0964-1726_23_1_015019

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Precise stiffness and damping emulation with MR dampers and its application to semi-active

tuned mass dampers of Wolgograd Bridge

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2014 Smart Mater. Struct. 23 015019

(http://iopscience.iop.org/0964-1726/23/1/015019)

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Smart Mater. Struct. 23 (2014) 015019 (18pp) doi:10.1088/0964-1726/23/1/015019

Precise stiffness and damping emulationwith MR dampers and its application tosemi-active tuned mass dampers ofWolgograd Bridge

F Weber1,3 and M Maslanka2

1 Empa, Swiss Federal Laboratories for Materials Science and Technology, Structural EngineeringResearch Laboratory, Uberlandstrasse 129, CH-8600 Dubendorf, Switzerland2 AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics,Department of Process Control, aleja A. Mickiewicza 30, 30-059 Krakow, Poland

E-mail: [email protected] and [email protected]

Received 24 September 2013, in final form 1 November 2013Published 10 December 2013

AbstractThis paper investigates precise stiffness and damping emulation with MR dampers whenclipping and a residual MR damper force constrain the desired control force. It is shown thatthese force constraints lead to smaller equivalent stiffness and greater equivalent damping ofthe constrained MR damper force than desired. Compensation methods for precise stiffnessand damping emulations are derived for harmonic excitation of the MR damper. Thenumerical validation of both compensation methods confirms their efficacy. The precisestiffness emulation approach is experimentally validated with the MR damper basedsemi-active tuned mass damper (MR-STMD) concept of the Wolgograd Bridge . Theexperimental results reveal that the precise stiffness emulation approach enhances theefficiency of the MR-STMD significantly when the MR-STMD is operated at reduced desireddamping, where the impact of control force constraints becomes significant.

Keywords: control, damping, MR damper, stiffness, semi-active, TMD, Wolgograd Bridge

(Some figures may appear in colour only in the online journal)

Nomenclature

Abbreviations

C1 control of MR-STMD without stiffness/dampingcorrection

C2 control of MR-STMD with stiffness correctionC3 control of MR-STMD with damping correctionFRF Frequency response functionFTE Force tracking errorMR MagnetorheologicalMR-STMD

Semi-active tuned mass damper based on

controlled MR damper

3 Author to whom correspondence should be addressed.

TMD Tuned mass damper

SymbolsEF cycle energy of friction damperFc-des Desired friction forceFc-des Corrected desired friction forceFw Disturbing force amplitudeF0 Residual forceV Potential energyXd Relative displacement amplitude of mass damperX1 Displacement amplitude of primary structureXstatic1 Static deflection of primary structurecdes Desired viscous damping coefficientcdes Corrected desired viscous damping coefficientcequiv Equivalent viscous damping coefficient

10964-1726/14/015019+18$33.00 c 2014 IOP Publishing Ltd Printed in the UK

Smart Mater. Struct. 23 (2014) 015019 F Weber and M Maslanka

cequiv Equivalent viscous damping coefficient of cor-rected control force

c2 Viscous damping coefficient of TMDfact Actual MR damper forcef act Corrected actual MR damper forcefcl, fcu Frequency lower, upper bounds without control

force constraintsfdes Desired control forcef clippeddes Clipped desired control forcef des Corrected desired control forcefw Disturbing frequencyf1 First resonance frequency of primary structureiact Actual MR damper currentides Desired MR damper currentkdes Desired stiffnesskdes Corrected desired stiffnesskequiv Equivalent stiffnesskequiv Equivalent stiffness of corrected control forcek1 Stiffness coefficient of primary structurek2 Passive spring stiffness of mass damperms Shaking mass of shakerm1 Modal mass of primary structurem2 Mass of mass damperuw Command voltage of shaker amplifierxd Relative displacement of mass damperxd Relative velocity of mass damperxw Acceleration of shaking mass of shakerx1 Displacement of primary structure1E Energy dissipation due to control force con-

straints Damping gain1 Damping ratio of primary structure2 Frequency dependent damping ratio of mass

damper Mass ratio Frequency dependent mass ratiod Natural radial frequency of mass damperw Disturbing radial frequency1 First resonance radial frequency of primary

structure.

