Application of Tuned Mass Dampers To ControlVibrations of Composite Floor Systems

ANTHONY C. WEBSTER and RIMAS VAICAITIS

INTRODUCTION

Although the incidence of floor vibration problems appearsto be on the rise,1.2 the use of mechanical damping devices tocontrol vibrations is limited. In a recent survey of vibrationcontrol methods, Murray3 reports that passive-mechanicaldamping methods, including viscous damping, visco-elasticdamping, and tuned-mass dampers, have often gone untriedoutside the laboratory or have had marginal impact in actualbuildings. This is particularly unfortunate becausemechanical dampers can sometimes control floor vibrationsmore cheaply than structural stiffening, and are often the onlyviable means of vibration control in existing structures.

This paper details the successful implementation of atuned-mass damping system to reduce the steady-statevibrations of the longspan, cantilevered, composite floorsystem at the Terrace on the Park Building in New YorkCity. The experience with this implementation suggests thattuned mass dampers (TMDs) can be successfully employed tocontrol steady-state vibration problems of other compositefloor systems. The potential for general application of TMDsin composite floor systems is discussed, and areas for furtherresearch are suggested.

BACKGROUND

The Terrace on the Park Building was designed by The PortAuthority of New York and New Jersey as its exhibitionbuilding for the 1964 Worlds Fair (Fig. 1). The buildingfeatures elliptical promenade and roughly-rectangularballroom levels, both suspended six floors above the groundon four steel supercolumns. The columns support a cross-shaped pattern of floor-girders and an elliptical ring girder,which in turn support a radial set of cantilevered floorbeams(Fig. 2). The floorbeams span between the floor and ringgirders, and cantilever from the ring girder to the face of thebuilding (Fig. 3). The ballroom sub-floor is a reinforcedconcrete deck-formed slab, resting on top of and periodicallywelded to the floor-beams.

Anthony C. Webster is assistant professor of architecture/Director ofBuilding Technologies, Columbia University, New York, NY.Rimas Vaicaitis, is professor of civil engineering, ColumbiaUniversity, New York, NY.

At the close of the Fair, the Authority turned over thebuilding to the New York City Department of Parks andRecreation, which leased it to a private caterer to generateincome for the city. The caterer partitioned the ballroom levelsymmetrically into four dining/dancing halls at the corners ofthe building, each served by an existing, central kitchen area.Individual halls were arranged with dining tables near thekitchen (and the center of the building); bandstands anddance floors were located at the tip of the cantilevered floors(Figs. 2 and 3).

As soon as the building's cantilevered main floors wereused as dining and dance halls, guests complained about thestructure's vibrations. Preliminary vibration tests performedduring dance events showed that the floor accelerations anddisplacements sometimes reached 0.07G* and 0.13 inches,respectively. Observations of sloshing waves in cocktailglasses and chandeliers that bounced to the beat of the bandgave credence to these measurements. Observations madeand complaints logged aside, the measured vibrationasinterpreted by the modified Reiher-Meister scale4 or morerecent work by Allen1are generally recognized asunacceptable for dining/dance floors. Floor displacements of0.13 inches are considered "Strongly Perceptible," asmeasured on the modified Reiher-Meister scale; Allen'srecommendations

* A "G" is equal to the acceleration of a body in a vacuum due to the forceof gravity. One G = 32.2 ft/second.2

Fig. 1. Terrace on the Park Buildinggeneral view.

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limit acceptable floor accelerations in combineddining/dancing environments to about 0.03G.

Preliminary free vibration tests of the structure found thefirst natural frequency of a typical quadrant of the ballroomlevel floor (corresponding to one dining/dance hall) to beabout 2.3 Hz. This very low frequency is well below therecommended levels for floors whose vibrations arecontrolled by structural stiffness,1,3 and corresponds closelyto the beat of many dances.5

Besides the low frequency of the ballroom-level floors,their vibrations were being exacerbated by the location of thedance floors, which maximized the amount of vibrations thatdancers were causing (Figs. 3 and 5). Moving the location ofthe dance floors toward the center of the building clearlywould reduce the structure's vibrations. This remedy wascompletely unacceptable to the caterer, who made thesensible point that, located between the kitchen and diningareas, the dance floors would block movement between thetwo and obstruct the exits.

