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Research ArticleSeismic Response Reduction of Structures Equipped with aVoided Biaxial Slab-Based Tuned Rolling Mass Damper

Shujin Li1 Liming Fu12 and Fan Kong1

1School of Civil Engineering and Architecture Wuhan University of Technology 122 Luoshi Road Wuhan 430070 China2Hubei Synthetic Space Building Technology Co Ltd Wuhan 430070 China

Correspondence should be addressed to Fan Kong kongfanwhuteducn

Received 25 February 2015 Accepted 14 May 2015

Academic Editor Kumar V Singh

Copyright copy 2015 Shujin Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a novel tuned mass damper (TMD) embedded in hollow slabs of civil structures The hollow slabs in thiscontext also referred to as ldquovoided biaxial reinforced concrete slabsrdquo feature a large interior space of prefabricated voided modulesthat are necessary in the construction of this special structural system In this regard a tuned rolling mass damper system(ldquoTRoMaDaSrdquo) is newly proposed in combination with hollow slabs to act as an ensemble passive damping device mitigatingstructural responses The main advantage of this TMD configuration lies in its capacity to maintain architectural integrity Tofurther investigate the potential application of the proposed TRoMaDaS in seismic response mitigation theoretical and numericalstudies including deterministic and stochastic analyses were performed They were achieved by deterministic dynamic modelingusing Lagrangersquos equation and the statistical linearization method Finally the promising control efficacy obtained from thedeterministicstochastic analysis confirmed the potential application of this newly proposed control device

1 Introduction

Civil structures exposed to ambient dynamic excitation oftenexhibit excessive responses that need to be mitigated fromconsiderations of safety andor serviceability issues of thestructures Traditionally relying on the enhancement ofmaterial strength and cross-sectional area structureelementresponses can be reduced to some level However thistreatment of suppressing structureelement responses is noteconomically feasible especially for structures located in thedisastrous earthquake- or gust-prone areas An alternativeand smarter measure to suppress structural response is so-called structural control

Begun by Yao [1] structural control techniques areintended to mitigate structure responses to random ambientexcitation They have been developed for decades and someof them have matured with large advances in both theoreticaland practical aspects The central concept of structuralcontrol theory in fact relies on the changing of structuralparameters or external excitations appropriately passivelyactively semiactively or with combinations of them leadingto mitigated structure responses To date many research

papers and several important textbooks have been publishedin this area such as [2 3] Among these structural controlmethods the most widely used and also most mature ispassive control whereas active control and semiactive controlsuffer from various practical or theoretical challenges thatneed further investigation

Tuned mass dampers (TMDs) attached to the primarystructure consist of a block of mass a stiffness-restoringelement and an energy dissipation element These are nowone of the most widely used passive control devices espe-cially against wind-induced vibration in high-rise buildingsbecause of their advantages in terms of operation mainte-nance and fabrication (see [4] for details)

However the traditional configuration of the TMDsrequires dedicated large space to accommodate the huge scaleof the mass and the corresponding functional componentsproviding damping and stiffness As an example the ChifleyTower in Sydney a 52-storey 209-m tall steel structure isfitted with a single pendulum-type TMD suspended by steelcables at the 44th floor [4] The 400 t block of steel withdimensions of 4 times 4 times 4m and the stroke (about plusmn910mm)of the mass indicates the size of the dedicated space needed

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 760394 15 pageshttpdxdoiorg1011552015760394

2 Shock and Vibration

Moreover the suspension cables need to stretch across twofloors (from the 44th to the 46th floors) to tune the naturalfrequency of the damper to the frequency of the tower Otherexamples include the platform-type TMD system consistingof a concrete mass of 373 t (about 150m3) with a stroke ofplusmn114m employed at the 63rd floor of the 278m tall CiticorpCenter in New York City [4] Even for the more compactconfiguration of the inverted pendulum-type TMD installedat the 100m level of the 134m high Sky Tower in Nagoya [5]Japan the dimensions of the mass block are 25m square times3m with a stroke of 150mm Furthermore the installation ofa TMD as an additional component leads to in some sensean inconsistency in architectural styles

Recently a structure system with a voided biaxial slabhas been used widely in practical engineering especiallyin Europe [6 7] and China [8] Originating from one-spanning hollow-core slabs recent developments on voidedslab technology have been aimed at reducing the deadweight and thus enhancing the span of the structure floorsSpecifically this is achieved by laying less heavy materialsuch as polystyrene or polypropylene made hollow modulesbetween the reinforcement of beamsribbed beams and thusdisplacing concrete with less structural benefit A typicaltechnology used in Europe is the so-called ldquoBubbleDeckrdquobiaxial voided slab [9 10] in which spherical hollowmodulescomposed of recycled industrial plastic are used allowing thehollow slab to act as a normal monolithic two-way spanningconcrete slab For another form of hollowmodule see the ldquoU-bootrdquo [7] technology as an example One of themain featuresof these hollow modules is the large voided interior space

In this context we propose a novel form of distributedTMD that we have named the tuned rolling mass dampersystem (ldquoTRoMaDaSrdquo) for this specific structure takingadvantage of the large space in the voided module that hasnot before been considered for structural control Specificallythe rolling ball TMD first suggested by Pirner [11] andNaprstek and colleagues [12] is for the first time proposedto be distributed in the hollow floors of the structures Thisarrangement of the TMDs is expected to not only suppressthe structure response but also maintain the consistency ofthe architectural style in the sense of architectural esthetics

