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Engineering Structures 33 (2011) 33173328

Contents lists available at SciVerse ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Development of passive viscoelastic damper to attenuate excessivefloor vibrations

I. Saidi a,, E.F. Gad a,b, J.L. Wilson a, N. Haritos ba Faculty of Engineering and Industrial Sciences, Swinburne University of Technology, Australiab Department of Civil and Environment Engineering, Melbourne University, Australia

a r t i c l e i n f o

Article history:

Received 27 July 2009Received in revised form3 May 2011Accepted 12 May 2011Available online 28 July 2011

Keywords:

Floor vibrationsViscoelastic materialsTuned mass dampers

a b s t r a c t

Recent changes in the construction of building floors have included the use of light material compositesystems and long span floor systems. Although these changes have many advantages, such floor systemscansuffer fromexcessivevibrationdue to human activities.This problem is exacerbated in officebuildingsdue to the reduction in inherent damping associated with modern fit outs. Excessive floor vibrations areoften realised after the completion of construction or following structural modifications and normallyarise due to inadequate knowledge of the damping values in the design process. Thus rectificationmeasures are normally required to reduce floor accelerations. This paper proposes a new innovativepassive viscoelasticdamper to reduce floorvibrations. Thisdamper can be easily tunedto the fundamentalfrequency of the floor and can be designed to achieve various damping values. The paper discusses theanalyticaldevelopment of the damper withexperimental results presentedon a prototype to demonstrateits effectiveness.

Crown Copyright 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Building floors are subjected to dynamic loads from peoplewhen they walk, run, dance or engage in aerobics activities. Suchexcitation forces cannot be easily isolated from the structure andthey occur frequently [1]. Typical pacing rates for walking arebetween 1.6 and 2.4 Hz (slow to fast walk) whilst for joggingthe rate is about 2.5 Hz and running occurs at rates up to about3 Hz.

Although the excitation from pedestrians is dominated by thepacing rate, it also includes higher harmonic components withfrequenciescorrespondingto an integer multiple of the pacing rate.Since annoying vibration amplitudes are caused by a coincidenceof the natural frequency of the floor (f1) with one of the harmonics

of the walking excitation, the problem may be avoided by keepingthese frequencies away from each other. For this reason, engineersmay aim to design floor systems to have a fundamental frequencygreater than three times the walking frequency (i.e. above about6 Hz) [2]. This is a simple and effective approach for design butit does not necessarily guarantee acceptable floor performancesince it does not take account of damping. Indeed compositefloors with very low damping (2%), can experience high levelsof vibration even if their fundamental natural frequency is above7.5 Hz [3].

Corresponding author.E-mail address: [email protected] (I. Saidi).

The reaction of people who experience floor vibration dependson the activity they are engaged in, as reflected in the commonlyused acceptance criteria as illustrated in Fig. 1. For example, officesand residences are normally designed to have a maximum peakacceleration of about 0.5% gravity (g) whereas pedestrian bridgescan be designed for acceleration levels 10 times greater (5% g) [4].In addition to acceleration amplitude, peoples perception is alsoaffected by the characteristics of the vibration response includingfrequency and duration [1]. Comfort studies for automobilesand aircraft have found that humans are especially sensitive tovibration in the frequency range of 48 Hz. This is explained bythe fact that many organs in the human body resonate at thesefrequencies [5] whilst outside this frequency range, people accepthigher vibration acceleration levels [4] as shown in Fig. 1.

There are several design models for predicting the maximumresponse of a floor due to walking excitation. One of the mostcommonly used method is that documented in the AmericanInstitute of Steel Construction Design Guide 11 (AISC DG11) [5,4].This is the most popular method used by Australian designers.This method is based on reducing the floor structure to a SingleDegree of Freedom (SDOF) system. The peak acceleration responseis calculated using Eq. (1) (the full derivation of this expression canbe found in [4]).

ap

g= P0Exp(0.35f1)

1W(1)

where ap/g is estimated peak acceleration in units of gravityacceleration (g), f1 is the fundamental frequency of the floor

0141-0296/$ see front matter Crown Copyright 2011 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.05.017
http://dx.doi.org/10.1016/j.engstruct.2011.05.017http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructmailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.05.017http://dx.doi.org/10.1016/j.engstruct.2011.05.017mailto:[email protected]://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://dx.doi.org/10.1016/j.engstruct.2011.05.017

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3318 I. Saidi et al. / Engineering Structures 33 (2011) 33173328

Fig. 1. Acceptance criteria for floor vibrations.

structure, P0 is a constant force based on a persons weight andtaken as 0.29 kN, 1 is the damping ratio of the floor and W isthe effective weight of the floor which oscillates because of thewalking. The ap/g value has to satisfy the values in Fig. 1 forsatisfactory performance.

Other methods for calculating the floor peak response and as-sessing the performance include the recently published EuropeanCommission guide [68]. In this method, the total damping of thefloor is taken as the sum of contributions from structural damping,furnishing and finishes. Similar to the AISC DG11, this method di-rectly relates the peak response to the total damping which has tobe assumed during the design phase. However, damping in prac-tical structures is seldom fully understood as it cannot be deter-mined directly based on the structural properties, as is the case forstiffness and mass. Damping is generally determined based on ex-perimental and historical data. Therefore, in applying assessmentmethods design engineers have to estimate the damping based onavailable knowledge during the design phase. However, the de-signers in many cases may not know the details of the fit out whichare specified by the client or architect. Consequently, overestima-tion of damping or altering the fit out of the floor can lead to ex-cessive vibrations.

