mean and best values of fitness function for finding actuator-sensor locations when the first two modes arecontrolled while the next three modes are the residual modes (a) one actuator-sensor and (b) a pair of actuator-sensors

35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)

Figure 3 Evolution of the mean and best values of fitness function for finding actuator-sensor locations when the first three modes arecontrolled while the next two modes are the residual modes (a) one actuator-sensor and (b) a pair of actuator-sensors

In next subsection several simulations are presentedfor illustrating the influence of optimization procedure andactive control algorithm

52 Simulation 1 In the first simulation the first two lowermodes should be controlled and the remaining three modesare considered as residual modes The transducers should belocated at random and optimal positions that are listed inTable 8 For exciting a larger number of systemmode shapeswe have applied the step function concentrated load at alocation off-center of plate The frequency responses of theplate when excited by a concentrated force and located at119909 = 043m and 119910 = 021m for different number of sensorsand actuators are plotted in Figure 4 As can be seen theoptimization procedure is more effective for the amplitudevibration reduction as contributed to arbitrary location ofactuatorssensors By locating the actuators and sensors inthe optimal locations more damping is developed in theplate When the sensors are located at random locationswe could not observe the first two modes in the response

and when the actuators are positioned at random locationsthe maximum mechanical energy cannot be transmitted tothe first two modes However if the sensors are locatedat the optimal locations the contributed modes could beobserved Furthermore if the actuators are located at theiroptimal positions themaximummechanical energy could betransmitted to the contributed modes causing the amplitudeof vibration could be reduced more The values of spatialcontrollability (40) are shown in Tables 10 and 11 Thetransducers are located in position of Table 6 The first twomodes are the controlled modes and the modes 3ndash5 are theresidual modes By comparing the values of these two tablesit can be seen that for the optimal locations the values ofthe system controllability for the first two modes are greaterthan the values of an arbitrarily selected location Thusthe optimization procedure would seem to be effective InFigure 5 two actuators are used for active vibration control ofthe simply supported rectangular plate under considerationIn this figure we would compare the optimal configurationsin Table 8 and use the optimal locations of one actuator


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)

Optimal locationArbitrarily located

Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)

Optimal locationArbitrarily locatedWithout control

(b)

Figure 4 Frequency response of the plate controlled by the piezoelectric actuator-sensor patches when the first two modes are controlledwhile the next three modes are the residual modes (a) one actuator-sensor and (b) a pair of actuator-sensors

Table 3 Material properties of piezoelectric transducers made of PZT-5H from [28]

Mechanical properties 119888119863

11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3

Electrical property 120573119904

33= 148 times 10

8 mFCoupling coefficients ℎ

13= ℎ23= minus272 times 10

9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)

GA locations (Table 8)Modal locations

101

102

Frequency (Hz)

Figure 5 Comparison of the optimal configurations in Table 8 anduse of the optimal locations of one actuator activated for the firstmode and the other actuator for the second mode

activated for the first mode and the other actuator forthe second mode As can be seen the results based onmathematical procedure used based on GA and representedin Table 8 are more effective than the individual activationsof each actuator for each mode

53 Simulation 2 In the second simulation the first threelower modes must be controlled and the remaining twomodes are residual modesThe transducers should be locatedat randomand optimal positions that are listed inTable 9Thefrequency responses of the plate excited by a concentratedforce and located at 119909 = 043mand119910 = 021m for single anddouble transducer configurations are presented in Figure 6As can be seen the optimization criteria are effective inhavingmore damping effect In Figure 7 the influences of the

Table 4 Natural frequencies of the plate (before active control)

Mode number 1 2 3 4 5Longitudinal andcircumferential modes (120581 120580) (1 1) (2 1) (3 1) (1 2) (2 2)

Natural frequencies [Hz] 2388 3822 6211 8123 9557

number of sensors and actuators are compared in the controlof vibration In this simulation the sensors and actuators arelocated in the optimal positions and the first two lowermodesmust be controlled It can be concluded that when a pair ofactuators and sensors are used the damping effect increasesand the amplitude vibration of plate reduces more

6 Conclusions

In the present study the active vibration control and optimalposition of piezoelectric patches attached to a thin platewere analyzed For deriving the equation of motion and sen-sor output equation Hamiltonrsquos principle and Rayleigh-Ritzmethodwere used In the next step the optimal positions andorientations of piezoelectric actuators and sensors attached toa rectangular plate were determined based on the concept ofspatial controllabilityobservability and considering residualmodes for reducing the spillover effect GA was utilized foroptimizing the locations and orientations of piezoelectricdevices The results indicated that by locating piezoelectricsensors and actuators in the optimal positions the dampingeffect could increase and the amplitudes of plate vibrationwere reduced more effectively Furthermore in general theactive vibration control was effective in the vibration controlof the thin plate


Table 5 The effect of values of the weight coefficients on the controllability for a single actuator for the case that only the first mode iscontrolled

Actuator longitudinalcoordinate 119909119886 Actuator lateral coordinate 119910119886 Actuator orientation 120579119886 Weight coefficient 120582 Values of controllability of 119894th mode

1 2 3 4 50487m 0207m 0818 rad 0500 0022 0000 0057 0000 00000538m 0226m 2114 rad 0100 0022 0000 0056 0000 00000494m 0455m 1188 rad 1000 0000 0000 0000 0000 00000469m 0457m 1350 rad 2000 0000 0000 0000 0000 00000501m 0455m 1121 rad 3000 0000 0000 0000 0000 0000

Table 6 The effect of values of the weight coefficients on the controllability for a single actuator for the case that only the second mode iscontrolled



Table 7 The effect of values of the weight coefficients on the controllability for a single actuator for the case that the first two modes arecontrolled