1. Introduction

Passive tuned mass dampers, commonly abbreviated asTMDs, are well known damping devices in the field of civiland mechanical engineering. Their working behaviour anddesign are well described in Den Hartogs book [1]. Due totheir robustness [2], TMDs are widely used for the mitigationof the structural vibrations of, for example, tall buildingsand slender bridges [3, 4]. Passive TMDs have also beendeveloped based on friction dampers [5] and designed forlarge tuned masses [6]. In some cases, the efficiency ofpassive TMDs might not be sufficient, for example, whenmodal parameters of the primary structure change duringdifferent construction phases [7], when life loads significantlychange the modal mass of the primary structure, as can be thecase for footbridges [8], or when environmental parametersinfluence the primary structure and TMD properties [9]. Then,

adaptive TMDs are seen to be the appropriate solution to theproblem [7, 8]. The large variety of adaptive TMDs may besplit into active (ATMD) and semi-active devices (STMD)athorough overview can be found in [10, 11]. For the mitigationof civil engineering structures, STMDs represent a veryattractive compromise between passive TMDs and ATMDssince they allow the adjustment of their properties to theactual vibration state of the primary structure, to someextent they are fail safe due to their semi-active nature andtheir power requirement is low [12]. In [13], a controllableSTMD is developed based on a controllable viscous damper.In [1416] the frequency adaptation of the STMD is realizedby actively controlled passive spring systems. Also, differentsemi-active materials and devices, respectively, are used forthe development of SMTDs: shape memory alloys are adoptedto make the stiffness of passive TMDs controllable [17],while in [18] an STMD based on piezoelectric materialsis presented, and controlled friction dampers are used forthe adaptation of STMDs [19]. Another class of STMDsis based on real-time controlled MR dampers, which arefound to be very effective [20, 21]. Many different controlapproaches are used for the control of the MR damper in theSTMD, e.g. fuzzy logic is adopted in [22] and a bangbangapproach is presented in [23]. A slightly different approachhas been developed by the authors of this paper, where the MRdamper is used to emulate the superposition of a controlledstiffness force and a controlled damping force [24]. Thestiffness force adjusts the natural frequency of the STMDand the friction force controls the energy dissipation in thedevice as a function of the actual frequency of the primarystructure [2527]. This STMD concept, which is subsequentlyabbreviated as MR-STMD, is numerically and experimentallyvalidated for steady state [25, 26] and transient operatingconditions [27] for single harmonic, narrow band excitationof the primary structure. In fall 2011, the MR-STMD systemwas installed by the industrial partner of this R&D projecton the Wolgograd Bridge to prevent this bridge from severesingle harmonic vibrations such as those observed in May2010. Several videos of this very impressive vibration eventare available on YouTube.

The basic working principle of the MR-STMD is asfollows: if the primary structure vibrates at nominal resonancefrequency, the MR damper is used to emulate amplitudeproportional friction damping that dissipates the same amountof energy as the viscous damper of a passive TMD. However,if the primary structure vibrates at frequencies lower/higherthan the nominal resonance frequency because (i) anothermode is vibrating, (ii) environmental effects change theresonance frequency, (iii) additional loads such as snow orlife loads change the modal mass and thereby the resonancefrequency, or (iv) forced vibrations not at resonance frequencyoccur, then the MR damper emulates a negative/positivestiffness force besides the friction force to adjust the naturalfrequency of the MR-STMD to the actual frequency ofthe primary structure. The control approach to combine thecontrolled stiffness force with a controlled friction force andnot with a controlled viscous force is chosen because thestiffnessfriction combination does not yield active desired

2


Figure 1. MR-STMD concept (a); MR-STMD prototype (b).

control forces as long as the maximum stiffness forceis not greater than the friction force [28, 29], while thestiffnessviscous combination always leads to active desiredcontrol forces [3032]. However, with a large frequency shiftbetween the nominal resonance frequency and the actualfrequency of the primary structure, the stiffnessfrictionapproach also includes active forces which are clipped to zerodue the semi-active constraint of the MR damper. In addition,the residual force of the MR damper at 0 A constrains thedesired stiffnessfriction force. Both control force constraintsend up in imprecise stiffness and damping emulation in theMR damper and thereby imprecise stiffness and dampingtuning of the MR-STMD.

This paper therefore develops compensation methodsboth for precise stiffness and precise damping emulationswhen the desired stiffnessfriction force is constrained byclipping and/or a residual force. Both methods are numericallytested and the precise stiffness emulation approach isexperimentally validated with the MR-STMD concept on the15.6 m long Empa bridge. The structure of the paper isas follows: section 2 describes the working behaviour andcontrol of the MR-STMD and introduces the problem ofimprecise stiffness and damping emulations when controlforce constraints are present. Section 3 develops the stiffnessand damping compensation methods, section 4 shows thenumerical verification of both methods and section 5 describesthe experimental validation. The paper is closed with asummary and conclusions.