In 1988, after studying various structural stiffeningschemes they could not afford to construct, the ParksDepartment decided to explore solving the vibration problemwith mechanical damping devices. The tuned mass damper(TMD) solution was developed by Weidlinger Associates andProfessor Vaicaitis after we performed a detailed study of thestructure's dynamic characteristics, the forcing functionshaking it, and an assessment of various nonstructuralremedies.

DYNAMIC CHARACTERISTICS OF THESTRUCTURE

First, we began analytical studies of the building's floorsystem to determine its dynamic characteristics. Apreliminary calculation of the first resonant frequency of thelongest cantilevered floorbeams (shown on Fig. 3), wasperformed, using an equation by Murray and Hendrick:6

[ ]f K gEI WLt= 3 12 , (Hz) (1)

Fig. 2. Ballroom (6th floor) plan.

where:f = the frequency of vibration of the floorK = a coefficient depending on ratio of overhang to

backspan [tabulated in Ref. 6]g = 386.4 in/s2

E = modulus of elasticityIt = transformed moment of inertiaW = weight supported by tee beamL = length of cantilever

Assuming composite action of the floorbeam andconcrete deck, Eq. 1 agreed with the earlier roughmeasurements taken at the structure, which showed that thefloor's first natural frequency of vibration was about 2.3 Hz.Although for most of their length, the bottom flanges of thefloorbeams are in compression, the composite floorbeamassumption made sense because the deck was significantlyreinforced, its steel underside was frequently welded to thefloorbeams, and the ratio of live load to dead load was verysmall, reducing the tendency for the concrete to crack and actindependent of the floorbeams.

Next, a detailed, finite element model of a typical floorquadrant (corresponding to one dining/dance hall) wascreated, to determine the fundamental floor frequency moreaccurately, compute the associated mode shape, and see ifhigher floor frequencies and mode shapes were being excited.The floorbeams were modeled with composite bendingproperties and the concrete deck was modeled with plateelements. The mass included all the structural loads,nonstructural loads such as windows, mullions, partitions,and hung ceilings, and about 15 percent of the 100 psf, code-prescribed live load.

Free vibration analysis of this model showed that thereinforced concrete deck and ring girder tied the floortogether, making an entire quadrant of the ballroom levelvibrate as a unit. The fundamental mode shape described acontinuously deformed floor, with maximum deflection at theextreme cantilevered corner, and monotomically decreasingin deformations toward the ring and floor girders. The firstfrequency of the floor system was predicted at 2.22 Hz. Thesecond resonant floor frequency was found at 3.9 Hz.

While the structure was being examined analytically, wealso measured the natural frequencies of each floor quadrant(corresponding to one dining/dance hall) of the actualstructure, the mode-shape associated with the first naturalfrequency,

Fig. 3. Section through ballroom floor (Section 1 on Fig. 2).

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Table 1.Experimentally Determined Floor Frequencies and Damping

Fundamental SecondQuadrant Frequency Frequency Damping(Dining/Dance all) (Hz) (Hz) % of CriticalRose 2.23 3.75 2.8Paradise 2.31 3.0Crystal 2.27 3.75 3.0Regency 2.46 3.6Computer model 2.22 3.91

and the damping in the first mode. Using a variable speed,largemass shaker, our prediction of the floor's resonantfrequencies was confirmed. By simultaneously recordingaccelerations at a number of locations along the floor, wealso confirmed the computer model's prediction of the firstmode shape. Using the half power method,7 the damping inthe first mode was determined. The measured frequencies andexperimentally obtained damping values for each floorquadrant are given in Table 1. The floors were typicallycovered with wood, and supported a lightweight steel-panelbuilding-envelope system from the bottom flanges of thefloor-beams.

The most important empirical data was obtained duringactual dancing. Spectral transforms of the acceleration time-histories obtained during dancing showed that each floorquadrant was vibrating almost exclusively at its first mode(Fig. 4). This result substantially simplified our lateranalyses and helped us determine an appropriate dampingmethod.

The peak root mean square (RMS) acceleration wemeasured at the extreme cantilevered corner of adining/dance hall was 0.06 G, recorded during a rock and rolldance. Assuming the floor to be vibrating in its first mode,we used this measured peak acceleration to determine themaximum floor displacement at the same location, With thefloor vibrating in its first mode, both the displacement, y2(t),and acceleration, ( )y t2 , of the tip of the floor are essentially

Fig. 4. Typical spectral response (floor excited by dancing).

sinusoidal functions in time. Their maximums are related by:

y y2 22

max max /= w (2)where:

w = the frequency of vibration of the floor, in radians persecond

y2 max = the maximum tip displacement at this frequency

maxy2 = the measured RMS floor acceleration

This gave an estimated maximum floor displacement ofabout 0.11 inches corresponding to the measured 0.06G peakRMS acceleration.