Table 8 GA-optimal and random configurations of piezoelectric transducers attached to the rectangular plate The first two modes are thedominant modes while the next three modes are the residual modes

Location typeOptimal locations from GA Random locations (no optimization)

Actuator longitudinalcoordinate 119909119886

Actuator lateralcoordinate 119910119886

Actuatororientation 120579119886




Transducer dataSingle transducer 0657m 0206m 09 rad 0757m 0386m 0905 rad

Double transducers 0673m0378m

0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad

Table 9 GA-optimal and random configurations of piezoelectric transducers attached to the rectangular plate The first three modes are thedominant modes while the next two modes are the residual modes










0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)

Figure 6 Frequency response of the plate controlled by the piezoelectric actuator-sensor The first three modes are the controlled modeswhile the next two modes are the residual modes (a) one actuator-sensor and (b) two actuator-sensors

Table 10 The effect of values of controllability for optimal location whenever the first two modes are controlled

Values of controllability of 119894th modeNumber of actuators 1 2 3 4 5One actuator 0033 0053 0000 0000 0000Two actuators 0016 0026 0000 0000 0000

Table 11 The effect of values of controllability for random location whenever the first two modes are controlled


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)

Single transducerDouble transducersWithout control

Figure 7 Comparison of frequency response of the plate controlledby different number of piezoelectric actuator-sensors

Nomenclature

119860 The area of piezoelectric patchesBmU B

mU The strain-displacement transformation

matricescst The matrix of elastic constants of structurecD The elasticity matrix of piezoelectric patches

D The electric charge density vectorf The concentrated external mechanical

forces119866(119909 119910 119904) The transfer function of the plate

resulting from Laplace transform1198662

2 The 2-norm of a transfer function matrix

Gc Thematrix of the amplifier constant gainsGCo The matrix of gains of the amplifier used

in the feedback controlh The matrix of piezoelectric constantsH The eigenvector matrixise The electric current vector accumulated

and passed through the surface of eachsensor

119869119886 119869se The fitness function

K The total activepassive matrix of systemstiffness

119896 119897 The total number of mode shapes in thelongitudinal 119909 and lateral 119910 directions

KE PE The kinetic and potential energy ofsystem

KUU The passive stiffness matrix of the system(structurepiezopatches)

KstKse

D andKaD The stiffness matrices of the structure

piezoelectric sensors and actuators


KseUqK

aUq The electromechanical coupling

matrices of sensors and actuatorsKse

qqKaqq The electric capacitance matrices of

sensors and actuators119897119909119904 119897119910119904 The length and width of the

rectangular plateM The total mass matrix of the systemMst

Mse andMa The mass matrices of the plate

piezoelectric sensors and actuators119899 The number of mode shapesNu1 Nu2 andNu3 The shape function matrices119873119862 Number of controlled modes

119873119877 The number of residual modes

119873119886 119873se The number of actuator and sensor

patchesq The generalized coordinates of

electric chargeR The generalized displacement vectorRjSR

jD The strain and electric charge density

transformation matrices119878 The strain vector119878119909119909 119878119910119910 and 119878

119909119910 The components of in-plane strain

U1U2 andU3 The generalized coordinates of platemechanical responses

1199061 1199062 and 119906

3 The midplane displacements of the

plate along 119909 119910 and 119911 directions1 2 and

3 The velocity components in the 119909 119910

and 119911 directionsk The vector of applied voltages to

piezoelectric patches119881

st 119881119886 and119881se The volumes of structure

piezoelectric actuatorssensors120573s The matrix of inverse of dielectric

constantsΦ

se The open circuit sensor voltage vector120581 120580 The number of half-waves in the

longitudinal 119909 and lateral 119910 directions 120582 The weighting constantsΛ n The diagonal matrix of natural

frequenciesΛ2n The diagonal matrix of the squares of

natural frequencies120588st 120588119886 and 120588se The mass densities of the host

structure and piezoelectric actuatorsand sensors

Υa The influence matrix of input voltagesapplied across actuator patches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Boudaoud S Belouettar E M Daya and M Potier-FerryldquoA shell finite element for active-passive vibration control ofcomposite structures with piezoelectric and viscoelastic layersrdquo

Mechanics of AdvancedMaterials and Structures vol 15 no 3-4pp 208ndash219 2008

[2] A Ajit K K Ang and C M Wang ldquoShape control of staticallyindeterminate laminated beams with piezoelectric actuatorsrdquoMechanics of Advanced Materials and Structures vol 10 no 2pp 145ndash160 2003

[3] R Kumar B K Mishra and S C Jain ldquoStatic and dynamicanalysis of smart cylindrical shellrdquo Finite Elements in Analysisand Design vol 45 no 1 pp 13ndash24 2008

[4] M K Kwak and S Heo ldquoActive vibration control of smartgrid structure by multiinput and multioutput positive positionfeedback controllerrdquo Journal of Sound and Vibration vol 304no 1-2 pp 230ndash245 2007

[5] P Cupiał ldquoCalculation of the optimum controls of transientvibrations of smart beams and platesrdquo Mechanics of AdvancedMaterials and Structures vol 15 no 3-4 pp 258ndash268 2008

[6] M Yiqi and F Yiming ldquoNonlinear dynamic response and activevibration control for piezoelectric functionally graded platerdquoJournal of Sound and Vibration vol 329 no 11 pp 2015ndash20282010

[7] X Q He T Y Ng S Sivashanker and K M Liew ldquoActivecontrol of FGM plates with integrated piezoelectric sensors andactuatorsrdquo International Journal of Solids and Structures vol 38no 9 pp 1641ndash1655 2001

