research article semiactive vibration control using a

Research ArticleSemiactive Vibration Control Usinga Magnetorheological Damper and a MagnetorheologicalElastomer Based on the Bouc-Wen Model

Wei Zhu and Xiao-ting Rui

Institute of Launch Dynamics Nanjing University of Science and Technology Nanjing 210094 China

Correspondence should be addressed to Wei Zhu zhuweicqueducn

Received 10 April 2014 Revised 21 May 2014 Accepted 24 May 2014 Published 15 June 2014

Academic Editor Ahmet S Yigit

Copyright copy 2014 W Zhu and X-t RuiThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A vibration control system is put forward using a magnetorheological damper (MRD) and a magnetorheological elastomer (MRE)connected in series In order to model the hysteresis of the MRD a Bouc-Wenmodel and a corresponding parameter identificationmethod are developed for theMRDThe experimental results validate the proposed Bouc-Wenmodel that can predict the hystereticbehavior of theMRDaccuratelyThe role of theMRE is illustrated by an example of a single degree-of-freedom system A semiactivevibration control strategy of the proposed vibration control system is proposed To validate this new approach experiments areconducted and the results highlight significantly improved vibration reduction effect of the proposed vibration control system thanthe vibration control system only using the MRD

1 Introduction

Magnetorheological dampers (MRDs) hold promise forvibration control since their properties can be adjusted inreal time and unlike active devices they do not inject energyinto the system being controlled and have relatively lowpower requirementsTheMRDs usingMR fluids that exhibitcontrollable yield characteristics produces sizeable dampingforce for small input current Being an energy dissipationdevice that cannot add mechanical energy to the structuralsystem an MR damper is also very stable and fail safe MRfluid contains a suspension of iron particles in a carrier fluidsuch as oil [1]

The Bouc-Wen model [2] describes the hysteretic behav-ior of MR dampers except near small velocities This short-coming was rectified by Spencer et al [3] in their modifiedBouc-Wen model where in additional damping and stiffnesselements were used to model the low-velocity behaviorand the accumulator respectively and voltage dependentparameters were introduced Dominguez et al [4] developeda current-frequency-amplitude dependent Bouc-Wen modeland an identification method Other damper models includethe phase-transition model of Wang and Kamath [5] and

modified LuGre friction model of Jimenez and Alvarez-Icaza[6] and Sakai et al [7]

Various controller designs have been used with MRdampers Predicting the applied voltage that produces adesired damper force is difficult due to the noninvertibleforce-voltage dynamics Hence different voltage laws havebeen considered Xu and Shen [8] used bistate controlstrategies with a Bingham damper model on-off currentlaw and neural network response prediction Dyke et al [9]implemented acceleration feedback linear quadratic Gaus-sian (LQG) control using the modified Bouc-Wen modelto obtain the desired optimal damper force using measuredaccelerations displacements and damper force Based onthe classical sky-hook damping a novel semiactive controlstrategy well suited for use in drive systems is presented byFrey et al [10] Prabakar et al [11] applied a half car model forsimulating the semiactive suspension system They modeledthe parameters of a MRD by the modified Bouc-Wen modeland determined that they fit the hysteretic behavior andput forward optimal semiactive preview control Weber [12]presented a Bouc-Wen model-based control scheme whichallows tracking the desired control force in real-time with

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 405421 10 pageshttpdxdoiorg1011552014405421

2 Shock and Vibration

Spring

MRD

MRE

Mass 1

Mass 2

(a) Photograph

Controller

MRE

Spring+ MRD

m2 x2(t)

kMRE

dMRE

m1 x1(t)

k1c1 FMRD

x0(t)

(b) Schematic diagram

Figure 1 Vibration control system

magnetorheological (MR) dampers without feedback from aforce sensor

However the MRD can only change its damping andthe response time is generally slower than 20ms whichcould make the high-frequency performance of the vibrationcontrol system decrease [9]

Magnetorheological elastomers (MREs) like MR fluidsexploit magnetic forces between dispersed micron-sizedferromagnetic particles to produce a material with instanta-neously adjustable properties However in MR fluids the par-ticles are dispersed within a liquid and operate in a postyieldregime while in MREs the particles are part of a structuredelastomer matrix in a preyield regime [13] Rigbi did theearliest work with what could be considered MRE materialsbut dealt mainly with the magnetic properties of a strainedisotropic sample [14] Jolly after considerable experience withMR fluids developed an anisotropic MRE where sphericaliron particles were aligned by an external magnetic field intolong parallel chains within the curing rubber [15] MREscan be used to make up for the shortcomings of MRDs invibration control system due to the adjustable properties oftheir stiffness

The paper is organized as follows Section 1 contains abrief introduction literature review and aims and scope ofthe paper Section 2 describes a vibration control device usinga MRD and a MRE connected in series Section 3 describesthe Bocu-Wen model and parameter identification methodfor MRDs a MRE device is put forward in Section 4 asemiactive vibration control strategy is proposed in Section 5Section 6 presents experimental results and discussionsincluding comparisons with available results and Section 7contains the conclusions and future scope

2 Vibration Control System

Figure 1 shows a vibration control system using a MRD anda MRE connected in series In the design the MRD is used

to reduce the large-range and low-frequency vibration theMRE is used to reduce the small-range and high-frequencyvibration

The dynamic equation of the vibration control device canbe given by

Mx +Dx + Kx = Γ119891MRD + K0x0 (1)

where x = [

1199091

1199092] The system matrices are

M = [

119898

10

0 119898

2

] D = [

119889MRE minus119889MREminus119889MRE 119889MRE

]

K = [

119896

1+ 119896MRE minus119896MRE0 119896MRE

] Γ = [

minus1

0

]

K0= [

119896

1119888

1

0 0

] x0= [

119909

0

0

]

(2)

where 119891MRD is the force of the MRD 119896MRE and 119888MRE are thestiffness and viscous damping coefficient respectively