To validate the control efficacy of the proposed passivecontrol system a physical and mathematical model of thecontrolled chain-likeMDOF structure was developed In thisregard Lagrangersquos equation was used to derive the coupledequations of motion of the controlled system in which thestroke of the rolling mass is considered a small quantityand nonlinearity is exhibited in the damping force Thederivation of the dynamic equation from the physical modelis quite general in which any TRoMaDaS with arbitraryparameters can be considered The Runge-Kutta method ofthe fourth order was used to solve the dynamic equationsnumerically for cases with different TRoMaDaS parametersor distributions We conclude that for all cases with theequipment of the proposed TRoMaDaS the input energyof an ambient excitation is absorbed to a large degreeFurthermore it has been found that because of the ldquostick-sliprdquo behavior of friction-type TMDs [13] the peak value ofresponse cannot be efficiently reduced whereas the postpeak

response can in the context of earthquake excitations Inthis study we also show that different vertical distributionsand parameter configurations of the TRoMaDaS lead todifferent control efficiencies necessitating a more completeinvestigation of the optimum design of TRoMaDaS in futurestudies Finally to investigate the control efficiency from aprobabilistic perspective a statistical linearization methodalong with a pertinent Monte Carlo simulation was used tocalculate the standard deviation of the response displacementof a structure with and without the proposed TRoMaDaSThe stochastic analysis of the structure revealed that not onlywas the displacement standard deviation reduced to a largedegree but also the stochastic response of the structure readilybecame stationary because of the TRoMaDaS

2 Voided Biaxial Slab-Based Tuned RollingMass Damper

21 Structural System with Voided Biaxial Slabs The voidedbiaxial reinforced concrete slab constructed using prefabri-cated voidedmodules and cast in situ reinforced concrete hasgained wide attention since its emergence as a new structuralsystem [6ndash10] These ecofriendly voided biaxial slabs havemany advantages over conventional solid concrete slabs suchas lower total cost reduced material use enhanced structuralefficiency and decreased construction time that have con-tributed to their wide application in civil construction

Taking one kind of voided biaxial slab used widely inChina as an example the construction procedures for thisstructural system are as follows First the prefabricated hol-low box-like modules are located between the reinforcementgrids of the main beams and the ribbed beams Next box-like voided modules are used as the side formwork whensite-casting the concrete beams In this way only the bottomformworks of the concrete slab are needed thus savingconsiderable construction costs To guarantee the integritybetween the prefabricated hollow modules and the circum-jacent reinforced concrete nonstructural measures such asextending the reinforcement bar into the main beamsribbedbeams may be implemented The vertical dimension of aprefabricated hollow module according to the spans of mainbeams ranges from 200 to 900mm Figure 1 shows the con-struction procedures of a kind of voided biaxial slab in detail

22 Motivation of Voided Biaxial Slab-Based Tuned RollingMass Damper Wenoted the unique large space in the hollowmodules and thus proposed to locate certain structural orarchitectural elements within these modules Specificallythese additional elements can be designed as a part of heatingventilation and air conditioning water supply and sewerageand structural control systems without altering the architec-tural appearance excessively Among these architectural andstructural requirements structural safety and serviceabilityare the most important issues In this regard we proposedto install passive control devices in the hollow modules tomitigate the dynamic responses of the structure

Among the passive control devices the passive vibra-tion absorber or the tuned mass damper (TMD) has been

Shock and Vibration 3

(a) (b)

Figure 1 Construction procedures for a hollow-ribbed floor (a) Locate prefabricated hollow box-like modules between the reinforcementgrids (b) Cast reinforced concrete beams between the box-like hollow modules

researched widely and used extensively in newly constructedbuildings and in the retrofitting of existing structuresbecause of its simplicity relatively low construction andmaintenance costs and safety In this regard TMDs of thependulum type [4] are the most widely used especiallyfor slender towers and high-rise buildings However thisconfiguration requires dedicated large vertical and horizontaldimensions to accommodate the extra weight as mentionedin Section 1 especially for high-rise buildings with a lownatural fundamental frequency Moreover safety measuressuch as mechanical stops or other braking systems are alsorequired to limit the excessive travel of the mass block

An alternative TMD configuration is the type mountedon a large complex mechanical platform [14 15] whichprovides damping and stiffness using hydrostatic bearingshydraulic cylinders pneumatic springs or coil springs It canbe argued that compared with TMDs of the pendulum typethe platform-type has increased complexity and thus costsmore in terms of daily operations and maintenance Thuswe concluded that neither of these TMD configurations wassuitable for a vibrationmitigation device to be installed in themodules of the voided biaxial slab

Another more compact and also simple configuration isthe onewith an object rolling on a tridimensional surfaceThedampingmechanism and restoring force of this configurationare provided by the rolling friction and the resulting forceof the gravity of the rolling mass respectively Specificallythe so-called tuned rolling mass damper (TRMD) or the ballvibration absorber (BVA) consists of a ball rolling along anarch path located in hollow modules to absorb the structuralkinetic and potential energy as the result of the ambientexcitations In fact the TRMD has been applied in severalother engineering structures such as long-span bridges [16]TV towers [11] and wind turbines [17 18] However to ourknowledge very few applications of TRMDs or BVAs inbuilding structures to suppress seismic vibrations have beenreported except those proposed recently by Fisher and Pirner[19] and Matta et al [20] Moreover compared with theldquotraditionalrdquo configuration of TMDs in high-rise buildingswhere the TMDs are always installed as an additional energy-absorbing element the most remarkable advantage of theinstallation of TRMDs in hollow modules embedded in slabs

Figure 2 Voided biaxial slab-based tuned rolling mass damper

lies in its architectural integrity By locating the passivecontrol device in an existing space (shown in constructionsite artwork Figure 1) installation of the voided biaxial slab-based tuned rollingmass damper system (ldquoVBS-TRoMaDaSrdquoor ldquoTRoMaDaSrdquo) does not alter the architectural interiorappearance excessively A diagram of a TRoMaDaS with asingle rectangular hollow module is shown in Figure 2