2. Control of floor vibrations

There area few remedial options available to rectify a floor withexcessive levels of vibration, including increasing the stiffness andhence the frequency or increasing the damping. The installationof tuned mass dampers can be performed more cheaply thanstructural stiffening, and often offer the only practical means ofvibration control in existing structures [9]. In new constructionsviscoelastic materials can be incorporated within the floor systemto increase its damping. Both embedded viscoelastic materials andtuned mass dampers represent typical passive damping optionswhilstmore sophisticated and expensive solutions may include theaddition of active dampers.

2.1. Viscoelastic materials

Embedded viscoelastic materials (VEM) offer the advantage of

reducing vibrations over a broad range of frequencies comparedwith TMDs. However, viscoelastic damping works optimally only

Fig. 2. Illustration for Resotec product in composite floor (after [11]).

for a specific mode of vibration. Nevertheless use of VEMs is acheap method of increasing the damping if incorporated duringconstruction [10].

An example of viscoelastic damping is the Resotec systemwhich is illustrated in Fig. 2. This product comprises a thin layerof high-damping viscoelastic material with an overall thicknessof about 3 mm. Resotec is sandwiched between the top flange ofthe floor steel beams and concrete slab for a proportion of thebeam near each end where the shear stresses are the greatest. It isreported that the damping of a fitted out floor is typically doubledby the incorporation of Resotec [11]. However, this product needsto be incorporated within the floor during construction and is notsuitable as a rectification measure.

2.2. Tuned mass dampers (TMD)

The principle of a TMD was initially utilised when Den Hartogin 1947 reintroduced the dynamic absorber invented by Frahm in1909 [1214]. Generally, a TMD consists of a mass, spring, anddashpot and is tuned to the natural frequency of the primarysystem. When the primary system begins to oscillate it excitesthe TMD into motion and hence the TMD absorbs energy from

the vibrating floor [15]. The TMD inertia forces produced by thismotionare anti-phase to the excitation force. The first useof a TMDfor floor vibration application was reported by Lenzen [16] whoused small TMDs with a total mass of about 2% of the floor mass.The TMDs were made of steel hung by springs and dashpots fromthe floor beams. Lenzen reported floors with annoying vibrationcharacteristicsbecame satisfactoryby tuningthe TMDs to a naturalfrequency of about 1.0 Hz less than that of the floor and usinga damping ratio of 7.5% [17]. An example of a more recent TMDis a Pendulum Tuned Mass Damper (PTMD) shown in Fig. 3.Experiments were undertaken to test theperformanceof thePTMDandit is reported that thedamper reduced thefloorvibration in therange of 50%70% [17].

Floor vibrations due to walking excitation typically produce

very small floor displacements which are generally less than0.1 mm. A TMD would typically have a maximum displacementaround ten times larger than the floor (i.e. in the order of 1 mm).In reality, it is difficult to produce a practical viscous damperthat provides a reasonable level of damping given this verysmall displacement. Viscous dampers were used in some floorapplications such as in the Terrace on the Park building in NewYork (1992). This problematic floor was cantilevered with a lownatural frequency of 2.3 Hz and responded to footfall vibrationswith 7% g acceleration and 3.3 mm maximum displacement. Inthis application the damper used was large and extended from thelower floor to the point of maximum response of the problematicfloor. Indeed such access is not always available for office floors.OtherapplicationsforviscousTMDcanbefoundinstadia.However

such structures tend to have long span cantilevers with largerdisplacements associated with crowd activities especially from

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I. Saidi et al. / Engineering Structures 33 (2011) 33173328 3319

Fig. 3. Pendulum tuned mass damper for floor vibrations (after [ 17]).

concerts. Therefore viscous dampers can be effective in these casesdue to the large motion associated with vibrations and in turnenergy can be dissipated through their typical dashpot systems.

For floor retrofitting applications there are usually physicallimitations associated with access and presence of mechanicalservices attached to the soffit of the slab and beams. Hence for afloor damper to be practical, it needs to be sufficiently small tobe accommodated in the available ceiling space and it should alsoallow for easy adjustment of frequency for tuning. Given the abovementioned limitations, an alternative novel viscoelastic damperspecifically for floor vibration is proposed in this paper.

3. Fundamentals of a tuned mass damper

In floor design, it is common to idealise the floor as a singledegree of freedom (SDOF) system. A floor can be represented withan equivalent mass (m1), stiffness (k1) and damping (c1) by a

SDOF system as shown in Fig. 4. The maximum response occurswhen the frequency of the excitation force coincides with thenatural frequency of the system, hence the maximum accelerationresponse is given by Eq. (2) [18]:

| X1| =F0

21m1. (2)

The addition of a TMD converts the floor into a two degrees offreedom system as illustrated in Fig.5. In such a systemthe dampermass (m2) and its spring stiffness (k2) are tuned so that the TMDhas the same natural frequency as the primary (floor) system.The addition of damping (c2) to the TMD reduces the overallresponse of the combined system as shown in Fig. 6. One methodto assess the efficiency of the TMD is to compare the response of

the original SDOF system and the two degrees of freedom system.The formulation of response of a generic two degrees of freedomsystem is presented in Section 3.2.

TMDs are typically effective over a narrow frequency band andmust be tuned to a particular natural frequency. They are noteffective if the structure has several closely spaced frequencies andthey can potentially increase the vibration if they are off-tuned[9]. A TMD splits the natural frequency of the primary systeminto a lower (f1) and higher frequency (f

2) as shown in Fig. 6. If

there is zero damping then resonance occurs at the two undampedresonant frequencies of the combined system (f1 & f

2). The other

extreme case occurs when there is infinite damping, which has theeffect of locking the spring (k2). In this case the system becomesonedegreeoffreedomwithstiffnessof(k1)andamassof(m1+m2).Using an intermediatevalueof damping such as optimum damping(opt), it is possible to control the vibration of the primary system

Fig. 4. Schematic of an SDOF system.

