review article modelling of hysteresis in vibration...

Review ArticleModelling of Hysteresis in Vibration Control Systems by meansof the Bouc-Wen Model

Chia-Ming Chang1 Salvatore Strano2 and Mario Terzo2

1Department of Civil Engineering National Taiwan University Taipei Taiwan2Department of Industrial Engineering University of Naples Federico II Naples Italy

Correspondence should be addressed to Mario Terzo mterzouninait

Received 24 July 2015 Revised 19 October 2015 Accepted 25 October 2015

Academic Editor Chao Tao

Copyright copy 2016 Chia-Ming Chang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The review presents developments concerning the modelling of vibration control systems with hysteresis In particular the reviewfocuses on applications of the Bouc-Wenmodel that describes accurate hysteretic behaviour in vibration control devicesThe reviewconsists of theoretical aspects of the Bouc-Wen model identification procedures and applications in vibration control

1 Introduction

Vibration control systems are adopted for the suppressionor at least the attenuation of undesirable vibrations thatcan affect systems and structures such as buildings vehiclesaircraft and bridges These systems often show nonlinearbehaviour due to variable material properties changeablegeometry and additional nonlinear devices resulting inhysteresis phenomena To predict system responses manyhysteretic models have been developed

Hysteresis is a nonlinear behaviour encountered in a widevariety of processes in which the relation between inputand output variables involves memory effects The detailedphysical modelling of these systems is an arduous task andthe models obtained are often too complex to be used in real-world applications [1] For this reason alternative models ofsystems with hysteresis have been proposed These modelscombine some physical understanding of the hysteretic sys-tem with some black-box modelling named ldquosemiphysicalrdquomodels

Over the years various semiphysical models of hysteresishave been proposed One of the most widely adopted ones isthe Bouc-Wen hysteresis model [2 3] The best feature of thismodel is its versatility that is through appropriate choices ofmodel parameters it can represent a wide variety of softeningor hardening smoothly varying or nearly bilinear hystereticbehaviours

The Bouc-Wen model has been extensively used in thecurrent literature to mathematically describe componentsand devices with hysteretic behavioursThus the objective ofthis literature review is to provide engineers and researcherswith an overview of the work that addresses the modellingof vibration control systems by means of the Bouc-WenmodelThis review is divided into threemajor parts Section 2focuses on the mathematical and physical properties of theBouc-Wen model whereas Section 3 focuses on identifica-tion of the Bouc-Wen model parameters Section 4 providesresults concerning several vibration control case studies inwhich the effectiveness of the Bouc-Wen model for theaccurate description of hysteresis is verified

2 Mathematical Properties ofthe Bouc-Wen Model

In this section a description of themathematical formulationof the Bouc-Wen model is presented

Consider as an example the equation of motion of asingle-degree-of-freedom (SDOF) mechanical system

119898 (119905) + 119888 (119905) + 119865 (119905) = 119891 (119905) (1)

where 119898 is the mass 119906(119905) is the displacement 119888 is the linearviscous coefficient 119865(119905) is the restoring force and 119891(119905) is the

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3424191 14 pageshttpdxdoiorg10115520163424191

2 Shock and Vibration

Elastic postyielding spring

Fel

akiu

Fel u

Hysteretic spring

Fhmax Fh

minusFhmaxu Fh u

(1a)ki

F u

F

Fy

ki =Fy

uy uuy

kf = aki

Figure 1 Bouc-Wen model [4]

excitation force the overdot indicates derivative with respectto time

The restoring force 119865(119905) based on the Bouc-Wen modelis

119865 (119905) = 119886

119865119910

119906119910

119906 (119905) + (1 minus 119886) 119865119910119911 (119905) (2)

where 119886 = 119896119891119896119894 is the ratio of postyield stiffness 119896119891

to preyield stiffness 119896119894 = 119865119910119906119910 119865119910 is the yield force119906119910 is the yield displacement and 119911(119905) is a nonobservabledimensionless hysteretic variable that obeys the followingnonlinear differential equation with zero initial condition(119911(0) = 0)

(119905) = (119905) 119860 minus [120574 + 120573 sgn ( (119905) 119911 (119905))] |119911 (119905)|119899 (3)

The coefficients 119860 120573 120574 and 119899 are dimensionless quantitiesthat control the behaviour of the model and sgn(sdot) is thesignum function For small values of the positive exponentialparameter 119899 the transition from elastic to postelastic branchis smooth whereas for large values the transition becomesabrupt approaching that of a bilinear model The parameters120573 and 120574 control the size and shape of the hysteretic loop Thenotation varies from paper to paper and very often the placesof 120573 and 120574 are exchanged

From (2) it follows that the restoring force 119865(119905) can bedivided into an elastic and a hysteretic part as follows

119865el(119905) = 119886

119865119910

119906119910

119906 (119905) = 119886119896119894119906 (119905) = 119896119891119906 (119905)

119865ℎ(119905) = (1 minus 119886) 119896119894119911 (119905)

(4)

Thus the model can be visualized as two springs connectedin parallel [4] (Figure 1)

The parameters of the Bouc-Wen model have the follow-ing criteria

119886 isin [0 1]

119896119894 gt 0

119896119891 gt 0

119888 gt 0

119860 gt 0

119899 gt 1

120573 gt 0

120574 isin [minus120573 120573]

(5)

In [23] it has been proved that the parameters of the Bouc-Wen model are functionally redundant and indeed multipleparameter vectors can produce an identical response undera given excitation This redundancy can be then removed byfixing parameter 119860 to unity [23]

Constantinou and Adnane [24] suggested the constraint119860(120573+120574) = 1 in order to reduce the total number of unknownparameters to six 120574 119899 119886 119865119910 119906119910 and 119888

In [25] an asymmetrical Bouc-Wen model has beenobtained adjusting the velocity as

(119905) larr997888 ( (119905) minus sgn (119906 (119905))) (6)

where 120583 is the scale factor for the adjustmentModification of the Bouc-Wen model presented in [26ndash

28] included strength stiffness and pinching degradationeffects by means of suitable degradation functions

(119905) =

ℎ (119911 (119905))

120578 (120576)

(119905) 119860 (120576)

minus ] (120576) [120573 sgn ( (119905)) |119911 (119905)|119899minus1

119911 (119905) + 120574 |119911 (119905)|119899]

(7)

where 120576 is the absorbed hysteretic energy and the functions](120576) 120578(120576) and ℎ(119911) are associated with strength stiffness and

Shock and Vibration 3

pinching degradation effects respectively ](120576)119860(120576) and 120578(120576)

are linearly increasing functions defined as

] (120576) = ]0 + 120575]120576 (119905)

119860 (120576) = 1198600 minus 120575119860120576 (119905)

120578 (120576) = 1205780 + 120575120578120576 (119905)

(8)

The pinching function ℎ(119911) is

ℎ (119911) = 1 minus 1205891 (120576) exp(minus

(119911 (119905) sgn () minus 119902119911119906)2

(1205892 (120576))2

) (9)

where

1205891 (120576) = (1 minus exp (minus119901120576 (119905))) 120589

1205892 (120576) = (1205950 + 120575120595120576 (119905)) (120582 + 1205891 (120576))

(10)

and 119911119906 is the ultimate value of 119911 given by

119911119906 =119899

radic

1

] (120573 + 120574)

(11)

The additional model parameters are 120575] gt 0 120575119860 gt 0 120575120578 gt

0 ]0 1198600 1205780 1205950 120575120595 120582 119901 and 120589 When 120575] = 0 120575120578 =

0 or ℎ(119911) = 1 respectively no strength degradation stiffnessdegradation or pinching effect is considered in the model