2. System description and problem formulation

2.1. MR-STMD concept

The MR-STMD is a semi-active tuned mass damper whosefrequency and damping are adjusted in real-time to the actualfrequency of the primary structure using a real-time controlledMR damper that replaces the passive oil damper of passiveTMDs. As figure 1(a) sketches, the MR damper force consistsof:

a controlled stiffness force to augment or diminish the totalstiffness of the MR-STMD and thereby to adjust the naturalfrequency of the MR-STMD to the actual frequency of theprimary structure

a controlled friction force that dissipates the same amountof energy per cycle as the energy equivalent viscousdamper of the passive TMD; however, the viscouscoefficient is not constant as for passive TMDs butadjusted in real-time to the actual frequency of the primarystructure.

The controlled stiffness force is combined with acontrolled friction force because this does not result inactive forces as long as the maximum stiffness force isnot greater than the friction force. This feature reduces theamount of clipped control forces, whereby a high efficiencyof the MR-STMD is guaranteed. The MR-STMD prototype isrealized with a rotational MR damper (figure 1(b)), where acantilever beam is used to transmit the MR damper torque tothe MR-STMD mass. The passive spring is realized by fourcompression springs. All single parts of the MR-STMD areinstalled within a rack to easily attach the MR-STMD to theprimary structure.

2.2. Control of the MR-STMD

The desired control force fdes to be tracked in real-time by theMR damper is the superposition of the desired stiffness forceand the desired friction force

fdes = kdes xd + Fcdes sgn(xd) (1)where kdes denotes the desired stiffness coefficient, xd is therelative MR-STMD motion, Fc-des is the desired friction forceamplitude and xd is the relative MR-STMD velocity. kdesis computed in real-time as a function of the actual radialfrequency w of the primary structure as follows

kdes = k1 (+ 1)2 k2 (2)

where k1 denotes the stiffness of the primary structure, k2 isthe stiffness of the passive spring of the MR-STMD derivedby Den Hartogs design [1] and is the frequency dependentmass ratio

= (w

1

)2(3)

3


Figure 2. Force displacement (a) and force velocity (b) trajectories not constrained by clipping and/or a residual force(F0 Fc-des |kdes|Xd).

where = m2/m1 is the mass ratio, m2 is the MR-STMDmass and m1 and 1 are the target modal mass and radialresonance frequency, respectively, of the primary structure.Note that has its physical meaning only if w = 1.The desired friction force is controlled in proportion to theMR-STMD relative motion amplitude Xd to dissipate the samecycle energy as the viscous damper of the passive TMD withdesired viscous coefficient cdes, thus

Fc-des = pi4 cdes w Xd (4)

where cdes and the damping ratio 2 are given by the commonformula of passive TMDs but formulated as a function of theactual frequency w of the primary structure

cdes = (2 2 m2 w) (5)

2 =

3

8 (+ 1)3 . (6)

The parameter in (5) represents the damping gain, whichallows a reduction of the desired damping in the MR-STMDto increase its relative motion amplitude. This approachaugments the efficiency of the MR-STMD because it increasesthe force amplitude of the passive spring force which workswith a phase shift of almost 180 at w = 1 against thedisturbing force. Note that this approach may only be adoptedin the environment of real-time frequency adaptation of theSTMD and must not be applied to passive TMDs, whichwould then result in larger responses of the structure. w isdetermined from the zero crossing of xd, since the MR-STMDvibrates at the same frequency as the primary structure(forced vibration). This frequency estimation method worksin real-time for single harmonic vibrations of the primarystructure; in the case of multi-harmonic vibrations, frequencyestimation methods, such as described, for example, in [33],should be used.

2.3. Real-time force tracking control scheme of MR-STMD

The real-time tracking of the desired control force withthe MR damper of the MR-STMD prototype is realized by

a combined feed forward/feedback force tracking controlscheme. The feed forward is used to estimate the MR dampercurrent for a given desired control force adopting an inversemodel of the MR damper. For this, the forward MR damperbehaviour is modelled by the Bingham approach, which isinvertible since it does not model the pre-yield behaviourbut describes the total MR damper force as a superpositionof a strongly current dependent Coulomb friction force andan almost current independent viscous force [34]. The forcetracking error that results from the model-based feed forwardonly is further reduced by a feedback correction of themodel-based estimated current. The feedback is based onthe measured actual MR damper force using the 500 Nload cell, see figure 1(b), and only includes a proportionalfeedback gain. In addition to the feed forward and feedback,negative current is applied each half cycle when the desiredcontrol force is far below the MR damper residual force toreduce remanent magnetization effects that were generated bythe previous magnetization time history. The achieved forcetracking accuracy is shown and discussed in the experimentalpart of the paper in sections 5.3 and 5.4. More detailedinformation on the real-time force tracking scheme can befound in [2527].