Fig. 5. Schematic of a two DOF system.

Fig. 6. Example showing the effects of attaching a TMD to an SDOF system.

over a wider frequency range [19]. In the optimum damper the

values of the dampers natural frequency and damping ratio (opt)are specified to obtain minimum and equal height peaks at f1 &f2 [20].

3.1. Properties of an optimum damper

The first step in the design of a TMD is to determine the desiredmass ratio () as defined by Eq. (3):

= m2m1

. (3)

Thelargerthemassofthedamper (m2) the larger the separationbetween the two new frequencies (f1 & f

2 in Fig. 6) which

are created by the damper. This would normally increase the

effectiveness of the damper over a broader range of frequenciesand also decreases the vibration level of the primary system.

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However, there are normally some structural and physicallimitations on the size of the damper and its mass. For mostpractical cases a mass ratio of 0.010.02 is recommended.

The optimum natural frequency of the damper (f2) is defined byEq. (4):

f2 =f1

1+

(4)

wheref1 is the natural frequency of the primary system, which canbe obtained from Eq. (5):

f1 =1

2

k1

m1. (5)

The optimum damping ratio of the damper 2 = opt (is whereresponse peaks atf1 &f

2 are equal) can be found using Eq. (6) [13]:

2 =

3

8(1 + ) . (6)

3.2. Response of retrofitted system

To determine the response of the two DOF system, the equationof motion as expressed in Eq. (7) needs to be solved. The values ofm2, k2 and 2 of the damper are obtained from optimum damperequations (3)(6).[

m1 00 m2

] [x1x2

]+[

c1 + c2 c2c2 c2

] [x1x2

]

+[

k1 + k2 k2k2 k2

] [x1x2

]=[

F1(t)

0

](7)

where c1 = 21

k1m1 and c2 = 22

k2m2 .Eq. (7) consists of two coupled second-order ordinary differen-

tial equations. There are several approaches to solve this equation,and a convenient method of solving this system is to use vectorsand matrices. This method requires the damping matrix to satisfyEq. (8). Otherwise for the case of arbitrary damping one of twooptions needs to be considered. The first is to assume the modaldamping based on experience and the second is to convert thedamping matrix to a proportional damping model using a propor-tional method [21].

[C] = 1[M] + 2[K] (8)where [C], [M] and K are damping, mass and stiffness matricesrespectively, and 1 & 2 are mass and stiffness multipliersrespectively.

The two splitting frequencies (f1

&f2) in Fig. 6 can be calculated

using Eq. (9) (see Box I) [22].The response acceleration of the primary system | X1| can be foundby solving Eq. (7) using the Mechanical Impedance Method (x1 =

X1eit,x2 = X2eit and F1(t) = F0eit), resulting in the expression

presented by Eq. (10) (see Box II):The system resonates at both splitting frequencies and the

maximum acceleration can be calculated using Eq. (10) (see Box II)by substituting either = 2f1 or = 2f2 as obtained fromEq. (9) (see Box I).

4. Equivalent viscoelastic damper

In Section 3 of the paper, the required properties of a typical

viscous TMD to dampen floor vibration are discussed. FromSection 3, the values of classical mass, spring stiffness and dashpot

Fig. 7. Typical sandwich beam.

are determined. In this section of the paper, the equivalentproperties aredevelopedfor a viscoelasticTMD ratherthan viscousTMD. The proposed viscoelastic TMD is in the form of a cantileverbeam with a sandwiched rubber layer.

The use of viscoelastic materials in reducing the effect ofvibrations is common in mechanical engineering applicationsespecially in machine vibration. Recently it also became a solution

for floor vibrations as illustrated in Fig. 2. Indeed an effectiveway to increase damping and reduce transient and steady statevibration is to add a layer of viscoelastic material, such as rubber,to an existing structure. The combined systemwould have a higherdamping level and thus reduces unwanted vibration [23].

The simplest form of a viscoelastic damper is a constrainedviscoelasticlayerin a beam. This could be made of twoconstrainingmetal plates bonded together with high damping rubber. In thiscomposite sandwich beam, the viscoelastic material experiencesconsiderable shear strain as it bends, dissipating energy andattenuating vibration response [24]. The proposed damper in thispaper has the form of a sandwich beam.

There are many factors which affect the damping performancesof viscoelastic materials in sandwich beams including materialtype, thickness, temperature and bonding. The viscoelastic damper

proposed in this paper is for internal use so variation in thetemperature is notsignificant. Theresinused forbondingthe layerscan be easily designed so that it does not allow a slip to occur at theinterfaces of thelayers. Hence thetwo main remainingfactors to betaken into account for the design of the damper are the viscoelasticmaterial type and thickness.

In order for the viscoelastic sandwich beam (as shown inFig. 7) to be used as a damper its natural frequency and dampingneed to be estimated. There are two methods for obtaining asolution, normally, an exact solution and an approximate methodas discussed below.