3 MRDs

31 Bouc-Wen Model of MRDs In order to control the forceof the MRD it is essential to propose a model for MRDsTheBouc-Wen model of MRDs is given by [1]

119891MRD = 119888 (119894) + 119896 (119894) (119909 minus 119909

0) + 120572 (119894) 119911

= 119860 minus 120573|119911|

119899minus 120574 || |119911|

119899minus1119911

(3)

where 119909 and are the damper displacement and velocity119911 is the Bouc-Wen hysteresis operator ldquosdotrdquo at the top ofvariables represents the first order derivative of the variableswith respect to time 119894 is the current applied to the MRD119896(119894) and 119888(119894) are the stiffness and damping function of theefficient current respectively 120572(119894) is function related to theMR material yield stress 119909

0is the initial displacement which

Shock and Vibration 3

can be measured 119860 120573 120574 and 119899 are the parameters of theBouc-Wen hysteresis operator

Let ℎ = 120572119911 ℎ is the hysteresis force Equation (3) can berewritten as

119891MRD = 119888 (119894) + 119896 (119894) (119909 minus 119909

0) + ℎ (4)

ℎ = 119860 (119894) minus 120573 (119894) |ℎ|

119899minus 120574 (119894) || |ℎ|

119899minus1ℎ (5)

In order to use the Bouc-Wen model given by (4) and(5) to simulate the hysteretic behaviour of the MRD thefunctions 119888(119894) 119896(119894) 119860(119894) 120573(119894) and 120574(119894) and the parameters 119909

0

and 119899 need to be identified

32 Parameter Identification Method The initial displace-ment 119909

0can be obtained by measuring the displacement of

the rod Let the current 119894 be a constant The functions 119888(119894)119896(119894) 119860(119894) 120573(119894) and 120574(119894) are the constants 119888 119896 119860 120573 and120574 respectively Consider the corresponding forces 119865

119909(119905) and

119865

119910(119905) from the MRD with two periodical displacements 119909(119905)

and 119910(119905) which are related by

119910 (119905) = 119909 (119905) + 119902 (6)

where 119902 is a constant We have

119865

119909(119905) = 119888 (119905) + 119896119909 (119905) + ℎ

119909(119905) (7)

119865

119910 (119905) = 119888

1 (119905) + 119896 [119909

1 (119905) + 119902] + ℎ

119910 (119905) (8)

where ℎ

119909(119905) and ℎ

119910(119905) are the hysteresis force According to

(5) and (6) ℎ119909(119905) = ℎ

119910(119905)

Consider a set of 119873

1points (119909

1 119865

1199091) (119909

119895 119865

119909119895)

(119909

1198731

119865

1199091198731

) in the hysteresis curve force 119865

119909(119905) against

displacement 119909(119905) determined by (7) and (119910

1 119865

1199101)

(119910

119895 119865

119910119895) (119910

1198731

119865

1199101198731


119910(119905)

against displacement 119910(119905) determined by (8) With the least-squares method the parameter 119896 is given by

119896 =

sum

1198731

119895=1(119865

119910119895minus 119865

119909119895)

119873

1119902

(9)

The Bouc-Wen hysteresis operator given by (5) possessesthe symmetrical characteristic [17 18]Therefore we have [19]

119865

119909119895(min) + 119865

119909119895(max) = 119888 (

119895(min) +

119895(max))

+ 119896 (119909

119895 (min) + 119909

119895 (max))

(10)

where 119909(min) and 119909(max) are the minimum and maximumvalue of the displacement 119909 in the 119895th (119895 = 1 2 119873

2)

period respectively 119865119909119895(min) and 119865

119909119895(max) are correspond-

ing force 119895(min) and

119895(max) are corresponding velocity

With the least-squares method the parameter 119888 can begiven by

119888 =

1

119873

2

1198732

sum

119895=1

119865

119909119895(min) + 119865

119909119895(max) minus 119896 (119909

119895(min) + 119909

119895(max))

119895(min) +

119895(max)

(11)

According to (6) we have

dℎ119909

d119909= 119860 minus [120574 + 120573 sgn (ℎ

119909)]

1003816

1003816

1003816

1003816

ℎ

119909

1003816

1003816

1003816

1003816

119899

(12)

where

sgn (119909) =

1 119909 ge 0

minus1 119909 lt 0

(13)

According to (4) the hysteresis operator can be written as

ℎ

119909= 119865

119909minus 119896119909 minus 119888 (14)

Let ℎ119909= 0 According to (14) we have

119865

119909minus 119896119909 minus 119888 = 0 (15)

Assume that (119909lowast 119865

lowast) is the solution of (15) According to (12)

and (14) we have

119860 =

dℎ119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909lowast

=

d119865119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=119906lowast

minus 119896V minus 119888

1

d119905 (16)

When (119905) gt 0 consider a set of points1199091 119909

119895 119909

119873(119895 =

1 2 119873

3) in 119909(119905) which ensures that the corresponding

hysteresis displacement ℎ1 ℎ

119895 ℎ

1198733is larger than zero

Then (12) can be rewritten as

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

= 119860 minus (120573 + 120574) ℎ

119899

1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

= 119860 minus (120573 + 120574) ℎ

119899

119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733

= 119860 minus (120573 + 120574) ℎ

119899

1198733

(17)

Let minus(120573 + 120574) gt 0 According to (17) we have

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

minus 119860) = 119899 ln ℎ

1+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) = 119899 ln ℎ

119895+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733

minus 119860) = 119899 ln ℎ

1198733

+ ln [minus (120573 + 120574)]

(18)


According to (18) and the least-squares method we have

119899 = (119873

3

1198733

sum

119895=1

ln ℎ

119895ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860)

minus

1198733

sum

119895=1

ln ℎ

119895

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860))

times (119873

3

1198733

sum

119895=1

(ln ℎ

119895)

2

minus (

1198733

sum

119895=1

ln ℎ

119895)