ASSESSMENT OF MECHANICAL VIBRATIONCONTROL SYSTEMS

The decision to employ tuned mass dampers was influencedby the functional layout and geometry of the structure; theclient's budget; the fact that the floors were being excitedprimarily at their first resonant frequencies; the largeamplitudes of floor motion; and the light structural floordamping.

Simple Passive Dampers

Simple passive dampers, including viscous, friction, andvisco-elastic systems, rely on a damper mounted between avibrating structure and a stationary object to dissipatevibration energy as heat. As the two systems move relative toeach other, the simple passive damper is stretched andcompressed, reducing the vibrations of the structure byincreasing its effecting damping. At the Terrace, there was nonon-moving element nearby to attach a damper to, so thesesystems were rejected.

Tuned Mass Dampers

Tuned mass dampers (TMDs) work by fastening a mass-block to a structural component (such as a floor) via a spring(Fig. 3). This system is set up so that, when the floor vibratesat a resonant frequency (which could be caused by dancing,for example), it induces analogous movement of the mass

Fig. 5. Floor deflection in first mode shape (Section 1 in Fig. 2).

where:Ff(t) = idealized, periodic forcing function on dance floorYt= deflection of tip of floor in first modeYf = deflection of floor under forcing function

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block and spring. By the conservation of energy, the TMDmotion in turn reduces the amplitude of the floor's vibration.A damping device (dashpot) is usually connected in parallelwith the spring between the mass-block and floor, increasingthe TMD's effectiveness over a range of frequencies andtaking a small amount of mechanical energy out of the systemas heat.

Because each TMD is "tuned" to a particular resonantfrequency, individual TMDs need to be installed for eachexcited floor frequency. Because they rely only on floorvibrations to operate, they do not need to be fastened to anearby stationary object. By the same token, TMDs are mosteffective when located where the floor's amplitudes are thegreatest.

TMDs were considered the only viable passive dampingsystem to employ at the Terrace because they did not requirefastening to a nearby stationary object. They were alsoparticularly well suited to the Terrace because there was onlyone floor frequency per ballroom to damp, reducing therequired number of TMDs, and the TMDs could be installedat locations where the floor amplitudes were largest (Fig. 5),maximizing their efficiency.

Active Mass Dampers

Active mass dampers, which are computer controlled and canalso be configured to work without relying on the relativemotion between the floor and a stationary object, were alsoconsidered. These systems, currently the subject of muchresearch for controlling wind and earthquake inducedvibrations,8 are a generally attractive solution to vibrationproblems because they are so effective. These systems wererejected for the Terrace on the basis of their high installationcost, and their need for regular continuing maintenance,which could not be ensured over the life of the structure.

DESIGN OF THE TUNED MASS DAMPERS

The TMD design process began by creating an "equivalent-displacement" one-degree-of-freedom system, representingthe dynamic behavior of one point of a typical floor quadrantwhen vibrating in its first mode. The one-mode model wasjustified by the experimental data taken in each floorquadrant, which (as noted above) showed that the ballroomfloors were vibrating almost exclusively in their first mode.A TMD was then added to this model, creating a two degreeof freedom system. The performance of this system,representing an actual floor quadrant and TMD, was used tooptimize each TMD's mass, spring stiffness, and damping.

Equivalent Displacement, One Degree ofFreedom Floor Model

Figure 5 shows, for a typical quadrant, the line of maximumfloor deflection in the first mode (cut at section 1 in Fig. 2).This characteristic mode shape and its associated frequencyprovided the basis for the equivalent, one degree of freedom(1 DOF) floor-vibration model shown in Fig. 6. To calibratethe 1 DOF system, we required that its free vibrations have

the same period as a typical floor quadrant's, vibrating in itsfirst mode. This requirement is stated mathematically by:

( / )k m f2 2 1= w (3)

where k2 and m2 are as defined in Fig. 6 and wf1 is the firstresonant frequency of the floor, in radians per second(rad/sec).