4.1. Exact solution

The equation of motion for sandwich beams has been consid-ered by a number of researchers. Kerwin [25] analysed the threelayer system and derived an expression for flexural stiffness ofsandwich beams DiTaranto [26] derived a sixth order differentialequation governing the motion of sandwich beams. In contrast,Eq. (11) wasderivedbyMeadandMarkus[ 27] fora sandwich beamwith arbitrary boundary conditions subjected to forced vibration.The solution of Eq. (11) is complex as it involves solving a sixthorder differential equation. This solution can be complicated fur-ther byboundary conditions such as the addition of an end mass tothe tip of the cantilever. Classical exact solutions are discussed byMead and Markus [28,29].

Wvin g (1 + j) (1 + Y) Wivn 2n (1 + jn)

ADt Wiin gWn (1 + j) = 0 (11)

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f1,2 =1

2

m1k2 + m2(k1 + k2)

[m1k2 + m2(k1 + k2)]2 4k1k2m1m2

2m1m2(9)

Box I.

| X1| = 2F0 2c22 + (k2 2m2)2

2

c2k1 + c1k2 2 (c1m2 + c2 (m1 + m2))2 + k1k2 (c1c2 + k1m2 + k2 (m1 + m2)) 2 + m1m242 (10)

where F0 is the excitation force and is the excitation force circular frequency.

Box II.

where W, n,g, , Y,,,,A and Dt are mode shape function,mode number, shear parameter, viscoelastic material loss factor,geometric parameter, circular natural frequency, overall dissipa-tion loss factor, density, cross-sectionalarea of the beam andflexu-ral rigidity of the constraining layers respectively, while j = 1.

Eq. (11) is based on the following realistic assumptions relatedto the behaviour of the damper:

(i) theviscoelastic core resists shear butnot directflexuralstress;(ii) shear strains in the constraining plates are negligible;(iii) transverse direct strains in both core and constraining plates

are neglected; and(iv) no slip occurs at the interfaces of the core and constraining

plates.

4.2. Approximate analytical method

In this alternative approximate analyticalmethod developed byMead [30], the flexural rigidity EItotal and the overall dissipationloss factor of the composite system are estimated based onthe dissipation loss factor of the viscoelastic material , thicknessof viscoelastic layer, geometric parameters and Young moduli of

the top and bottom plates constraining the viscoelastic material.This method can be applied to any composite beam configurationsuch as simply supported or cantilever beams as illustrated in thefollowing subsections.

4.2.1. Flexural rigidity (EItotal) of viscoelastic cantilever beam

The flexural rigidity EItotal of the viscoelastic cantileverbeam canbe calculated using Eq. (12) together with Eqs. (13)(15) [30];

EItotal =

1 + gY(1 + g(1 + 2))

1 + 2g + g2(1 + 2)

E1I

1 + E3I3

(12)

where is the dissipation loss factor of the rubber and Y is ageometric parameter calculated using Eq. (13):

Y = (E1A1)(E3A3)d2

(E1A1 + E3A3)(E1I1 + E3I3)(13)

E1 and E3 are the moduli of elasticity of the top and bottomconstraining plates, A1 and A3 are the cross-sectional area of thetop and bottom constraining plates, I1 and I

3 are the moment of

inertia of top and bottom constraining plates about their neutralrespective axes and d is the distance between their respectivecentroids.

The shear parameter g is calculated using Eq. (14):

g = Gbh2K

2B

1

E1A1+ 1

E3A3

(14)

where G, b and h2 are shear modulus, width and thickness of the

viscoelastic core respectively and KB is the wave number of thebeam.

When the core shear stiffness is very low, the constraininglayersdominatethe flexuralstiffnessof thebeamand thesandwichbeam vibrates in the same mode as a EulerBernoulli beam [27].

The wave number for a cantilever sandwich beam without anend mass can be calculated using Eq. (15) [23]:

KB = 1.875/L (15)where L is the length of the viscoelastic cantilever beam.

4.2.2. Viscoelastic cantilever beam with an end mass

The mass of the damper for a given mass ratio must satisfyEq. (3). Therefore, an end mass would normally be required toattain the mass ratio of the optimum damper.

The natural frequency of a viscoelastic cantilever beam dampercan be estimated as:

f2 = 2

k2

m2(16)

where k2 and m2 are the effective stiffness and effective mass ofthe cantilever beam respectively.

The stiffness k2 can be calculated based on basic beam theoryasshown in Eq. (17):

k2 =3EItotal

L3(17)

where EItotal and L are the flexural rigidity obtained from Eq. (16)andthe length of the viscoelastic cantilever beam respectively. Theeffective mass of a uniform viscoelastic cantilever beam can becalculated from Eq. (18):

m2 =33

140AL + mend (18)

where andA are total mass density and total cross-sectional areaof the viscoelastic cantilever beam respectively and mend istheendmass. The addition of an end mass at the free tip of the cantileverallows easy adjustment of the mass ratio and fundamental

frequency of the damper in order to achieve optimum design.The addition of the end mass will change the wave number as aresult of corresponding change in the system natural frequency asshown in Eq. (19) [30,31]:

K2B = 2f2

A

EItotal(19)

where andA are mass density and overall cross-sectional area ofthe sandwich beam, respectively.

Substitution of Eq. (19) into Eq. (14) yields:

ge =Gb

2h2f2

EItotal

A

1

E1A1+ 1

E3A3

(20)

wherege is the shearparameter for the viscoelastic cantileverbeamwith end mass.

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Finally, the overall dissipation loss factor of the viscoelasticdamper (viscoelastic cantilever sandwich beam) can be estimatedusing Eq.(21). The dissipation loss factor isequal totwo times thecritical damping ratio of the damper, and is directly proportionalto the material dissipation loss factor .