2

)

minus1

(19)

120573 + 120574

=

[

[

minus exp(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

(20)

When (119905) lt 0 consider a set of points 119909

1198733+1 119906

1198733+119895

119909

21198733

(119895 = 1 2 119873) in 119909(119905) which ensures that the cor-responding hysteresis displacement ℎ

119873+1 ℎ

119873+119895 ℎ

2119873

is larger than zero Then (12) can be rewritten as

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+1

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+119895

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

= 119860 minus (minus120573 + 120574) ℎ

119899

21198733

(21)

Let 120573 minus 120574 gt 0 According to (21) we have

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+1

minus 119860) = 119899 ln ℎ

1198733+1

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) = 119899 ln ℎ

1198733+119895

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

minus 119860) = 119899 ln ℎ

21198733

+ ln (120573 minus 120574)

(22)

With the least-squares method (22) can be rewritten as

120573 minus 120574

= exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

1198733+119895)

times (119873

3)

minus1]

]

(23)

According to (20) and (23) the parameters 120573 and 120574 aregiven by

120573 =

1

2

minus exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

120574 = minus

1

2

exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

(24)

According to (9) (11) (16) (19) and (24) under thesingle current the parameters 119896 119888 119860 120573 120574 and 119899 canbe identified if the periodical displacements 119909(119905) and 119910(119905)


Figure 2 Photograph of the test setup

and the corresponding forces 119909(119905) and 119910(119905) are knownApplying the various currents to theMRD the correspondingparameters 119896 119888 119860 120573 and 120574 can be obtained With the least-squares method the functions 119888(119894) 119896(119894) 119860(119894) 120573(119894) and 120574(119894)

can be identified

33 Modeling Results The MRD is subjected to sinusoidalexcitations on an electrohydraulic servo fatigue machine(type LFV 150 kN theW+BGmbH Switzerland) to validatethe Bouc-Wen model and the corresponding parameteridentification method The primary components of the testsetup are shown in Figure 1 The fatigue machine has its ownsoftware to collect the data from the data card and use themto plot force versus displacement and force versus velocitygraphs for each test A programmable power (type IT6122the ITECH Electronic Co Ltd) supply is used to feed currentto the MRDThe damper is fixed to the machine via grippersas shown in the Figure 2 The machine excites the damperrsquospiston rod sinusoidally while a load cell measures the forceon the damper and a linear variable displacement transducermeasures the displacement of the piston rod as well as therelative velocity Since the identification method uses thevalues of the derivatives at some points of the experimentaldata it is necessary to filter the data before applying theidentification algorithm To this end a second order filter ofthe form 120596

2

119899(119904

2+ 2120585120596

119899+ 120596

2

119899) is used with 120585 = 07 and

120596

119899= 40120596

119904 where 120596

119904= 2120587 is the frequency of the input

signalA comparison between the predicted responses and the

corresponding experimental data is provided in Figure 3TheBouc-Wen model predicts the force-displacement behaviorof the damper well and it possesses force-velocity behaviorthat also closely resembles the experimental data Thereforeit is reasonable to believe that the Bouc-Wen model and thecorresponding parameter identification method can predictthe hysteretic behavior of the MRD accurately

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500minus12 minus8 minus4 0 4 8 12

Displacement (mm)

ExperimentalBouc-Wen model

1A075A05A025A0A

Forc

e (N

)

(a) Force versus displacement

1500

1000

500

0

minus500

minus1000

minus1500

minus2000minus80 minus60 minus40 minus20 0 20 40 60 80

Velocity (mms)

1A075A05A025A0A

Forc

e (N

)


(b) Force versus velocity

Figure 3 Comparison between the predicted and experimentallyobtained responses for the Bouc-Wenmodel under a 1Hz sinusoidalexcitation

4 MRE Device

The MRE device which is composed of two MREs coil andtwo magnetic conductors is shown in Figure 4 The size ofthe MRE device can be given by

119903

1= 35mm 119903

2= 127mm

119903

3= 177mm 119903

4= 215mm 119897 = 3mm

(25)

According to (25) 11990322minus119903

2

1= 119903

2

4minus119903

2

3Therefore the areas of

twoMREs are equal which canmake themagnetic inductionintensity of two MREs be consistent

So the total area of MREs can be given by

119860 = 120587 (119903

2

2minus 119903

2

1) + 120587 (119903

2

4minus 119903

2

3) = 936 times 10

minus4m2 (26)

Themiddle hole with screw can play the roles of fixed andlimited displacement

The average values of tensioncompression modulus andloss factor with current at different loading frequencies (1Hz


Magneticconductor MRE Magnetic

circuit Coil

l

Hole

r4r3r2

r1

(a) Structure (b) Photograph

Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)

Youngs modulusLoss modulusComplex modulus

(a) Tensioncompression modulus

2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor

Figure 5 Average values of the tensioncompression modulus and loss factor with the various current at different loading frequencies (1Hz10Hz 20Hz and 30Hz)

10Hz 20Hz and 30Hz) are shown in Figure 5 119864 representsthe complex tensioncompressive modulus which can beexpressed as

119864 = 119864

1015840+ 119894119864

10158401015840 (27)

where 1198641015840 is the storage modulus and 119864

10158401015840 is the loss modulusThe loss factor 120578 can be expressed as

120578 =

119864

10158401015840

119864

1015840

(28)

From Figure 5 we have

119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)

where 119864min and 119864max are the minimum and maximum valueof the storage modulus respectively 119896min and 119896max are theminimum and maximum value of the stiffness respectively

The stiffness and damping of theMREdevice can be givenby

119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)

where 119870(119894) is a stiffness function of the efficient current 119894Observing Figure 5 we have

119870 (119894) = minus01391119894

2+ 07649119894 (31)

Figure 6 shows a single-DOF mass-damper-spring semi-active vibration control system composed of the MRE deviceand a mass Let mass = 5 kg The bode diagram of the system