2.4. Resulting force displacement trajectories

2.4.1. Zero frequency shift. In case of zero frequency shiftbetween the primary structure and the MR-STMD passivemassspring-system, i.e. w = 1, the control code (1)(6)ends up in pure friction damping with zero stiffness, sincede-tuning between w of the primary structure and the passiveelements k2 and m2 of the MR-STMD is not present. Thedamping generation by controlled friction damping (4) doesnot end up in nonlinear motion of m2 because the maximumpassive stiffness force k2 Xd is approximately eleven timesgreater than Fc-des whereby k2 Xd dominates the motion of m2.

2.4.2. Small frequency shifts. If the shift between w and1 is small, fdes requires the superposition of a controlledfriction force with a controlled stiffness force. Figure 2 plots

4


Figure 3. Force displacement (a) and force velocity (b) trajectories constrained by clipping and/or residual force for F0 < Fc-des andF0 > Fcdes |kdes|Xd.

this situation for w > 1, which uses kdes > 0 to augmentthe total stiffness of the MR-STMD and thereby to adjust theMR-STMD natural frequency to w in real-time. fdes can befully tracked by the MR damper and thereby the actual MRdamper force fact is equal to fdes because:

fdes is fully dissipative since the maximum desired stiffnessforce is not greater than the desired friction force,i.e. |kdes|Xd Fc-des |fdes| is always greater than the residual force F0 of the MR

damper, which is simplified here as a pure Coulomb frictionforce. The authors are aware of the fact that the residualforce of MR dampers is a combination of a predominanthysteretic force with a small superimposed viscous forcedue to the MR fluid and a friction force due to themechanical bearings [35, 36]; however, this simplificationand control-oriented modelling approach, respectively, willlater be used for control purposes.

2.4.3. Large frequency shifts. In contrast to the two casesdiscussed above, large frequency shifts end up in |kdes|Xd >Fc-des which invokes active desired forces in (1). Due tothe semi-active nature of MR dampers, the active forces areclipped to zero, which yields the clipped desired control forceas follows

f clippeddes ={

fdes: fdesxd > 0

0: otherwise.(7)

However, as figure 3 displays, the residual force F0 furtherconstrains f clippeddes , which is seen from |xd0| < |xdc|. Theresulting actual MR damper force then becomes

fact =

F0: f

clippeddes < F0, xd > 0

F0: f clippeddes > F0, xd < 0f clippeddes : otherwise.

(8)

Note that (8) represents the optimal case where the MRdamper current control does not generate any force trackingerrors in those regions where (1) can be tracked, i.e. where (1)

outputs dissipative forces and is not constrained by F0. Thedeviations between fact and fdes end up in:

(1) a smaller equivalent stiffness coefficient kequiv thandesired, which is caused by the horizontal parts in theforce displacement trajectory of fact versus xd with zeroequivalent stiffness

(2) a higher equivalent damping coefficient cequiv than desireddue to clipping and/or F0, which is highlighted by thedashed areas.

Drawback (1) ends up in imprecise frequency tuningand drawback (2) in imprecise damping tuning of theMR-STMD. Both drawbacks will lower the efficiency of theMR-STMD. Stiffness and damping compensation methodsare therefore developed that ensure precise stiffness anddamping emulations with MR dampers even when the desiredcontrol force (1) is constrained by clipping and/or a residualforce. These compensation methods are presented in section 3.

3. Compensation methods for precise stiffness anddamping emulations

3.1. Approach

This section derives closed-form solutions for:

(i) the precise stiffness emulation with MR dampers

(ii) the precise damping emulation with MR dampers

when the desired control force (1) is constrained by clippingand/or a residual force. The closed-form solutions are validfor the following assumptions:

Single harmonic motion of the MR-STMD mass, which isa reasonable assumption considering that large amplitudevibrations in the primary structure most likely occurwhen all the external loading energy, e.g. wind load, istransferred to one mode of vibration; exactly this situationwas observed in May 2010 on the Wolgograd Bridge.

5


The residual force of the MR damper can be simplifiedby the Coulomb friction model, although the residualforce is given by bearing friction of the mechanicalMR damper parts and the hysteretic force and viscousforce of the MR fluid at 0 A [35, 36]. However, thestrongly simplified, i.e. control-oriented Coulomb frictionmodelling approach allows the derivation of closed-formsolutions for both compensation methods and thereby thereal-time applicability of these methods is guaranteed.