=2

=

geY1

+g

e(2

+Y)

+g

2

e(1

+Y)(1

+2)

. (21)

4.3. Design of viscoelastic damper

Having developed a mathematical solution for determining thedynamic properties of the viscoelastic damper, a simple procedurefor the design of an optimum damper can now be specified asfollows.

(i) Determine the basic dynamic properties of the primarysystem (floor) to be retrofitted with a damper (i.e. f1 , 1 andk1 for an equivalent SDOF system). The properties are to bebased on the as built conditions.

(ii) Determine a suitable mass ratio (Eq. (3)) for the damper

based on given physical limitations and the requiredreductionin the response of the primary system. A mass ratioof 1%2% is practical in most cases.

(iii) The optimum natural frequency of the TMD, f2, can becalculated using Eq. (4).

(iv) The optimum damping ratio of the damper 2 can now becalculated using Eq. (6).

(v) Trial dimensions for the damper can be proposed to suitany physical limitations (i.e. h1, h2, h3, L and b illustratedin Fig. 7). In addition, the material properties for theconstraining layers and viscoelastic material need to bespecified (E1 & E3 for the constraining layers and & G forthe viscoelastic material). For simplicity the top and bottomlayers can be of the same material (i.e. E1

=E3).

(vi) Calculate the overall flexural rigidity (EItotal) of the beamwithout an end mass using Eqs. (12)(15).

(vii) Calculate the natural frequency of the damper with an endmass using Eqs. (16)(18). The damper frequency is criticalin reducing the overall floor vibrations and the end mass canbe used to fine tune the damper frequency.

(viii) Calculate the overall dissipation loss factor and then theestimated damping ratio 2 of the damper using Eqs. (12),(20) and (21).

This procedure is required to be repeated by altering thematerial and dimensions of the viscoelastic damper until theoptimum damper properties are achieved.

One of the main factors that affect the performance of aviscoelastic damper is the material dissipation loss factor . Thismaterial property needs to be established for the type of rubber tobe used which is discussed in the following section.

5. Determination of viscoelastic material properties

Many types of commercially available rubbers do not have ad-equate technical specifications concerning their material proper-ties and consequently it becomes necessary to undertake specifictests on the acquired rubber to determine the shear modulus G andloss factor . This can be achieved using one of two types of tests,(i) by direct measurements using a Dynamic Mechanical Analyser(DMA) [32,33] or (ii) by back calculation from experimental re-sults performed on a prototype damper. These methods were both

utilised to find the properties of the rubber used in developing theviscoelastic cantilever damper proposed in this paper.

Dynamic mechanical analysis is a testing technique thatmeasures the mechanical properties of materials as a functionof time, temperature and frequency. In DMA testing, a smalldeformation is applied to a sample in a cyclic manner withthe measured response providing information on the stiffnessand damping properties for the material. For the rubber used indeveloping the damper presented in this paper, three samplesmeasuring 35

10.8

5.2 mm were tested. The average loss

factor () was 0.12 and the average measured shear modulus (G)was 690 kPa based on the assumption that E = 3G for elastomericmaterials [31].

The rubber dissipation loss factor can also be back calculatedfrom vibration tests if access to a Dynamic Mechanical Analyser isnot available. This method requires the construction of a prototypesandwich beam damper with the designed rubber. This is thentested to obtain the overall damping ratio of the damper (2) andthe total flexural rigidity (EItotal) from basic vibration testing. Thesetwo measured properties along with the geometric parametersand other material properties of the damper are substituted intoEqs. (12)(21) to back calculate the and G values of the rubber.The flexural rigidity EItotal can be obtained from the measurednatural frequency using Eqs. (16)(18). The damping ratio 2 of

the viscoelastic beam can be estimated from the time domain ofthe excited viscoelastic beam using the Logarithmic DecrementMethod (LDM). As a comparison, based on this back calculationmethodthe estimatedaverage valuesofG and of therubber werefound to be 640 kPa and 0.10 respectively which were in very goodagreement with the values of 690 kPa and 0.12 obtained from theDMA test.

It shouldbe noted that thedissipationloss factorof therubberis the factor that determines the upper limit of overall dissipationloss factor of the composite system. In other words, the value(=2) of the composite system cannot exceed the value of therubber [33,31].

6. Validation of damper properties

The analytical model described in Section 4.2 for determiningthe viscoelastic dynamic properties of the damper was thenvalidated by experimental testing as well as by comparison withfinite element (FE) results.

6.1. Experimental test

Two prototype viscoelastic cantilever dampers of length 500and750mmwereconstructedwithoutanendmassandtwootherswith an end mass. For all four dampers, the constraining layerswere 1 mm thick steel plates and the rubber core was either 12 or32 mm thick. Tables 1 and 2 show the details of the four dampersalong with their predicted fundamental frequency and dampingratio. The rubber loss factor and shear modulus were obtained

using a Dynamic Mechanical Analyser (DMA) machine discussedin Section 5. Each cantilever was fixed at one end to a rigid supportand subjected to pluck tests. A non-contact accelerometer was setto measure the time domain response of the excited viscoelasticcantilever beam. Using the recorded response time history, thedamping ratio was calculated using the log decay method whilethe natural frequency was obtained from Fast Fourier Transform(FFT) analysis.

Oneofthekeyinfluencingpropertiesofthedamperisitsnaturalfrequency as this affects its performance as a TMD. As can be seenfrom Tables 1 and 2, the approximate analytical model predictedthe natural frequency well with an average error of approximately8%. The error islargerfor the cases withend massdue tothe greaterapproximation for the wave number KB in Eq. (19). The averageerror in damping ratio between the predictedand measured valueswas in the vicinity of 12%.

