research article experimental verifications of vibration

Research ArticleExperimental Verifications of Vibration Suppression for a SmartCantilever Beam with a Modified Velocity Feedback Controller

Ting Zhang1 Hong Guang Li1 Guo Ping Cai2 and Fu Cai Li1

1 The State Key Laboratory of Mechanical System and Vibration Shanghai Jiao Tong University Shanghai 200240 China2Department of Engineering Mechanics Shanghai Jiao Tong University Shanghai 200240 China

Correspondence should be addressed to Hong Guang Li hglisjtueducn

Received 5 August 2013 Accepted 21 January 2014 Published 27 February 2014

Academic Editor Mohammad Elahinia

Copyright copy 2014 Ting Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents various experimental verifications for the theoretical analysis results of vibration suppression to a smart flexiblebeam bonded with a piezoelectric actuator by a velocity feedback controller and an extended state observer (ESO) During thestate feedback control (SFC) design process for the smart flexible beam with the pole placement theory in the state feedback gainmatrix the velocity feedback gain is much more than the displacement feedback gain For the difference between the velocityfeedback gain and the displacement feedback gain a modified velocity feedback controller is applied based on a dynamical modelwith the Hamilton principle to the smart beam In addition the feedback velocity is attained with the extended state observerand the displacement is acquired by the foil gauge on the root of the smart flexible beam The control voltage is calculated by thedesigned velocity feedback gain multiplied by the feedback velocity Through some experiment verifications for simulation resultsit is indicated that the suppressed amplitude of free vibration is up to 6213 while the attenuated magnitude of its velocity is upto 6131 Therefore it is demonstrated that the modified velocity feedback control with the extended state observer is feasible toreduce free vibration

1 Introduction

In recent years a smart system which consists of a cantileverbeam bonded with a piezoelectric actuator has drawn muchinterest of many researchers [1ndash5] At the same time thecontroller designs based on positioning control and vibrationsuppression of the smart system have attracted wide attentionaround the world [6] Especially for dynamic modeling taskand vibration reduction work of a smart cantilever beamwith piezoelectric materials there have been lots of studieson suppressing vibration with a designed controller [7] Forexample a full-order model is developed using assumedmodel expansion and the Lagrangian approach for a flexiblecantilever beam bonded with a PZT patch to control a basemotion [8] a finite element model of the three-layered smartbeam is utilized to reduce vibration by a velocity feedbackcontroller [9] and the multimodal vibration suppression ofa smart flexible cantilever beam with piezoceramic actuatorand sensor by using a pole placement method is proposed[10]

As for adopted control law to suppress vibrationof mechanical system in the past several years manyresearchers have committed to the work on vibration controlof a smart beam with piezoelectric sensor and actuator by avariety of controllers [11] An acceleration sensor based onproportional feedback control algorithm and sliding modevariable structure control algorithm with phase shiftingtechnology is proposed for suppressing the first two bendingmode vibrations of a beam [12] A modified accelerationfeedback control method is developed for active vibrationcontrol of the aerospace structures [13] A linear quadraticregulator (LQR) controller is adopted to achieve vibrationsuppression of the laminated smart beam [14] A controlleris designed using proportion-integral-derivative theory withoutput feedback to control the vibrations of any real-lifesystem [15] A slidingmode control (SMC)with backsteppingtechnique is employed to control the attitude motion of aspacecraft [16]The vibration suppression is achieved througha combined scheme of PD-based hub motion control anda PZT actuator controller that is a composite of linear and

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 172570 9 pageshttpdxdoiorg1011552014172570

2 Shock and Vibration

PZTFoil gaugeBeam

xF

xp

lb

wp

wb

lp

z(3)x(1)

y(2)

Figure 1 A cantilevered beam bonded with a piezoelectric actuator

angular velocity feedback controllers [17] Nonaxisymmetricvibrations of a clamped-free cylindrical shell partiallytreated with a laminated PVDF actuator are controlledusing an adaptive filtered-X least mean square algorithm[18] A modal velocity feedback control method is appliedto suppress the undesirable vibration [19] An improvedversion of the previously developed synchronized switchdamping on voltage (SSDV) approach [20] and an adaptivesemiactive SSDV method through the LMS algorithm [21]are proposed and are applied to the vibration control of acomposite beam An adaptive law is adopted for the purposeof providing an additional force to control frequency changescaused by broadband vibrations [22] A novel approach isdeveloped for achieving a high performance piezoelectricvibration absorber [23] A robust adaptive sliding modeattitude controller is designed to control system for rotationmaneuver and vibration suppression of a flexible spacecraft[24] However there is lack of a simplified model to suppressvibration for a smart cantilever beam with an effectivecontroller

In this paper a dynamical model of a smart beam is pro-posed by combining the Hamilton principle and the assumedmode method [25] In addition based on the dynamicalmodel a controller with the velocity feedback control [26ndash28]through the pole placement theory is designed to suppressfree vibration of the smart beam In the velocity feedbackcontrol design process the feedback velocity is observedthrough the extended state observer [29ndash31] with the outputdisplacement of the smart beam sensed by foil gauge In aword a dynamicalmodel is constructed to design the velocityfeedback gain and suppress the first-order free vibration ofthe smart beam with the extended state observer

The rest of this paper is organized as follows A dynamicalmodel for the smart system is constructed in Section 2 Anextended state observer and a velocity feedback controllerare designed for the purpose of vibration suppression inSection 3 Some simulations and experiments are performedto verify the vibration reduction effectiveness for the smartbeam by the velocity feedback controller in Section 4 Finallysome conclusions are given in Section 5

2 Dynamic Modeling for Smart System

A cantilevered beam bonded with a piezoelectric actuatorand a foil gauge is shown in Figure 1 The piezoelectricactuator and the foil gauge are placed on the middle andthe root of the smart beam respectively When the endtip of the smart beam is subjected to an external distur-bance the piezoelectric actuator is activated with the control

voltage generated by a designed controller to suppress thedisturbance or vibrationHowever before that the dynamicalmodel for the actuating system must be proposed usingenergy conservation law

As shown in Figure 1 the displacement 119906 of the smartbeam can be written as

1199061= 119906119909(119909 119911 119905) minus 119911120595

119909(119909 119911 119905)

1199063= 119906119911(119909 119911 119905)

(1)

where 120595119909(119909 119911 119905) = (120597119906

119911(119909 119911 119905))120597119909

At the same time the strain 119904 and stress 119878 should beexpressed as

1199041=

120597119906119909

120597119909

minus 119911

120597120595119909

120597119909

1198781= 1198881199041

(2)

where 119888 represents the elastic stiffness constantMoreover the linear constitutive equations of piezoelec-

tric actuator have been widely used as

1198781= 119888119863

111199041minus ℎ311198633

1198643= minusℎ311199041+ 120573119909

331198633

(3)

where 119864 119863 and 120576 represent electric field electric displace-ment and free dielectric constant respectively ℎ and 120573

represent piezoelectric stiffness constant and free dielectricisolation rate respectively