The calibration for mass and stiffness was completed bydictating that the maximum dynamic displacement of the 1DOF system would be the same as the tip of the floorconstrained to vibrate in its first mode shape, while beingforced by a periodic, concentrated load at its tip; i.e., y2 max(Fig. 6) = yt max (Fig. 5). Using the free-vibration computermodel, the 1 DOF system's mass, m2, was found by:

m u d t22= / , ( /kips sec in)2 (4)

where u is the mass-normalized generalized mass of the firstmode of the floor system, and dt is the associated modaldisplacement at the tip of the floor. (This equation is derivedin Appendix A.)

As calculated by Eq. 4, m2 is called the "equivalent-displacement generalized floor mass." Using this value form2, k2 was found from Eq. 3.

We also computed k2 and m2 from our experimental data.First, we assumed the floor would respond only in its firstmode when shaken by a harmonic forcing function of a

Fig. 6. 1 DOF floor model.

where:m2 = displacement normalized generalized mass of floor system

in first modek2 = displacement normalized generalized stiffness of floor

system in first modec2 = damping in first mode

F2(t) = idealized, periodic, dance-floor equivalent forcing functiony2 = yt = deflection of tip of floor in first mode

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Table 2.Stiffness and Mass of 1 DOF Floor Models

k2 m2 c2 zz2(kips/in.) (kips) (kips*s/in.) (% of Critical)

Equations 4, 3, 6 197 389 .874 3.1%(Computer) (Determined from (Chosen to match

choice of z2) experimental data)

Equations 12, 10 205 406 .912 3.1%(Experimental) (Determined by z2)

frequency equal to the floor's first resonant frequency [F(t)=Fosin(wf1t), where wf1 is the first resonant frequency of thefloor quadrant (rad/sec) and t = time (sec)]. In this case, thefloor behaves as a one degree of freedom system, whosesteady-state response is given by:

y t F t h ko f2 1 22( ) sin( ) / ( )= +w z (Ref. 7) (5)

where:Fo = the amplitude of the forcing function driving the floor at its

cantilevered tipy2 = the peak floor response measured at the same locationwf1 = the resonant frequency of the floorh = a phase anglek2 = the equivalent displacement generalized stiffness of the

floorz = the measured damping of the floor, expressed as a percent of

the floor's critical damping, cc*

from which:

y F ko2 22max / ( )= z , (in.) (8)

and:

k F yo2 22= / ( )maxz , (kips/in.) (9)

and m2 is then found from Eq. 3.

The damping included in the 1 DOF model (z) was 3.1percent, corresponding to the average of the fourexperimentally determined values given in Table 1. This is abit lower than what would be expected based on publishedvalues.4,9 Using Eqs. 6 and 7, the absolute floor damping, c2,was found to be 0.874 kip-seconds/in.

k2, c2 and m2, computed both analytically andexperimentally, are given in Table 2. The computer generatedvalues were used in the subsequent analysis and design work.

The 1 DOF system's forcing function, F2(t), was alsocalibrated to approximate the effect of dancing on the actual

* For the 1 DOF floor model, z and cc are related by:z=c2/cc (Ref. 7) (6)

wherec k mc = 2 2 2( / ) (kip * sec/in.) (7)

structural floor. The function was assumed to be sinusoidal(which is arguably a fair approximation for dancing1), i.e.:

F2(t) = Fo sin(w t), (kips) (10)

The force amplitude (Fo) was adjusted so that atfrequencies (w ) close to the beat of previously measureddancing at the Terrace, the maximum steady-stateacceleration of the 1 DOF model would match the RMS peakacceleration at the tip of the actual floor during aninstrumented dance event.

Two Degree of Freedom, Floor-TMD Model

After the equivalent-displacement 1 DOF system wasdeveloped, tuned mass dampers were added, creating a twodegree of freedom (2 DOF) system (Fig. 7). Using thissystem, the TMD parameters of mass (m1), stiffness (k1), anddamping (c1), were optimized to reduce the dynamicdisplacement of the floor (y2), due to the forcing functionF2(t), representing dancers on the real structural floor.