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Table 1

Dynamic properties of a damper without an end mass as obtained from analytical model, FE analysis and experimental testing.

Dimensions (mm) Analytical FE model Experiment

h1 h2 h3 b L f2 (Hz) 2 (%) f2 (Hz) 2 (%) f2 (Hz) 2 (%)

1 32 1 25 750 10.2 5.8 9.1 5.5 10.4 4.91 12 1 25 500 12.2 5.5 10.6 4.5 12.0 4.5

Table 2Dynamic properties of a damper with an end mass as obtained from analytical model, FE analysis and experimental testing.

Dimensions (mm) Analytical FE model Experiment

h1 h2 h3 b L f2 (Hz) 2 (%) f2 (Hz) 2 (%) f2 (Hz) 2(%)

1 32 1 25 750 4.8 5.1 5.1 5.1 5.7 5.31 12 1 25 500 4.1 4.8 4.1 4.9 4.6 5.3

6.2. Damper FE models

For the FE modelling, the commercially available softwarepackage ANSYS was used. The constraining layers and rubber corewere modelled with solid elements. The constraining layers wereassumed to be linear elastic, while the rubber layer was assumed

to be hyperelastic. Modal analysis was performed using ANSYSto obtain the vibration natural frequencies and correspondingmode shapes. The damping was modelled for each material as astiffness multiplier. In addition to the modal analysis, harmonicand transient dynamic analyses were performed so that theperformance of the damper could be investigated in both the timeand frequency domains.

The overall damping of the damper was obtained from theFE analysis using the conventional methods of logarithmic decayin the time domain and the half power bandwidth method inthe frequency domain. The damping values from the FE analysesobtained from the two methods were in very good agreement.Logarithmic decay values are presented in Tables 1 and 2 for boththe FE model and the experimental tests. The FE results for both

frequency and damping are in good agreement with analyticaland experimental results. This indicated that the approximateanalytical method for determining the damper properties can beused with confidence. Furthermore, FE models can also be utilisedfor determining damper properties with good accuracy.

7. Application of viscoelastic TMD

Prototypes of the proposed viscoelastic damper were con-structed to demonstrate the application of the new damper andillustrate its efficiency.Furthermore, sensitivity analyses were per-formed to examine the performance of the damper if it were not tohave optimum properties.

Two case studies are presented below which cover bothexperimental and analytical work.

7.1. Case study 1steel beam

This case study is for a 3 m long simply supported steel beamhaving properties as shown in Table 3. This beam was initiallytested as a bare beam. The beam was then retrofitted with aviscoelastic cantilever damper as shown in Fig. 8 with a mass ratioof 1%. The aim of these tests was to examine the effectiveness ofthe designed viscoelastic cantilever damper.

7.1.1. Bare steel beam test

The effective mass and natural frequency of the steel beamwere calculated using standard classical expressions for simply

Table 3

Simply supported steel beam properties for Case Study-1.

Length (mm) 3000Width (mm) 100Thickness (mm) 25Natural Frequency f1 (Hz) 6.3Damping ratio 1 0.3%

supported beams. The total mass of the steel beam was approxi-mately 59 kg, hence the effective mass of the beam is 29.5 kg (50%of the total mass) and the stiffness of the beam can be calculatedusing Eq. (22) [34]:

K = 6144EI125L3

. (22)

The natural frequency of the steel beam was obtained as 6.3 Hzusing Eq. (5).

In order to measure the natural frequency and the dampingratio of the steel beam experimentally, the bare beam wassubjected to pluck tests to record the free vibration of the steel

beam. The natural frequency of the steel beam was then extractedfrom the time history using the Fast Fourier Transform technique.It was found that thenatural frequency wasin excellent agreementwith value obtained from Eq. (5). The damping ratio obtained fromthe pluck tests using the log decay method was about 0.3%.

Harmonic excitation was imposed using a mechanical shakerlocated at a distance of about one third of the span from one ofthe ends to measure the maximum response of the steel beam. Theresponseofthesteelbeamduetothisexcitationinthetimedomainis shown in Fig. 9.

7.1.2. Retrofitted steel beam test

The optimum damper properties were obtained from Eqs. (3)(6) and a viscoelastic damper was designed according to the

procedure outlined in Section 4.3. The damper was attached atthe mid-span of the simply supported beam using a rigid bracket.The mass, stiffness, and damping properties of the cantilever weretuned to satisfy the optimum design, with the properties listed inTable 4 including opt = 6%. It should be noted that the bendingmoment dueto additionof thedamper at themid-spanof thebeamis negligible because it just produces an additional moment ofabout 1% of the bending moment produced by the primary system.

The responses of the bare and retrofitted beam were predictedfrom Eqs. (2) and (10) (see Box II for Eq. (10))usingaforceof1Nasshown in Fig.10. The predicted reduction factor from the analyticalmodel was about 12.5.

The bare and retrofitted beams were experimentally testedwith a harmonic excitation using a mechanical shaker located

at a distance of about one third of the span from one of theends. Fig. 9 shows the acceleration responses of the bare and

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Fig. 8. Viscoelastic damper attached to a vibrating beam.

0 10 20 30 40 50 60

0 10 20 30 40 50 60

-4

-2

0

2

4

6

-4

-2

0

2

4

6

Fig. 9. Case Study 1steel beam response with and without damper attached.

Table 4

Viscoelastic damper properties for Case Study 1.