The compensation methods are deduced in the followingtwo steps:

(1) Derivation of equivalent stiffness and equivalent dampingcoefficients of the actual MR damper force (8) when thedesired control force (1) is constrained by clipping and/ora residual force.

(2) Derivation of the compensation methods for the precisestiffness and precise damping emulations when thedesired control force (1) is constrained by clipping and/ora residual force.

3.2. Equivalent stiffness and equivalent viscous damping

3.2.1. Desired control force not constrained. The desiredcontrol force fdes is not constrained by clipping and/or aresidual force if F0 Fc-des |kdes|Xd (figure 2). Theequivalent stiffness kequiv and equivalent viscous dampingcoefficient cequiv generated by fact are equal to their desiredcounterparts kdes and cdes, respectively.

3.2.2. Desired control force constrained, case 1. The caseF0 < Fc-des and F0 > Fcdes|kdes|Xd is displayed in figure 3,where the control force constraints, i.e. clipping and/or F0,result in xd0 < 0 for fact > 0. The equivalent stiffness of theconstrained force displacement trajectory of fact is derivedfrom the potential energy that is stored and released by factduring half a cycle [37]. The potential energy V1 that is storedin the damper at maximum displacement is equal to the workdone from point (A) to point (B). For positive desired stiffnesskdes > 0,V1 becomes

V1 = Xd

0fact dxd = Fc-des Xd + 0.5 X2d kdes (9)

and the potential energy release V2 from point (B) to point (D)is

V2 = 0

Xdfact dxd = F0 Xd (Fc-des F0)

2

2 kdes. (10)

The sought equivalent stiffness is kequiv = (V1 + V2)/X2d ,which yields

kequiv = 12kdes + sgn(kdes)Fc-des F0

Xd (Fc-des F0)

2

2 kdes X2d(11)

where the signum function is introduced such that (11) isvalid for positive and negative desired stiffness coefficients.

The equivalent viscous damping coefficient is derived fromthe energy dissipation of fact during one vibration cycle. Theenergy dissipation due to Fc-des is [38, 39]

EF = 4 Fc-des Xd. (12)However, fact dissipates more energy due to clipping and/orthe residual force that invoke the additional energy dissipation1E

1E = |kdes|X2d +(F0 Fc-des)2|kdes| + 2(F0 Fc-des)Xd

(13)

which is highlighted by the two dashed areas in figure 3.The sought equivalent viscous damping coefficient is cequiv =(EF +1E)/(pi d X2d), which yields

cequiv =|kdes|Xd + 2(F0 + Fc-des)+ (F0Fc-des)2|kdes|Xd

pi d Xd. (14)

3.2.3. Desired control force constrained, case 2. The caseF0 Fc-des and F0 < Fc-des + |kdes|Xd is shown in figure 4,where the desired control force (1) is strongly constrained byclipping and/or F0 such that xd0 0 for fact > 0. For kdes > 0

V1 + V2 = Fc-desXd + 12X2d kdes +

(F0 Fc-des)22 kdes

F0 Xd(15)

which yields

kequiv = 12kdes + sgn(kdes)Fc-des F0

Xd+ (Fc-des F0)

2

2 kdes X2d(16)

and cequiv is given by (14).

3.2.4. Desired control force constrained, case 3. The caseof F0 Fc-des + |kdes|Xd results in zero equivalent stiffnesskequiv = 0 and the equivalent damping coefficient cequiv is onlygiven by F0, i.e. cequiv = 4 F0/(pi d Xd).

3.2.5. General expressions for equivalent stiffness andequivalent damping. Comparing the expressions (11) and(16), the only difference is the sign in front of the lastterm. The equivalent stiffness and the equivalent dampingcoefficients of fact (8) in their general forms therefore become

kequiv =

kdes: F0 Fc-des |kdes|Xd12

kdes + sgn(kdes)Fc-des F0Xd

sgn(Fc-des F0) (Fcdes F0)2

2 kdes X2d

:

Fc-des |kdes| Xd < F0< Fc-des + |kdes|Xd

0: F0 Fc-des + |kdes|Xd

(17)

6


Figure 4. Force displacement (a) and force velocity (b) trajectories constrained by clipping and/or a residual force for F0 Fc-des andF0 < Fcdes + |kdes|Xd.

Figure 5. Example of force displacement trajectories before (a) and after (b) stiffness correction.

cequiv =

cdes: F0 Fc-des |kdes|Xd|kdes|Xd + 2(F0 + Fc-des)+ (F0Fc-des)2|kdes|Xd

pi d Xd:

Fc-des |kdes| Xd < F0< Fc-des + |kdes|Xd

4F0pi d Xd

: F0 Fc-des + |kdes| Xd.