It is important to note that a Bouc-Wen model couldpresent a good match with the real data for a specificinput while it could not necessarily keep significant physicalproperties for different exciting inputs On the basis of thisconsideration the physical and mathematical properties ofthe Bouc-Wen model are comprehensively discussed in [1]and it has been found that if the system parameters respectthe constraints

119899 ge 1

119906119910 gt 0

119860 gt 0

120573 + 120574 gt 0

120573 minus 120574 ge 0

(12)

then the model is valid independently of the exciting inputWhen (12) are satisfied (3) can be expressed in a normalizedform

Defining the parameters

120588 =

119860

1199061199101199110

gt 0

120590 =

120573

120573 + 120574

ge

1

2

1199110 =119899

radic

119860

120573 + 120574

(13)

it follows

(119905)

= 120588 (119905) 1 + |119908 (119905)|119899120590 [1 minus sgn ( (119905) 119908 (119905)) minus

1

120590

]

(14)

with

119908 (119905) =

119911 (119905)

1199110

(15)

In [29] it is demonstrated that 119908(119905) is bounded in the range[minus1 1]

Substituting (14) and (15) in (2) it follows

119865 (119905) = 119896119891119906 (119905) + 119896119908119908 (119905) (16)

where

119896119908 = (1 minus 119886) 1198651199101199110 gt 0 (17)

Consequently the unknown parameters of the normalizedform of the Bouc-Wen model are 120588 120590 119899 119896119891 and 119896119908 with thefollowing constraints

120588 gt 0

120590 ge

1

2

119899 ge 1

119896119891 gt 0

119896119908 gt 0

(18)

The model parameters can be determined by system identifi-cation techniques using experimental input and output data

3 Bouc-Wen Model Parameter Identification

The identification of the Bouc-Wen model parameters isperformed by adopting an identification algorithm thatcompares model output signals and those measured for thesame input signals in order to determine the unknownmodelparameters Nonlinearity of the Bouc-Wen model introducesthe complexity for parameter identification Severalmethodsbased on different approaches have been then proposed Inthis section an overview of different identification algorithmsis presented

An iterative least-squares procedure based on amodifiedGauss-Newton approach was presented in [5] Identificationwas carried out to estimate the parameters of an extendedBouc-Wen model that accounts for strength and stiffnessdegradation in accordance with [26] The system identifica-tion procedure is shown in Figure 2

In [30] a vibration control system composed of a mag-netorheological damper in series with a magnetorheolog-ical elastomer was presented The Bouc-Wen model wasadopted to reproduce the hysteresis of the magnetorheologi-cal damper and the parameters were identified using a least-squares method


Read experimental data

Determine envelope pointsfrom max values for

each cycle

Fit smooth curvethrough envelopeDetermine

initial stiffness

Determineyield force displacement

Start modifiedGauss-Newton fit

Set up initialestimates for parameters

and modify 120581 120582as needed

Call hysteresismodel

Compare experimentaland simulated results

NoErrorwithin

tolerance

YesPlot force-deformation history

(a)

(b)

Set up new Si middot a

Figure 2 Flow diagram for complete identification process [5]

On-line parameter identification of the Bouc-Wen modelhas also drawn a lot of attention to researchers Studies in[31 32] presented an on-line identification method using aleast-squares adaptive law Another study in [33] employed anadaptive on-line identification methodology with a variabletrace method to adjust the adaptation gain matrix In [34] alinear parameterized estimatorwas established for the on-lineestimation of the hysteretic Bouc-Wen model with unknowncoefficients (including the parameter 119899) In [35] an adaptiveon-line identification algorithmwas proposed for parametricand nonparametric identification of structural models andwas applied to a generalized Bouc-Wen model The proposedidentification methodology was a recursive least-squaresalgorithm that required only acceleration measurements

In addition to the least-squares regression method agenetic based identification algorithm was proposed in [36]The reproduction procedure adopts the roulette wheel selec-tion and themethod of crossover and uniformmutation [37]

To account for asymmetric behaviour a modified Bouc-Wen model [6] was adopted for modelling a PZT actuatorA modified particle swarm optimization algorithm [38]was proposed in order to identify and optimize the modelparameters (see Figure 3) The fitness function in Figure 3 isevaluated as the root-mean-square error between the actualmeasured displacement and the model output

Start

(i) Determine the spatial dimension d(ii) Set the population size(iii) Initial position(iv) Initial velocity(v) Set maximum number of iteration G

Set i = 1

Calculate the hysteresismodel output using

individual i

i + 1

Evaluate the fitness function

f(xi) = radic 1

N

Nsuml=1

[y(l) minus y(l)]2

kg + 1

Yes

Noi lt size

Find individual best position pbestand find global position gbest

Update position and velocity using(9) and (10)

Yes

No

kg lt G

Get the optimal parameters ofhysteresis model

End

Part I

Part II

Part III

Figure 3 Flowchart of the identification hysteretic model parame-ters [6]

The particle swarm optimization was also adopted in [39]to identify the parameters of Bouc-Wen model In this studythe identified model was used to describe the dynamics ofa large-scale magnetorheological damper for seismic hazardmitigation

Gauss-Newton iterations were used as a method ofestimating the parameters of hysteretic system with slip onthe basis of input-output data [40] Reference [41] proposeda frequency domain parametric identification method ofnonlinear hysteretic isolators In [42 43] an identificationmethod for the normalized Bouc-Wen model was devel-oped Using the analytical description of the hysteresis loopdeveloped in [29] an algorithm was proposed along withits analytical proof It consisted in exciting the Bouc-Wenmodel with two periodic signals with a loading-unloadingshape (wave periodic) which gives rise asymptotically to ahysteretic periodic response The obtained two limit cycles


Initialization of food sourcesand the related nectar amounts

Update the new food positionfor the employed bees

Calculate nectar amounts

Determine a neighborfood source position for

the onlooker

All onlookersdistributed

Select a food source forthe onlooker bees

No

No

Yes

Yes

Save the best food source

Find abandoned food source

Determine new exhaustedfood source

Stop criteriasatisfied

Final food position

Figure 4 Flowchart of the parameter identification algorithmpresented in [7]

were then used as an input to exactly determine the unknownparameters

The limit cycle approach was also adopted in [10] toidentify the parameters of the Bouc-Wen model adopted toreproduce the hysteresis of a wire-cable vibration isolator In[44] the limit cycle approach was adopted to identify theparameters of a large-scale magnetorheological fluid dampermodel

Another optimization method based on artificial beecolony algorithm [7 45] was developed to determine theoptimum parameters of Bouc-Wen hysteretic systems Theproposed flowchart is shown in Figure 4

In [46 47] a constrained nonlinear optimization wasexploited in parametric identification In [48] the Bouc-Wen nonlinear hysteresis term was approximated by a powerseries expansion of suitable basis functions and then thecoefficients of the functions were determined using thestandard least-squares method

Bouc-Wen model parameters could be also identifiedwith procedures based on nonlinear filtering using forexample the extended Kalman filter (EKF) or the unscentedKalman filter (UKF) Yang and Ma [49] proposed a con-strained EKF with a global weighted iteration strategy whichwas effective in estimating all the parameters of the Bouc-Wen model of hysteresis Zhang et al [50] also applied theEKF for the identification of hysteretic systems that exhibit

degradation and pinching all the parameters of the BW-model were identified without problem

In [51ndash53] the UKF was used for the identification ofnondegrading and degrading hysteretic systems The iden-tification results show that the UKF is well suited for theidentification of complicated nonlinear systems and that thismethodology can yield accurate estimates of the parametersof the Bouc-Wen models The results also show that the UKFoutperforms the EKF with regard to computational efficiencyand robustness to measurement noise levels