[8] R R Orszulik and J Shan ldquoActive vibration control usinggenetic algorithm-based system identification and positiveposition feedbackrdquo Smart Materials and Structures vol 21 no5 Article ID 055002 2012

[9] S Julai andM O Tokhi ldquoVibration suppression of flexible platestructures using swarm and genetic optimization techniquesrdquoJournal of Low Frequency Noise Vibration and Active Controlvol 29 no 4 pp 293ndash318 2010

[10] M D Kulkarni G Kumar P M Mujumdar and A JoshildquoActive control of vibrationmodes of awing box by piezoelectricstack actuatorsrdquo in Proceedings of the 51st AIAAASMEASCEAHSASC Structures Structural Dynamics and Materials Con-ference Orlando Fla USA April 2010

[11] Y H Zhao andH Y Hu ldquoActive control of vertical tail buffetingby piezoelectric actuatorsrdquo Journal of Aircraft vol 46 no 4 pp1167ndash1175 2009

[12] K R Kumar and S Narayanan ldquoActive vibration control ofbeams with optimal placement of piezoelectric sensoractuatorpairsrdquo Smart Materials and Structures vol 17 no 5 Article ID055008 2008

[13] 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

[14] A Hac and L Liu ldquoSensor and actuator location in motioncontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 167 no 2 pp 239ndash261 1993

[15] I Bruant F Pablo and O Polit ldquoActive control of lami-nated plates using a piezoelectric finite elementrdquo Mechanics ofAdvancedMaterials and Structures vol 15 no 3-4 pp 276ndash2902008

[16] D Halim and O R Moheimani ldquoAn optimization approachto optimal placement of collocated piezoelectric actuators andsensors on a thin platerdquo Mechatronics vol 13 no 1 pp 27ndash472003

[17] S M Yang and Y J Lee ldquoOptimization of noncollocatedsensoractuator location and feedback gain in control systemsrdquo


Smart Materials and Structures vol 2 no 2 article 005 pp 96ndash102 1993

[18] A A Rader F F Afagh A Yousefi-Koma and D G ZimcikldquoOptimization of piezoelectric actuator configuration on aflexible fin for vibration control using genetic algorithmsrdquoJournal of IntelligentMaterial Systems and Structures vol 18 no10 pp 1015ndash1033 2007

[19] P Ambrosio F Resta and F Ripamonti ldquoAnH2norm approach

for the actuator and sensor placement in vibration control of asmart structurerdquo Smart Materials and Structures vol 21 no 12Article ID 125016 2012

[20] M R Sajizadeh and I Z M Darus ldquoOptimal location ofsensor for active vibration control of flexible square platerdquo inProceedings of the 10th International Conference on InformationSciences Signal Processing and their Applications (ISSPA rsquo10) pp393ndash396 Kuala Lumpur Malaysia May 2010

[21] F Qian J Wang and L Qu ldquoOptimal placements of piezo-electric patch using genetic algorithm in structure vibrationcontrolrdquo Acta Mechanica Solida Sinica vol 32 no 4 pp 398ndash404 2011

[22] I Bruant L Gallimard and S Nikoukar ldquoOptimization ofpiezoelectric sensors location and number using a geneticalgorithmrdquo Mechanics of Advanced Materials and Structuresvol 18 no 7 pp 469ndash475 2011

[23] Z-C Qiu X-M Zhang H-XWu and H-H Zhang ldquoOptimalplacement and active vibration control for piezoelectric smartflexible cantilever platerdquo Journal of Sound and Vibration vol301 no 3-5 pp 521ndash543 2007

[24] A M Sadri J R Wright and R J Wynne ldquoModelling andoptimal placement of piezoelectric actuators in isotropic platesusing genetic algorithmsrdquo Smart Materials and Structures vol8 no 4 pp 490ndash498 1999

[25] I Bruant L Gallimard and S Nikoukar ldquoOptimal piezoelectricactuator and sensor location for active vibration control usinggenetic algorithmrdquo Journal of Sound and Vibration vol 329 no10 pp 1615ndash1635 2010

[26] J-H Han and I Lee ldquoOptimal placement of piezoelectricsensors and actuators for vibration control of a composite plateusing genetic algorithmsrdquo Smart Materials and Structures vol8 no 2 pp 257ndash267 1999

[27] I Bruant andL Proslier ldquoOptimal location of actuators and sen-sors in active vibration controlrdquo Journal of Intelligent MaterialSystems and Structures vol 16 no 3 pp 197ndash206 2005

[28] D J Leo Engineering Analysis of Smart Material Systems JohnWiley and Sons New York NY USA 2007

[29] D J Inman Vibration with Control John Wiley and Sons NewYork NY USA 2006

[30] S O Reza Moheimani D Halim and A J Fleming SpatialControl of Vibration Theory and Experiments World ScientificPublishing New York NY USA 2002

[31] S O R Moheimani and M Fu ldquoSpatial H2 norm of flexiblestructures and its application in model order selectionrdquo inProceedings of the 37th IEEE Conference on Decision and Control(CDC rsquo98) pp 3623ndash3624 Tampa Fla USA December 1998

[32] S O R Moheimani and T Ryall ldquoConsiderations on placementof piezoceramic actuators that are used in structural vibrationcontrolrdquo in Proceedings of the 38th IEEE Conference on Decisionand Control (CDC rsquo99) pp 1118ndash1123 Phoenix Ariz USADecember 1999

[33] S N Sivanandam and S N Deepa Introduction to GeneticAlgorithms Springer New York NY USA 2008

[34] M Biglar H RMirdamadi andMDanesh ldquoOptimal locationsand orientations of piezoelectric transducers on cylindricalshell based on gramians of contributed and undesired Rayleigh-Ritz modes using genetic algorithmrdquo Journal of Sound andVibration vol 333 no 5 pp 1224ndash1244 2014