MRE

m y(t)

kMRE

dMRE

x(t)

Figure 6 Single-DOF mass-damper-spring semiactive vibrationcontrol system

102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)

Figure 7 Bode diagram of the system shown in Figure 5 with thevarious applied current

with the applied various current is shown in Figure 7 FromFigure 7 the vibration characteristics of the system can bechanged by the various currents Therefore the MRE devicecan reduce vibration when the semiactive vibration control isproposed

5 Semiactive Vibration Control Strategy

According to (1) the state-space equation of the system canbe given by

X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin

Figure 8 Sky-hook control model [16]

where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2

minusMminus1119904K minusMminus1

119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)

The sky-hook control [10 20 21] is widely used semiactivevibration control The sky-hook control model of the vibra-tion control system is shown in Figure 8 From Figure 8 andaccording to (32) we have

119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)

where 119888sky1 and 119888sky2 are the sky-hook damped coefficients


Consider the actual dynamic responses of the system canbe obtained by the sensors The Bouc-Wen model of theMRD is computed in real time for the constant currents 119894 =0 01 119894max119863A for the actual displacement and velocityThe corresponding estimated forces of the MRD can beobtained theoretically by calculation Based on the estimatedforces and the desired control force 119891MRD the control current119894control is derived by piecewise linear interpolation [12 22]Therefore (34) can be rewritten as

119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)

where 119894MRD is the actual control current of the MRDAccording to (30) (31) and (35) the actual control

current of the MRE can be expressed by

119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

minus1(119896) is the inverse function of 119870(119894) and its value

range is [0 119894max119864] 119894max119864 is the maximum control current ofthe MRE

According to (36) and (37) the actual control currents ofthe MRD and the MRE can be obtained

6 Experimental Implementation

In order to experimentally validate the proposed vibrationcontrol system and the semiactive vibration control strategythe schematic and photograph of the experimental setup areshown in Figures 9(a) and 9(b) respectively According toFigure 9 the experimental setup is composed of the proposedvibration control system accelerometers (type CA-YD-109B range 0ndash50ms2 linearity 998) charge amplifier(type 5018 Kistler Corporation) current source (linearity992) vibration table (type MPA407G334A force range0ndash6000 kgf frequency range 0ndash2500Hz ETS Solutions Ltd)DSP (type MS320F2812 TI Corporation) 12 bit AD 16 bitDA and data acquisition

The acceleration responses of the mass 2 of the vibrationcontrol system with different control methods under a 10Hz

Table 1 Acceleration responses of the mass 2

Vibration excitation Control method |119886|max (ms2) E

10HzUncontrolled 1013 534Control MRD 321 256

Control MRD +MRE 267 169



NonperiodicalUncontrolled 296 1025Control MRD 129 483


sinusoidal 40Hz sinusoidal and nonperiodic accelerationexcitations are shown in Figure 10 The uncontrolled methodmeans that the input currents of both the MRD and theMREare zero

The whole evaluation vibration acceleration response isdefined as

119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)

where 119886(120591) is the acceleration of the mass 2 in the time 120591According to (38) and Figure 10 responses of the vibra-

tion control system with different control method are listedin Table 1 The acceleration responses of mass 2 droppedsignificantly with the semiactive control which indicates thevibration control system with the semiactive control strategyis very effective in reducing the vibration Interestingly thecontrol MRD can effectively reduce the peak accelerationresponses but inspire some of the high-frequency vibrationsTheMRE can not onlymarginally reduce the amplitude of theacceleration responses but can also play an important role inhigh-frequency vibration reduction Therefore MREs can beused to make up for the shortcomings of MRDs in vibrationcontrol system which is consistent with the design idea inSection 2

7 Conclusions and Future Scope

The vibration control systemwas put forward using theMRDand the MRE connected in series In order to modeling thehysteresis of the MRD the Bouc-Wen model and the corre-sponding parameter identification method were developedfor the MRD The role of the MRE was illustrated by anexample of a single degree-of-freedom system the semiactivevibration control strategy of the proposed vibration controlsystem was proposed To validate this new approach exper-iments were conducted The following conclusions can bedrawn

(1) The experiments results validate the proposed Bouc-Wen model and the corresponding parameter identi-ficationmethod can predict the hysteretic behavior ofthe MRD accurately


Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA

CurrentsourceAccelerometer

AD

Spring + MRD

(a) Schematic diagram (b) Photograph

Figure 9 Experimental setup

20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz

UncontrolledControl MRDControl MRD + MRE

Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical

Figure 10 Acceleration responses of the mass 2 with different control

(2) The vibration control system with the semiactivecontrol strategy is very effective in reducing thevibration

(3) The MRD can reduce the large-range and low-frequency vibration and the MRE can reduce thesmall-range and high-frequency vibration

According to the above conclusions the future workinvolves

(1) researching their applications such as suspensionsystem

(2) researching the vibration control system using theMRD and the MRE connected in parallel

Conflict of Interests

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


Acknowledgment

The authors wish to acknowledge the financial support byNatural Science Foundation of China (NSFC Grant no61304137)

References

[1] N K Chandiramani and S P Purohit ldquoSemi-active controlusing magnetorhelogical dampers with output feedback anddistributed sensingrdquo Shock andVibration vol 19 no 6 pp 1427ndash1443 2012

[2] Y-K Wen ldquoMethod for random vibration of hysteretic sys-temsrdquo ASCE Journal of Engineering Mechanics Divison vol 102no 2 pp 249ndash263 1976

[3] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[4] A Dominguez R Sedaghati and I Stiharu ldquoA new dynamichysteresis model for magnetorheological dampersrdquo SmartMaterials and Structures vol 15 no 5 pp 1179ndash1189 2006

[5] L X Wang and H Kamath ldquoModelling hysteretic behaviour inmagnetorheological fluids and dampers using phase-transitiontheoryrdquo Smart Materials and Structures vol 15 no 6 pp 1725ndash1733 2006