4 Vibration Control System Modelling Usingthe Bouc-Wen Model

The main scope of the vibration control is the suppressionor at least the attenuation of undesirable vibrations thatcan affect systems and structures such as buildings vehiclesaircraft and bridges Vibration control is typically realizedusing passive semiactive or active [54ndash58] systems andconsiderable hysteretic behaviours can be found in each ofthese

Passive vibration control such as the passive base isola-tion is one solution that has proven effective for enhancingstructural performance against seismic events The mainconcept behind passive base isolation is to increase the struc-ture flexibility thus avoiding potentially dangerous seismicground motions [59ndash62] Base isolation bearings have beeninstalled in many buildings for seismic protection [63] how-ever large base displacements resulting from the increasedflexibility of the passive isolation system can potentiallyexceed the prescribed limit of structural designs under severeseismic excitations [64ndash66]

Semiactive vibration control consists of a passive isolationsystem combined with a controllable semiactive device [67ndash71] Semiactive vibration control is one control techniquethat consumes less power to change the features of theisolation system but no mechanical energy is introducedinto the structural system Differently from passive controltechniques semiactive control systems have higher variabilitydue to the different capabilities of energy dissipation from thecontrol devices when the power levels are changed such asvariable stiffness andor damping values

Active vibration control is another control techniquewhich uses the energy generated from the active controldevices supplied by means of an external power source toimprove the vibrational system performance The mitigationof the vibration phenomena is based on the employmentof suitable actuators that transmit mechanical energy to thestructure and in recent years a great number of studieshave been accomplished on active vibration control of flexiblestructures using piezoelectric actuators Among the severalkinds of actuation systems piezoelectric ceramic materialshave received much diffusion in the active vibration controlbecause of their mechanical simplicity small volume lightweight large useful bandwidth efficient conversion betweenelectrical energy andmechanical energy and easy integrationwith various metallic and composite structures [72]


Figure 5 Isolator prototype [8]

41 Passive Systems The Bouc-Wen model has been widelyadopted for the numerical modelling of passive hystereticcontrol devices In this section an overview of variouscomponents for different applications is presented

Passive seismic isolators aim to reduce the earthquakeinput energy to the structure to keep linear structural vibra-tion Many devices are strongly nonlinear showing differenthysteretic behaviours In this context the Bouc-Wen modelhas been used extensively for its intrinsic ability in describingthe behaviour of a wide range of real-world passive seismicisolators

In [8] a prototype seismic isolator composed of a wirerope spring and a ball transfer unit was proposed as shown inFigure 5 The nonlinear behaviour of the restoring force wasrepresented by the Bouc-Wen model

The Bouc-Wen model was used in [73] to model thedissipated energy by the wire rope isolators for seismicprotection of equipment in buildings In [9 10] modifiedBouc-Wen models were adopted to numerically reproducehardening behaviours [9] (see Figure 6) or asymmetricalhysteresis cycles [10] (see Figure 7) of wire rope isolators

A combined energy dissipation system was presented in[11] As shown in Figure 8 a lead rubber damper (LRD)and its parallel connection with an oil damper (OD) wereused in the braces of a structural frame The restoring forcecharacteristic of the LRD was then simulated by the Bouc-Wen hysteretic model

In [12] the Bouc-Wen model was employed to mathe-matically represent two different elastomeric seismic isola-tors (see Figure 9) characterized by different characteristicsbecause of the elastomeric layers and reinforcements (inthe following indicated with ldquoIUT ardquo and ldquoIUT brdquo resp)The experimental and simulated hysteresis cycles for IUT aand IUT b were demonstrated in Figures 10(a) and 10(b)respectively

In [74] the Bouc-Wen model was utilized to mathemati-cally model a frictional behaviour of Teflon sliding bearingsfor a base isolation application The study in [13] facilitatedthe Bouc-Wen model to describe the behaviour of hystereticdampers that interconnect two adjacent structures subjectedto seismic excitation as shown in Figure 11

Effectiveness of dissipative passive devices used as con-nections between two structures was investigated [75] Ananalytical model for the response of a reinforced concrete

minus20 minus10 0 10 20minus150

minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)

Figure 6 Hysteresis cycle with hardening [9]

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10

Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)

Figure 7 Asymmetrical cycles for tension-compression loading ofa wire rope isolator [10]

Beam

LRD

Stress transducercolumn

OD

Steel braces

Figure 8 The configuration of the combined energy dissipationsystem in a frame [11]

Figure 9 Seismic isolators presented in [12]


minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)

TheoreticalExperimental

(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)

Figure 10 Experimental and simulated hysteretic cycles (a) IUT a (b) IUT b [12]

c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y

Figure 11 Schematic of the hysteretic device that interconnects twoadjacent structures [13]

panel with friction energy dampers was presented Themodelling of the hysteretic device was developed using anextended Bouc-Wenmodel In [14] the study verified that theBouc-Wen model well predicted the responses of suspensionseats of the off-road machines to transient inputs as shownin Figure 12 The Bouc-Wen model used in [14] is shownschematically in Figure 13 The Bouc-Wen coefficients wereobtained by minimizing the difference between the predictedandmeasured acceleration of a load supported in the seatThemeasured hysteresis force-deflection cycles for the bottombuffers are shown in Figure 14 The study in [14] concludedthat the Bouc-Wen model can provide a useful simulation ofan existing seat

42 Semiactive Systems Themagnetorheological (MR) fluidsconsist of suspensions of micron-sized ferrous particlesimmersed in a carrier fluid their rheological behaviourcan be changed by means of an applied magnetic field

Top end-stop buffer

Suspension spring

Suspension damper

Cushion

Bottom end-stop buffer

Seat base

Figure 12 Schematic of the industrial truck seat [14]

Researchers have used the controllable variation in yieldstress to develop various smart devices [76ndash79] In recentyears magnetorheological dampers have been widely studiedas a controllable engineering component because of theircontinuously controllable mechanical properties and rapidresponse [80] As shown in Figure 15 MR dampers canoperate under three different fluid working modes [15] shear[81 82] flow [83ndash85] and squeeze [86ndash89] The shear modeoccurs when one wall of a gap translates or rotates relative tothe other wall In the shear mode the fluid is sheared parallelto thewallsThe flowmode occurswhen twowalls of a gap arefixed as in a valve system and the fluid flows through the gapand along the longitudinal axis of thewallsThe squeezemodeoccurs when the walls move toward each other squeezing outthe fluidThe fluid in the squeeze mode flows orthogonally tothe direction of wall motion According to the motion of the


Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)

Bouc-Wen force (FBW)

z-axis

Load mass

Figure 13 Schematic of the Bouc-Wen model [14]


Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)

Figure 14 Force-displacement diagrammdashindustrial truck seatexposed to random excitation [14]

constitutive elements MR dampers can be categorized as lin-earMR dampers [16] (see Figure 16) and rotaryMR dampers

The modelling of MR dampers represents an importantrole to accurately describe their behaviours A validateddynamic model allows executing several fundamental stepssuch as performance prediction design numerical simula-tions and control synthesis To this end it is important tohighlight the hysteretic phenomenon that occurs in thesedevices Indeed as it can be observed from experimental datain Figure 17 a hysteretic loop appears in the force-velocitydiagram

Several approaches have been established to model theMR dampers and among these the parametric approachdrew considerable attention to researchers The parametricmodels were developed using the schematization of thedevice as a combination of different physical elements and

a typical example was constituted by the Bouc-Wen hysteresisoperator-based dynamic model [20 90ndash93]

The Bouc-Wen model has been extensively appliedto simulate the hysteresis loops since it possesses theforce-displacement and force-velocity behaviour of theMR dampers In the following the most widely adoptedapproaches will be presented Spencer Jr et al [15 17]employed the Bouc-Wen hysteretic operator to represent thehysteretic behaviour ofMRdampers and the schematic of theproposed simple Bouc-Wenmodel for MR dampers is shownin Figure 18 [15 17]The damping force in this system is givenby