Furthermore the smart systemrsquos dynamics model canthen be derived using Hamiltonrsquos principle and the assumedmode method In this paper the former first mode of theelastic beam modes is adopted for discretization Thereforethe displacement of 119911 direction can be expressed as

119906119911(119909 119905) = 120601 (119909) sdot 119902 (119905) (4)

wherein 120601(119909) represents the assumedmode vector and 119902(119905) isthe generalized mechanical displacement vector

And in (4) the assumed mode is

120601 (119909) = cosh (119904119887) minus cos (119904

119887119909)

minus

sinh (119904119887119897119887) minus sin (119904

119887119897119887)

cosh (119904119887119897119887) + cos (119904

119887119897119887)

(sinh (119904119887119909) minus sin (119904

119887119909))

(5)

where 119904119887is the eigenvalue of the elastic beam 119897

119887is the length

of beam and 119904119887119897119887= 18751

Shock and Vibration 3

Owing to (1) (2) and (4) the strain 1199041is expressed as

1199041= minusℎ119901

1205972120601

1205971199092119902 = 119861119902 (6)

where 119861 is specified in the appendixTherefore the kinetic energy 119864

119896 the potential energy 119864

119901

and the virtual work 119882 of the smart cantilevered beam aredescribed as follows

First the kinetic energy term 119864119896is written as

119864119896=

1

2

119902 (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911) 119902

+

1

2

119902 (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911) 119902 =

1

2

119898( 119902)2

(7)

wherein subscripts 119887 and 119901 refer to the beam and thepiezoelectric actuator respectively ℎ 119897 120588 119908 and 119909

119901are

height length density width and the location of piezoelec-tric actuator with respect to the fixed end of beam

Then assuming that 119863 = 119876119860119901 119876 is charge and 119860

119901is

the cross-section area of piezoelectric actuator the elasticelectrical and thermal potential energy term 119864

119901is given as

119864119901=

1

2

int

119881

(11987811199041+ 11986431198633) 119889119881 +

1

2

119902int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119902

=

1

2

int

119881

[(119888119863

111199041minus ℎ311198633) 1199041+ (minusℎ

311199041+ 120573119909

331198633)1198633] 119889119881

+

1

2

(int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119889119909)1199022

=

1

2

1198961199021199021199022+ 119876119896119876119902119902 +

1

2

119896119876119876

1198762

(8)

where 119864119887and 119868119887are the elastic modulus and the moment of

inertia of beam respectively and the corresponding matricesor vectors are specified in the appendix

Finally the virtual work 119882 done by external force 119891the beamrsquos damping 119888

119887 and the voltage 119881 applied on the

piezoelectric actuator are expressed as

120575119882 = int

119897119887

0

119891120575119906119911119889119909 minus int

119897119887

0

119888119887119911120575119906119911119889119909 + int

119909119901+119897119887

119909119901

119881

ℎ119901

120575119876119889119909

= (int

119897119887

0

119891120601119889119909)120575119902 minus (int

119897119887

0

1198881198871206012119889119909) 119902120575119902

+ (int

119909119901+119897119901

119909119901

119881

ℎ119901

119889119909)120575119876

= 120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876

(9)

where 119860119887is the cross-section area of beam and the corre-

sponding matrices or vectors are specified in the appendix

Moreover Hamiltonrsquos principle is

120575119869 (119902 120601) = 120575int

119905119891

1199050

(119871 (119902 119876 119905) + 119882) 119889119905

= 120575int

119905119891

1199050

(119864119896minus 119864119901+119882)119889119905 = 0

(10)

Substituting (7) (8) and (9) into (10) then

119898 119902120575119902 + 119896119902119902119902120575119902 + 119896

119876119902119902120575119876 + 119896

119876119902119876120575119902

+ 119896119876119876

119876120575119876 minus (120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876)

= (119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 minus 120572119891) 120575119902

+ (119896119876119902119902 + 119896119876119876

119876 minus 120574119881) 120575119876 = 0

(11)

Therefore from (11) the dynamicalmodel of the smart systemis obtained as

119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 = 120572119891

119896119876119902119902 + 119896119876119876

119876 = 120574119881

(12)

where the corresponding parameters are specified in theappendix

Due to (6) the output displacement 119906119911at 119909119891along the 119911

direction in Figure 1 is derived as

119906119911= 1198971198871199041= 119861 (119909

119891) 119897119887119902 (13)

Finally when external force119891 is zero (12) is transformed into

119911= minus2120585120596

119911minus 1205962119906119911+ 119887119881 (14)

where 120596 = radic(119896119902119902

minus 1198962

119876119902119896119876119876

)119898 120585 = 1198882119898120596 and 119887 =

(120574119896119876119902119897119887119861(119909119891))119896119876119876

The verification analyses of model (14) are implemented

with some necessary simulations and experiments throughapplying a sinusoidal voltage at different frequency (150 times

sin(2120587119891 times 119905) and f is frequency and 119905 is time) In additionthe related geometric and mechanical parameters aboutsimulation are given in Table 1 From Figure 2 it is obviousthat the simulation curve from (14) fits in well with theexperiment data Furthermore it is informed that the firstnatural frequency is about 1556Hz and its correspondingmaximum amplitude of harmonic vibration is 1518 times

10minus5m Therefore the theoretical model (14) can almost

describe the harmonic characteristics of smart beam fromexperiment

3 Observer and Controller Design

Based on above the proposed dynamical model for thesmart beam in the section an observer and a controllerare designed to suppress free vibration from the end tip ofbeam Moreover the observer is employed with the extendedstate observer (ESO) The ESO has advantages of not onlyobserving high-order states of the system but also filtering


Table 1 System parameters used in simulations

Beam parameters119897119887= 092m 119908

119887= 004m ℎ

119887= 00032m

120588119887= 784 times 10

3 kgm3119884119887= 20 times 10

3 Nm2119888119887= 001

Piezoelectric parameters119897119901= 005m 119908

119901= 0034m ℎ

119901= 000025m

120588119901= 78 times 10

3 kgm3119884119901= 21 times 10

3 Nm211988931

= 22 times 10minus11 CN

119888119863

11= 1064 times 10

10 Nm2ℎ31

= minus135 times 109 Vm 119909

119901= 0085m

Table 2 Corresponding parameters of observer and controller

Observer Controller119903 ℎ ℎ

11198961

119896

1000 0001 003 23439 times 103 753672

0 05 1 15 20

05

1

15

2

Frequency (Hz)

Am

plitu

de (m

)

SimulationExperiment

times10minus5

Figure 2 Amplitude-frequency property

noise which is caused by measurement of experiment Inaddition the control law is adopted with the state feedbackcontrol through the pole placement theory However in thedesign process it is found that the velocity feedback gain ismuch more than the displacement feedback one Thereforea simplified state feedback controller namely a modifiedvelocity feedback controller is designed to attenuate the freevibration of the smart beam