[6] R Jimenez and L Alvarez-Icaza ldquoLuGre friction model fora magnetorheological damperrdquo Structural Control and HealthMonitoring vol 12 no 1 pp 91ndash116 2005

[7] C Sakai H Ohmori and A Sano ldquoModeling of MR damperwith hysteresis for adaptive vibration controlrdquo in Proceedings ofthe 42nd IEEE Conference on Decision and Control pp 3840ndash3845 Maui Hawaii USA December 2003

[8] Z-D Xu and Y-P Shen ldquoIntelligent bi-state control for thestructure with magnetorheological dampersrdquo Journal of Intel-ligent Material Systems and Structures vol 14 no 1 pp 35ndash422003

[9] S J Dyke B F Spencer Jr M K Sain and J D CarlsonldquoModeling and control of magnetorheological dampers forseismic response reductionrdquo Smart Materials and Structuresvol 5 no 5 pp 565ndash575 1996

[10] S Frey K Groh and A Verl ldquoSemi-active damping of drivesystemsrdquo JVCJournal of Vibration and Control vol 19 no 5pp 742ndash754 2013

[11] R S Prabakar C Sujatha and S Narayanan ldquoOptimal semi-active preview control response of a half car vehicle model withmagnetorheological damperrdquo Journal of Sound and Vibrationvol 326 no 3-5 pp 400ndash420 2009

[12] F Weber ldquoBouc-Wen model-based real-time force trackingscheme for MR dampersrdquo Smart Materials and Structures vol22 no 4 Article ID 045012 12 pages 2013

[13] G Y Zhou and F Weber ldquoComplex shear modulus of amagnetorheological elastomerrdquo Smart Materials and Structuresvol 13 no 5 pp 1203ndash1210 2004

[14] Z Rigbi and L Jilken ldquoThe response of an elastomer filled withsoft ferrite to mechanical and magnetic influencesrdquo Journal ofMagnetism and Magnetic Materials vol 37 no 3 pp 267ndash2761983

[15] M R Jolly J D Carlson and B C Munoz ldquoA model of thebehaviour of magnetorheological materialsrdquo Smart Materialsand Structures vol 5 no 5 pp 607ndash614 1996

[16] A Preuont Vibration Control of Active Structures KluwerAcademic Publishers 2002

[17] W Zhu and D-H Wang ldquoNon-symmetrical Bouc-Wen modelfor piezoelectric ceramic actuatorsrdquo Sensors and Actuators APhysical vol 181 pp 51ndash60 2012

[18] C-J Lin and S-R Yang ldquoPrecise positioning of piezo-actuatedstages using hysteresis-observer based controlrdquo Mechatronicsvol 16 no 7 pp 417ndash426 2006

[19] A Rodrıguez N Iwata F Ikhouane and J Rodellar ldquoModelidentification of a large-scalemagnetorheological fluid damperrdquoSmart Materials and Structures vol 18 no 1 Article ID 0150102009

[20] C Collette and A Preumont ldquoHigh frequency energy transferin semi-active suspensionrdquo Journal of Sound and Vibration vol329 no 22 pp 4604ndash4616 2010

[21] C Spelta S M Savaresi F Codeca M Montiglio and MIeluzzi ldquoSmart-bogie semi-active lateral control of railwayvehiclesrdquo Asian Journal of Control vol 14 no 4 pp 875ndash8902012

[22] F Weber ldquoSemi-active vibration absorber based on real-timecontrolledMR damperrdquoMechanical Systems and Signal Process-ing vol 46 pp 272ndash288 2014

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



Spring

MRD

MRE

Mass 1

Mass 2

(a) Photograph

Controller

MRE

Spring+ MRD

m2 x2(t)

kMRE

dMRE

m1 x1(t)

k1c1 FMRD

x0(t)

(b) Schematic diagram

Figure 1 Vibration control system

magnetorheological (MR) dampers without feedback from aforce sensor

However the MRD can only change its damping andthe response time is generally slower than 20ms whichcould make the high-frequency performance of the vibrationcontrol system decrease [9]

Magnetorheological elastomers (MREs) like MR fluidsexploit magnetic forces between dispersed micron-sizedferromagnetic particles to produce a material with instanta-neously adjustable properties However in MR fluids the par-ticles are dispersed within a liquid and operate in a postyieldregime while in MREs the particles are part of a structuredelastomer matrix in a preyield regime [13] Rigbi did theearliest work with what could be considered MRE materialsbut dealt mainly with the magnetic properties of a strainedisotropic sample [14] Jolly after considerable experience withMR fluids developed an anisotropic MRE where sphericaliron particles were aligned by an external magnetic field intolong parallel chains within the curing rubber [15] MREscan be used to make up for the shortcomings of MRDs invibration control system due to the adjustable properties oftheir stiffness

The paper is organized as follows Section 1 contains abrief introduction literature review and aims and scope ofthe paper Section 2 describes a vibration control device usinga MRD and a MRE connected in series Section 3 describesthe Bocu-Wen model and parameter identification methodfor MRDs a MRE device is put forward in Section 4 asemiactive vibration control strategy is proposed in Section 5Section 6 presents experimental results and discussionsincluding comparisons with available results and Section 7contains the conclusions and future scope

2 Vibration Control System

Figure 1 shows a vibration control system using a MRD anda MRE connected in series In the design the MRD is used

to reduce the large-range and low-frequency vibration theMRE is used to reduce the small-range and high-frequencyvibration

The dynamic equation of the vibration control device canbe given by

Mx +Dx + Kx = Γ119891MRD + K0x0 (1)

where x = [

1199091

1199092] The system matrices are

M = [

119898

10

0 119898

2

] D = [

119889MRE minus119889MREminus119889MRE 119889MRE

]

K = [

119896

1+ 119896MRE minus119896MRE0 119896MRE

] Γ = [

minus1

0

]