119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)

where 1198880 and 1198960 are the viscous damping and stiffnessrespectively 1199090 represents an initial displacement due to thepresence of an accumulator 119911 is the evolutionary variablegoverned by (3) By adjusting the parameter values 120572 120573 120574and 119899 the force-velocity relationship is characterized

Later Spencer Jr et al [15 17] proposed a modified Bouc-Wen model to predict the behaviour of MR dampers over awide range of inputs as shown in Figure 19Themodel is givenby the following equations

119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1

119911 minus 120573 ( minus ) |119911|119899+ 119860 ( minus )

(21)

and 1198961 is the accumulator stiffness 1198880 and 1198881 are the viscousdamping observed at large and low velocities respectively 1198960is the stiffness at large velocities 1199090 is the term that accountsfor the presence of an accumulatorThe scale and shape of thehysteresis loop can be adjusted by 120574 120573 119860 and 119899

To model the behaviour of the shear mode MR damper(see Figure 20(a)) a Bouc-Wen hysteresis operator-based


PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)

Figure 15 MR fluid operational modes [15] (a) the flow mode (b) the direct shear mode and (c) the squeeze mode

electromagnetWires to

Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator

Figure 16 Typical scheme of a linear MR damper [16]

dynamic model as shown in Figure 20(b) was proposed[18 19] The equation governing the damping force is givenby

119865 = 1198880 + 120572119911 (22)

minus10 minus5 0 5 10

Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A

Figure 17 The damping force versus velocity [16]

where 119911 is the evolutionary variable given by (3) The Bouc-Wen model in Figure 21 was developed for large-scale MRdampers [20] and the damper force is given by

119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)

In (23) 119898 is the equivalent mass 1198960 is the accumulatorstiffness 1198910 is the damper friction force 119911 is the evolutionaryvariable governed by (3) 1198880() is given by

1198880 () = 1198861119890minus(1198862||)119901

(24)

where 1198861 1198862 and 119901 are positive constants


Bouc-Wen

x

Fk0

c0

Figure 18 Simple Bouc-Wen model for MR dampers [15 17]

Bouc-Wen

x

F

k0

c0

y

c1

k1

Figure 19 Modified Bouc-Wen model for MR dampers [15 17]

43 Active Systems Piezoelectric actuators (PEAs) representa functional tool in the field of the active vibration controland in the recent years extensive researches have beenconducted for modelling and control PEAs utilize the con-verse piezoelectric effect of piezoelectricmaterials to generatedisplacement and force Indeed a piece of piezoelectricmaterial will be mechanically strained if subject to an electricfield (by placing it into the electric field or applying voltagesto its surfaces)

The hysteresis nonlinearity constitutes one of the princi-pal key issues to be solved in PEAs and its modelling is offundamental importance for their control In PEAs hysteresisexists in both the electric field- (voltage-) polarization rela-tionship and the electric field- (voltage-) strain (deformationor displacement) relationship (Figure 22) with the latterbeing mostly of concern in micro- and nanopositioningsystems and it is caused by the nonlinearities in the conversepiezoelectric effect of the unit cells and the switching andmovement of domain walls [21 94]

The hysteresis trajectory of a PEA can be treated asbeing composed of three types of components (1) the majorloop which is the hysteresis loop that spans the whole input(voltage) range (2) the minor loops which are the hysteresis

loops that only span portions of the input range and (3) theinitial ascending curveAs hysteresis is themajor nonlinearityof PEAs and possesses detrimental effects on the positioningaccuracy and stability margins of feedback control systems[95] compensation of hysteresis has always been a majorconcern in modelling and control of PEAs

In [96] a PEA was modelled by means of the Bouc-Wen model and a PID control was applied The dynamicrelationship governing the actuator is given by

119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)

where 119908 is the evolutionary variable governed by (14) 119898 isthe equivalent mass 119888 is the damping coefficient 119896119886 and 119896119887

are elastic constants 119909 is the actuator displacement 119896119909 and119896119908 are constant gains 119899 120588 and 120590 are the Bouc-Wen modelparameters

A Bouc-Wen based approach was introduced in [22]to compensate the hysteresis of piezoelectric actuators (seeFigure 23) via a feedforward control Indeed feedback con-trollers for small systems such as micronanoactuators arestrongly limited by the difficulty integrating the sensorAccording to the multiplicative-inverse structure the pro-posed compensator scheme was adapted to hysteresis withan advantage that no more computation was required for thecompensator

The hysteresis compensation is established using thefollowing relationship

119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)

119880 being the applied electrical voltage 119889119901 the piezoelectriccoefficient 119910119903 the target displacement of the actuator 119867(119880)

a nonlinear operator due to the Bouc-Wen model Equation(26) is used as a compensator that employs 119910119903 as an input and119880 as an output with respect to the scheme in Figure 24

To linearize the hysteresis behaviour of stack piezoelectricceramic actuators the feedforward linearization methodbased on the Bouc-Wen model and the hybrid linearizationmethod combining the feedforwardmethod and PI feedbackloop were proposed and explored in [97] The rapid controlprototypes of the linearization controllers were establishedand tested and the results showed that both the feedforwardand hybrid linearization methods can linearize the hysteresisbehaviour Meanwhile PEAs can exhibit an asymmetrichysteretic behaviour so that a modified Bouc-Wen modelwas proposed in [6] by introducing an input bias and anasymmetric factor into the standard Bouc-Wen hysteresismodel

5 Conclusion

This review reported the literature related to the utilization ofthe Bouc-Wen model for modelling hysterical behaviours ofseveral vibration control systems The review was organizedinto three sections that address specific issues mathematicalproperty of the Bouc-Wen model identification of the modelparameters and applications of the Bouc-Wen model for


Direction of motion

Front view Coil Side view

MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)

Figure 20 (a) Schematic diagram of shear mode MR damper (b) mechanical model of the parallel plate MR damper [15 18 19]

Magnetic flux

Flow

MR fluid

accumulator3-stage piston

LORD

MRD-9000Rheonetic seismic damper

Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)

Figure 21 (a) Schematic of large-scale magnetorheological fluid damper (b) Bouc-Wen model for large-scale MR dampers [15 20]

0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops

Initial ascendingcurve

Figure 22 Hysteresis of a PEA [21]

the dynamical description of different types of vibrationcontrol systems Section 2 described the theoretical basis ofthe Bouc-Wen model its first formulation and its successive

Displacement sensor Cantilevered piezoelectricactuator

Figure 23 Photography of the piezoelectric actuator [22]

modifications Section 3 presented different approaches usedfor the identification of the Bouc-Wen model parameters


Hysteresis compensator

H(middot)

H(middot)

yyr 1

dp

+ ++U

Bouc-Wen hysteresis model

dp minus

h

Figure 24 Bouc-Wen based compensator [22]

Section 4 provided several applications of the Bouc-Wenmodel for the modelling of devices extensively used invibration control Each section has presented what from theauthorsrsquo point of view are the main contributions for thespecific issue

Conflict of Interests

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

References

[1] M Ismail F Ikhouane and J Rodellar ldquoThe hysteresis Bouc-Wen model a surveyrdquo Archives of Computational Methods inEngineering vol 16 no 2 pp 161ndash188 2009

[2] R Bouc ldquoModele mathematique drsquohysteresisrdquo Acustica vol 21pp 16ndash25 1971

[3] Y K Wen ldquoMethod of random vibration of hysteretic systemsrdquoJournal of the EngineeringMechanics Division vol 102 no 2 pp249ndash263 1976