[35] L Costa P Oliveira I N Figueiredo and R Leal ldquoActuatoreffect of a piezoelectric anisotropic plate modelrdquo Mechanics ofAdvanced Materials and Structures vol 13 no 5 pp 403ndash4172006

[36] DMarinova ldquoFinite element formulation for analysis and shaperegulating of plates with laminated piezoelectric materialrdquoMechanics of Advanced Materials and Structures vol 16 no 3pp 224ndash235 2009

[37] P Vannucci ldquoA new general approach for optimizing the perfor-mances of smart laminatesrdquo Mechanics of Advanced Materialsand Structures vol 18 no 7 pp 548ndash558 2011

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014


Shock and Vibration


Mechanical Engineering

Advances in


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of


Distributed Sensor Networks


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Advances inOptoElectronics


Volume 2014

RoboticsJournal of


Chemical EngineeringInternational Journal of


Control Scienceand Engineering

Journal of


Antennas andPropagation




Navigation and Observation



method based onminimizing the mechanical energy integralof the system for obtaining the optimal actuator locations andmaximizing the energy of the state outputs for obtaining theoptimal sensor locations on the flexible beam Hac and Liu[14] derived analytical modeling of a smart plate equippedwith piezoelectric sensors and actuators In their worktwo criteria for the optimal configuration of piezoelectricactuators were suggested using modal controllability and thecontrollability grammian

Bruant et al [15] developed a simple tool to simulatethe active control of laminated plates In a next step theyoptimized the geometry and the number of sensors andactuators Halim and Moheimani [16] suggested a criterionfor finding the optimal placement of collocated piezoelectricactuatorsensor pairs on a thin flexible plate However inthis work not only the equations of motion were not ofderived analytically but also the control algorithm had notbeen used for suppressing the plate vibration Yang and Lee[17] developed an integer-real-encoded GA to search forthe optimal placement and size of the piezoelectric patchesRader et al [18] presented an approach to optimizing theconfiguration of piezoelectric actuators for vibration controlof a flexible aircraft fin However not only they derivedthe equations of motion of smart structure explicitly butalso the residual modes were not considered in their workAmbrosio et al [19] presented an H

2norm approach for

the actuator and sensor placement Sajizadeh and Darus[20] determined the optimal locations of sensors based onobservability concept Qian et al [21] formulated the optimalactuatorsensor location problem in a framework of a zero-one optimization problem which could be solved using GAHowever in this work residual modes were neglected Thusspillover effects could distort the system performance Bruantet al [22] formulated the optimal locations and number ofpiezoelectric sensors for active vibration control However in[20 22] they did not find the optimal locations of actuators

The spillover effect could be among one of the most chal-lenging issues that might spoil the active vibration control ofstructure Indeed when feedback controllers would be usedfor active vibration control of systems there would be noassurance that control algorithm could excite the residualmodes Thus for enhancing control algorithm operationa consideration of residual modes should be necessaryHowever in [14 23 24] residualmodes were neglectedThusspillover effects could distort the system operation Bruantet al [25] and Han and Lee [26] determined the sensorand actuator locations with the consideration of controlla-bility observability and spillover reduction Furthermorethey used GA to find efficient locations of piezoelectricsensorsactuators However in their papers the dynamiccharacteristics of both rectangular plate and piezoelectricsensorsactuators were not derived explicitly Bruant andProslier [27] presented an optimization problem based onoptimization criteria of ensuring either good observability orgood controllability of the structure as well as consideringresidual modes to limit the spillover effects However in [27]the dynamic characteristics of both rectangular plate andpiezoelectric sensorsactuators were not derived explicitly

u3 z

u1 x

u2 y

hst

lx

ly

120579

lys

lxs

Figure 1 Flexible plate and orientation of piezoelectric patches

In the present paper an integrated and consistent formu-lation is derived based on energy principles and variationalmethods for both the host plate structure and the patchtransducers together with a GA optimization algorithm foroptimal locations and orientations of these patches usingspatial controllability-observability-based optimality crite-ria In addition in this study the effects of residual modesare incorporated in the fitness function of optimization algo-rithm Based on the CPT and linear piezoelectric theory theequations ofmotion and sensor output equations of the smartplate are derived by using Hamiltonrsquos principle Rayleigh-Ritzapproximation procedure and the assumed modes methodFor increasing the system efficiency the optimal locationsand orientations of piezoelectric transducers are determinedbased on spatial controllability and observability and con-sidering residual modes to reduce the spillover effect Foractive vibration control a negative velocity feedback controlalgorithm is used

The remainder of the paper is organized as follows Wederive the basic equation in Section 2 In Section 3 we discussthe optimization of the location of sensors and actuatorsmounted on a rectangular plate The GA is used to findthe optimal location of sensors and actuators in Section 4In Section 5 designing controller for active vibration ofplate and several simulations for showing the influenceof optimization and active control algorithm are outlinedFinally in Section 6 we draw conclusions

2 Basic Equations21 Potential and Kinetic Energies and Virtual Work Con-sider a flexible plate (Figure 1) with119873

119886piezoelectric actuators

and 119873se piezoelectric sensors The following formulationcould be used for any plate geometry and any boundaryconditions The total potential energy of the structure andpiezoelectric patch is expressed as [28]

PE = int119881

st

1

2STcstSd119881

st

+

119873119886

sum

119895=1

int119881119886

119895

(1

2STcDS minus SThDj

+1

2DjT120573SD

j) d119881119886119895

+

119873se

sum

119895=1

int119881

se119895

(1

2STcDS minus SThDj

+1

2DjT120573SD

j) d119881se119895

(1)