K0= [

119896

1119888

1

0 0

] x0= [

119909

0

0

]

(2)

where 119891MRD is the force of the MRD 119896MRE and 119888MRE are thestiffness and viscous damping coefficient respectively

3 MRDs

31 Bouc-Wen Model of MRDs In order to control the forceof the MRD it is essential to propose a model for MRDsTheBouc-Wen model of MRDs is given by [1]

119891MRD = 119888 (119894) + 119896 (119894) (119909 minus 119909

0) + 120572 (119894) 119911

= 119860 minus 120573|119911|

119899minus 120574 || |119911|

119899minus1119911

(3)

where 119909 and are the damper displacement and velocity119911 is the Bouc-Wen hysteresis operator ldquosdotrdquo at the top ofvariables represents the first order derivative of the variableswith respect to time 119894 is the current applied to the MRD119896(119894) and 119888(119894) are the stiffness and damping function of theefficient current respectively 120572(119894) is function related to theMR material yield stress 119909

0is the initial displacement which




119891MRD = 119888 (119894) + 119896 (119894) (119909 minus 119909

0) + ℎ (4)

ℎ = 119860 (119894) minus 120573 (119894) |ℎ|

119899minus 120574 (119894) || |ℎ|

119899minus1ℎ (5)


0





119909(119905) and

119865



119910 (119905) = 119909 (119905) + 119902 (6)


119865

119909(119905) = 119888 (119905) + 119896119909 (119905) + ℎ

119909(119905) (7)

119865

119910 (119905) = 119888

1 (119905) + 119896 [119909

1 (119905) + 119902] + ℎ

119910 (119905) (8)

where ℎ

119909(119905) and ℎ


(5) and (6) ℎ119909(119905) = ℎ

119910(119905)


1points (119909

1 119865

1199091) (119909

119895 119865

119909119895)

(119909

1198731

119865

1199091198731


119909(119905) against


1 119865

1199101)

(119910

119895 119865

119910119895) (119910

1198731

119865

1199101198731


119910(119905)


119896 =

sum

1198731

119895=1(119865

119910119895minus 119865

119909119895)

119873

1119902

(9)


119865

119909119895(min) + 119865

119909119895(max) = 119888 (

119895(min) +

119895(max))

+ 119896 (119909

119895 (min) + 119909

119895 (max))

(10)


2)






119888 =

1

119873

2

1198732

sum

119895=1

119865

119909119895(min) + 119865

119909119895(max) minus 119896 (119909

119895(min) + 119909

119895(max))

119895(min) +

119895(max)

(11)


dℎ119909

d119909= 119860 minus [120574 + 120573 sgn (ℎ

119909)]

1003816

1003816

1003816

1003816

ℎ

119909

1003816

1003816

1003816

1003816

119899

(12)

where

sgn (119909) =

1 119909 ge 0

minus1 119909 lt 0

(13)


ℎ

119909= 119865

119909minus 119896119909 minus 119888 (14)


119865

119909minus 119896119909 minus 119888 = 0 (15)



and (14) we have

119860 =

dℎ119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909lowast

=

d119865119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=119906lowast


1

d119905 (16)


119895 119909

119873(119895 =

1 2 119873



119895 ℎ



dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

= 119860 minus (120573 + 120574) ℎ

119899

1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

= 119860 minus (120573 + 120574) ℎ

119899

119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733

= 119860 minus (120573 + 120574) ℎ

119899

1198733

(17)


ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

minus 119860) = 119899 ln ℎ

1+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) = 119899 ln ℎ

119895+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733

minus 119860) = 119899 ln ℎ

1198733

+ ln [minus (120573 + 120574)]

(18)



119899 = (119873

3

1198733

sum

119895=1

ln ℎ

119895ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860)

minus

1198733

sum

119895=1

ln ℎ

119895

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860))

times (119873

3

1198733

sum

119895=1

(ln ℎ

119895)

2

minus (

1198733

sum

119895=1

ln ℎ

119895)

2

)

minus1

(19)

120573 + 120574

=

[

[

minus exp(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

(20)


1198733+1 119906

1198733+119895

119909

21198733


119873+1 ℎ

119873+119895 ℎ

2119873


dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+1

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+119895

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

= 119860 minus (minus120573 + 120574) ℎ

119899

21198733

(21)


ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+1

minus 119860) = 119899 ln ℎ

1198733+1

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) = 119899 ln ℎ

1198733+119895

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

minus 119860) = 119899 ln ℎ

21198733

+ ln (120573 minus 120574)

(22)


120573 minus 120574

= exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

1198733+119895)

times (119873

3)

minus1]

]

(23)


120573 =

1

2

minus exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

120574 = minus

1

2

exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

(24)





can be identified


2

119899(119904

2+ 2120585120596

119899+ 120596

2

119899) is used with 120585 = 07 and

120596

119899= 40120596

119904 where 120596




2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000


Displacement (mm)


1A075A05A025A0A

Forc

e (N

)


1500

1000

500

0

minus500

minus1000

minus1500


Velocity (mms)

1A075A05A025A0A

Forc

e (N

)




4 MRE Device


119903

1= 35mm 119903

2= 127mm

119903

3= 177mm 119903

4= 215mm 119897 = 3mm

(25)


2

1= 119903

2

4minus119903

2




119860 = 120587 (119903

2

2minus 119903

2

1) + 120587 (119903

2

4minus 119903

2

3) = 936 times 10

minus4m2 (26)





circuit Coil

l

Hole

r4r3r2

r1


Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)



2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor



119864 = 119864

1015840+ 119894119864

10158401015840 (27)



120578 =

119864

10158401015840

119864

1015840

(28)


119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)



119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)


119870 (119894) = minus01391119894

2+ 07649119894 (31)



MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119891MRD = 119888 (119894) + 119896 (119894) (119909 minus 119909

0) + ℎ (4)

ℎ = 119860 (119894) minus 120573 (119894) |ℎ|

119899minus 120574 (119894) || |ℎ|

119899minus1ℎ (5)