[4] A E Charalampakis and V K Koumousis ldquoIdentification ofBouc-Wen hysteretic systems by a hybrid evolutionary algo-rithmrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 571ndash585 2008

[5] S K Kunnath J B Mander and L Fang ldquoParameter identifi-cation for degrading and pinched hysteretic structural concretesystemsrdquo Engineering Structures vol 19 no 3 pp 224ndash232 1997

[6] H Qin N Bu W Chen and Z Yin ldquoAn asymmetric hysteresismodel and parameter identification method for piezoelectricactuatorrdquo Mathematical Problems in Engineering vol 2014Article ID 932974 14 pages 2014

[7] S TalatahariM Nouri and F Tadbiri ldquoOptimization of skeletalstructures using artificial bee colony algorithmrdquo InternationalJournal of Optimization in Civil Engineering vol 2 no 4 pp557ndash571 2012

[8] S Pagano and S Strano ldquoWire rope springs for passive vibrationcontrol of a light steel structurerdquo WSEAS Transactions onApplied and Theoretical Mechanics vol 8 no 3 pp 212ndash2212013

[9] G Di Massa S Pagano E Rocca and S Strano ldquoSensitiveequipments on WRS-BTU isolatorsrdquo Meccanica vol 48 no 7pp 1777ndash1790 2013

[10] H-X Wang X-S Gong F Pan and X-J Dang ldquoExperimentalinvestigations on the dynamic behaviour of O-type wire-cable

vibration isolatorsrdquo Shock and Vibration vol 2015 Article ID869325 12 pages 2015

[11] X Lu and Q Zhou ldquoDynamic analysis method of a combinedenergy dissipation system and its experimental verificationrdquoEarthquake Engineering amp Structural Dynamics vol 31 no 6pp 1251ndash1265 2002

[12] S Strano and M Terzo ldquoA first order model based control of ahydraulic seismic isolator test rigrdquo Engineering Letters vol 21no 2 pp 52ndash60 2013

[13] M Basili and M De Angelis ldquoOptimal passive control ofadjacent structures interconnected with nonlinear hystereticdevicesrdquo Journal of Sound and Vibration vol 301 no 1-2 pp106ndash125 2007

[14] T P Gunston J Rebelle andM J Griffin ldquoA comparison of twomethods of simulating seat suspension dynamic performancerdquoJournal of Sound and Vibration vol 278 no 1-2 pp 117ndash1342004

[15] D H Wang and W H Liao ldquoMagnetorheological fluiddampers a review of parametric modellingrdquo Smart Materialsand Structures vol 20 no 2 Article ID 023001 2011

[16] S J Dyke B F Spencer Jr M K Sain and J D CarlsonldquoAn experimental study of MR dampers for seismic protectionrdquoSmart Materials and Structures vol 7 no 5 pp 693ndash703 1998

[17] 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

[18] L M Jansen and S J Dyke ldquoSemiactive control strategiesfor MR dampers comparative studyrdquo Journal of EngineeringMechanics vol 126 no 8 pp 795ndash803 2000

[19] F Yi S J Dyke J M Caicedo and J D Carlson ldquoExperimentalverification of multi-input seismic control strategies for smartdampersrdquo Journal of Engineering Mechanics vol 127 no 11 pp1152ndash1164 2001

[20] G Yang B F Spencer Jr H-J Jung and J D CarlsonldquoDynamic modeling of large-scale magnetorheological dampersystems for civil engineering applicationsrdquo Journal of Engineer-ing Mechanics vol 130 no 9 pp 1107ndash1114 2004

[21] P Jingyang andCXiongbiao ldquoA survey ofmodeling and controlof piezoelectric actuatorsrdquoModernMechanical Engineering vol3 no 1 pp 1ndash20 2013

[22] M Rakotondrabe ldquoBouc-Wen modeling and inverse multi-plicative structure to compensate hysteresis nonlinearity inpiezoelectric actuatorsrdquo IEEE Transactions on Automation Sci-ence and Engineering vol 8 no 2 pp 428ndash431 2011

[23] F Ma H Zhang A Bockstedte G C Foliente and P PaevereldquoParameter analysis of the differential model of hysteresisrdquoJournal of Applied Mechanics vol 71 no 3 pp 342ndash349 2004

[24] M C Constantinou andMA Adnane ldquoDynamics of soil-base-isolated structure systems evaluation of twomodels for yieldingsystemsrdquo Report to NSAF Department of Civil EngineeringDrexel University Philadelphia Pa USA 1987

[25] NM Kwok Q P Ha M T Nguyen J Li and B Samali ldquoBouc-Wen model parameter identification for a MR fluid damperusing computationally efficient GArdquo ISA Transactions vol 46no 2 pp 167ndash179 2007

[26] T T Baber and Y K Wen ldquoRandom vibration of hystereticdegrading systemsrdquo Journal of Engineering Mechanics vol 107no 6 pp 1069ndash1087 1981

[27] T T Baber and M N Noori ldquoRandom vibration of degradingpinching systemsrdquo Journal of EngineeringMechanics vol 111 no8 pp 1010ndash1026 1985


[28] T T Baber and M N Noori ldquoModeling general hysteresisbehavior and random vibration applicationrdquo Journal of Vibra-tion Acoustics Stress and Reliability in Design vol 108 no 4pp 411ndash420 1986

[29] F Ikhouane and J Rodellar ldquoOn the hysteretic Bouc-Wenmodel Part I forced limit cycle characterizationrdquo NonlinearDynamics vol 42 no 1 pp 63ndash78 2005

[30] W Zhu and X-T Rui ldquoSemiactive vibration control using amagnetorheological damper and a magnetorheological elas-tomer based on the bouc-wen modelrdquo Shock and Vibration vol2014 Article ID 405421 10 pages 2014

[31] A G Chassiakos S F Masri A W Smyth and T K CaugheyldquoOn-line identification of hysteretic systemsrdquo Journal of AppliedMechanics vol 65 no 1 pp 194ndash203 1998

[32] A W Smyth S F Masri A G Chassiakos and T K CaugheyldquoOnline parametric identification of MDOF nonlinear hys-teretic systemsrdquo Journal of Engineering Mechanics vol 125 no2 pp 133ndash142 1999

[33] J-W Lin R Betti A W Smyth and R W Longman ldquoOn-lineidentification of non-linear hysteretic structural systems using avariable trace approachrdquo Earthquake Engineering and StructuralDynamics vol 30 no 9 pp 1279ndash1303 2001

[34] C-H Loh L Y Wu and P Y Lin ldquoDisplacement control ofisolated structures with semi-active control devicesrdquo Journal ofStructural Control vol 10 no 2 pp 77ndash100 2003

[35] J-W Lin and R Betti ldquoOn-line identification and damagedetection in non-linear structural systems using a variable for-getting factor approachrdquo Earthquake Engineering amp StructuralDynamics vol 33 no 4 pp 419ndash444 2004

[36] J-L Ha Y-S Kung R-F Fung and S-C Hsien ldquoA comparisonof fitness functions for the identification of a piezoelectrichysteretic actuator based on the real-coded genetic algorithmrdquoSensors and Actuators A Physical vol 132 no 2 pp 643ndash6502006

[37] R L Haupt and S E Haupt Practical Genetic Algorithms JohnWiley amp Sons New York NY USA 1998

[38] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[39] Y Chae J M Ricles and R Sause ldquoModeling of a large-scalemagneto-rheological damper for seismic hazard mitigationPart I passive moderdquo Earthquake Engineering and StructuralDynamics vol 42 no 5 pp 669ndash685 2013

[40] S J Li H Yu and Y Suzuki ldquoIdentification of non-linearhysteretic systems with sliprdquo Computers amp Structures vol 82no 2-3 pp 157ndash165 2004