Length (L) 500 mmWidth (b) 25 mmThickness of steel top constraining layer (h1) 1 mmThickness of rubber (h2) 12 mmThickness of steel bottom constraining layer (h3) 1 mmDissipation loss factor of rubber () 0.12Rubber shear modulus 690 kPaEnd mass (mend) 220 gNatural frequency of damper (f2) 6.2 HzDamping ratio (2) of damper 5.4%

Fig.10. Steel beam responsein thefrequencydomain(Eqs. (2) and (10)) (see BoxIIfor Eq. (10)).

the retrofitted beams. It is clear that the response of the beam

is reduced by a factor of 10 which is in good agreement withthe predicted reduction factor of about 12.5. The overall damping

of the retrofitted system was estimated using the half powerbandwidth method using curve fitting [35] and was found to beabout 3%, which is a significant increase from the original 0.3%damping.

7.1.3. Finite element model

The bare steel beam and retrofitted beam with the TMD wereboth modelledin ANSYS using themethod describedin Section 6.2.

The properties of the damper were as built with rubber propertiesbeing the same as described earlier with G = 690 kPa and =0.12. In order to assess the effects of the damper, a harmonicexcitation was imposed on each beam and the peak responsecompared.

The FE results for the cases with and without dampers showedthat the peak acceleration response would be reduced by a factorof 12 due to the TMD. This reduction factor is in good agreementwith experimental results (reduction factor of 10). This furtherdemonstrates that an FE analysis can be used for predicting theperformance of the proposed damper which would be particularlyuseful for complex floor systems.

7.1.4. Sensitivity analysis

Investigations of the performance of the viscoelastic damperdue to variation in damping ratio, mass ratio and location of thedamper along the steel beam were undertaken analytically andexperimentally.

In order to assess the effectiveness of the damper for when itdoes not have the optimum damping value, the damper dampingratio (2) was varied from approximately 1%9%. The optimumdamping value opt was calculated earlier to be 6%. Fig. 11 showsthe predicted reduction factor in the acceleration response of thesteelbeamforthisrangeof2.Itcanbeseenthatthedampercanbequite effective over the range of 4.5%7.5% for2. This correspondsto a variation of approximately 25% from the optimum value.However, it shouldbe noted that if thedampingratioof thedamperis not at the optimum the response amplitude at the two splitfrequencies (f1 & f

2) would not be equal as shown in Fig. 12.

The sensitivity to damper performance of the variation in thedampers point of attachment along the length of the beam wasinvestigated as shown in Fig. 13(a). The study indicates that thedamper would remain effective in reducing the vibration if it islocated within the central one third of the length of the beam.

Another experimental sensitivity analysis was performed to in-vestigate the effect of variation of the damper end mass and hencedamper frequency on the system response. The experimental re-sults are summarised in Fig. 13(b) and clearly show that a varia-tion in the end mass of up to 20% (10% of the damper frequency)from the optimum value has little effect on the efficiency of theviscoelastic damper due to the very low damping of the primarysystem. The natural frequency of the damper due to the varia-tion in the end mass was also calculated using Eqs. (16)(18). Itwasfoundthata frequencywithin

10%of theoptimum frequency

does not degrade the effectiveness of the damper.

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Fig.11. Calculatedresponse reductionfactorfor retrofitted steel beamwitha TMDhaving different damping ratio (Eq. (10), see Box II).

Fig. 12. Steel beam response with a TMD having different damping ratios whensubjected to harmonic excitation (Eq. (10), see Box II).

7.2. Case study 2reinforced concrete T beam

In order that the effectiveness of the new viscoelastic dampercould be fully assessed another prototype was developed for an

Fig. 14. RC T-beam cross-section.

experimental floor. The experimental floor used in this case studyis essentially a segmentof a reinforced concrete floor systemwith areinforced concrete beam and composite slab. The cross section ofthe experimental floor is shown in Fig. 14 and is referred to hereinas a T beam.

7.2.1. Bare reinforced concrete T beam

The T beamhas a span of9.5 m,a total weightof 6000kg and issimply supported at the ends. The long span of the T beam and its

geometry, makes it relatively flexible and easily excited by foot fallexcitation. Hencethe T beam wasa prime candidatefor retrofittingusing the newly developed viscoelastic damper. Prior to the designof the damper, the T beam floor was tested using various forms ofexcitation including heel drop, walking and impulse loading usinga modal impact hammer. From the tests, the natural frequencies,mode shapes and damping ratios were determined using modalanalysis. In addition, simpler assessment techniques which wouldnormally be used in the field such as log decay and half powerbandwidth were used to estimate the apparent overall damping.Based on theexperimental results it wasfoundthat the Tbeam hada fundamental natural frequency of 4.2 Hz, and apparent dampingof 2.9%. The higher than expected damping is simply due to thepresence of cracks in the beam because of earlier load tests.

The reduction in the response of the T beam due to the additionof the damper was predicted to be around a factor of 2 usingEqs. (2) and (10).

7.2.2. Viscoelastic damper design

Fora simplysupported beam with a uniformlydistributed mass,theeffectivemassishalfofthetotalmass.Hence,fortheequivalentSDOF system, the effective mass of the T beam can be taken as3000 kg. For a tuned mass viscoelastic damper, with a mass ratio

Fig. 13. Results from sensitivity analysis for steel beam in Case Study 1: (a) Effect of location of damper along the beam. (b) Effect of change of mass of damper.

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Fig. 15. Response of T beam due to the variation in damping ratio of the TMD(Eq. (10)).

Fig. 16. Tuned mass viscoelastic cantilever damper attached to the experimentalT beam.

of 1%, the required mass is 30 kg. Using Eqs. (8)(11) (see Box I forEq. (9) and Box II for Eq. (10)) the optimum damper is required tohave a natural frequency of 4.2 Hz and damping ratio of 6%.