23 Characteristics of TRoMaDaS By appropriately selectingparameters including the radius of the arc path and theoscillator (the natural frequency of the TRoMaDaS dependson these parameters as shown in Section 3) the frictioncoefficient between the arc path and the oscillator (thedamping capacity of the TRoMaDaS depends on this) andthe materialdensity of the oscillator an optimum con-trolled structure can be obtained From the review aboveit can be observed that compared with pendulum-typeand platform-type TMDs the energy dissipation mechanismof the TRoMaDaS is more compact and simpler Thesepromising features naturally lead to a more durable passivecontrol system and reduce considerably the cost of fabrica-tion maintenance and operation Moreover the proposedTRoMaDaS is expected to suppress the coupled lateral andtorsional motions of structures due to accidental or intendedeccentricities between their mass and stiffness centers Inthis regard Singh et al [21] investigated the efficiency offour TMDs of the dashpot-spring type placed along twoorthogonal directions in pairs tomitigate the torsionally cou-pled responses of irregular multistorey buildings Jangid andDatta [22] conducted a parametric study of the effectivenessof multi-TMDs along the width of an eccentric structurein reducing torsionally coupled responses However both

4 Shock and Vibration

TRoMaDaS Voided

biaxial slab

Fn

Fnminus1

F2

F1

mn1

mnminus11

m21

m11

cn

c2

c1

Mn

kn

Mnminus1

M2

k2

M1

k1

middot middot middot

middot middot middot

middot middot middot

middot middot middot

mnn119899

m2n2

m1n1

Figure 3 Lumped-mass model of a shear-type structure equippedwith a TRoMaDaS

investigations focused on a theoretical analysis of mitigatingtorsionally coupled responses using ideal physical modelswhereas the practical feasible multi-TMDs configuration hasnot been considered

In this aspect the proposed voided biaxial slab-basedTRoMaDaS can be used as a platform to realize opti-mally controlled structures with various damper distri-butionsarrangement and parameters Specifically becausethere is no restriction on the direction of the rolling motionor the locationdistribution of the oscillator the proposedTRoMaDaS can in fact be fabricated as a type of spatialmulti-TMDs system that can be used to control structuremotion in every direction and even torsional motion causedby eccentric structures or bidirectional base excitationsMoreover to fully use the spatial movement of the oscillatora three-dimensional TRoMaDaS with a nonaxially symmet-rical three-dimensional path surface can be used to controlmotions of spatial structures with different natural modefrequencies in twoorthogonal directions simultaneouslyThisconcept was initially proposed by Matta et al [20] in thecontext of rolling-pendulumTMDs and can be used to extendthe possible application of the TRoMaDaS in structures withdifferent frequencies in two orthogonal directions

3 Equation of Motion of a ControlledStructure with TRoMaDaS

31 Simplified Model of a TRoMaDaS-Controlled MDOFStructure Consider a mass-lumped model of a shear-typestructure equipped with a TRoMaDaS in each hollow floor(Figure 3) where (11987211198722 119872119899) (1198961 1198962 119896119899) and(1198881 1198882 119888119899) are the mass stiffness and the damping coef-ficient of the main structure respectively and 119865

119895 119895 =

1 2 119899 are the external ambient excitations includingthe equivalent force of base excitation or wind loading For

Initial state

y

A

A

x

B

Rij

120579ij

120579ij

120596ij

rij

120595ij

fij

mijg Nij

Figure 4 Free-body diagram of a TRoMaDaS

simplicity the TRoMaDaS in each floor contains severaloscillators with different masses (119898

1198941 1198981198942 119898119894119899119894) rollingalong the axial-symmetrical sphere surfaces where 119898

119894119895

represents the 119895th oscillator located on the 119894th floor FromFigure 3 it can be seen that the total degrees of freedomare 119899 + (1198991 + 1198992 + sdot sdot sdot + 119899

119899) with 119899 degrees of freedom of

translationalmotion of the lumpedmass and (1198991+1198992+sdot sdot sdot+119899119899)degrees of freedom of the oscillatorsrsquo rotation with respectto the centers of the arch paths respectively A free-bodydiagram of the TRoMaDaS is shown in Figure 4 where 119877

119894119895

119903119894119895 119894 = 1 2 119899

119894 119895 = 1 2 119899

119894 are the radii of the (119894 119895)th

arc path and oscillator respectively 120579119894119895 120595119894119895are the rotation

angles of the oscillator with respect to the center of the pathand of the oscillator ball itself and 120596

119894119895= 119894119895

denotes theangular velocity of the oscillator

32 Governing Equation of the TRoMaDaS-Controlled Struc-ture Thus the kinetic and potential energy of the controlledstructure can be written as

119879 =

119899

sum

119894=1

119899119894

sum

119895=1

12119898119894119895[119894+ 120588119894119895

120579119894119895cos 120579119894119895]

2

+

12119898119894119895[120588119894119895

120579119894119895sin 120579119894119895]

2+

121198691198941198951205962119894119895+

121198721198942119894

119894 = 1 2 119899 119895 = 1 2 119899119894

(1)

119881 =

119899

sum

119894=1

12119896119894(119909119894minus119909119894minus1)

2+

119899

sum

119894=1

119899119894

sum

119895=1119898119894119895119892120588119894119895(1 minus cos 120579

119894119895)

119894 = 1 2 119899 119895 = 1 2 119899119894

(2)

respectively where 119909119894 119894are the translational displacement

and velocity of the lumped mass 120588119894119895

= 119877119894119895minus 119903119894119895is the radius

difference between the arc path and the oscillator 120579119894119895is the

angular velocity of the oscillator with respect to the center ofthe arc path and 119869

119894119895= 211989811989411989511990321198941198955 is the rotational inertia of

Shock and Vibration 5

the 119895th oscillator on the 119894th floor Furthermore (1)-(2) can besimplified to

119879 =

119899

sum

119894=1

119899119894

sum

119895=1

12119898119894119895(

2119894+ 2119894120588119894119895

120579119894119895+120588

2119894119895

1205792119894119895)