31 Observer Design To achieve the first-order state (veloc-ity) for the smart system the designed observer must havethe capacity of obtaining the high-order state through outputdisplacement of systemThe extended state observer (ESO) isa kind of state observer that can getmultiorders state variablesof system First assume that a second-order system is writtenas (15) from (14)

119911(119905) = 119865 (

119911(119905) 119906119911(119905)) + 119887119881 (15)

where 119865(119911(119905) 119906119911(119905)) = minus2120585120596

119911minus1205962119906119911 And (15) is written as

a state equation

1199011(119905) = 119906

119911+ 119899 (119905)

1(119905) = 119901

2(119905)

2(119905) = 119901

3(119905) + 119887119881

3(119905) = (119901)

(16)

where 1199011(119905) and 119901

2(119905) are states of the smart system and 119899(119905)

is measurement noiseThen assume the two observed states (displacement and

velocity) of system as V1and V

2and transform them into

discrete variables as

V1(119905 + ℎ) = V

1(119905) + ℎV

2(119905)

V2(119905 + ℎ) = V

2(119905) + ℎ sdot fst (V

1(119905) V2(119905) 1199011(119905) 119903 ℎ

1)

(17)

where ℎ is time increment and the function fst is the opticalcontrol synthesis function [32] written as

119889 = 119903 sdot ℎ1

1198890= 119889 sdot ℎ

1

119890 = V1minus 1199011+ ℎ1sdot V2

1198860= radic1198892+ 8119903 |119890|

1198861=

V2+

119890

ℎ1

|119890| le 1198890

V2+ sgn (119890) sdot

1198860minus 119889

2

|119890| gt 1198890

fst (V1 V2 119903 ℎ1) =

minus119903 sdot

1198861

119889

10038161003816100381610038161198861

1003816100381610038161003816le 119889

minus119903 sdot sgn (1198861)

10038161003816100381610038161198861

1003816100381610038161003816gt 119889

(18)

where 119903 is the initial state parameter and ℎ1is the time

increment in the fst functionFinally with the parameters of the designed observer

in Table 2 the observed states can track the correspondingpractical states of system in

V1(119905) 997888rarr 119901

1(119905) V

2(119905) 997888rarr 119901

2(119905) (19)

The principle diagram of ESO is shown in Figure 3 Theextended state 119906

119911(119905) is thought as the displacement sensed

by foil gauge By the designed observer the online observedresults are shown in Figure 4 Furthermore Figures 4(a) and4(b) show actual acquired displacement and the observed dis-placement respectively Compared with the actual collected


Target+

++

+

minus

minus

k

k1

Controller

Control voltage

1

2

Smart

Observer

fst(1 2 uz)uz

beam Output displacement

Figure 3 Control block

displacement signal in Figure 4(a) the observed result signalin Figure 4(b) is filtered and becomes smoother In additionFigure 4(c) shows the observed velocity by the extendedstate observer The amplitude of the observed velocity V

2(its

maximum value is 4404 times 10minus5ms) is about 10 times of the

amplitude of observed displacement V1(its maximum value

is 3959 times 10minus6m)

32 Controller Design In the paper a state feedback control(SFC) is adopted to suppress free vibration for the smartbeam Figure 3 shows that the velocity feedback gain 119896 ismuch greater than the displacement feedback gain 119896

1in

the design process of SFC Therefore a velocity feedbackcontroller is simplified from the SFC In addition the mod-ified velocity feedback controller can increase the systemrsquosdamping to attenuate the vibration faster The control law isdesigned from the principle of pole placement theory Firstthe target damping ratio is assumed as 120585

119900

120585119900= 120585 + Δ (20)

where 120585 lt 120585119900lt 1 and Δ is the quantity compensated by the

velocity feedback controllerThen from (14) the objective systemmodel is attained as

119911= minus (2120585

119900120596119900119911+ 1205962119906119911) + 119887119881 (21)

As a result the eigen matrix 119860119900of (21) is written as

119860119900= 120582119868 minus [

0 1

minus1205962

119900minus2120585119900120596119900

] (22)

where 120582 is the eigenvalue and 119868 is the unit matrixTo achieve the objective damping ratio 120585

119900 the input

control voltage 119906119888in Figure 3 is designed through the state

feedback control law and is expressed as

119906119888= 119881 = minus (119896

119911+ 1198961119906119911) (23)

Therefore the close loop system is described as

119911= minus (2120585120596

119911+ 1205962119906119911) minus 119887119896

119911minus 1198871198961119906119911 (24)

The eigen matrix of (24) is calculated as

119860119898= 120582119868 minus [

0 1

minus1205962minus 1198871198961

minus2120585120596 minus 119887119896

] (25)

0 5 10 15 20 25 30 35 40 45 50

024

Time (s)

minus4minus2

times10minus6

p1

(m)

(a) Collected

0 5 10 15 20 25 30 35 40 45 50Time (s)

024

minus4minus2

times10minus6

1(m

)

(b) Observed

0 5 10 15 20 25 30 35 40 45 50

0

5

minus5

times10minus5

2(m

s)

Time (s)

(c) Observed

Figure 4 Observed and collected displacement and velocity with-out control

0 5 10

0

5

10

Real

Imag

e

minus10minus10

minus5

minus5

Figure 5 Stability analysis of velocity feedback control for smartsystem


Stain amplifier(YE3817C)

Gauge foil Piezoelectric actuator Iron weight

Computer

Power amplifier(HPV-3C0150A0300D)

SEED-DEC2812

Beam

Figure 6 Experimental setup

Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment

Figure 7 Control voltage

Then by pole placement theory the eigenvalues in (22) and(25) are equal yielding

1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)

where | ∙ | is determinant of the matrixFinally due to (26) the state feedback gains are derived as

119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)

The calculated state feedback gains 119896 and 1198961are given in

Table 2 The control voltage is attained through 119896 and 1198961

Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment

Figure 8 Control results of displacement

multiplied by the displacement and velocity respectivelyHowever the term (119896

1timesV1) is much less than the term (119896timesV

2)

In other words the control voltage is composed mainly of(minus119896 times V

2) Therefore a modified velocity feedback controller

is simplified from the SFC to suppress vibration for the smartbeam

The stability analysis of close loop system is shown inFigure 5 The actual damping ratio 120585 of the close loop system


is 00055 without control When the damping ratio 120585 isdesigned from 00055 to 1 through the modified feedbackcontroller the real parts of the two conjugate poles in theleft side of imaginary axis increase negatively Therefore it isindicated that the controlled system is stable However theapplied control voltage by the power equipment is restrictedat le150V the designed damping ratio cannot be achieved upto 0707 or even is less