(18)

The expressions (17) and (18) demonstrate that the controlforce constraints of clipping and/or F0 result in |kequiv| cdes, which ends up in imprecisefrequency and damping tunings of the MR-STMD. Thefollowing subsection therefore derives stiffness and dampingcompensation methods which are formulated as closed-formsolutions to ensure precise stiffness and damping emulationsin MR dampers in real-time.

3.3. Compensation method for precise stiffness emulation

As seen from figures 3 and 4, the control force constraintsof clipping and/or the residual force result in reducedequivalent stiffness, i.e. |kequiv| < |kdes|, due to the horizontal

trajectory parts. The basic idea therefore is to replace thedesired stiffness kdes in (1) by the corrected desired stiffnesskdes which must be greater than kdes (for positive kdes) tocompensate for the horizontal trajectory parts with zeroequivalent stiffness such that the equivalent stiffness kequivof the entire force displacement trajectory of the correctedactual MR damper force f act is equal to the originally desiredstiffness kdes. This procedure is illustrated in figure 5. Thecorrected desired control force becomes

f des = kdesxd + Fc-dessgn(xd) (19)which yields f act that is constrained by clipping and/or F0. Theequivalent stiffness of f act is

kequiv =12

kdes + sgn(kdes)Fc-des F0

Xd

sgn(Fc-des F0) (Fc-des F0)2

2 kdes X2d(20)

which must be equal to the originally desired stiffness kdes,i.e. kequiv = kdes. Solving (20) with kequiv = kdes yields aquadratic equation whose first root is not feasible. The second

7


Figure 6. Corrected desired friction force versus desired frictionforce.

root yields the sought corrected desired stiffness

(kdes) root2 = sgn(kdes)b+b2 4 a c

2 a(21)

with a = X2d, b = 2 Xd (Fcdes F0 Xd|kdes|) and c =sgn(Fc-des F0)(Fc-des F0)2. Since (21) is only neededwhen f act is constrained by clipping and/or F0, i.e. if Fcdes (Fc-des F0)/Xd.

(22)

Figure 5 depicts the force displacement trajectoriesbefore and after stiffness correction for the case of F0 =10 N and d = 19.85 rad s1. It is seen that the stiffnesscorrection method (22) increases the slope of the forcedisplacement trajectory, due to |kdes| > |kdes|, and therebycompensates for the horizontal trajectory parts with zeroequivalent stiffness such that the equivalent stiffness kequiv ofthe entire corrected force displacement trajectory is equal tothe originally desired stiffness kdes. In contrast, the equivalentstiffness due to (1) with kdes, i.e. without stiffness correction,results in an error of (kdes kequiv)/kdes = 22.3%! Whereasthe stiffness correction generates precise stiffness emulationas targeted, the amplification of the corrected stiffness resultsin augmented cequiv, which is observed from the larger areasurrounded by the force displacement trajectory of f act andfrom the displayed numbers of cequiv and cequiv, respectively.This drawback cannot be avoided because the stiffness forcecannot be controlled independently of the friction force in MRdampers when control force constraints such as clipping andF0 are present.

3.4. Compensation method for precise damping emulation

The control force constraints of clipping and/or a residualforce yield higher energy dissipation in the MR damper thandesired, i.e. cequiv > cdes, which can be seen from the dashedareas in figures 3 and 4. To emulate the correct energydissipation in the MR damper, cdes in (4) must be replacedby the corrected value cdes, which must be smaller than cdesto compensate for the too high energy dissipation of factdue to clipping and/or F0 such that the equivalent dampingcoefficient cequiv of the force displacement trajectory of thecorrected actual MR damper force f act is equal to the originallydesired damping coefficient cdes. Thus, the corrected desiredcontrol force is computed with the corrected friction forceamplitude Fc-des

f des = kdes xd + Fc-des sgn(xd) (23)which yields the equivalent viscous damping coefficient of f actas follows

cequiv =|kdes|Xd + 2(F0 + Fc-des)+

(F0Fcdes)2|kdes|Xd

pi d Xd(24)

which must be equal to the originally desired viscousdamping coefficient cdes, i.e. cequiv = cdes. Solving (24) forthe corrected friction force amplitude Fc-des with cequiv = cdesleads to a quadratic equation with the sought solution asfollows (first root not feasible)