S = RjSS Dj

= RjDD

j (2)

where RjS and R



KE = 1

2int119881

st120588st(2

1+ 2

2+ 2

3) d119881st

+1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(2

1+ 2

2+ 2

3) d119881119886119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(2

1+ 2

2+ 2

3) d119881se119895

=1

2int119881

st120588st(uTu) d119881st +

1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(uTu) d119881119886

119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(uTu) d119881se

119895

(3)

where 1 2 and



120575119882ext=

119873119886

sum

119895=1

V119886119895120575119902119886

119895+

119873se

sum

119895=1

Vse119895120575119902

se119895+

119873119891

sum

119895=1

120575u(119909119895 119910119895)119879

119891119895 (4)



1199061(119909 119910 119905) = Nu

1

(119909 119910)U1(119905)

1199062(119909 119910 119905) = Nu

2

(119909 119910)U2(119905)

1199063(119909 119910 119905) = Nu

3

(119909 119910)U3(119905)

(5)

where 1199061 1199062 and 119906




and Nu1

Nu2

and Nu3


u =

1199061

1199062

1199063

= [

[

Nu1 0 00 Nu

2

00 0 Nu

3

]

]

U1U2U3

= NuU (6)


119878119909119909

119878119910119910

119878119909119910

=

1205971199061

120597119909

1205971199062

120597119910

1205971199061

120597119910+1205971199062

120597119909

+ 119911

minus12059721199063

1205971199092

minus1205971199063

120597119909

minus212059721199063

120597119909120597119910

(7)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[

[

120597Nu1

1205971199090 0

0120597Nu

2

1205971199100

120597Nu1

120597119910

120597Nu2

1205971199090

]]]]]]]]

]

U1U2U3

+

[[[[[[[[[[

[

0 0 minus119911

1205972Nu3

1205971199092

0 0 minus119911

1205972Nu3

1205971199102

0 0 minus119911

1205972Nu3

120597119909120597119910

]]]]]]]]]]

]

U1U2U3

= BmUU + Bb

UU

(8)

where BmU and Bb





D119895 = B119895qq119901997904rArr B1q =

[[[[[

[

0 0 0


1

1198601119901

0 0

]]]]]

]

B2q =[[[[[

[

0 0 0


01

1198602119901

0

]]]]]

]

B119873119901

q =

[[[[[

[

0 0 0


0 01

119860119873119901

119901

]]]]]

]

(9)


q119901 =

[[[[[[[[[[

[

119902119901

1

119902119901

2

119902119901

3

119902119901

119873119901

]]]]]]]]]]

]

(10)


PE = 1

2UTKstU + 1

2UTKse

DU +1

2UTKa

DU

minus UTKseUqq

seminus UTKa

Uqqa

+1

2qseTKse

qqqse+1

2qaTKa

qqqa

(11)

whereKstKseD andK







Kst= int119881

st(Bm

UTcstB

mU + Bb

UTcstB

bU + Bm

UTcstB

bU

+BbUTcstB

mU) d119881

st

KpD =

119873119901

sum

119895=1

int119881119901

119895

(BmUTRj

STcDRj

SBmU + Bb

UTRjSTcDRj

SBbU

+BmUTRj

STcDRj

SBbU + Bb

UTRjSTcDRj

SBmU)

times d119881119901119895

K119901Uq =119873119901

sum

119895=1

(int119881119901

119895

BmUTRj

SThRj

DBjqd119881119901

119895

+int119881119901

119895

BbUTRjSThRj

DBjqd119881119901

119895)

Kpqq =

119873119901

sum

119895=1

int119881119901

119895

BjqTRjDT120573sR

jDB

jqd119881119901

119895

(12)


KE = 12UMstU + 1

2UMseU + 1

2UMaU (13)


Mst= int119881

st120588stNT

uNud119881st

Mp=

119873119901

sum

119895=1

int119881119901

119895

120588119901

119895NT

uNud119881119901

119895

(14)


as

120575119882ext= 120575qaTBa

VVa+ 120575qseTBse

VVse+ 120575UTFc (15)

where Fc is

Fc =119873119891

sum

119895=1

NTu (119909119895 119910119895) 119891119895 (16)


V are defined as

Bak = INa

Bsek = INse

(17)





seminus Ka

Uqqa= F (18)

minusKseqUU + Kse

qqqse= Bse

k k(119905)se (19)

minusKaqUU + Ka

qqqa= Ba

kk(119905)a (20)


+ Mse+ Ma and KUU =

Kst+ Kse



k kse+ Kse

qqminus1Kse

qUU (21)


qa = Kaqqminus1Ba

kka+ Ka

qqminus1Ka

qUU (22)



where

Υa = KaUqK

aqqminus1Ba

k (24)



seqqminus1Kse

qU minus KaUqK

aqqminus1Ka

qU (25)


ise (119905) = dqse (119905)d119905

(26)


Φse(119905) = Gci (119905) (27)


Φse(119905) = GcK

seqqminus1Kse

qUU = CU (28)





2

1205962




U = HR (30)


R + Λ2nR = Υ

1015840

aka+HTFc Φ

se(t) = C1015840R (31)




1015840

aka+HTFc (32)


2

n are written as

Z =[[[[[[

[


0 1205772

0 0

d

0 0 120577119899minus1

0

0 0 sdot sdot sdot 0 120577119899

]]]]]]

]

Λ n =

[[[[[[

[


0 1205962

0 0

d

0 0 120596119899minus1

0

0 0 sdot sdot sdot 0 120596119899

]]]]]]

]

(33)



2norm is


1199061= 0 119906

2= 0 119906

3= Nu3U3 (34)




]119879

(35)



119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)


3 Υ119886


Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063




as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)


119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)


119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)