A commercial rubber with a dissipation loss factor of 0.15was used to develop the viscoelastic damper. The geometry andnatural frequencyof thedamper were obtained from Eqs. (13)(21)and the resulting damper properties are listed in Table 5. It shouldbe noted that the dissipation loss factor of this rubber is notsufficient to provide the optimum damping ratio of 6% for theTMD with the given thickness, width and length of the rubber andplates. A rubber with a higher dissipation loss factor would beneededto increase the damping (such rubbers can be sourced from

specialist suppliers but are not readily available). However a lessthan optimum damper can still provide a major improvement [36].The sensitivity of damper performance to the damping ratio wasinvestigated using Eq. (14) and Fig. 15 shows that the coupledsystem is not very sensitive to the damping ratio of the damperin the range 4.5%7.5% as discussed previously in Section 7.1.4.

The viscoelastic damper was attached to the T beam as shownin Fig. 16 and tuned to the natural frequency of the optimumdamper. Theeffectiveness of thedamper wasthen testedaccordingto the response of the T beam due the heel drop and walkingexcitations.

7.2.3. Performance of the damper due to heel drop excitation

Fig. 17 shows the response of the T beam in the frequency

domaindue to heel drop excitation forboth cases with andwithoutdamper. The value of reduction in the acceleration is about 2.

Table 5

Viscoelastic damper properties for Case Study 2.

Length (L) 510 mmWidth (b) 100 mmThickness of top constraining layer (steel) 6 mmThickness of rubber (h2) 38 mmThickness of bottom constraining layer (steel) 6 mmDissipation loss factor of rubber () 0.15Rubber shear modulus 637 kPa

End mass (mend) 29 kgNatural frequency of damper 4.2 HzDamping ratio (2) of damper 4.5%

Fig. 17. T beam response due to heel drop excitation.

Fig. 18. Time domain decay due to heel drop excitation for cases of T beam withand without damper.

Fig. 18 clearly shows the increase of damping in the time history

response of the T beam, the overall damping ratio of the T beamwith damper was found to be 6.1% using the logarithmic method.

The bare T beam and the T beam with the attached damperwere also modelled using ANSYS as discussed in Section 6. Basedon the ANSYS results the expected reduction factor accelerationresponse was predicted to be 1.9 which is in good agreement withthe experimental reduction factor of 2.

7.2.4. Performance of the damper due to walking excitation

The effectiveness of the damper in reducing the T beamresponse due to walking excitation was also investigated in thisexperiment. It was found that the reduction factor was about 1.4as shown in Fig. 19. This apparent reduction in the effectivenessof the damper for walking excitation compared with heel drop is

due to the fact that the T beam did not reach the full steady stateresonant motion [4].

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Fig. 19. T beam response due to walking excitation based on averaging of 24records for each of the two cases.

Fig.20. Sensitivity of thedamperdue to variation in itsnatural frequency by 10%from the optimum value.

The averaged maximum acceleration of the T beam for 24 time

history samples due to walking excitation was about 3.6% g asshown in Fig. 19. The maximum acceleration obtained from Eq. (1)using Design Guide 11 assessment method [4] was about 3.8% g. Itis quite evident that the value of maximum acceleration obtainedfrom experiment is in good agreement with value obtained fromEq. (1). Clearly, the damper will perform better if the T beamwere to be excited to full steady state resonance which conditionmay not happen in floors for office use. However the damper isstill effective in producing a reasonable reduction in the T beamresponse. An option to further reduce the T beam response wouldbe to specify a higher mass ratio for the damper.

7.2.5. Sensitivity of the damper performance to its natural frequency

and damping ratio of the T beam

In order to assess the sensitivity of the viscoelastic damper tovariation in its natural frequency, the end mass of the damper wasexperimentally changed to alter its natural frequency by 10%from its optimum frequency. The T beam with damper was sub-

jected to heel drop tests for each new frequency. The experimentalresults shown in Fig. 20 illustrate that the efficiency of the damperwas considerably reduced. This highlights the importance of tun-ning the damper to its optimum natural frequency. It should benoted that the efficiency of the damper due to the variation in thefrequency (10%changefrom theoptimum frequency) in this caseis quite differentfrom that of thesteel beam (Case Study 1) becausethe damping ratio of the T beam is significantly higher.

In addition to the experimental results presented in Fig. 20, ananalytical analysis was also performed. In this sensitivity analysis

using Eqs. (2) and (10), two parameters were investigated. Thenatural frequencyof the damperwas changed in the range of10%

Fig. 21. Effect of T beam damping ratio on the performance of damper withdifferent damper natural frequency (Eqs. (2) and (10)).

from the optimum value and the damping ratio of the T beam wasalso changed from 0.5% to 5%. The damping ratio of the damperitself was kept constant at 6%. The results from this study arepresented in Fig. 21. It is clear from Fig. 21 that the effectiveness ofthe damper is significantly degraded when the natural frequencyof the damperdepartedfrom theoptimum value. The investigationindicates that the damper has to be tuned in order to attain theoptimum performance. This is particularly important for floorsystems which have relatively high damping. For floor systemswith low damping (

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The developed viscoelastic damper can be easily tuned to givenproperties and optimised to fit in available spaces. The testingand analyses conducted demonstrate clear advantages of such aviscoelastic damper over conventional viscous dampers for floorvibration applications where displacements are small. By usingseveral dampers in one location or in a distributed system a largefrequency range of effectiveness can be addressed.

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