+

151198981198941198951205882119894119895

1205792119894119895+

121198721198942119894

119894 = 1 2 119899 119895 = 1 2 119899119894

119881 =

119899

sum

119894=1

119899119894

sum

119895=1[

12119896119895(119909119895minus 119909119895minus1)

2+

121198981198941198951198921205881198941198951205792119894119895]

119894 = 1 2 119899 119895 = 1 2 119899119894

(3)

when the angular motion of the oscillator (120579119895) is considered

to be small Note in (3) that the compatibility condition ofdisplacement 119877

119894119895120579119894119895

= (120579119894119895

+ 120595119894119895)119903119894119895

is used It may beargued that the small-quantity assumption for the oscillatorresponse and (3) tend to be reasonable in cases of lower-levelexternal excitations However it should be noted that notonly may the structure undergo nonlinear behavior but alsothe oscillator response may surpass a certain small-quantitylevel in the case of intense excitations Although the small-quantity assumption is important and often used in the sim-plification of governing equations of pendulum- or rolling-pendulum-type TMDs further examination of the applica-bility of this assumption during intense external excitation isnecessary

The nonconservative force 119876nc119895 including the external

force the damping force of the structure and the rollingfriction force between the sphere surface and the oscillatorcan be derived by the virtual work principle that is

120575119882 =

119899

sum

119894=1119865119894120575119909119894

+ [minus (119894minus 119894minus1) 119888119894 + (

119894+1 minus 119894) 119888119894+1] 120575119909119894

minus

119899

sum

119894=1

119899119894

sum

119895=1119891119894119895

120575119904119894119895

119903119894119895

119894 = 1 2 119899 119895 = 1 2 119899119894

(4)

where 120575119882 denotes the total virtual work of the nonconserva-tive force 119904

119894119895= 119877119894119895120579119894119895

= 119903119894119895(120595119894119895+120579119894119895) is the arc length of the

oscillator movements between the initial state and state beingconsidered and119891

119894119895is themoment of the rolling friction force

with respect to center of the oscillator and can be determinedby

119891119894119895

= minus

120579119894119895

10038161003816100381610038161003816

120579119894119895

10038161003816100381610038161003816

120583119894119895119873119894119895

= minus

120579119894119895

10038161003816100381610038161003816

120579119894119895

10038161003816100381610038161003816

120583119894119895119898119894119895(119892 cos 120579

119894119895+120588119894119895

1205792119894119895)

(5)

In (5) 119873119894119895

is the 119895th reaction force perpendicular to thetangent of the contact point on the 119894th floor and 120583

119894119895is the

coefficient of the rolling friction (in meters) between the 119895thoscillator and the sphere surface on the 119894th floor Combining(4)-(5) the nonconservative force can be written as

119876nc119904119894

= 119865119894+ [minus (

119894minus 119894minus1) 119888119894 + (

119894+1 minus 119894) 119888119894+1]

119894 = 1 2 119899(6)

119876nco119894119895

= minus

120579119894119895119877119894119895

10038161003816100381610038161003816

120579119894119895

10038161003816100381610038161003816119903119894119895

120583119894119895119898119894119895119892

119894 = 1 2 119899 119895 = 1 2 119899119894

(7)

In fact (6) is the expression for the viscous damping forceand the external excitation of main structures whereas (7)denotes the rolling friction force between the oscillator andthe sphere surface

The governing equation of the controlled structure can bederived by invoking the Lagrange equation

dd119905

(

120597119879

120597 119902119894

)minus

120597119879

120597119902119894

+

120597119881

120597119902119894

= 119876nc119894

119894 = 1 2 2119899 + (1198991 + 1198992 + sdot sdot sdot + 119899119899)

(8)

where 119902119894 119902119894are the generalized displacement and velocity of

the 119894th coordinate Combining (3) and (8) and consideringthe angular displacement and velocity (120579

119895

120579119895) to be small

quantities one can obtain

(119872119894+

119899119894

sum

119895=1119898119894119895) 119894+

119899119894

sum

119895=1119898119894119895120588119894119895

120579119894119895+ (119909119894minus119909119894minus1) 119896119894

minus (119909119894+1 minus119909

119894) 119896119894+1 + (

119894minus 119894minus1) 119888119894 minus (

119894+1 minus 119894) 119888119894+1

= 119865119894

(9a)

120579119894119895+

5120583119894119895119892119877119894119895

71205882119894119895119903119894119895

sgn ( 120579119894119895) +

5119892120579119894119895

7120588119894119895

= minus

57120588119894119895

119894119895

119894 = 1 2 119899 119895 = 1 2 119899119894

(9b)

where sgn(sdot) denotes the sign function Equations (9a) and(9b) are a set of coupled nonlinear ordinary differentialequations even with the small-quantity assumption of therollingmotionOn the left side of (9a) the terms

119894sum119899119894

119895=1 119898119894119895+

sum119899119894

119895=1 119898119894119895120588119894119895

120579119894119895can be regarded as the control force produced

by the translational and rotational motion of the oscillatorcounteracting part of the external force 119865

119894 Moreover the

right side of (9b) shows that the motion of the oscillatorsis caused by the supported base acceleration which is theacceleration of the main structureThe left side of (9b) showsthe undamped natural frequency of the 119895th oscillator on the119894th floor depends only on the radius difference between thearch path and the oscillator and can be written as

120596119894119895

= radic

51198927120588119894119895

(10)

6 Shock and Vibration

This expression is consistent with that derived by Zhang andcolleagues [17] and Chen and Georgakis [18] Equations (9a)and (9b) can be rewritten in a compact form

Mz+Cz+Kz+ f (z z) = w (119905) (11)

where

M = [

M1 M2

M3 M4]

C = [

C1 00 0

]

K = [

K1 0

0 K2]

(12)

are the mass damping and stiffness coefficients the aug-mented displacement is z = [x119879 120579119879

1 120579119879

2 120579

119879

119899]119879 with x =

(1199091 1199092 119909119899)119879 1205791 = (12057911 12057912 12057911198991)