(Fc-des) root2 =b+b2 4 a c

2 a(25)

with a = 1, b = 2(|kdes|Xd F0), c = |kdes|2X2d |kdes|pi d X2d cdes + 2|kdes|XdF0 + F20 and for F0 Fc-des 0. However, the maximum current constraintof 2 A and the limited power of the current driver make itimpossible to achieve zero FTE during this force step. Also,the pre-yield stiffness of the MR damper under considerationlimits how fast the actual force can be increased at thedisplacement extreme. This is indicated by the approximateslope of the pre-yield stiffness in figure 15(a). The pre-yieldstiffness thereby explains that fact cannot track the verticalstep of fdes when MR dampers are operated in the pre-yieldregime and when the maximum current is limited. Furtherinformation on the force tracking accuracy limitation due tothe pre-yield stiffness can be found in [41].

The second main source of FTE is of type FTE(b), whenthe desired control force decreases from a large value tozero, i.e. it comes close to or below the residual force. In

Figure 14. Measured FRFs of the nominal bridge with the MR-STMD and passive TMD at = 1 (a) and = 0.7 (b).

13


Figure 15. Measured real-time force tracking in the MR damper at 2.92 Hz due to C1 ((a)(c)) and C2 ((d)(f)) at = 0.7 for the nominalbridge.

Figure 16. Measured real-time force tracking in the MR damper at 3.28 Hz due to C1 ((a)(c)) and C2 ((d)(f)) at = 0.7 for the nominalbridge.

14


Figure 17. Measured FRFs of lighter bridge with MR-STMD and passive TMD at = 1 (a) and = 0.7 (b).

this case, the FTE(b) is caused by remanent magnetizationeffects due to the previous magnetization time history whenfact was increased from small values to its maximum andthen was lowered again. The remanent magnetization, whichis caused by the magnetic hysteresis, then invokes largerMR damper forces for a certain current level than withoutremanent magnetization. The approach chosen here to reducethis nonlinear and undesirable effect is the application of anegative current during a fairly short time interval and withlimited magnitude, with the goal to cancel the remanent mag-netization as far as possible. The negative current applicationmust be limited in time and magnitude only to cancel theremanent magnetization but not to magnetize the MR fluid anddamper housing in the opposite direction, which would resultin the same dissipative MR damper force as with a positivecurrent. The application of negative current is also modulatedby the proportional feedback, as seen in figures 15(c), (f)and 16(c), (f). In the case of negative stiffness with friction,a negative current can be applied during a fairly large timeinterval, which then results in very low FTE(b), see figure 15,since fdes is slowly reduced. In the case of positive stiffnesswith friction (figure 16), the desired force steps down to zeroat the displacement extreme. Due to the large magnetizationat maximum force, a force overshoot in fact results when fdeschanges its sign. This FTE is also reduced as far as possibleby the application of a negative current. However, the forceovershoot is not fully removed because the available timeinterval of negative current application between the previousforce maximum and step down is simply very short.

Despite the existence of FTEs, the displayed numbers ofthe equivalent stiffness show that C2 generates more precisestiffness emulation than C1, which explains the improvedperformance of C2 compared to C1, as seen from figure 14(b).

5.3.3. Behaviour during transient excitation. When the dis-turbing frequency and amplitude are changed to measure thesteady state response of the bridge at the distinct frequenciesof the plotted FRFs (figure 14), both the precise stiffness anddamping approaches run stable during these transients despitethe fact that both methods rely on the real-time detection ofXd, which is delayed by half a period because Xd is the value

of the previous amplitude, and fw which includes a time delayof approximately 5 s due to the low pass filtering to make C2and C3, respectively, insensitive to measurement noise [27].Consequently, the emulation of the desired stiffness anddamping values during transients include small errors, butthe errors vanish rapidly when steady state vibrations arereached. Large amplitude vibrations of real civil structuresare often predominantly single harmonic vibrations withslowly changing frequencies and amplitudes; as observed, forexample, on the Franjo Tudjman Bridge, where stay cablesoscillated at their fundamental frequencies with mid-spanamplitudes of up to 1 m [42], or the Wolgograd Bridge,where the bridge deck vibrated at 0.45 Hz with amplitudesof up to 40 cm [43]. Thus, C2 improves the frequency tuningof the MR-STMD for most vibration scenarios of real civilstructures, where both frequency and amplitude of the primarystructure will change slowly over time.