[41] Y Q Ni J M Ko and C W Wong ldquoIdentification of non-linear hysteretic isolators from periodic vibration testsrdquo Journalof Sound and Vibration vol 217 no 4 pp 737ndash756 1998

[42] F Ikhouane and O Gomis-Bellmunt ldquoA limit cycle approachfor the parametric identification of hysteretic systemsrdquo Systemsamp Control Letters vol 57 no 8 pp 663ndash669 2008

[43] F Ikhouane and J Rodellar ldquoOn the hysteretic Bouc-Wenmodel Part II robust parametric identificationrdquo NonlinearDynamics vol 42 no 1 pp 79ndash95 2005

[44] 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

[45] S Talatahari H Mohaggeg K Najafi and A ManafzadehldquoSolving parameter identification of nonlinear problems by

artificial bee colony algorithmrdquo Mathematical Problems inEngineering vol 2014 Article ID 479197 6 pages 2014

[46] P-Q Xia ldquoAn inverse model of MR damper using optimalneural network and system identificationrdquo Journal of Sound andVibration vol 266 no 5 pp 1009ndash1023 2003

[47] D H Wang and W H Liao ldquoModeling and control of mag-netorheological fluid dampers using neural networksrdquo SmartMaterials and Structures vol 14 no 1 pp 111ndash126 2005

[48] S F Masri J P Caffrey T K Caughey A W Smyth andA G Chassiakos ldquoIdentification of the state equation incomplex non-linear systemsrdquo International Journal of Non-Linear Mechanics vol 39 no 7 pp 1111ndash1127 2004

[49] Y Yang and F Ma ldquoConstrained Kalman filter for nonlinearstructural identificationrdquo Journal of Vibration and Control vol9 no 12 pp 1343ndash1357 2003

[50] H Zhang G C Foliente Y Yang and F Ma ldquoParameteridentification of inelastic structures under dynamic loadsrdquoEarthquake Engineering and Structural Dynamics vol 31 no 5pp 1113ndash1130 2002

[51] E N Chatzi andAW Smyth ldquoThe unscentedKalman filter andparticle filter methods for nonlinear structural system identifi-cation with non-collocated heterogeneous sensingrdquo StructuralControl and Health Monitoring vol 16 no 1 pp 99ndash123 2009

[52] E N Chatzi A W Smyth and S F Masri ldquoExperimentalapplication of on-line parametric identification for nonlinearhysteretic systems with model uncertaintyrdquo Structural Safetyvol 32 no 5 pp 326ndash337 2010

[53] Z Xie and J Feng ldquoReal-time nonlinear structural systemidentification via iterated unscented Kalman filterrdquoMechanicalSystems and Signal Processing vol 28 pp 309ndash322 2012

[54] R I Skinner W H Robinson and G H McVerryAn Introduc-tion to Seismic Isolation Wiley Chichester UK 1993

[55] F Naeim and J M Kelly Design of Seismic Isolated StructuresFromTheory to Practice Wiley Chichester UK 1999

[56] H Yoshioka J C Ramallo and B F Spencer Jr ldquolsquoSmartrsquo baseisolation strategies employing magnetorheological dampersrdquoJournal of Engineering Mechanics vol 128 no 5 pp 540ndash5512002

[57] C-M Chang and B F Spencer Jr ldquoActive base isolation ofbuildings subjected to seismic excitationsrdquo Earthquake Engi-neering and Structural Dynamics vol 39 no 13 pp 1493ndash15122010

[58] J Fei ldquoActive vibration control of flexible steel cantileverbeam using piezoelectric actuatorsrdquo in Proceedings of the 37thSoutheastern Symposium on SystemTheory (SST rsquo05) pp 35ndash39IEEE March 2004

[59] J M Kelly G Leitmann and A G Soldatos ldquoRobust control ofbase-isolated structures under earthquake excitationrdquo Journalof Optimization Theory and Applications vol 53 no 2 pp 159ndash180 1987

[60] S Strano andMTerzo ldquoAmulti-purpose seismic test rig controlvia a sliding mode approachrdquo Structural Control and HealthMonitoring vol 21 no 8 pp 1193ndash1207 2014

[61] A Calabrese G Serino S Strano andM Terzo ldquoExperimentalinvestigation of a low-cost elastomeric anti-seismic device usingrecycled rubberrdquoMeccanica vol 50 no 9 pp 2201ndash2218 2015

[62] J C Ramallo E A Johnson and B F Spencer Jr ldquolsquoSmartrsquo baseisolation systemsrdquo Journal of Engineering Mechanics vol 128no 10 pp 1088ndash1099 2002

[63] I Buckle S Nagarajaiah andK Ferrell ldquoStability of elastomericisolation bearings experimental studyrdquo Journal of StructuralEngineering vol 128 no 1 pp 3ndash11 2002


[64] J M Kelly ldquoThe role of damping in seismic isolationrdquo Earth-quake Engineering and Structural Dynamics vol 28 no 3 pp3ndash20 1999

[65] S Nagarajaiah and K Ferrell ldquoStability of elastomeric seismicisolation bearginsrdquo Journal of Structural Engineering vol 125no 9 pp 946ndash954 1999

[66] B F Spencer Jr E A Johnson and J C RamalloldquolsquoSmartrsquoisolation for seismic controlrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 43 no 3 pp 704ndash711 2000

[67] N Makris ldquoRigidity-plasticity-viscosity can electrorheologi-cal dampers protect base-isolated structures from nearsourceearthquakesrdquo Earthquake Engineering amp Structural Dynamicsvol 26 no 5 pp 571ndash591 1997

[68] S Nagarajaiah S Sahasrabudhe and R Iyer ldquoSeismic responseof sliding isolated bridges with MR dampersrdquo in Proceedingsof the American Control Conference vol 6 pp 4437ndash4441Chicago Ill USA June 2000

[69] D Shook P-Y Lin T-K Lin and PN Roschke ldquoA comparativestudy in the semi-active control of isolated structuresrdquo SmartMaterials and Structures vol 16 no 4 pp 1433ndash1446 2007

[70] S Sahasrabudhe S Nagarajaiah and C Hard ldquoExperimentalstudy of sliding isolated buildingswith smart dampers subjectedto near-source groundmotionsrdquo in Proceedings of the ASCE 14thEngineering Mechanics Conference (EM rsquo00) Austin Tex USA2000

[71] J Zhang L He E Wang and R Gao ldquoA LQR controller designfor active vibration control of flexible structuresrdquo in Proceedingsof the Pacific-Asia Workshop on Computational Intelligence andIndustrial Application (PACIIA rsquo08) pp 127ndash132 IEEE WuhanChina December 2008

[72] APC International Piezoelectric Ceramics Principles and Appli-cations APC International 2011

[73] G F Demetriades M C Constantinou and A M ReinhornldquoStudy of wire rope systems for seismic protection of equipmentin buildingsrdquo Engineering Structures vol 15 no 5 pp 321ndash3341993

[74] M Constantinou AMokha andA Reinhorn ldquoTeflon bearingsin base isolation IIModelingrdquo Journal of Structural Engineeringvol 116 no 2 pp 455ndash474 1990

[75] M Sasani and E P Popov ldquoSeismic energy dissipators for RCpanels analytical studiesrdquo Journal of EngineeringMechanics vol127 no 8 pp 835ndash843 2001

[76] A Lanzotti M Russo R Russo F Renno and M TerzoldquoA physical prototype of an automotive magnetorheologicaldifferentialrdquo in Proceedings of theWorld Congress on Engineering(WCE 2013) vol 3 of LectureNotes in Engineering andComputerScience pp 2131ndash2135 2013