Fig. 7. 2 DOF floor-TMD model.

where:m1 = mass of TMDk1 = TMD spring stiffnessc1 = TMD dampingy1 = displacement of TMD

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Table 3.Summary of TMD Parameters *

Trial Optimum Initial Construction Final TunedQuantity Value Value ValueMass, m1 (kips) 18.0 19.0 18.4

(Controlled by 20 kipfloor beam capacity)

Damping, C1 0.19 0.19 0.15(kips* s/in.) Equation 13Spring stiffness, k1 8.3 8.8 8.8(kips/in.) Equations 12, 11 (Cannot be field

adjusted)* Values are presented for the Rose floor quadrant whose measured natural frequency without TMDsinstalled was 2.23 Hz. Results for other quadrants are similar.

The TMDs needed to minimize the floor's vibrationswithout using so much mass that the existing floorbeamswould be overstressed. Although to a point TMDs becomemore effective with increased mass,10 calculations showedthat the floorbeams supporting the TMDs would beoverstressed with masses greater than about 20 kips locatedat tips. Therefore, 18 kips became our trial-optimal TMDmass. This corresponds to a mass ratio (m1 / m2) of about 4.6percent.

Because each actual ballroom floor was respondingprimarily in its first mode shape, the TMDs needed to beoperating near the associated resonant frequency to maximizethe amount of energy shifted from the vibrating floor tothemselves. Various approaches to optimizing a TMD'snatural frequency have been reported.11,12 As a start point, weused the approach outlined by Reed,12 in which the naturalfrequency of the TMD attached to a fixed base is denoted w1.Then:

w1 1 1= ( / )k m , (rad/sec) (11)

where k1 and m1 are the spring-stiffness and mass,respectively, of the TMD.

And Reed's optimum value for w1 is given by:

w1,optimum = 1 / [1 + (m1 / m2)], (rad/sec) (12)

where m2 is the equivalent-displacement generalized floormass defined above.

With m1 and m2, determined, w1optimum was found by Eq.12, and k1 was determined by Eq. 11. We also used Reed'smethod for obtaining a trial value of optimum damping, c1:

[ ]c m k m moptimum1 1 2 1 22 1, / ( / )= + , (kipsec/in.) (13)The trial-optimum values, k1, c1, and m1, computed usingEqs. 11 through 13, are summarized in Table 3.

Starting with the maximum safe mass and predicted-optimum values for c1 and k1, the 2 DOF model of the floor-TMD system (Fig. 7) was analyzed. The system's equationsof motion are:

mm

yy

cc

cc c

yy

kk

kk k

yy

1

1

1

2

1

1

1

1 2

1

2

1

1

1

1 2

1

200

+

--+

+

--+

= =

F t F to2

0( ) sin( )w kips (14)

Where w is the frequency of the forcing function (rad/sec).These equations were used to: check the validity of the

TMD parameters given in Eqs. 12 and 13; predict thereduction in floor acceleration caused by the TMDs; andestimate the maximum accelerations and relativedisplacements of the TMD mass (m1). Because Eq. 14 cannotbe solved modally (due to the high damping in the system),they were integrated numerically with the Runge-Kutta fourthorder method.13 For values of w between 1 and 8 Hz, timehistories were produced and maximum values of y2, y2 , andy2 y1 were recorded.

Because it was our experience that TMDs needed to beadjusted in the field, we designed the actual TMDs to be"tuned" for frequency and damping after installation. Thiswas done by varying the TMDs' mass (m1) with 200 poundsteel plates, and adjusting its damping (c1) with variableenergy dissipation dashpots. Two types of variable energydissipation viscous dashpots were tested at the Carleton Labof Columbia University's Engineering School (Fig. 8), andfound to need a minimum stroke (in the form of enoughrelative floor-TMD mass movement) of about 0.05 inches tobe effective. In practice, the relative motion between theTMD and floor (y2 y1 in Fig. 7) is reduced with increasingTMD mass (m1) and increased damping (c1). To obtain adesired stroke, it was found by manipulating m1 and c1 in Eq.14 that the TMDs performed better if their damping wasslightly decreased than if their mass was reduced. Thus,ensuring the stroke of the TMDs was large enough effectivelyput an upper bound on their damping.

The TMD stiffness, k1, was limited by the properties ofcommercially available springs. The spring stiffness, ofcourse, could not be modified in the field, which did not posemuch of a problem because the natural frequency of the

THIRD QUARTER / 1992 121

TMDs was controlled by adjusting their mass, as describedabove.