119879 1205792 = (12057921 12057922

12057921198992)119879 and 120579

119899= (120579

1198991 1205791198992 120579119899119899119899)119879 the external

excitation on the structure at the right side of (11) is w =

[w11987911990400 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198991+1198992+sdotsdotsdot+119899119899

]119879 with w

119904= (1198651 1198652 119865119899)

119879 and the nonlin-

ear damping force can be written as

119891 (z z) = [

[

0 0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

51205831111989211987711

712058821111990311

sdot sgn ( 12057911)

51205831211989211987712

712058821211990312sgn (

12057912) 51205831119899111989211987711198991712058821119899111990311198991

sdot sgn ( 12057911198991)

512058311989911198921198771198991

7120588211989911199031198991

sgn ( 1205791198991)

512058311989921198921198771198992

7120588211989921199031198992

sdot sgn ( 1205791198992)

5120583119899119899119899

119892119877119899119899119899

71205882119899119899119899

119903119899119899119899

sgn ( 120579119899119899119899

)]

]

(13)

Furthermore submatrices in (12) are

M1 = diag(1198721 +1198991

sum

119895=111989811198951198722 +

1198992

sum

119895=11198982119895 119872119899

+

119899119899

sum

119895=1119898119899119895)

M2 = diag (M12M

22 M

119899

2)

M1198942 = (11989811989411205881198941 11989811989421205881198942 119898119894119899119894120588119894119899119894)1times119899119894

M3 = diag (M13M

23 M

119899

3)

M1198943 =57(

11205881198941

11205881198942

1120588119894119899119894

)

119879

119899119894times1

M4 = I

K2 = diag (K12K

22 K

119899

2)

K1198942 =51198927

sdot diag( 1

1205881198941

11205881198942

1120588119894119899119894

)

K1 = (

1198961 + 1198962 minus1198962

minus1198962 1198962 + 1198963 minus1198963

d

minus119896119899

119896119899

)

C1 = (

1198881 + 1198882 minus1198882

minus1198882 1198882 + 1198883 minus1198883

d

minus119888119899

119888119899

)

(14)

where diag( ) denotes the diagonal matrixThe nonlinear dif-ferential equation (11) can be solved by numerical algorithmsfor example a fourth-order Runge-Kutta method

33 Numerical Examples Consider a six-floor structure withthe lumped-mass model (Figure 1) subject to earthquakeexcitations In the present numerical example the externalexcitation in this case can be written as 119865

119894= minus(119872

119894+

sum119899119894

119895=1 119898119894119894)119892(119905) where 119892(119905) is the earthquake acceleration

each lumped mass is assumed to be the same at 1632 t thelinear stiffness coefficients of each storey are 1351 1286 1158965 707 and 386 times 103 kNm and the damping coefficientsof each storey are selected as 279 2376 2139 1787 1307 and1004 kNsdotsm To reduce the dynamic response of a MDOFstructure efficiently the natural frequency of the damperneeds to be resonant with the structurersquos natural frequencies(628 1538 2432 3323 4213 and 5102 rads) In thisrespect the natural frequency of the TRoMaDaS can be tunedto the first-mode frequency by which the structure responseis dominated In fact the optimum physical parameters ofthe tuned mass damper including the frequency and thedamping coefficient in this case relative to the rolling frictioncoefficient of the oscillator are functions of the mass ratiobetween the main structure mass to the damper mass andother structural parameters It can be argued that by onlytuning the frequency of the control device to the first-modefrequency of the structure preferable control efficiency canbe expected to result in the case of a low mass ratio [23]Research on control parameter optimizationmdashin this casethe selection of the optimum oscillator mass rolling frictioncoefficient and the radii of the arc path and the oscillatormdashisbeyond the scope of the present paper Invoking (10) and thefirst-model frequency of the structure the radius differencebetween the arch paths and the oscillator can be obtainedThe oscillators located in certain floors of the structure areassumed to be composed of iron with a mass density of7800 kgm3 the rolling friction coefficients between the archpaths and the oscillators are chosen to be identical 001m

Shock and Vibration 7

0 10 20 30 40 50

0

2

4

6

8

Time (s)

Disp

lace

men

t (m

)

Without TRoMaDaSWith TRoMaDaS

minus8

minus6

minus4

minus2

times10minus3

(a)

0 10 20 30 40 50

0

2

4

6

8

Time (s)

Disp

lace

men

t (m

)

Without TRoMaDaSWith TRoMaDaS

minus8

minus6

minus4

minus2

times10minus3

(b)

Figure 5 Comparison between interstorey drift ((a) first storey (b) sixth storey) of structures with (solid line) and without (dashed line)TRoMaDaS Structure subjected to scaled El Centro acceleration (peak acceleration = 0513ms2) TRoMaDaS oscillators with a mass ratioof 5 are distributed equally on every floor

The mass ratio of the TRoMaDaS mass to the total massof main structure according to engineering considerationscould be selected to be 5 Four cases with different oscillatordistributions or different levels of earthquake excitationswereconsidered

(a) Oscillators are equally distributed on six floors ofstructures subject to El Centro record NS componentof the 1940 Imperial Valley earthquake (El Centroexcitation for short) and JMA record NS componentof the 1995 Kobe earthquake (Kobe excitation forshort) with a peak value of 0513ms2

(b) The damper and structure parameters are the same asin case (a) whereas the peak values of El Centro andJMA Kobe excitation increase to 1026ms2

(c) The structure and excitation parameters are the sameas in case (b) except that oscillators with mass ratiosof 3 and 2 are installed on the third and the fourthfloors respectively

(d) All the parameters are the same as in case (c) exceptthat the oscillator on the fourth floor is split into 10smaller oscillators with equal mass