4 Experimental Verifications

Based on the designs of the velocity feedback control lawand the extended state observer the theoretical simulationanalyses with the corresponding physical test parameters areverified by the practical experiments Figure 6 presents theexperimental setup When the end tip of beam suffers froman initial deformation (the iron weight behind the smartbeam is used for calibrating the initial disturbance loadedon the end tip of beam in each experiment) the responsedisplacement is transformed from the strain through thegauge foil and is transferred to the control processor (SEED-DEC2812) through strain amplifier (YE3817C)The processornot only sends and saves the collected data to the computerbut also receives the commands from the computer to realizethe designed controller Then by the control law the velocityfeedback gain 119896 is obtained Moreover the calculated controlvoltage is outputted from the processor and is applied on thepiezoelectric actuator of smart beam after power amplifier(HPV-3C0150A0300D) to damp the free vibration in shortertime

Figure 7 shows the control voltages with simulation andexperiment Before the time about 2 s the control voltagesare more than the limited voltage 150V Figure 8 gives thecontrol results of displacement in simulation and experimentAfter the time about 15 s the free vibration amplitudes ofdisplacement are all reduced up to a small value Comparedwith the free vibrations without control the control effects insimulation and experiment are obvious Figure 9 presents thecontrol results of velocity with simulation and experiment Itis demonstrated that the amplitudes of velocity with controlare damped faster than the ones without control

In Figures 10 and 11 in simulation and experimentrespectively the spectrum analyses for the control results intime domain of Figures 8 and 9 are presented by Fast FourierTransform (FFT) in MATLAB Figures 10(a) and 10(b) givethe amplitude reduction results of displacement and velocityin simulation respectively The two peaks with no controland with control are 1616 times 10

minus6m and 05604 times 10minus6m

in Figure 10(a) while Figure 10(b) presents the peak 1576 times

10minus5m of curve with no control and the peak 05562 times

10minus5m of curve with control By theoretical simulation for

the smart beam it is indicated that the damping amplitudesare up to 6532 and 6471 in displacement and velocityrespectively

As shown in Figure 11(a) the peaks with no controland with control in experiment are 15106 times 10

minus6m and57215 times 10

minus7m respectively which indicates that the ampli-tudes of vibration are attenuated up to 6213 Figure 11(b)

Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment

Figure 9 Control results of velocity

Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)

Figure 10 Spectrum analysis of control results in simulation

shows the spectrum analysis of velocity without control andwith control with experiment In addition the peaks withno control and with control are 15060 times 10

minus5ms and58273 times 10

minus6ms respectively Therefore it is obvious thatthe reduced amplitude quantities of velocity are 6131 inexperiment


05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)

Figure 11 Spectrum analysis of control results in experiment

In short the simulation and experiment verificationsdemonstrate that the velocity feedback control with theextended state observer is feasible to suppress free vibrationFurthermore the control effects of displacement and velocityare 6532 and 6471 respectively in simulation whilethe control result of displacement in experiment is up to6213 and the control result of velocity in experiment isup to 6131 It is verified that the control effectiveness isconsiderable

5 Conclusions

In this paper the suppression vibration of a cantileveredbeam bonded with a piezoelectric actuator by a velocityfeedback controllerwith an extended state observer is focusedon The dynamical mathematical model for a smart beamis constructed using the Hamilton principle Based on thedynamical model the velocity feedback control is designedthrough the pole placement theory The feedback velocityis obtained by an extended observer with the output dis-placement of the smart beam Finally some simulationsand experiments prove that the velocity feedback controlis feasible to control free vibration Moreover the reducedamplitudes of displacement and velocity for the smart beamare 6532 and 6471 respectively in simulation At thesame time through the experiment verifications the controlresult of displacement is up to 6213 while the controlresult of velocity is up to 6131 It is clear that the controleffectiveness is considerable and the extended state observeris useful to obtain the high-order state for the smart system

Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)

Conflict of Interests

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

Acknowledgment

This work was funded by the National Natural ScienceFoundation of China under Grant no 11072148

References

[1] V Fakhari and A Ohadi ldquoNonlinear vibration control offunctionally graded plate with piezoelectric layers in thermalenvironmentrdquo Journal of Vibration and Control vol 17 no 3pp 449ndash469 2011

[2] H Gu and G Song ldquoActive vibration suppression of a flexiblebeam with piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4 pp 1453ndash1459 2007

[3] M Marinaki Y Marinakis and G E Stavroulakis ldquoFuzzy con-trol optimized by amulti-objective particle swarm optimizationalgorithm for vibration suppression of smart structuresrdquo Struc-tural and Multidisciplinary Optimization vol 43 no 1 pp 29ndash42 2011

[4] O BilgenMAminKaramiD J Inman andM I Friswell ldquoTheactuation characterization of cantilevered unimorph beamswith single crystal piezoelectric materialsrdquo Smart Materials andStructures vol 20 no 5 Article ID 055024 2011


[5] D Ezhilarasi M Umapathy and B Bandyopadhyay ldquoDesignand experimental evaluation of simultaneous periodic outputfeedback control for piezoelectric actuated beam structurerdquoStructural Control andHealthMonitoring vol 16 no 3 pp 335ndash349 2009

[6] Z C Qiu ldquoAdaptive nonlinear vibration control of a Carte-sian flexible manipulator driven by a ballscrew mechanismrdquoMechanical Systems and Signal Processing vol 30 pp 248ndash2662012

[7] G Meng L Ye X Dong and K Wei ldquoClosed loop finiteelement modeling of piezoelectric smart structuresrdquo Shock andVibration vol 13 no 1 pp 1ndash12 2006

[8] MDadfarnia N Jalili Z Liu andDMDawson ldquoAn observer-based piezoelectric control of flexible Cartesian robot armstheory and experimentrdquo Control Engineering Practice vol 12no 8 pp 1041ndash1053 2004

[9] C M A Vasques and J Dias Rodrigues ldquoActive vibrationcontrol of a smart beam through piezoelectric actuation andlaser vibrometer sensing simulation design and experimentalimplementationrdquo Smart Materials and Structures vol 16 no 2pp 305ndash316 2007

[10] V Sethi andG Song ldquoMultimodal vibration control of a flexiblestructure using piezoceramic sensor and actuatorrdquo Journal ofIntelligentMaterial Systems and Structures vol 19 no 5 pp 573ndash582 2008

[11] D Ezhilarasi M Umapathy and B Bandyopadhyay ldquoDesignand experimental evaluation of piecewise output feedbackcontrol for structural vibration suppressionrdquo Smart Materialsand Structures vol 15 no 6 pp 1927ndash1938 2006

[12] Z-C Qiu J-D Han X-M Zhang YWang and ZWu ldquoActivevibration control of a flexible beam using a non-collocatedacceleration sensor and piezoelectric patch actuatorrdquo Journal ofSound and Vibration vol 326 no 3ndash5 pp 438ndash455 2009

[13] S N Mahmoodi and M Ahmadian ldquoModified accelerationfeedback for active vibration control of aerospace structuresrdquoSmart Materials and Structures vol 19 no 6 Article ID 06501510 pages 2010