5.4. Test results for lighter and heavier bridges

The control laws C1 and C2 at nominal ( = 1) and reduced( = 0.7) damping in the MR-STMD are also tested for alighter bridge and a heavier bridge, which is achieved bychanging the additional local masses on the bridge deck. Theresonance frequency of the lighter bridge is 3.487 Hz, whichcorresponds to a frequency shift of +10.87% relative to thenominal resonance frequency of f1 = 3.145 Hz for which thepassive massspring-system of the MR-STMD is designed.The resonance frequency of the heavier bridge is 2.760 Hz,which corresponds to a frequency shift of 12.24% relativeto f1.

The FRFs of the lighter and heavier bridges are depictedin figures 17 and 18; the real-time force tracking at theselected disturbing frequencies 3.56 and 2.70 Hz is plottedin figure 19. As for the nominal bridge, C2 outperforms C1 bymore when the MR-STMD is operated at reduced dampingwith = 0.7. The measured improvements of C1 and C2relative to the passive TMD at 1.0c2 are 42% and 52% forthe lighter bridge and 40% and 50% for the heavier bridge.The improvements between C1 and C2 with = 0.7 arelarger than those obtained for the nominal bridge because

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Figure 18. Measured FRFs of heavier bridge with MR-STMD and passive TMD at = 1 (a) and = 0.7 (b).

Figure 19. Measured real-time force tracking in the MR damper at 3.56 and 2.70 Hz due to C1 ((a), (d)), C2 ((b), (e)) and emulation ofviscous damping for tests with passive TMD ((c), (f)) for lighter and heavier bridges.

precise frequency tuning becomes especially important whende-tuning between the actual resonance frequency of thebridge and the nominal resonance frequency is present, asfor the lighter and heavier bridges. The measured real-timeforce tracking shows the same two main sources of FTEs asobserved in figures 15 and 16 for the nominal bridge.

6. Summary and conclusions

The MR damper in the semi-active tuned mass damperconcept of the Wolgograd Bridge (MR-STMD) is used to

adjust in real-time both the natural frequency and dampingof the MR-STMD to the actual bridge frequency by theemulation of a combined stiffnessfriction control force.Control force constraints such as clipping and a residualMR damper force lead to a lower equivalent stiffness andhigher equivalent damping of the actual MR damper forcethan desired, and thereby to imprecise frequency and dampingtunings of the MR-STMD.

In order to overcome this drawback, two independentcompensation methods that yield precise stiffness anddamping emulations with MR dampers in the presence of

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control force constraints are presented. Both methods areformulated as closed-form solutions to ensure their real-timeapplicability, and are valid for single harmonic excitationof the MR damper. The numerical verification of bothmethods by the MR-STMD concept demonstrates that theprecise stiffness approach enhances the efficiency of theMR-STMD due to the precise frequency tuning of theMR-STMD, especially when the MR-STMD is operated atreduced damping, where control force constraints becomemore relevant. The precise damping emulation approach turnsout to be inappropriate for the control of the MR-STMD dueto the resulting imprecise frequency tuning of the MR-STMD.The experimental validation of the precise stiffness emulationapproach with the prototype MR-STMD on the 15.6 m longEmpa bridge confirms the numerical findings. It is thereforeconcluded that the precise stiffness emulation approach is apowerful tool to make the MR-STMD more efficient.

Acknowledgments

The authors gratefully acknowledge the financial support ofEmpa, Swiss Federal Laboratories for Materials Science andTechnology, Dubendorf, Switzerland, the financial support ofAGH University of Science and Technology, Department ofProcess Control, Krakow, Poland (statutory research fundsNo. 11.11.130.560), and the technical support of the industrialpartner Maurer Sohne GmbH & Co. KG, Munich, Germany.

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Precise stiffness and damping emulation with MR dampers and its application to semi-active tuned mass dampers of Wolgograd BridgeIntroductionSystem description and problem formulationMR-STMD conceptControl of the MR-STMDReal-time force tracking control scheme of MR-STMDResulting force displacement trajectoriesZero frequency shiftSmall frequency shiftsLarge frequency shifts

Compensation methods for precise stiffness and damping emulationsApproachEquivalent stiffness and equivalent viscous dampingDesired control force not constrainedDesired control force constrained, case 1Desired control force constrained, case 2Desired control force constrained, case 3General expressions for equivalent stiffness and equivalent damping

Compensation method for precise stiffness emulationCompensation method for precise damping emulation

Numerical validationControl laws under considerationSystem parametersDynamic simulationSimulation results for F0=2 NImpact of residual force on the performances of C1 and C2

Experimental validationEmpa bridge with prototype MR-STMDMeasurement and control hardwareTest results for nominal bridgeBridge response due to harmonic excitationForce tracking accuracyBehaviour during transient excitation

Test results for lighter and heavier bridges

Summary and conclusionsAcknowledgmentsReferences

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