The TMD parameters, m1, c1, and k1, which were usedfor their initial construction, are given in Table 3. Thesevalues were adjusted from the trial-optimum values asrequired by the constraints on spring stiffness, damping, andmass noted above. The corresponding predicted performanceof the TMDs is shown in Fig. 9. Each point on the graphrepresents maximum steady state floor displacementcorresponding to the calibrated forcing function operating atfrequency w. The curve predicted that the TMDs wouldreduce dance-induced floor vibration by a maximum of 70percent, corresponding to dancing at about 2.2 Hz.

PERFORMANCE OF THE AS-BUILT SYSTEM

In 1991, one TMD was installed in the corner closet of eachdining/dance hall (Fig. 3). A typical system is shown in Fig.10. Each TMD was tuned for optimum frequency anddamping by using a variable-speed, large mass shaker toexcite the floor at a range of frequencies while monitoringboth floor and TMD accelerations. During an actual danceevent, floor accelerations were monitored first with thedampers locked into place, then free to move. The final,"tuned" TMD parameters are summarized for one floorquadrant in Table 3. Results for other quadrants are similar.The results of the shaker and dance-event tests are given inFigs. 11 and 12 respectively. Our measurements of TMDperformance during dance events showed that the TMDsreduced ballroom floor vibrations by at least 60 percent. Thedifference between the

Fig. 8. Viscous dashpot work per stroke at various energyabsorption settings.

("Kinechek" and "Cushioneer" refer to the manufacturer'sproprietary names of tested models. The energy absorbed by the

dashpots per stroke is adjustable. Different "preset" curvescorrespond to different dashpot settings.)

predicted 70 percent reduction and the 60 percent in-situperformance is ascribed to the difference between the actualand analytical forcing functions (dancers and a sine-wave,respectively), and the floor's vibrations in its second modeshape, which the TMDs were not designed to reduce. Nofloor vibration complaints have been reported to us since theTMDs were installed.

The cost of constructing the four TMDs was $220,000.This is less than 15 percent of the estimated construction costof structural stiffening (with new columns between theballroom floors and the ground) recommended for the Terracein 1987.14

SUMMARYCONCLUSIONS

The TMD implementation described in this paperdemonstrates their successful use in substantially reducingthe vibrations of an existing composite floor system. Thecritical reasons for the success of the system are: itstunability, which helped ensure that the theoreticallypredicted performance could be approximated by the actualas-built system; and the cost of the system, which was aboutan order of magnitude less than the cost of recommendedstructural corrective measures.

Although the methods used to analyze the case-studyfloor system and design its TMDs are very general, and canbe applied in principle to many composite floor systems, theeffective use of TMDs in structures with higher dampingvalues and lower maximum floor displacements may provetroublesome. It has been claimed that it is generally difficultto make TMDs useful in structures with high naturaldamping. The adjustable viscous dashpots used in this case-study perform marginally at small strokes, suggesting theywould not perform adequately in floors whose amplitudes aresmall. However, other types of damping, which are fieldtunable and

Fig. 9. Maximum, steady-state floor amplitudes at tip of floor, aspredicted by Eq. 14. (TMD parameters correspond to "initial

construction values" in Table 3.)

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may perform well at small amplitudes, have been used inTMD applications,15 and warrant further study. It should alsobe noted that, although floor frequency itself should notimpact the viability of TMDs, most composite floors havefrequencies much higher than the fundamental floorfrequency at the Terrace. This may affect the choice ofhardware in other installations, including the types of springsand dashpots used.

The success of the field-tuned case-study systempresented in this paper, and the small number of mechanicaldamping systems installed in actual buildings today, suggestthat damping systems are not being used as often as theypossibly should be. Increased use of passive damping systemsrequires that structural engineers better understand theiroverall performance, and the limitations of their actualcomponents (such as dashpots). With this in mind, furtherresearch in the performance of passive damping devices inactual floor systems is recommended in the following areas:

In-depth studies of TMD dashpots, including linearviscous and Coulomb friction types.

Analysis of tuned-in-the-field TMD effectiveness infloor systems with smaller dynamic displacements.

Analysis of tuned-in-the-field TMD effectiveness inreducing transient vibrations.

Fig. 10. TMD elevation.

Comparison of the effectiveness of TMDs and otherpassive and active damping systems, in controllingboth transient and steady-state vibrations, in terms ofboth performance and cost.