[77] A Lanzotti F Renno M Russo R Russo and M TerzoldquoDesign and development of an automotive magnetorheolog-ical semi-active differentialrdquo Mechatronics vol 24 no 5 pp426ndash435 2014

[78] K Karakoc E J Park and A Suleman ldquoDesign considerationsfor an automotive magnetorheological brakerdquo Mechatronicsvol 18 no 8 pp 434ndash447 2008

[79] B M Kavlicoglu N C Kavlicoglu Y Liu et al ldquoResponse timeand performance of a high-torque magneto-rheological fluidlimited slip differential clutchrdquo Smart Materials and Structuresvol 16 no 1 pp 149ndash159 2007

[80] N M Wereley J U Cho Y T Choi and S B Choi ldquoMag-netorheological dampers in shear moderdquo Smart Materials andStructures vol 17 no 1 Article ID 015022 2008

[81] Z Lou R D Ervin F E Filisko and C B Winkler ldquoAnelectrorheologically controlled semi-active landing gearrdquo SAETechnical Paper Series 931403 SAE International 1993

[82] Z Lou R D Ervin and F E Filisko ldquoPreliminary parametricstudy of electrorheological dampersrdquo Journal of Fluids Engineer-ing Transactions of the ASME vol 116 no 3 pp 570ndash576 1994

[83] N M Wereley and L Pang ldquoNondimensional analysis of semi-active electrorheological and magnetorheological dampersusing approximate parallel plate modelsrdquo Smart Materials andStructures vol 7 no 5 pp 732ndash743 1998

[84] Y-T Choi and N M Wereley ldquoComparative analysis of thetime response of electrorheological and magnetorheologicaldampers using nondimensional parametersrdquo Journal of Intelli-gent Material Systems and Structures vol 13 no 7-8 pp 443ndash451 2002

[85] N M Wereley J Lindler N Rosenfeld and Y-T Choi ldquoBivis-cous damping behavior in electrorheological shock absorbersrdquoSmart Materials and Structures vol 13 no 4 pp 743ndash752 2004

[86] R Stanway J L Sproston and N G Stevens ldquoNon-linearmodelling of an electro-rheological vibration damperrdquo Journalof Electrostatics vol 20 no 2 pp 167ndash184 1987

[87] E W Williams S G Rigby J L Sproston and R StanwayldquoElectrorheological fluids applied to an automotive enginemountrdquo Journal of Non-Newtonian Fluid Mechanics vol 47 pp221ndash238 1993

[88] N D Sims R Stanway A R Johnson and J S Yang ldquoVibrationisolation using a magnetorheological damper in the squeeze-flowmoderdquo in Smart Structures andMaterials Smart Structuresand Integrated Systems vol 3668 ofProceedings of SPIE pp 520ndash527 Newport Beach Calif USA June 1999

[89] Y K Ahn J-Y Ha Y-H Kim et al ldquoDynamic characteristics ofsqueeze-type mount using magnetorheological fluidrdquo Proceed-ings of the Institution of Mechanical Engineers Part K Journal ofMulti-body Dynamics vol 219 no 1 pp 27ndash34 2005

[90] A Dominguez R Sedaghati and I Stiharu ldquoModelling the hys-teresis phenomenon of magnetorheological dampersrdquo SmartMaterials and Structures vol 13 no 6 pp 1351ndash1361 2004

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

[92] A Dominguez R Sedaghati and I Stiharu ldquoSemi-active vibra-tion control of adaptive structures using magnetorheologicaldampersrdquo AIAA Journal vol 44 no 7 pp 1563ndash1571 2006

[93] A Dominguez R Sedaghati and I Stiharu ldquoModeling andapplication of MR dampers in semi-adaptive structuresrdquo Com-puters and Structures vol 86 no 3ndash5 pp 407ndash415 2008

[94] I Mayergoyz and G Bertotti EdsThe Science of Hysteresis vol3 Elsevier St Louis Mo USA 2005

[95] J AMain and E Garcia ldquoPiezoelectric stack actuators and con-trol system design strategies and pitfallsrdquo Journal of GuidanceControl and Dynamics vol 20 no 3 pp 479ndash485 1997

[96] O Gomis-Bellmunt F Ikhouane and D Montesinos-MiracleldquoControl of Bouc-Wen hysteretic systems application to apiezoelectric actuatorrdquo in Proceedings of the 13th InternationalPower Electronics and Motion Control Conference (EPE-PEMCrsquo08) pp 1670ndash1675 Poznan Poland September 2008

[97] D H Wang W Zhu and Q Yang ldquoLinearization of stackpiezoelectric ceramic actuators based on Bouc-Wen ModelrdquoJournal of IntelligentMaterial Systems and Structures vol 22 no5 pp 401ndash413 2011

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DistributedSensor Networks



Elastic postyielding spring

Fel

akiu

Fel u

Hysteretic spring

Fhmax Fh

minusFhmaxu Fh u

(1a)ki

F u

F

Fy

ki =Fy

uy uuy

kf = aki

Figure 1 Bouc-Wen model [4]

excitation force the overdot indicates derivative with respectto time

The restoring force 119865(119905) based on the Bouc-Wen modelis

119865 (119905) = 119886

119865119910

119906119910

119906 (119905) + (1 minus 119886) 119865119910119911 (119905) (2)

where 119886 = 119896119891119896119894 is the ratio of postyield stiffness 119896119891

to preyield stiffness 119896119894 = 119865119910119906119910 119865119910 is the yield force119906119910 is the yield displacement and 119911(119905) is a nonobservabledimensionless hysteretic variable that obeys the followingnonlinear differential equation with zero initial condition(119911(0) = 0)

(119905) = (119905) 119860 minus [120574 + 120573 sgn ( (119905) 119911 (119905))] |119911 (119905)|119899 (3)

The coefficients 119860 120573 120574 and 119899 are dimensionless quantitiesthat control the behaviour of the model and sgn(sdot) is thesignum function For small values of the positive exponentialparameter 119899 the transition from elastic to postelastic branchis smooth whereas for large values the transition becomesabrupt approaching that of a bilinear model The parameters120573 and 120574 control the size and shape of the hysteretic loop Thenotation varies from paper to paper and very often the placesof 120573 and 120574 are exchanged

From (2) it follows that the restoring force 119865(119905) can bedivided into an elastic and a hysteretic part as follows

119865el(119905) = 119886

119865119910

119906119910

119906 (119905) = 119886119896119894119906 (119905) = 119896119891119906 (119905)

119865ℎ(119905) = (1 minus 119886) 119896119894119911 (119905)

(4)

Thus the model can be visualized as two springs connectedin parallel [4] (Figure 1)

The parameters of the Bouc-Wen model have the follow-ing criteria

119886 isin [0 1]

119896119894 gt 0

119896119891 gt 0

119888 gt 0

119860 gt 0

119899 gt 1

120573 gt 0

120574 isin [minus120573 120573]

(5)

In [23] it has been proved that the parameters of the Bouc-Wen model are functionally redundant and indeed multipleparameter vectors can produce an identical response undera given excitation This redundancy can be then removed byfixing parameter 119860 to unity [23]

Constantinou and Adnane [24] suggested the constraint119860(120573+120574) = 1 in order to reduce the total number of unknownparameters to six 120574 119899 119886 119865119910 119906119910 and 119888

In [25] an asymmetrical Bouc-Wen model has beenobtained adjusting the velocity as

(119905) larr997888 ( (119905) minus sgn (119906 (119905))) (6)

where 120583 is the scale factor for the adjustmentModification of the Bouc-Wen model presented in [26ndash

28] included strength stiffness and pinching degradationeffects by means of suitable degradation functions

(119905) =

ℎ (119911 (119905))