[14] A Zabihollah R Sedagahti and R Ganesan ldquoActive vibrationsuppression of smart laminated beams using layerwise theoryand an optimal control strategyrdquo Smart Materials and Struc-tures vol 16 no 6 pp 2190ndash2201 2007

[15] S M Khot N P Yelve R Tomar S Desai and S Vittal ldquoActivevibration control of cantilever beam by using PID based outputfeedback controllerrdquo Journal of Vibration and Control vol 18no 3 pp 366ndash372 2012

[16] Q Hu J Cao and Y Zhang ldquoRobust backstepping slidingmodeattitude tracking and vibration damping of flexible spacecraftwith actuator dynamicsrdquo Journal of Aerospace Engineering vol22 no 2 pp 139ndash152 2009

[17] K Gurses B J Buckham and E J Park ldquoVibration control ofa single-link flexible manipulator using an array of fiber opticcurvature sensors and PZT actuatorsrdquoMechatronics vol 19 no2 pp 167ndash177 2009

[18] Y Zhang X Zhang and S Xie ldquoAdaptive vibration controlof a cylindrical shell with laminated PVDF actuatorrdquo ActaMechanica vol 210 no 1-2 pp 85ndash98 2010

[19] Q Hu ldquoRobust adaptive attitude tracking control with L2-gainperformance and vibration reduction of an orbiting flexiblespacecraftrdquo Journal of Dynamic Systems Measurement andControl Transactions of the ASME vol 133 no 1 Article ID011009 11 pages 2011

[20] H Ji J Qio A Badel andK Zhu ldquoSemi-active vibration controlof a composite beamusing an adaptive SSDV approachrdquo Journalof Intelligent Material Systems and Structures vol 20 no 4 pp401ndash412 2009

[21] H Ji J Qiu A Badel Y Chen and K Zhu ldquoSemi-activevibration control of a composite beamby adaptive synchronizedswitching on voltage sources based on LMS algorithmrdquo Journalof Intelligent Material Systems and Structures vol 20 no 8 pp939ndash947 2009

[22] M Ahmadian and D J Inman ldquoAdaptive modified positiveposition feedback for active vibration control of structuresrdquoJournal of Intelligent Material Systems and Structures vol 21 no6 pp 571ndash580 2010

[23] J Lin and W S Chao ldquoVibration suppression control of beam-cart system with piezoelectric transducers by decomposedparallel adaptive neuro-fuzzy controlrdquo JVCJournal of Vibrationand Control vol 15 no 12 pp 1885ndash1906 2009

[24] Q L Hu ldquoRobust adaptive sliding mode attitude control andvibration damping of flexible spacecraft subject to unknowndisturbance and uncertaintyrdquo Transactions of the Institute ofMeasurement and Control vol 34 no 4 pp 436ndash447 2012

[25] X Xue and J Tang ldquoRobust and high precision control usingpiezoelectric actuator circuit and integral continuous slidingmode control designrdquo Journal of Sound and Vibration vol 293no 1-2 pp 335ndash359 2006

[26] D Sun J KMills J Shan and S K Tso ldquoA PZT actuator controlof a single-link flexible manipulator based on linear velocityfeedback and actuator placementrdquo Mechatronics vol 14 no 4pp 381ndash401 2004

[27] J Roos J C Bruch Jr J M Sloss S Adali and I S SadekldquoVelocity feedback control with time delay using piezoelectricsrdquoin Proceedings of the SPIE The International Society for OpticalEngineering smart Structures and Materials 2003 ModelingSignal Processing and Control vol 5049 pp 233ndash240 March2003

[28] P Gardonio and S J Elliott ldquoSmart panels with velocity feed-back control systems using triangularly shaped strain actuatorsrdquoJournal of the Acoustical Society of America vol 117 no 4 pp2046ndash2064 2005

[29] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[30] A J Hillis ldquoActive motion control of fixed offshore platformsusing an extended state observerrdquo Proceedings of the Institutionof Mechanical Engineers I vol 224 no 1 pp 53ndash63 2010

[31] R Zhang and C Tong ldquoTorsional vibration control of themain drive system of a rolling mill based on an extended stateobserver and linear quadratic controlrdquo Journal of Vibration andControl vol 12 no 3 pp 313ndash327 2006

[32] Y Dong M X Jun and C Hua ldquoRealization of DESO filter onDSP and its applicationrdquo Journal of Academy of Armored ForceEngineering vol 24 no 3 pp 57ndash61 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



PZTFoil gaugeBeam

xF

xp

lb

wp

wb

lp

z(3)x(1)

y(2)

Figure 1 A cantilevered beam bonded with a piezoelectric actuator

angular velocity feedback controllers [17] Nonaxisymmetricvibrations of a clamped-free cylindrical shell partiallytreated with a laminated PVDF actuator are controlledusing an adaptive filtered-X least mean square algorithm[18] A modal velocity feedback control method is appliedto suppress the undesirable vibration [19] An improvedversion of the previously developed synchronized switchdamping on voltage (SSDV) approach [20] and an adaptivesemiactive SSDV method through the LMS algorithm [21]are proposed and are applied to the vibration control of acomposite beam An adaptive law is adopted for the purposeof providing an additional force to control frequency changescaused by broadband vibrations [22] A novel approach isdeveloped for achieving a high performance piezoelectricvibration absorber [23] A robust adaptive sliding modeattitude controller is designed to control system for rotationmaneuver and vibration suppression of a flexible spacecraft[24] However there is lack of a simplified model to suppressvibration for a smart cantilever beam with an effectivecontroller

In this paper a dynamical model of a smart beam is pro-posed by combining the Hamilton principle and the assumedmode method [25] In addition based on the dynamicalmodel a controller with the velocity feedback control [26ndash28]through the pole placement theory is designed to suppressfree vibration of the smart beam In the velocity feedbackcontrol design process the feedback velocity is observedthrough the extended state observer [29ndash31] with the outputdisplacement of the smart beam sensed by foil gauge In aword a dynamicalmodel is constructed to design the velocityfeedback gain and suppress the first-order free vibration ofthe smart beam with the extended state observer

The rest of this paper is organized as follows A dynamicalmodel for the smart system is constructed in Section 2 Anextended state observer and a velocity feedback controllerare designed for the purpose of vibration suppression inSection 3 Some simulations and experiments are performedto verify the vibration reduction effectiveness for the smartbeam by the velocity feedback controller in Section 4 Finallysome conclusions are given in Section 5

2 Dynamic Modeling for Smart System

A cantilevered beam bonded with a piezoelectric actuatorand a foil gauge is shown in Figure 1 The piezoelectricactuator and the foil gauge are placed on the middle andthe root of the smart beam respectively When the endtip of the smart beam is subjected to an external distur-bance the piezoelectric actuator is activated with the control

voltage generated by a designed controller to suppress thedisturbance or vibrationHowever before that the dynamicalmodel for the actuating system must be proposed usingenergy conservation law

As shown in Figure 1 the displacement 119906 of the smartbeam can be written as

1199061= 119906119909(119909 119911 119905) minus 119911120595

119909(119909 119911 119905)