Figures 5 and 6 show the comparison of inter-storey driftin case (a) with and without the TRoMaDaS of structuressubject to El Centro and Kobe excitations From these plotsit can be seen that the peak responses (for the El Centrocase at about 3-4 s and for the Kobe case at about 7ndash9 s)were not reduced significantly whereas responses duringthe time interval after the peak value decreased to a largedegree because of the energy dissipation mechanism of the

TRoMaDaS This phenomenon of response reduction delaycan be also observed from the displacement data of a shakingtable experiment of a wind turbine structure controlled bya similar ball vibration absorber (BVA) rolling at the topof the nacelle [17] The failure in peak response reductionmay be attributable to several reasons the most significantis the stick-slip [13] behavior of ball oscillators The steel balloscillators initially resting at the bottomof the arch path onlybegin to move along the paths until the acceleration of themain structure exceeds the maximum acceleration of the balloscillator caused by the friction forcemoment That is theTRoMaDaS can be regarded as being ldquoactivatedrdquo only whenthe oscillator starts rolling along the arch path after strongstructure acceleration has already occurred thus leading toa delay in response reduction and the failure of peak-valuemitigation The maximum absolute values of the interstoreydrift of the controlled and uncontrolled structure and thecorresponding response reduction rates shown in Table 1further confirm the response reduction delay effect Also therotation angles of the oscillators shown in Table 1 do notexceed the limitation of the small quantity assumption whichis often regarded as 03 rad (or about ca 20∘)

The effect of the TRoMaDaS on mitigating the responseduring the postpeak interval may benefit the damage behav-ior of structures because of material fatigue One of thewidely used approaches to investigate damage behavior isbased on the amount of energy absorbed by the materialsTo further investigate the effectiveness of TRoMaDaS theenergy dissipation efficiency of the controlled structure wasinvestigated Figure 7 shows the amount of total energy inputby the earthquake excitation and dissipated by the structural

8 Shock and Vibration

Table 1 Peak response of inner-storey drifts of structures with and without TRoMaDaS with an earthquake acceleration peak value of0513ms2

Excitation Storey 1 2 3 4 5 6

El Centro

Max uncontr disp (mm) 629 652 681 706 713 769Max contr disp (mm) 602 630 677 698 692 673Disp red rate () 429 337 058 113 294 1248

Max ang disp (rad) 00073 00077 00087 00268 00714 01870

Kobe

Max uncontr (mm) 735 739 735 785 878 1002Max contr disp (mm) 617 624 645 691 783 864Disp red rate () 1605 1556 1224 1197 1082 1377

Max ang disp (rad) 00063 00069 00099 00180 00621 01865

0 10 20 30 40 50 60 70 80

0

2

4

6

8

Time (s)

Disp

lace

men

t (m

)

Without TRoMaDaSWith TRoMaDaS

minus8

minus6

minus4

minus2

times10minus3

(a)

0 20 40 60 80

0

0005

001

Time (s)

Disp

lace

men

t (m

)

Without TRoMaDaSWith TRoMaDaS

minus001

minus0005

(b)

Figure 6 Comparison between the interstorey drift ((a) first storey (b) sixth storey) of structures with (solid line) and without (dashed line)TRoMaDaS Structure subjected to scaled Kobe acceleration (peak acceleration = 0513ms2) TRoMaDaS oscillators with a mass ratio of 5are distributed equally on every floor

damping and by the TRoMaDaS It can be seen that for theEl Centro and Kobe excitation respectively about 375 and401 of the input energy were dissipated by the TRoMaDaS

To further investigate the response reduction efficiencyof the TRoMaDaS in case (b) of intensive earthquake exci-tations scaled base acceleration with a peak value 1026ms2was considered The maximum absolute interstorey drifts ofthe controlled and uncontrolled structures to the El Centroand Kobe excitation are shown in Table 2 However themaximum absolute rotation angle of the oscillator in thetop floor reached 105 rad for the El Centro excitation and101 rad for Kobe excitation These values obviously exceedthe limitation of the small quantity assumption made in thederivation of the governing equations and thus the dataobtained in this situation cannot be used to assess the per-formance of the TRoMaDaS in reducing structure responses

In the parameter optimization procedure of a ldquotradi-tionalrdquo tuned mass damper the stroke of the damper is an

important factor that should be considered In this situationthe stroke of the oscillator say the rotation angle of the ballneeds more investigation not only from the point of view ofthe small quantity assumption for rotation angle but moreimportantly also from the aspect of practical considerationsAccording to numerical investigations the rotation angle ofthe oscillators can be reduced by properly locating theseoscillators on certain floors For example in case (c) theTRoMaDaS is installed in the third and the fourth floor with3and 2of the structuremass respectively Comparedwithcase (b) the results of case (c) show that although the controlefficiency in mitigating the peak response deteriorated therotation angle was reduced greatly by about 50 The rota-tion angle of oscillators can be further reduced by splittingthe oscillator with excess angular displacement For examplein case (d) the oscillator in the fourth floor is split into 10smaller ball oscillators with equal mass In this situationwith a slight expense in terms of a deteriorated response

Shock and Vibration 9

Table 2 Peak response of inner-storey drifts of structures with and without TRoMaDaS with an earthquake acceleration peak value of1026ms2

Excitation Storey 1 2 3 4 5 6

El Centro

Max uncontr disp (mm) 1259 1304 1362 1411 1427 1538

Max contr disp (mm)(b) 1036 1079 1146 1181 1156 1314(c) 1150 1223 1274 1275 1233 1301(d) 1206 1279 1328 1324 1268 1300

Disp red rate ()(b) 1771 1725 1586 1630 1899 1456(c) 866 621 646 936 1360 1541(d) 421 192 250 617 1114 1547

Max ang disp (rad)(b) 00075 00270 00993 03191 06343 10499(c) mdash mdash 02409 05213 mdash mdash(d) mdash mdash 02730 02816 mdash mdash