119886






119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)


and 119873119877


119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))






119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)



Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)










119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of












S = RjSS Dj

= RjDD

j (2)

where RjS and R



KE = 1

2int119881

st120588st(2

1+ 2

2+ 2

3) d119881st

+1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(2

1+ 2

2+ 2

3) d119881119886119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(2

1+ 2

2+ 2

3) d119881se119895

=1

2int119881

st120588st(uTu) d119881st +

1

2

119873119886

sum

119895=1

int119881119886

119895

120588119886

119895(uTu) d119881119886

119895

+1

2

119873se

sum

119895=1

int119881

se119895

120588se119895(uTu) d119881se

119895

(3)

where 1 2 and



120575119882ext=

119873119886

sum

119895=1

V119886119895120575119902119886

119895+

119873se

sum

119895=1

Vse119895120575119902

se119895+

119873119891

sum

119895=1

120575u(119909119895 119910119895)119879

119891119895 (4)



1199061(119909 119910 119905) = Nu

1

(119909 119910)U1(119905)

1199062(119909 119910 119905) = Nu

2

(119909 119910)U2(119905)

1199063(119909 119910 119905) = Nu

3

(119909 119910)U3(119905)

(5)

where 1199061 1199062 and 119906




and Nu1

Nu2

and Nu3


u =

1199061

1199062

1199063

= [

[

Nu1 0 00 Nu

2

00 0 Nu

3

]

]

U1U2U3

= NuU (6)


119878119909119909

119878119910119910

119878119909119910

=

1205971199061

120597119909

1205971199062

120597119910

1205971199061

120597119910+1205971199062

120597119909

+ 119911

minus12059721199063

1205971199092

minus1205971199063

120597119909

minus212059721199063

120597119909120597119910

(7)


119878119909119909

119878119910119910

119878119909119910

=

[[[[[[[[

[

120597Nu1

1205971199090 0

0120597Nu

2

1205971199100

120597Nu1

120597119910

120597Nu2

1205971199090

]]]]]]]]

]

U1U2U3

+

[[[[[[[[[[

[

0 0 minus119911

1205972Nu3

1205971199092

0 0 minus119911

1205972Nu3

1205971199102

0 0 minus119911

1205972Nu3

120597119909120597119910

]]]]]]]]]]

]

U1U2U3

= BmUU + Bb

UU

(8)

where BmU and Bb





D119895 = B119895qq119901997904rArr B1q =

[[[[[

[

0 0 0


1

1198601119901

0 0

]]]]]

]

B2q =[[[[[

[

0 0 0


01

1198602119901

0

]]]]]

]

B119873119901

q =

[[[[[

[

0 0 0


0 01

119860119873119901

119901

]]]]]

]

(9)


q119901 =

[[[[[[[[[[

[

119902119901

1

119902119901

2

119902119901

3

119902119901

119873119901

]]]]]]]]]]

]

(10)


PE = 1

2UTKstU + 1

2UTKse

DU +1

2UTKa

DU

minus UTKseUqq

seminus UTKa

Uqqa

+1

2qseTKse

qqqse+1

2qaTKa

qqqa

(11)

whereKstKseD andK







Kst= int119881

st(Bm

UTcstB

mU + Bb

UTcstB

bU + Bm

UTcstB

bU

+BbUTcstB

mU) d119881

st

KpD =

119873119901

sum

119895=1

int119881119901

119895

(BmUTRj

STcDRj

SBmU + Bb

UTRjSTcDRj

SBbU

+BmUTRj

STcDRj

SBbU + Bb

UTRjSTcDRj

SBmU)

times d119881119901119895

K119901Uq =119873119901

sum

119895=1

(int119881119901

119895

BmUTRj

SThRj

DBjqd119881119901

119895

+int119881119901

119895

BbUTRjSThRj

DBjqd119881119901

119895)

Kpqq =

119873119901

sum

119895=1

int119881119901

119895

BjqTRjDT120573sR

jDB

jqd119881119901

119895

(12)


KE = 12UMstU + 1

2UMseU + 1

2UMaU (13)


Mst= int119881

st120588stNT

uNud119881st

Mp=

119873119901

sum

119895=1

int119881119901

119895

120588119901

119895NT

uNud119881119901

119895

(14)


as

120575119882ext= 120575qaTBa

VVa+ 120575qseTBse

VVse+ 120575UTFc (15)

where Fc is

Fc =119873119891

sum

119895=1

NTu (119909119895 119910119895) 119891119895 (16)


V are defined as

Bak = INa

Bsek = INse

(17)





seminus Ka

Uqqa= F (18)

minusKseqUU + Kse

qqqse= Bse

k k(119905)se (19)

minusKaqUU + Ka

qqqa= Ba

kk(119905)a (20)


+ Mse+ Ma and KUU =

Kst+ Kse



k kse+ Kse

qqminus1Kse

qUU (21)


qa = Kaqqminus1Ba

kka+ Ka

qqminus1Ka

qUU (22)



where

Υa = KaUqK

aqqminus1Ba

k (24)



seqqminus1Kse

qU minus KaUqK

aqqminus1Ka

qU (25)


ise (119905) = dqse (119905)d119905

(26)


Φse(119905) = Gci (119905) (27)


Φse(119905) = GcK

seqqminus1Kse

qUU = CU (28)





2

1205962




U = HR (30)


R + Λ2nR = Υ

1015840

aka+HTFc Φ

se(t) = C1015840R (31)




1015840

aka+HTFc (32)


2

n are written as

Z =[[[[[[

[


0 1205772

0 0

d

0 0 120577119899minus1

0

0 0 sdot sdot sdot 0 120577119899

]]]]]]

]