1199063= 119906119911(119909 119911 119905)

(1)

where 120595119909(119909 119911 119905) = (120597119906

119911(119909 119911 119905))120597119909

At the same time the strain 119904 and stress 119878 should beexpressed as

1199041=

120597119906119909

120597119909

minus 119911

120597120595119909

120597119909

1198781= 1198881199041

(2)

where 119888 represents the elastic stiffness constantMoreover the linear constitutive equations of piezoelec-

tric actuator have been widely used as

1198781= 119888119863

111199041minus ℎ311198633

1198643= minusℎ311199041+ 120573119909

331198633

(3)

where 119864 119863 and 120576 represent electric field electric displace-ment and free dielectric constant respectively ℎ and 120573

represent piezoelectric stiffness constant and free dielectricisolation rate respectively

Furthermore the smart systemrsquos dynamics model canthen be derived using Hamiltonrsquos principle and the assumedmode method In this paper the former first mode of theelastic beam modes is adopted for discretization Thereforethe displacement of 119911 direction can be expressed as

119906119911(119909 119905) = 120601 (119909) sdot 119902 (119905) (4)

wherein 120601(119909) represents the assumedmode vector and 119902(119905) isthe generalized mechanical displacement vector

And in (4) the assumed mode is

120601 (119909) = cosh (119904119887) minus cos (119904

119887119909)

minus

sinh (119904119887119897119887) minus sin (119904

119887119897119887)

cosh (119904119887119897119887) + cos (119904

119887119897119887)

(sinh (119904119887119909) minus sin (119904

119887119909))

(5)

where 119904119887is the eigenvalue of the elastic beam 119897

119887is the length

of beam and 119904119887119897119887= 18751



1199041= minusℎ119901

1205972120601

1205971199092119902 = 119861119902 (6)



119901



119864119896=

1

2

119902 (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911) 119902

+

1

2

119902 (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911) 119902 =

1

2

119898( 119902)2

(7)


119901are



119901is


119901is given as

119864119901=

1

2

int

119881

(11987811199041+ 11986431198633) 119889119881 +

1

2

119902int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119902

=

1

2

int

119881

[(119888119863

111199041minus ℎ311198633) 1199041+ (minusℎ

311199041+ 120573119909

331198633)1198633] 119889119881

+

1

2

(int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119889119909)1199022

=

1

2

1198961199021199021199022+ 119876119896119876119902119902 +

1

2

119896119876119876

1198762

(8)






120575119882 = int

119897119887

0

119891120575119906119911119889119909 minus int

119897119887

0

119888119887119911120575119906119911119889119909 + int

119909119901+119897119887

119909119901

119881

ℎ119901

120575119876119889119909

= (int

119897119887

0

119891120601119889119909)120575119902 minus (int

119897119887

0

1198881198871206012119889119909) 119902120575119902

+ (int

119909119901+119897119901

119909119901

119881

ℎ119901

119889119909)120575119876

= 120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876

(9)




120575119869 (119902 120601) = 120575int

119905119891

1199050

(119871 (119902 119876 119905) + 119882) 119889119905

= 120575int

119905119891

1199050

(119864119896minus 119864119901+119882)119889119905 = 0

(10)


119898 119902120575119902 + 119896119902119902119902120575119902 + 119896

119876119902119902120575119876 + 119896

119876119902119876120575119902

+ 119896119876119876

119876120575119876 minus (120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876)

= (119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 minus 120572119891) 120575119902

+ (119896119876119902119902 + 119896119876119876

119876 minus 120574119881) 120575119876 = 0

(11)


119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 = 120572119891

119896119876119902119902 + 119896119876119876

119876 = 120574119881

(12)




119906119911= 1198971198871199041= 119861 (119909

119891) 119897119887119902 (13)


119911= minus2120585120596

119911minus 1205962119906119911+ 119887119881 (14)

where 120596 = radic(119896119902119902

minus 1198962

119876119902119896119876119876

)119898 120585 = 1198882119898120596 and 119887 =

(120574119896119876119902119897119887119861(119909119891))119896119876119876











119887= 004m ℎ

119887= 00032m

120588119887= 784 times 10

3 kgm3119884119887= 20 times 10

3 Nm2119888119887= 001


119901= 0034m ℎ

119901= 000025m

120588119901= 78 times 10

3 kgm3119884119901= 21 times 10

3 Nm211988931


119888119863

11= 1064 times 10

10 Nm2ℎ31


119901= 0085m



11198961

119896

1000 0001 003 23439 times 103 753672

0 05 1 15 20

05

1

15

2

Frequency (Hz)

Am

plitu

de (m

)


times10minus5




119911(119905) = 119865 (

119911(119905) 119906119911(119905)) + 119887119881 (15)

where 119865(119911(119905) 119906119911(119905)) = minus2120585120596


a state equation

1199011(119905) = 119906

119911+ 119899 (119905)

1(119905) = 119901

2(119905)

2(119905) = 119901

3(119905) + 119887119881

3(119905) = (119901)

(16)

where 1199011(119905) and 119901






V1(119905 + ℎ) = V

1(119905) + ℎV

2(119905)

V2(119905 + ℎ) = V

2(119905) + ℎ sdot fst (V

1(119905) V2(119905) 1199011(119905) 119903 ℎ

1)

(17)


119889 = 119903 sdot ℎ1

1198890= 119889 sdot ℎ

1

119890 = V1minus 1199011+ ℎ1sdot V2

1198860= radic1198892+ 8119903 |119890|

1198861=

V2+

119890

ℎ1

|119890| le 1198890

V2+ sgn (119890) sdot

1198860minus 119889

2

|119890| gt 1198890

fst (V1 V2 119903 ℎ1) =

minus119903 sdot

1198861

119889

10038161003816100381610038161198861

1003816100381610038161003816le 119889


10038161003816100381610038161198861

1003816100381610038161003816gt 119889

(18)




V1(119905) 997888rarr 119901

1(119905) V

2(119905) 997888rarr 119901

2(119905) (19)





Target+

++

+

minus

minus

k

k1

Controller

Control voltage

1

2

Smart

Observer

fst(1 2 uz)uz




2(its





1in


119900

120585119900= 120585 + Δ (20)



119911= minus (2120585

119900120596119900119911+ 1205962119906119911) + 119887119881 (21)


119860119900= 120582119868 minus [

0 1

minus1205962

119900minus2120585119900120596119900

] (22)


119900 the input



119906119888= 119881 = minus (119896

119911+ 1198961119906119911) (23)


119911= minus (2120585120596

119911+ 1205962119906119911) minus 119887119896

119911minus 1198871198961119906119911 (24)


119860119898= 120582119868 minus [

0 1

minus1205962minus 1198871198961

minus2120585120596 minus 119887119896

] (25)

0 5 10 15 20 25 30 35 40 45 50

024

Time (s)

minus4minus2

times10minus6

p1

(m)