Kobe

Max uncontr disp (mm) 1470 1478 1471 1570 1756 2004

Max contr disp (mm)(b) 1203 1225 1271 1359 1532 1636(c) 1240 1255 1303 1387 1568 1301(d) 1245 1263 1311 1392 1567 1301

Disp red rate ()(b) 1816 1712 1360 1344 1276 1836(c) 1565 1509 1142 1166 1071 3508(d) 1531 1455 1088 1134 1076 3508

Max ang disp (rad)(b) 00061 00309 01199 03082 06185 10127(c) mdash mdash 02216 05352 mdash mdash(d) mdash mdash 02618 02593 mdash mdash

0 10 20 30 40 500

500

1000

1500

Time (s)

Ener

gy (J

)

Absorbed energy

InputStructure dampedStructure damped + kineticStructure damped + strain + kinetic

(a)

0 10 20 30 40 50 60 70 800

500

1000

1500

2000

Time (s)

Ener

gy (J

)

Absorbed energy

InputStructure dampedStructure damped + kineticStructure damped + strain + kinetic

(b)

Figure 7 Cumulative energy time history for TRoMaDaS-damped MDOF structure Structure subjected to scaled El Centro (shown in (a)with peak acceleration = 0513ms2) and Kobe (shown in (b) with peak acceleration = 0513ms2) acceleration TRoMaDaS oscillators with amass ratio of 5 are distributed equally on every floor

reduction the angular displacement is greatly reduced belowthe limitations of the small quantity assumption Figures 8and 9 show a comparison of interstorey drift of the first andthe top floor with and without the TRoMaDaS in case (d)Figure 10 shows the efficiency of the TRoMaDaS in case (d)

from the energy perspective From these plots it was furtherconfirmed that for both the El Centro and Kobe excitationthe peak values were not mitigated significantly whereasover 50 of the input seismic energy was dissipated by theproposed TRoMaDaS

10 Shock and Vibration

0 10 20 30 40 50

0

0005

001

0015

Time (s)

Disp

lace

men

t (m

)

Without TRoMaDaSWith TRoMaDaS

minus0015

minus001

minus0005

(a)

Without TRoMaDaSWith TRoMaDaS

0 10 20 30 40 50

0

0005

001

0015

002

Time (s)

Disp

lace

men

t (m

)

minus0015

minus001

minus0005

(b)

Figure 8 Comparison between interstorey drift ((a) first storey (b) sixth storey) of structures with (solid line) and without (dashed line)TRoMaDaS Structure subjected to scaled El Centro acceleration (peak acceleration = 1026ms2) TRoMaDaS oscillators with a mass ratioof 5 are installed on the third (one ball) and fourth floors (10 balls with equal mass) of 3 and 2 of the main structure mass respectively

Without TRoMaDaSWith TRoMaDaS

0 20 40 60 80

0

0005

001

0015

Time (s)

Disp

lace

men

t (m

)

minus0015

minus001

minus0005

(a)

Without TRoMaDaSWith TRoMaDaS

0 20 40 60 80

0

0005

001

0015

002

Time (s)

Disp

lace

men

t (m

)

minus002

minus0015

minus001

minus0005

(b)

Figure 9 Comparison between the interstorey drift ((a) first storey (b) sixth storey) of structures with (solid line) and without (dashed line)TRoMaDaS Structure subject to scaled Kobe acceleration (peak acceleration = 1026ms2) TRoMaDaS oscillators are installed on the third(one ball) and the fourth floor (10 balls with equal mass) of 3 and 2 of the main structure mass respectively

4 Stochastic Dynamic Analysis

It can be seen from the numerical analysis above that becauseof the uncertain input of the earthquake excitation the con-trol efficiency of the TRoMaDaS appears to differ for differentexcitations even with the same intensity (eg compare case(d) with the El Centro and Kobe excitations) Using thenonlinear random vibration theory one can investigate the

control efficiency of the proposed device in a different prob-abilistic manner Furthermore the optimization of controlparameters in future investigations should be implementedsimilarly based on a stochastic perspective

Stochastic dynamic analysis as an engineering applicationwas begun in 1958 by Crandall [24] for mechanical engineer-ing This theory was soon used to investigate the randomresponses of civil structures to quantify the uncertainty

Shock and Vibration 11

0 10 20 30 40 500

1000

2000

3000

4000

5000

6000

7000

Time (s)

Ener

gy (J

)

Absorbed energy

InputStructure dampedStructure damped + kineticStructure damped + strain + kinetic

(a) El Centro excitation

0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

Time (s)

Ener

gy (J

)

Absorbed energy

InputStructure dampedStructure damped + kineticStructure damped + strain + kinetic

(b) Kobe excitation

Figure 10 Cumulative energy time history for TRoMaDaS-damped MDOF structure Structure subjected to scaled El Centro (shown in (a)with peak acceleration = 1026ms2) and Kobe (shown in (b) with peak acceleration = 1026ms2) acceleration TRoMaDaS oscillators areinstalled in the third (one ball) and the fourth floor (10 balls with equal mass) of 3 and 2 of the main structure mass respectively

propagation of ambient stochastic excitations Later stochas-ticmethods considering nonlinear and even hysteretic behav-ior in civil structures were developed by researchers Thesemethods include stochastic averaging [25] statistical lin-earizationnonlinearization [26] moment closure [27] andMonte Carlo simulations [28] Among them the statisticallinearization andMonte Carlo methods are two that are usedwidely although the latter often suffers from computationalinefficiency problems especially for large-scale civil struc-tures In this section to consider the uncertainty propagationand to further extend the stochastic analysis of a TRoMaDaS-controlledMDOF structure statistical linearization was usedto determine the standard deviation of displacement in aclosed form of a SDOF structure with a controlling device

41 Analytical Solution for SDOF Structure with TRoMaDaSFor simplicity and for the purposes of this preliminaryinvestigation of the stochastic control efficiency of the pro-posed control system a single-degree-of-freedom (SDOF)structural system with a single oscillator ball subject to whitenoise excitation was considered The stochastic differentialequation of motion of this system can be written as