120578 (120576)

(119905) 119860 (120576)

minus ] (120576) [120573 sgn ( (119905)) |119911 (119905)|119899minus1

119911 (119905) + 120574 |119911 (119905)|119899]

(7)

where 120576 is the absorbed hysteretic energy and the functions](120576) 120578(120576) and ℎ(119911) are associated with strength stiffness and




] (120576) = ]0 + 120575]120576 (119905)

119860 (120576) = 1198600 minus 120575119860120576 (119905)

120578 (120576) = 1205780 + 120575120578120576 (119905)

(8)


ℎ (119911) = 1 minus 1205891 (120576) exp(minus

(119911 (119905) sgn () minus 119902119911119906)2

(1205892 (120576))2

) (9)

where

1205891 (120576) = (1 minus exp (minus119901120576 (119905))) 120589

1205892 (120576) = (1205950 + 120575120595120576 (119905)) (120582 + 1205891 (120576))

(10)


119911119906 =119899

radic

1

] (120573 + 120574)

(11)


0 ]0 1198600 1205780 1205950 120575120595 120582 119901 and 120589 When 120575] = 0 120575120578 =



119899 ge 1

119906119910 gt 0

119860 gt 0

120573 + 120574 gt 0

120573 minus 120574 ge 0

(12)



120588 =

119860

1199061199101199110

gt 0

120590 =

120573

120573 + 120574

ge

1

2

1199110 =119899

radic

119860

120573 + 120574

(13)

it follows

(119905)

= 120588 (119905) 1 + |119908 (119905)|119899120590 [1 minus sgn ( (119905) 119908 (119905)) minus

1

120590

]

(14)

with

119908 (119905) =

119911 (119905)

1199110

(15)



119865 (119905) = 119896119891119906 (119905) + 119896119908119908 (119905) (16)

where

119896119908 = (1 minus 119886) 1198651199101199110 gt 0 (17)


120588 gt 0

120590 ge

1

2

119899 ge 1

119896119891 gt 0

119896119908 gt 0

(18)









each cycle


initial stiffness







NoErrorwithin

tolerance


(a)

(b)






Start


Set i = 1


individual i

i + 1


f(xi) = radic 1

N

Nsuml=1

[y(l) minus y(l)]2

kg + 1

Yes

Noi lt size



Yes

No

kg lt G


End

Part I

Part II

Part III









the onlooker



No

No

Yes

Yes





Final food position

























minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)



Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)


Beam

LRD


OD

Steel braces




minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












] (120576) = ]0 + 120575]120576 (119905)

119860 (120576) = 1198600 minus 120575119860120576 (119905)

120578 (120576) = 1205780 + 120575120578120576 (119905)

(8)


ℎ (119911) = 1 minus 1205891 (120576) exp(minus

(119911 (119905) sgn () minus 119902119911119906)2

(1205892 (120576))2

) (9)

where

1205891 (120576) = (1 minus exp (minus119901120576 (119905))) 120589

1205892 (120576) = (1205950 + 120575120595120576 (119905)) (120582 + 1205891 (120576))

(10)


119911119906 =119899

radic

1

] (120573 + 120574)

(11)


0 ]0 1198600 1205780 1205950 120575120595 120582 119901 and 120589 When 120575] = 0 120575120578 =



119899 ge 1

119906119910 gt 0

119860 gt 0

120573 + 120574 gt 0

120573 minus 120574 ge 0

(12)



120588 =

119860

1199061199101199110

gt 0

120590 =

120573

120573 + 120574

ge

1

2

1199110 =119899

radic

119860

120573 + 120574

(13)

it follows

(119905)

= 120588 (119905) 1 + |119908 (119905)|119899120590 [1 minus sgn ( (119905) 119908 (119905)) minus

1

120590

]

(14)

with

119908 (119905) =

119911 (119905)

1199110

(15)



119865 (119905) = 119896119891119906 (119905) + 119896119908119908 (119905) (16)

where

119896119908 = (1 minus 119886) 1198651199101199110 gt 0 (17)


120588 gt 0

120590 ge

1

2

119899 ge 1

119896119891 gt 0

119896119908 gt 0

(18)









each cycle


initial stiffness







NoErrorwithin

tolerance


(a)

(b)






Start


Set i = 1


individual i

i + 1


f(xi) = radic 1

N

Nsuml=1

[y(l) minus y(l)]2

kg + 1

Yes

Noi lt size



Yes

No

kg lt G


End

Part I

Part II

Part III









the onlooker



No

No

Yes

Yes





Final food position

























minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)



Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)


Beam

LRD


OD

Steel braces




minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












each cycle


initial stiffness







NoErrorwithin

tolerance


(a)

(b)






Start


Set i = 1


individual i

i + 1


f(xi) = radic 1

N

Nsuml=1

[y(l) minus y(l)]2

kg + 1

Yes

Noi lt size



Yes

No

kg lt G


End

Part I

Part II

Part III









the onlooker



No

No

Yes

Yes





Final food position

























minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)



Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)


Beam

LRD


OD

Steel braces




minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














the onlooker



No

No

Yes

Yes





Final food position

























minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)



Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)


Beam

LRD


OD

Steel braces




minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















minus100

minus50

0

50

100

150

Forc

e (N

)

Displacement (mm)



Displacement x (mm)

minus400

minus200

0

200

400

600

800

Resto

ring

forc

ef(N

)


Beam

LRD


OD

Steel braces




minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(a)

minus004 minus002 0 002 004

d (m)

minus5

0

5times104

FBW

(N)


(b)


c1

y1

m1

k1

F F y2

m2

k2

c2

g(t)

Hystereticdevice

y




Top end-stop buffer

Suspension spring

Suspension damper

Cushion


Seat base




Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Top buffer (FT)

Bottom buffer (FB)

Damper (c)

Spring (ks)


z-axis

Load mass



Displacement (cm)

minus300

minus200

minus100

0

100

200

300

Forc

e (N

)







119865 = 1198880 + 1198960 (119909 minus 1199090) + 120572119911 (19)



119865 = 1198881 + 1198961 (119909 minus 1199090) (20)

where

=

1

1198880 + 1198881

[120572119911 + 1198880 + 1198960 (119909 minus 119910)]

= minus1205741003816100381610038161003816 minus

1003816100381610038161003816|119911|119899minus1


(21)




PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










PressureFlow Q

H

(a)

H

ForceSpeed

(b)

H

ForceDisplacement

(c)



Annularorifice

Bearing and seal

MR fluid

Coil

Diaphragm

Accumulator



119865 = 1198880 + 120572119911 (22)


Velocity (mms)

minus1000

0

1000

Forc

e (N

)

00A05A10A

15A20A



119865 = 119898 + 1198880 () + 1198960119909 + 120572119911 + 1198910 (23)


1198880 () = 1198861119890minus(1198862||)119901

(24)



Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bouc-Wen

x

Fk0

c0


Bouc-Wen

x

F

k0

c0

y

c1

k1







119898 + 119888 + 119896119886119909 = 119896119887119896119909119906 + 119896119887119896119908119908 (25)





119880 =

1

119889119901

(119910119903 + 119867 (119880)) (26)




5 Conclusion



Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Direction of motion


MR fluidsaturated

foam

(a)

Bouc-Wen

x

F

c0

(b)


Magnetic flux

Flow

MR fluid


LORD


Thermal expansion

(a)

Bouc-Wen

x

F minus f0k0

c0( )

m

x

(b)


0 20 40 60

Input voltage (V)

0

5

10

Out

put d

ispla

cem

ent (120583

m)

Major loop

Minor loops









H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





References








































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











H(middot)

H(middot)

yyr 1

dp

+ ++U


dp minus

h





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









review article modelling of hysteresis in vibration...

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