Λ n =

[[[[[[

[


0 1205962

0 0

d

0 0 120596119899minus1

0

0 0 sdot sdot sdot 0 120596119899

]]]]]]

]

(33)



2norm is


1199061= 0 119906

2= 0 119906

3= Nu3U3 (34)




]119879

(35)



119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)


3 Υ119886


Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063




as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)


119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)


119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)



119886






119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)


and 119873119877


119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))






119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)



Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)










119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of










D119895 = B119895qq119901997904rArr B1q =

[[[[[

[

0 0 0


1

1198601119901

0 0

]]]]]

]

B2q =[[[[[

[

0 0 0


01

1198602119901

0

]]]]]

]

B119873119901

q =

[[[[[

[

0 0 0


0 01

119860119873119901

119901

]]]]]

]

(9)


q119901 =

[[[[[[[[[[

[

119902119901

1

119902119901

2

119902119901

3

119902119901

119873119901

]]]]]]]]]]

]

(10)


PE = 1

2UTKstU + 1

2UTKse

DU +1

2UTKa

DU

minus UTKseUqq

seminus UTKa

Uqqa

+1

2qseTKse

qqqse+1

2qaTKa

qqqa

(11)

whereKstKseD andK







Kst= int119881

st(Bm

UTcstB

mU + Bb

UTcstB

bU + Bm

UTcstB

bU

+BbUTcstB

mU) d119881

st

KpD =

119873119901

sum

119895=1

int119881119901

119895

(BmUTRj

STcDRj

SBmU + Bb

UTRjSTcDRj

SBbU

+BmUTRj

STcDRj

SBbU + Bb

UTRjSTcDRj

SBmU)

times d119881119901119895

K119901Uq =119873119901

sum

119895=1

(int119881119901

119895

BmUTRj

SThRj

DBjqd119881119901

119895

+int119881119901

119895

BbUTRjSThRj

DBjqd119881119901

119895)

Kpqq =

119873119901

sum

119895=1

int119881119901

119895

BjqTRjDT120573sR

jDB

jqd119881119901

119895

(12)


KE = 12UMstU + 1

2UMseU + 1

2UMaU (13)


Mst= int119881

st120588stNT

uNud119881st

Mp=

119873119901

sum

119895=1

int119881119901

119895

120588119901

119895NT

uNud119881119901

119895

(14)


as

120575119882ext= 120575qaTBa

VVa+ 120575qseTBse

VVse+ 120575UTFc (15)

where Fc is

Fc =119873119891

sum

119895=1

NTu (119909119895 119910119895) 119891119895 (16)


V are defined as

Bak = INa

Bsek = INse

(17)





seminus Ka

Uqqa= F (18)

minusKseqUU + Kse

qqqse= Bse

k k(119905)se (19)

minusKaqUU + Ka

qqqa= Ba

kk(119905)a (20)


+ Mse+ Ma and KUU =

Kst+ Kse



k kse+ Kse

qqminus1Kse

qUU (21)


qa = Kaqqminus1Ba

kka+ Ka

qqminus1Ka

qUU (22)



where

Υa = KaUqK

aqqminus1Ba

k (24)



seqqminus1Kse

qU minus KaUqK

aqqminus1Ka

qU (25)


ise (119905) = dqse (119905)d119905

(26)


Φse(119905) = Gci (119905) (27)


Φse(119905) = GcK

seqqminus1Kse

qUU = CU (28)





2

1205962




U = HR (30)


R + Λ2nR = Υ

1015840

aka+HTFc Φ

se(t) = C1015840R (31)




1015840

aka+HTFc (32)


2

n are written as

Z =[[[[[[

[


0 1205772

0 0

d

0 0 120577119899minus1

0

0 0 sdot sdot sdot 0 120577119899

]]]]]]

]

Λ n =

[[[[[[

[


0 1205962

0 0

d

0 0 120596119899minus1

0

0 0 sdot sdot sdot 0 120596119899

]]]]]]

]

(33)



2norm is


1199061= 0 119906

2= 0 119906

3= Nu3U3 (34)




]119879

(35)



119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)


3 Υ119886


Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063




as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)


119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)


119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)



119886






119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)


and 119873119877


119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))






119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)



Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)










119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of











seminus Ka

Uqqa= F (18)

minusKseqUU + Kse

qqqse= Bse

k k(119905)se (19)

minusKaqUU + Ka

qqqa= Ba

kk(119905)a (20)


+ Mse+ Ma and KUU =

Kst+ Kse



k kse+ Kse

qqminus1Kse

qUU (21)


qa = Kaqqminus1Ba

kka+ Ka

qqminus1Ka

qUU (22)



where

Υa = KaUqK

aqqminus1Ba

k (24)



seqqminus1Kse

qU minus KaUqK

aqqminus1Ka

qU (25)


ise (119905) = dqse (119905)d119905

(26)


Φse(119905) = Gci (119905) (27)


Φse(119905) = GcK

seqqminus1Kse

qUU = CU (28)





2

1205962




U = HR (30)


R + Λ2nR = Υ

1015840

aka+HTFc Φ

se(t) = C1015840R (31)




1015840

aka+HTFc (32)


2

n are written as

Z =[[[[[[

[


0 1205772

0 0

d

0 0 120577119899minus1

0

0 0 sdot sdot sdot 0 120577119899

]]]]]]

]

Λ n =

[[[[[[

[


0 1205962

0 0

d

0 0 120596119899minus1

0

0 0 sdot sdot sdot 0 120596119899

]]]]]]

]

(33)



2norm is


1199061= 0 119906

2= 0 119906

3= Nu3U3 (34)




]119879

(35)



119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)


3 Υ119886


Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063




as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)


119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)


119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)



119886






119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)


and 119873119877


119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))






119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)



Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)










119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of











]119879

(35)



119866 (119904 119909 119910) =

119899

sum

119894=1

119866119894(119904 119909 119910) =

119899

sum

119894=1

119873119894

1199063

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

(36)


3 Υ119886


Υ119894

119886= [(119870

119886(1)

119880119902)119894(119870119886(1)

119902119902)minus1

(119870119886(2)

119880119902)119894(119870119886(2)

119902119902)minus1

sdot sdot sdot (119870119886(119873119886)

119880119902)119894(119870119886(119873119886)

119902119902)minus1

] (37)

and1198731198941199063




as [28 31 32]

1198662

2= int

+infin

minusinfin

int

119897119909119904

0

int

119897119910119904

0

trace 119866 (119895120596 119909 119910) lowast 119866 (119895120596 119909 119910)

times 119889119910 119889119909 119889120596

=

119899

sum

119894=1

10038171003817100381710038171003817119866119894

10038171003817100381710038171003817

2

2

(38)


119866119894=

Υ119894119886

1199042 + 2120577119894120596119894119904 + 1205962119894

119894 = 1 119899 (39)


119894119895 as

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

=10038171003817100381710038171003817119866119895

119894

100381710038171003817100381710038172

=

1003816100381610038161003816100381610038161003816(119870119886(119895)

119880119902)119894(119870119886(119895)

119902119902)minus11003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817100381710038171003817

Υ119894

119886

1199042 + 2120577119894120596119894119904 + 1205962

119894

100381710038171003817100381710038171003817100381710038171003817

119894 = 1 119899

(40)



119886






119873119886

sum

119895=1

radic

119873119862

sum

119894=1

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

119873119886

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(41)


and 119873119877


119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2 and

radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))






119869119886=

sum119873119886

119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862

119894=1120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

minus 120582

sum119873119886

119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

maxsum119873119886119895=1radicsum119873119862+119873119877

119894=119873119862

120595119894119895(119909119886(119895) 119910119886(119895) 120579119886(119895))

2

(42)



Φse(119905) = C1015840R = CU =

119899

sum

119894=1

119862119894119894 (43)










119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of
















119862119894= [1198661

119888(119870

se(1)119902119902

)minus1

(119870se(1)119880119902

)1198941198662

119888(119870

se(2)119902119902

)minus1

(119870se(2)119880119902


119873se119888(119870

se(119873se)119902119902

)minus1

(119870se(119873se)119880119902

)119894

]

119879

(44)


119894119895 as

Θ119894119895(119909

se(119895) 119910

se(119895) 120579

se(119895)) =

10038161003816100381610038161003816119862119895

119894

10038161003816100381610038161003816=100381610038161003816100381610038161003816119866119895

119888(119870

se(119895)119902119902

)minus1

(119870se(119895)119880119902

)119894

100381610038161003816100381610038161003816

(45)



119873se

sum

119895=1

radic

119873119862

sum

119894=1

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

119873se

sum

119895=1

radic

119873119862+119873119877

sum

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(46)



119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=1Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

minus

sum119873se119895=1radicsum119873119862+119873119877

119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2


119894=119873119862

Θ119894119895(119909se(119895) 119910se(119895) 120579se(119895))

2

(47)












119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of











119909119904(m) and 119897

119909(m) 1 008

119910-length 119897119910119904(m) and 119897

119910(m) 05 004


119901(m) 0002 00001



1199063(119909 119910 119905) =

119896

sum

120581=1

119897

sum

120580=1

119873120581120580

1199063

(119909 119910)119880120581120580(119905) (48)


1199063= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] [119880111198802111988031sdot sdot sdot 119880

119896119897]119879

(49)

We define

Nu3

= [11987311

1199063

11987321

1199063

11987331

1199063

sdot sdot sdot 119873119896119897

1199063

] (50)


1199061= 1199062= 1199063= 0 at 119909 = 0

119909 = 119897119909119904 119910 = 0 119910 = 119897

119910119904

(51)



119873120581120580

1199063

= sin(120581120587119909119897119909119904

) sin(120580120587119910

119897119910119904

) (52)

where 119897119909119904

and 119897119910119904







k(119905)a = minusGCoΦse(119905) (53)




2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of










2

15

1

05

0

minus05

minus1

minus15

minus2

minus25

minus3

Fitn

ess v

alue


(a)

45

4

35

3

25

2

15

1

05

0

minus05

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(b)


35

3

25

2

15

1

05

0

Fitn

ess v

alue

0 5 10 15 20 25 30 35 40 45 50



(a)

0 5 10 15 20 25 30 35 40 45 50


6

5

4

3

2

1

0

minus1

Fitn

ess v

alue


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


Without control

(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)




11= 119888119863

22= 1316GPa 119888

119863

12= 842GPa 119888

119863

66= 3GPa 120588 = 7800 kgm3


33= 148 times 10



9 NC

minus10

minus20

minus30

minus40

minus50

minus60

minus70

Mag

nitu

de (d

B)


101

102

Frequency (Hz)








6 Conclusions






















0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of




























0205m0251m

125 rad271 rad

0757m0278m

0386m0351m

0905 rad2706 rad











0215m0152m

18 rad149 rad

0337m0275m

0115m0352m

1796 rad1494 rad


20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(a)

20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)


(b)






20

0

minus20

minus40

minus60

minus80

minus100

Mag

nitu

de (d

B)

101

102

Frequency (Hz)



Nomenclature
















KstKse




KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









KseUqK
















1199061 1199062 and 119906








constantsΦ









References











































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of
































VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of









VLSI Design



RotatingMachinery





Shock and Vibration



Advances in







Journal of





SensorsJournal of







Volume 2014

RoboticsJournal of





Journal of








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