APPENDIX ADERIVATION OF EQ. 4

Applying free vibration analysis techniques to a finiteelement model of the floor system, the following quantitiescan be computed:16

w1 = the first resonant floor frequencyd = the associated mode-shape column vectorM = the mass matrix of the floor systemK = the stiffness matrix of the floor systemu = the generalized mass of the first mode = d MdT

z = the generalized stiffness of the first mode = d KdT

Leaving damping aside for simplicity, if the floor ismoving in only its first mode, forced by the function F(t), at aparticular node, n. then it can be shown7.17 that the floormovement at any point is described by the equations:

u z d F tn n ( )a a+ = (A1)

x d= a (A2)

Fig. 11. Peak RMS floor response at tip of floor, subject tosinusoidal forcing function. Test of actual floor system with field-

tuned TMD (Crystal Quadrant).

Fig. 12. Measured floor acceleration at tip of floor due to dancingwith field-tuned TMD.

THIRD QUARTER / 1992 123

Where a is called the generalized coordinate of the firstmode, and x is the vector of nodal coordinates from thefinite element formulation.

The equation of motion of the 1 DOF, equivalentdisplacement model (Fig. 6) is:

m2y2 + k2y2 = F(t) (A3)

By definition of the 1 DOF model, k2 / m2 = z / u. Byspecifying that y2 = xn when F(t) = fn(t), these constraintslead to:

y2 = dna (A4)

uy zy d F t d F tn n n( ) ( )2 22 2+ = = (A5)

Dividing this by dn2 , and comparing to Eq. A3 yields m2 = u

/ dn2 . Noting that, in the case of the Terrace, dn = dt, and

making this substitution yields Eq. 4.

REFERENCES

1. Allen, D. E., "Building Vibrations from Human Activities,"Concrete International: Design and Construction, AmericanConcrete Institute, 12:No.6 (1990) pp. 6673.

2. Ellingwood, B., "Structural Serviceability: Floor Vibrations,"Journal of Structural Engineering, Vol. 110, No. 2, February1984.

3. Murray, T. M., "Building Floor Vibrations," EngineeringJournal, 28:No. 3, Third Quarter, 1991, pp. 102109.

4. Murray, T., "Design to Prevent Floor Vibrations," EngineeringJournal, Third Quarter, 1975, pp. 8387.

5. Commentary on the National Building Code of Canada, Chapter4, 1985.

6. Murray, T. and Hendrick, W., "Floor Vibrations andCantilevered Construction," Engineering Journal, Third Quarter1977, pp. 8591.

7. Meirovitch, L., Analytical Methods in Vibrations, Macmillan,1967.

8. Masri, S., ed., Proceedings of the US National Workshop ofStructural Control Research, October 1990, Department of CivilEngineering, University of Southern California, Los Angeles,CA.

9. Allen, D., and Rainer, J., "Vibration Criterion for Long SpanFloors," Canadian Journal of Civil Engineering, Vol. 3, No. 2,June, 1976, pp. 165173.

10. Candir, B. and Ozguven, H., Dynamic Vibration Absorbers forReducinq Amplitudes of Hysteretically Damped Beams, pp.16281635.

11. Jacquot, R., "Optimal Dynamic Vibration Absorbers for GeneralBeam Systems," Journal of Sound and Vibration, 60(4), pp.535542, 1978.

12. Reed, F. E., "Dynamic Vibration Absorbers and Auxiliary MassDampers," Shock and Vibration Handbook, Harris, C., ed., 3rded., McGraw Hill, 1988.

13. Gerald, C., and Wheatley, P. Applied Numerical Analysis, 3rded., Addison-Wesley, 1984., pp. 311318.

14. Gandhi, K., Final Report: Study of the Structural Integrity ofthe Terrace on the Park Building. Prepared for the City of NewYork, Department of Parks and Recreation, April 10, 1987.

15. "Harmonizing with the Wind," Engineering News Record,October 25, 1984.

16. Gockel, M., ed., MSCINASTRANHandbook for DynamicAnalysis, MacNeil-Schwendler Corporation, Los Angeles, CA,1983.

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MAIN MENUSEARCH MENUINTRODUCTIONBACKGROUNDDYNAMIC CHARACTERISTICS OF THE STRUCTUREASSESSMENT OF MECHANICAL VIBRATION CONTROL SYSTEMSDESIGN OF THE TUNED MASS DAMPERSPERFORMANCE OF THE AS-BUILT SYSTEMSUMMARYCONCLUSIONSAPPENDIX ADERIVATION OF EQ. 4REFERENCES

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