(a) Collected

0 5 10 15 20 25 30 35 40 45 50Time (s)

024

minus4minus2

times10minus6

1(m

)

(b) Observed

0 5 10 15 20 25 30 35 40 45 50

0

5

minus5

times10minus5

2(m

s)

Time (s)

(c) Observed


0 5 10

0

5

10

Real

Imag

e

minus10minus10

minus5

minus5





Computer


SEED-DEC2812

Beam


Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment



1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)


119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)



Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment




2)



















Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1199041= minusℎ119901

1205972120601

1205971199092119902 = 119861119902 (6)



119901



119864119896=

1

2

119902 (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911) 119902

+

1

2

119902 (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911) 119902 =

1

2

119898( 119902)2

(7)


119901are



119901is


119901is given as

119864119901=

1

2

int

119881

(11987811199041+ 11986431198633) 119889119881 +

1

2

119902int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119902

=

1

2

int

119881

[(119888119863

111199041minus ℎ311198633) 1199041+ (minusℎ

311199041+ 120573119909

331198633)1198633] 119889119881

+

1

2

(int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092)

2

119889119909)1199022

=

1

2

1198961199021199021199022+ 119876119896119876119902119902 +

1

2

119896119876119876

1198762

(8)






120575119882 = int

119897119887

0

119891120575119906119911119889119909 minus int

119897119887

0

119888119887119911120575119906119911119889119909 + int

119909119901+119897119887

119909119901

119881

ℎ119901

120575119876119889119909

= (int

119897119887

0

119891120601119889119909)120575119902 minus (int

119897119887

0

1198881198871206012119889119909) 119902120575119902

+ (int

119909119901+119897119901

119909119901

119881

ℎ119901

119889119909)120575119876

= 120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876

(9)




120575119869 (119902 120601) = 120575int

119905119891

1199050

(119871 (119902 119876 119905) + 119882) 119889119905

= 120575int

119905119891

1199050

(119864119896minus 119864119901+119882)119889119905 = 0

(10)


119898 119902120575119902 + 119896119902119902119902120575119902 + 119896

119876119902119902120575119876 + 119896

119876119902119876120575119902

+ 119896119876119876

119876120575119876 minus (120572119891120575119902 minus 119888 119902120575119902 + 120574119881120575119876)

= (119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 minus 120572119891) 120575119902

+ (119896119876119902119902 + 119896119876119876

119876 minus 120574119881) 120575119876 = 0

(11)


119898 119902 + 119888 119902 + 119896119902119902119902 + 119896119876119902119876 = 120572119891

119896119876119902119902 + 119896119876119876

119876 = 120574119881

(12)




119906119911= 1198971198871199041= 119861 (119909

119891) 119897119887119902 (13)


119911= minus2120585120596

119911minus 1205962119906119911+ 119887119881 (14)

where 120596 = radic(119896119902119902

minus 1198962

119876119902119896119876119876

)119898 120585 = 1198882119898120596 and 119887 =

(120574119896119876119902119897119887119861(119909119891))119896119876119876











119887= 004m ℎ

119887= 00032m

120588119887= 784 times 10

3 kgm3119884119887= 20 times 10

3 Nm2119888119887= 001


119901= 0034m ℎ

119901= 000025m

120588119901= 78 times 10

3 kgm3119884119901= 21 times 10

3 Nm211988931


119888119863

11= 1064 times 10

10 Nm2ℎ31


119901= 0085m



11198961

119896

1000 0001 003 23439 times 103 753672

0 05 1 15 20

05

1

15

2

Frequency (Hz)

Am

plitu

de (m

)


times10minus5




119911(119905) = 119865 (

119911(119905) 119906119911(119905)) + 119887119881 (15)

where 119865(119911(119905) 119906119911(119905)) = minus2120585120596


a state equation

1199011(119905) = 119906

119911+ 119899 (119905)

1(119905) = 119901

2(119905)

2(119905) = 119901

3(119905) + 119887119881

3(119905) = (119901)

(16)

where 1199011(119905) and 119901






V1(119905 + ℎ) = V

1(119905) + ℎV

2(119905)

V2(119905 + ℎ) = V

2(119905) + ℎ sdot fst (V

1(119905) V2(119905) 1199011(119905) 119903 ℎ

1)

(17)


119889 = 119903 sdot ℎ1

1198890= 119889 sdot ℎ

1

119890 = V1minus 1199011+ ℎ1sdot V2

1198860= radic1198892+ 8119903 |119890|

1198861=

V2+

119890

ℎ1

|119890| le 1198890

V2+ sgn (119890) sdot

1198860minus 119889

2

|119890| gt 1198890

fst (V1 V2 119903 ℎ1) =

minus119903 sdot

1198861

119889

10038161003816100381610038161198861

1003816100381610038161003816le 119889


10038161003816100381610038161198861

1003816100381610038161003816gt 119889

(18)




V1(119905) 997888rarr 119901

1(119905) V

2(119905) 997888rarr 119901

2(119905) (19)





Target+

++

+

minus

minus

k

k1

Controller

Control voltage

1

2

Smart

Observer

fst(1 2 uz)uz




2(its





1in


119900

120585119900= 120585 + Δ (20)



119911= minus (2120585

119900120596119900119911+ 1205962119906119911) + 119887119881 (21)


119860119900= 120582119868 minus [

0 1

minus1205962

119900minus2120585119900120596119900

] (22)


119900 the input



119906119888= 119881 = minus (119896

119911+ 1198961119906119911) (23)


119911= minus (2120585120596

119911+ 1205962119906119911) minus 119887119896

119911minus 1198871198961119906119911 (24)


119860119898= 120582119868 minus [

0 1

minus1205962minus 1198871198961

minus2120585120596 minus 119887119896

] (25)

0 5 10 15 20 25 30 35 40 45 50

024

Time (s)

minus4minus2

times10minus6

p1

(m)

(a) Collected

0 5 10 15 20 25 30 35 40 45 50Time (s)

024

minus4minus2

times10minus6

1(m

)

(b) Observed

0 5 10 15 20 25 30 35 40 45 50

0

5

minus5

times10minus5

2(m

s)

Time (s)

(c) Observed


0 5 10

0

5

10

Real

Imag

e

minus10minus10

minus5

minus5





Computer


SEED-DEC2812

Beam


Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment



1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)


119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)



Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment




2)



















Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119887= 004m ℎ

119887= 00032m

120588119887= 784 times 10

3 kgm3119884119887= 20 times 10

3 Nm2119888119887= 001


119901= 0034m ℎ

119901= 000025m

120588119901= 78 times 10

3 kgm3119884119901= 21 times 10

3 Nm211988931


119888119863

11= 1064 times 10

10 Nm2ℎ31


119901= 0085m



11198961

119896

1000 0001 003 23439 times 103 753672

0 05 1 15 20

05

1

15

2

Frequency (Hz)