MZ+CZ+KZ+ f (Z Z) = w (119905) (15)

where Z is the stochastic response and

M =[[

[

119872 +119898 119898120588

57120588

1]]

]

C = [

119888 00 0

]

K =[[

[

119896 0

051198927120588

]]

]

w =

119908 (119905)

0

(16)

where 119908(119905) is the Gaussian white noise excitation withpower spectrum 1198780 The linearized equation of motion of thestructure with a TRoMaDaS (15) can be written in the form

MZ (119905) + (C+C119890) Z (119905) +KZ (119905) = minusw (119905) (17)

where Z = (119883Θ)119879 is the augmented displacement vector

and C119890is the additional equivalent damping matrix that can

be appropriately determined by minimizing the differencebetween (15) and (17) in the mean squared sense (see [26] fordetails) In this regard for a chain-like controlled structuralsystems C

119890can be determined by

119888119890

119894119895= 119864

120597119891119894

120597119895

119894 = 1 2 119895 = 1 2 (18)

12 Shock and Vibration

where f(Z Z) = (0 5120583119892119877(71205882119903)sgn(Θ))119879 is the nonlinear

damping in (15) Furthermore according to (18)

119862eq = [

0 00 119888

eq22] (19)

119888eq22 = 120573eq =

512058311989211987771199031205882120590

Θ

radic2120587

(20)

where 120590Θis the standard deviation of the angular velocity of

the oscillator Furthermore combining with (10) yields theequivalent damping ratio of the nonlinear equation ofmotionof the oscillator as

120577eq =120573eq

21205960= radic

5119892141205871205883

120583119877

119903120590Θ

(21)

where 1205960 is the natural frequency of the oscillatorNext the standard deviation of the response of an equiv-

alent linear system can be determined according to linearrandom vibration theory that is

120590X = int

infin

minusinfin

SXX (120596) d120596 = int

infin

minusinfin

1205962SXX (120596) d120596 (22)

where SXX(120596) is the power spectrum density matrix of thesystem displacement that can be determined by

SXX (120596) = H119879 (120596) SFF (120596)H (120596) (23)

with SFF(120596) being the power spectrum density matrix of theexcitation w(119905) = (119908(119905) 0) Clearly because of the specialform of the excitation vector in thematrixH(120596) only11986711(120596)and 11986721(120596) are needed for the autospectrum 119878

119883119883(120596) and

119878ΘΘ

(120596) In this regard these frequency transfer functions canbe solved by

M (p 120596) h (120596) = N (24)

where

M (p 120596)

=[[

[

minus1205962(119872 + 119898) + 119894120596119888 + 119896 minus120596

2119898119901

minus

51205962

7120588minus120596

2+ 119894120596120573eq +

51198927120588

]]

]

h (120596) =

11986711 (120596)

11986721 (120596)

N =

10

(25)

where 119901 = (119872119898 119888 119896 120588 120573eq) is parameters of equivalentlinear systems

At this point it can be argued that (20) and (22)ndash(24)constitute a mutually dependent relationship between 120590

Θ

and equivalent damping coefficient that can be solved in aniterative manner However (22) is always difficult to obtain

in an explicit closed form although numerical algorithms canbe used to obtain the relationship in a discrete form In thisregard a closed-form integration approach was developed byRoberts and Spanos [26] providing an accurate method fordetermining the standard deviation from the power spectrumdensity (see the Appendix for the standard deviation of theresponses in detail)

42 Numerical Example Consider a SDOF structure withthe lumped mass being 8 times 103 kg The stiffness and thedamping ratio are 2 times 105 Nm and 002 respectively Thestructure is initially at rest and subject to stochastic earth-quake excitation modeled by a zero-mean Gaussian whitenoise with the power spectrum strength being 3times10minus3 msdotsminus3The natural frequency of the structure is 1205960 = 5 rads and theratio between the mass of the oscillator ball and the primarystructure is selected to be 5 To obtain preferable controlefficiency the natural frequency of the oscillator ball is tunedto agree with the one of the structure In this regard the radiiof the oscillator ball and of the arc path are 119903 = 0337m and119877 = 119903 + 120588 = 0617 where 120588 = 0280m According to (10) theradius difference between these two radii is a factor on whichthe natural frequency of the oscillator depends

To validate the randomness in the control efficiency acomparison of the response displacements between uncon-trolled and controlled structures subjected to two differentsample excitations is plotted in Figure 11 From these plots itcan be concluded that the structure response is not mitigatedefficiently until the response surpasses a certain level Thisis probably according to the analysis in Section 33 due tothe stick-slip behavior of the TRoMaDaS Moreover it can beseen that the ldquorearrdquo part of the response displacement of thecontrolled structure shown in Figures 11(a) and 11(b) behavesdifferently confirming the need for a probabilistic investiga-tion of the control efficiency Figure 12 shows a comparison ofthe standard deviation of the response displacement betweenthe structures with and without TRoMaDaS It can be seenfrom this figure that not only is the displacement standarddeviation reduced to a large extent but also that the stochasticresponse of the structure readily becomes stationary becauseof the TRoMaDaS

5 Concluding Remarks

Anovel tunedmass damper system for civil structures namedtuned rolling mass damper system or TRoMaDaS has beenproposed in this paper Compared with the traditional instal-lation of tuned mass dampers the most promising feature ofthe proposed passive control system is that it does not requirea dedicated space to accommodate the oscillator and thusmaintains the integrity of civil buildings To investigate theseismic response reduction of the proposed passive controlsystem the equation of motion of the controlled system sub-ject to external dynamic excitationwas derived In this regarda physical model of the passively controlled structure wasmodeled as a chain-likeMDOF systemwith arbitrary numberof oscillators rolling within the hollow floors Lagrangersquosequation was used to derive the equation of motion of