0





119909(119905) and

119865



119910 (119905) = 119909 (119905) + 119902 (6)


119865

119909(119905) = 119888 (119905) + 119896119909 (119905) + ℎ

119909(119905) (7)

119865

119910 (119905) = 119888

1 (119905) + 119896 [119909

1 (119905) + 119902] + ℎ

119910 (119905) (8)

where ℎ

119909(119905) and ℎ


(5) and (6) ℎ119909(119905) = ℎ

119910(119905)


1points (119909

1 119865

1199091) (119909

119895 119865

119909119895)

(119909

1198731

119865

1199091198731


119909(119905) against


1 119865

1199101)

(119910

119895 119865

119910119895) (119910

1198731

119865

1199101198731


119910(119905)


119896 =

sum

1198731

119895=1(119865

119910119895minus 119865

119909119895)

119873

1119902

(9)


119865

119909119895(min) + 119865

119909119895(max) = 119888 (

119895(min) +

119895(max))

+ 119896 (119909

119895 (min) + 119909

119895 (max))

(10)


2)






119888 =

1

119873

2

1198732

sum

119895=1

119865

119909119895(min) + 119865

119909119895(max) minus 119896 (119909

119895(min) + 119909

119895(max))

119895(min) +

119895(max)

(11)


dℎ119909

d119909= 119860 minus [120574 + 120573 sgn (ℎ

119909)]

1003816

1003816

1003816

1003816

ℎ

119909

1003816

1003816

1003816

1003816

119899

(12)

where

sgn (119909) =

1 119909 ge 0

minus1 119909 lt 0

(13)


ℎ

119909= 119865

119909minus 119896119909 minus 119888 (14)


119865

119909minus 119896119909 minus 119888 = 0 (15)



and (14) we have

119860 =

dℎ119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909lowast

=

d119865119909

d119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=119906lowast


1

d119905 (16)


119895 119909

119873(119895 =

1 2 119873



119895 ℎ



dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

= 119860 minus (120573 + 120574) ℎ

119899

1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

= 119860 minus (120573 + 120574) ℎ

119899

119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733

= 119860 minus (120573 + 120574) ℎ

119899

1198733

(17)


ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091

minus 119860) = 119899 ln ℎ

1+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) = 119899 ln ℎ

119895+ ln [minus (120573 + 120574)]

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733

minus 119860) = 119899 ln ℎ

1198733

+ ln [minus (120573 + 120574)]

(18)



119899 = (119873

3

1198733

sum

119895=1

ln ℎ

119895ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860)

minus

1198733

sum

119895=1

ln ℎ

119895

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860))

times (119873

3

1198733

sum

119895=1

(ln ℎ

119895)

2

minus (

1198733

sum

119895=1

ln ℎ

119895)

2

)

minus1

(19)

120573 + 120574

=

[

[

minus exp(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

(20)


1198733+1 119906

1198733+119895

119909

21198733


119873+1 ℎ

119873+119895 ℎ

2119873


dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+1

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+119895

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

= 119860 minus (minus120573 + 120574) ℎ

119899

21198733

(21)


ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+1

minus 119860) = 119899 ln ℎ

1198733+1

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) = 119899 ln ℎ

1198733+119895

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

minus 119860) = 119899 ln ℎ

21198733

+ ln (120573 minus 120574)

(22)


120573 minus 120574

= exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

1198733+119895)

times (119873

3)

minus1]

]

(23)


120573 =

1

2

minus exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

120574 = minus

1

2

exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

(24)





can be identified


2

119899(119904

2+ 2120585120596

119899+ 120596

2

119899) is used with 120585 = 07 and

120596

119899= 40120596

119904 where 120596




2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000


Displacement (mm)


1A075A05A025A0A

Forc

e (N

)


1500

1000

500

0

minus500

minus1000

minus1500


Velocity (mms)

1A075A05A025A0A

Forc

e (N

)




4 MRE Device


119903

1= 35mm 119903

2= 127mm

119903

3= 177mm 119903

4= 215mm 119897 = 3mm

(25)


2

1= 119903

2

4minus119903

2




119860 = 120587 (119903

2

2minus 119903

2

1) + 120587 (119903

2

4minus 119903

2

3) = 936 times 10

minus4m2 (26)





circuit Coil

l

Hole

r4r3r2

r1


Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)



2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor



119864 = 119864

1015840+ 119894119864

10158401015840 (27)



120578 =

119864

10158401015840

119864

1015840

(28)


119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)



119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)


119870 (119894) = minus01391119894

2+ 07649119894 (31)



MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119899 = (119873

3

1198733

sum

119895=1

ln ℎ

119895ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860)

minus

1198733

sum

119895=1

ln ℎ

119895

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860))

times (119873

3

1198733

sum

119895=1

(ln ℎ

119895)

2

minus (

1198733

sum

119895=1

ln ℎ

119895)

2

)

minus1

(19)

120573 + 120574

=

[

[

minus exp(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

(20)


1198733+1 119906

1198733+119895

119909

21198733


119873+1 ℎ

119873+119895 ℎ

2119873


dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+1

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+1

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119906=1199061198733+119895

= 119860 minus (minus120573 + 120574) ℎ

119899

1198733+119895

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

= 119860 minus (minus120573 + 120574) ℎ

119899

21198733

(21)


ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+1

minus 119860) = 119899 ln ℎ

1198733+1

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) = 119899 ln ℎ

1198733+119895

+ ln (120573 minus 120574)

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=11990921198733

minus 119860) = 119899 ln ℎ

21198733

+ ln (120573 minus 120574)

(22)


120573 minus 120574

= exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

1198733+119895)

times (119873

3)

minus1]

]

(23)


120573 =

1

2

minus exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

120574 = minus

1

2

exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=119909119895

minus 119860) minus 119899

1198733

sum

119895=1

ln ℎ

119895)

times (119873

3)

minus1]