Am

plitu

de (m

)


times10minus5




119911(119905) = 119865 (

119911(119905) 119906119911(119905)) + 119887119881 (15)

where 119865(119911(119905) 119906119911(119905)) = minus2120585120596


a state equation

1199011(119905) = 119906

119911+ 119899 (119905)

1(119905) = 119901

2(119905)

2(119905) = 119901

3(119905) + 119887119881

3(119905) = (119901)

(16)

where 1199011(119905) and 119901






V1(119905 + ℎ) = V

1(119905) + ℎV

2(119905)

V2(119905 + ℎ) = V

2(119905) + ℎ sdot fst (V

1(119905) V2(119905) 1199011(119905) 119903 ℎ

1)

(17)


119889 = 119903 sdot ℎ1

1198890= 119889 sdot ℎ

1

119890 = V1minus 1199011+ ℎ1sdot V2

1198860= radic1198892+ 8119903 |119890|

1198861=

V2+

119890

ℎ1

|119890| le 1198890

V2+ sgn (119890) sdot

1198860minus 119889

2

|119890| gt 1198890

fst (V1 V2 119903 ℎ1) =

minus119903 sdot

1198861

119889

10038161003816100381610038161198861

1003816100381610038161003816le 119889


10038161003816100381610038161198861

1003816100381610038161003816gt 119889

(18)




V1(119905) 997888rarr 119901

1(119905) V

2(119905) 997888rarr 119901

2(119905) (19)





Target+

++

+

minus

minus

k

k1

Controller

Control voltage

1

2

Smart

Observer

fst(1 2 uz)uz




2(its





1in


119900

120585119900= 120585 + Δ (20)



119911= minus (2120585

119900120596119900119911+ 1205962119906119911) + 119887119881 (21)


119860119900= 120582119868 minus [

0 1

minus1205962

119900minus2120585119900120596119900

] (22)


119900 the input



119906119888= 119881 = minus (119896

119911+ 1198961119906119911) (23)


119911= minus (2120585120596

119911+ 1205962119906119911) minus 119887119896

119911minus 1198871198961119906119911 (24)


119860119898= 120582119868 minus [

0 1

minus1205962minus 1198871198961

minus2120585120596 minus 119887119896

] (25)

0 5 10 15 20 25 30 35 40 45 50

024

Time (s)

minus4minus2

times10minus6

p1

(m)

(a) Collected

0 5 10 15 20 25 30 35 40 45 50Time (s)

024

minus4minus2

times10minus6

1(m

)

(b) Observed

0 5 10 15 20 25 30 35 40 45 50

0

5

minus5

times10minus5

2(m

s)

Time (s)

(c) Observed


0 5 10

0

5

10

Real

Imag

e

minus10minus10

minus5

minus5





Computer


SEED-DEC2812

Beam


Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment



1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)


119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)



Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment




2)



















Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Target+

++

+

minus

minus

k

k1

Controller

Control voltage

1

2

Smart

Observer

fst(1 2 uz)uz




2(its





1in


119900

120585119900= 120585 + Δ (20)



119911= minus (2120585

119900120596119900119911+ 1205962119906119911) + 119887119881 (21)


119860119900= 120582119868 minus [

0 1

minus1205962

119900minus2120585119900120596119900

] (22)


119900 the input



119906119888= 119881 = minus (119896

119911+ 1198961119906119911) (23)


119911= minus (2120585120596

119911+ 1205962119906119911) minus 119887119896

119911minus 1198871198961119906119911 (24)


119860119898= 120582119868 minus [

0 1

minus1205962minus 1198871198961

minus2120585120596 minus 119887119896

] (25)

0 5 10 15 20 25 30 35 40 45 50

024

Time (s)

minus4minus2

times10minus6

p1

(m)

(a) Collected

0 5 10 15 20 25 30 35 40 45 50Time (s)

024

minus4minus2

times10minus6

1(m

)

(b) Observed

0 5 10 15 20 25 30 35 40 45 50

0

5

minus5

times10minus5

2(m

s)

Time (s)

(c) Observed


0 5 10

0

5

10

Real

Imag

e

minus10minus10

minus5

minus5





Computer


SEED-DEC2812

Beam


Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment



1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)


119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)



Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment




2)



















Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Computer


SEED-DEC2812

Beam


Time (s)

Con

trol v

olta

ge (V

)

0 5 10 15 20 25 30 35 40 45 50

0

100

minus100

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

100

Time (s)

Con

trol v

olta

ge (V

)

minus100

(b) Experiment



1003816100381610038161003816119860119900

1003816100381610038161003816=1003816100381610038161003816119860119898

1003816100381610038161003816 (26)


119896 = minus

2 (120596119900120585119900minus 120596120585)

119887

1198961=

120596119900minus 120596

119887

(27)



Time (s)

Disp

lace

men

t (m

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus6

(a) Simulation

0 5 10 15 20 25 30 35 40 45 50

0

5

Time (s)

Disp

lace

men

t (m

)

No controlControl

minus5

times10minus6

(b) Experiment




2)



















Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























Time (s)

Velo

city

(ms

)

0 5 10 15 20 25 30 35 40 45 50

0

5

No controlControl

minus5

times10minus5

(a) Simulation

0

5

Velo

city

(ms

)

Time (s)0 5 10 15 20 25 30 35 40 45 50

No controlControl

minus5

times10minus5

(b) Experiment


Frequency (Hz)0 05 1 15 2 25 3

005

115

2

Disp

lace

men

t (m

)

No controlControl

times10minus6

(a)

0

1

2

Frequency (Hz)

Velo

city

(ms

)

0 05 1 15 2 25 3

No controlControl

times10minus5

(b)






05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










05 1 15 2 25 30

0

1

2

Disp

lace

men

t (m

)

Frequency (Hz)

No controlControl

times10minus6

(a)

0 05 1 15 2 25 30

1

2

Frequency (Hz)

Velo

city

(ms

)

No controlControl

times10minus5

(b)



5 Conclusions


Appendix

Consider

119861 = minusℎ119901

1205972120601

120597119909

119898 = (int

ℎ119887

0

int

119897119887

0

1205881198871199081198871206012119889119909 119889119911)

+ (int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

1205881199011199081199011206012119889119909 119889119911)

119896119902119902

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

119888119863

11119908119861119879119861119889119909119889119911 + int

119897119887

0

119864119887119868119887(

1205972120601

1205971199092

)

2

119889119909

119896119876119902

= minusint

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

ℎ31

ℎ

119861 119889119909 119889119911

119896119876119876

= int

ℎ119887+ℎ119901

ℎ119887

int

119909119901+119897119901

119909119901

120573119909

33119908119890

1

1198602

119909

119889119909 119889119911

120572 = (int

119897119887

0

120601119889119909) 119888 = (int

119897119887

0

1206012119889119909) 119888

119887

120574 = (int

119909119901+119897119901

119909119901

1

ℎ119901

119889119909)

(A1)



Acknowledgment


References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article experimental verifications of vibration

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