+ exp[

[

(

1198733

sum

119895=1

ln(

dℎd119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1199091198733+119895

minus 119860)

minus119899

1198733

sum

119895=1

ln ℎ

1198733+119895)(119873

3)

minus1]

]

(24)





can be identified


2

119899(119904

2+ 2120585120596

119899+ 120596

2

119899) is used with 120585 = 07 and

120596

119899= 40120596

119904 where 120596




2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000


Displacement (mm)


1A075A05A025A0A

Forc

e (N

)


1500

1000

500

0

minus500

minus1000

minus1500


Velocity (mms)

1A075A05A025A0A

Forc

e (N

)




4 MRE Device


119903

1= 35mm 119903

2= 127mm

119903

3= 177mm 119903

4= 215mm 119897 = 3mm

(25)


2

1= 119903

2

4minus119903

2




119860 = 120587 (119903

2

2minus 119903

2

1) + 120587 (119903

2

4minus 119903

2

3) = 936 times 10

minus4m2 (26)





circuit Coil

l

Hole

r4r3r2

r1


Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)



2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor



119864 = 119864

1015840+ 119894119864

10158401015840 (27)



120578 =

119864

10158401015840

119864

1015840

(28)


119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)



119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)


119870 (119894) = minus01391119894

2+ 07649119894 (31)



MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












can be identified


2

119899(119904

2+ 2120585120596

119899+ 120596

2

119899) is used with 120585 = 07 and

120596

119899= 40120596

119904 where 120596




2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000


Displacement (mm)


1A075A05A025A0A

Forc

e (N

)


1500

1000

500

0

minus500

minus1000

minus1500


Velocity (mms)

1A075A05A025A0A

Forc

e (N

)




4 MRE Device


119903

1= 35mm 119903

2= 127mm

119903

3= 177mm 119903

4= 215mm 119897 = 3mm

(25)


2

1= 119903

2

4minus119903

2




119860 = 120587 (119903

2

2minus 119903

2

1) + 120587 (119903

2

4minus 119903

2

3) = 936 times 10

minus4m2 (26)





circuit Coil

l

Hole

r4r3r2

r1


Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)



2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor



119864 = 119864

1015840+ 119894119864

10158401015840 (27)



120578 =

119864

10158401015840

119864

1015840

(28)


119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)



119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)


119870 (119894) = minus01391119894

2+ 07649119894 (31)



MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











circuit Coil

l

Hole

r4r3r2

r1


Figure 4 MRE device

25

2

15

1

05

02151050

Tens

ion

com

pres

sion

mod

ulus

E(M

Pa)

Current (A)



2151050

Current (A)

05

04

03

02

01

0

Loss

fact

or

(b) Loss factor



119864 = 119864

1015840+ 119894119864

10158401015840 (27)



120578 =

119864

10158401015840

119864

1015840

(28)


119896min = 119864min119860

119897

= 337 times 10

5Nm

119896max = 119864max119860

119897

= 624 times 10

5Nm

(29)



119896MRE = 119896min + 119870 (119894) 119889MRE =

120578119896MRE120596

(30)


119870 (119894) = minus01391119894

2+ 07649119894 (31)



MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










MRE

m y(t)

kMRE

dMRE

x(t)


102

100

10minus2

10minus4

100 101 102 103 104

100 101 102 103 104

Frequency (rads)

Frequency (rads)

0

minus50

minus100

minus150

minus200

0A2A

Am

plitu

de ra

tioPh

ase s

hift

(∘)





X = AX + B119891MRD + E1199090 Y = CX (32)

x2(t)

x1(t)

x0(t)

m2

m1

k1

csky2

csky1

kmin

dmin


where

X =

[

[

[

[

119909

1

119909

2

1

2

]

]

]

]

A =

[

[

02times2

I2


119904C]

]

B = [

02times1

Mminus1Γ] E =

[

[

0

0

Mminus1119904K0

]

]

C =

[

[

[

[

1

1

0

0

]

]

]

]

(33)


119891MRD

=

119865max

1(

1minus

0) ge 0 119865max lt 119888sky11

119888sky11

1(

1minus

0) ge 0 119865max ge 119888sky11

0

1(

1minus

0) lt 0

(34)

119896MRE

=

119896max

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1) 119896max

119888sky22

119909

2minus 119909

1

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1) 119896max

119896min

2(119909

2minus 119909

1) lt 0

(35)




119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119894MRD

=

119894max119863

1(

1minus

0) ge 0 119894control gt 119894max119863

119894control

1(

1minus

0) ge 0 119894control le 119894max119863

0

1(

1minus

0) lt 0

(36)



119894MRE

=

119894max119864

2(119909

2minus 119909

1) ge 0

119888sky22 gt (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

119870

minus1(

119888sky22

119909

2minus 119909

1

)

2(119909

2minus 119909

1) ge 0

119888sky22 le (119909

2minus 119909

1)

times (119896min + 119870 (119894maxE))

0

2(119909

2minus 119909

1) lt 0

(37)

where 119870

















119864 =

1

119905

radic

int

119905

0

119886

2(120591) d120591 (38)







Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Vibrationtable

Vibrationexcitation

Mass 1 Mass 2

AD

Accelerometer

MRE

Accelerometer

AD

DSP

DA

Currentsource

Dataacquisition

DA


AD

Spring + MRD



20

10

0

minus10

minus200 005 01 015 02 025 03 035 04 045 05

Acce

lera

tion

(ms2)

Time (s)

(a) 10Hz

Acce

lera

tion

(ms2) 30

20

10

0

minus10

minus20

minus300 002 004 006 008 01 012 014 016 018 02

Time (s)

(b) 40Hz


Acce

lera

tion

(ms2)

30

20

10

0

minus10

minus20

minus300 02 04 06 08 1 12 14 16 18 2

Time (s)

(c) Nonperiodical










Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










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 semiactive vibration control using a

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