Tuned Vibration Absorbers for "Lively" Structures

Summar's'

In the first part of this paper, the theoretical background and a design procedurefor tuned vibration absorbers are presented. In the second part, practical experi-ences with two footbridges and a diving platform are described. Experimentsconfirm both the applicability of the proposed design procedure and the effec-tiveness of the absorbers.

Introduction

Human activities such as walking andjumping can produce dynamic forceswith a predominant frequency contentaround 2 Hz. Coincidentally, 2 Hz isalso the fundamental frequency ofmany structures, especially foot-bridges. In many cases these structuresare lightly damped, a condition de-scribed as "lively." and can undergolarge vibrations. These vibrations cancreate a serviceability problem be-cause users can become frightened bythe structure's readily perceived mo-tions. Structures that are easily excitedare also tempting targets for acts ofvandalism. One measure against ex-cessive vibrations is the application oftuned vibration absorbers. They havebeen known for some time, but havenot been widely used, perhaps becausecivil engineers feel uncomfortable withsuch machine-like devices. This paperillustrates the theoretical and practicalexperiences with three structures sub-jected to human-induced vibrations.whose dynamic behavior has been im-proved by tuned vibration absorbers.References to examples with wind-and traffic-induced vibrations are giv-en in [I].

(2)

Structural Engineering International 1/95 Science and Technology 31

Hugo BachmannProf.Swiss Federal Inst. of Technology (ETH)Zurich. Switzerland

Hugo Bachmann. born in 1935. receivedhis civil engineering and doctorate degreesfrom the ETH Zurich. After working withconsulting and contracting firms, he re-turned to the ETH as Professor of CivilEngineering. His teaching and researchactivities concentrate on reinforced andprestressed concrete, structural vibrationsand earthquake engineering.

Using a dimensionless form, the para-meters m11, CH, kH, fllj CT, kT, F andthe two variables u11, UT can be re-duced to the five parameters rnT / rn11,fT / fH. i,. r and the two dime nsion-less variables tH/UHQ, lIT! UJ.

Optimal Absorber Parameters

The behavior of the two-degrees-of-freedom system can be visualized con-sidering the dynamic amplificationcurves in Fig. 2 ('flT!IflH = 1/100, =1%). The excitation frequencyf is nor-malized by the eigenfrequency of thestructure. Without the absorber, thewell-known curve for a single-degree-of-freedom system results. For a struc-ture with an absorber. i.e., for a two-degrees-of-freedom system, the ampli-fication curve typically exhibits twohumps, one below the eigenfrequencyof the absorber and one above theeigenfrequency of the structure. Theoptimal absorbing capacity is obtainedif both humps have the same height. asshown by the curve for fT = fe,,,,. For astructure with no damping, the optimalparameters have been determined in[2] as

f1 = fi (1)1 + rn7 /

I 3m / '11H3

\ 8(1 /mH)

To see whether they are also applica-ble to damped structures, the optimaltuning can be investigated by calculat-ing amplification curves for differentabsorber parameters and comparingtheir maximum values. The maximumamplifications constitute a surfacewith a minimum at the optimal para-meters. Fig. 3 shows the contours of

Benedikt WeberResearcherSiss Federal Inst. of Technology (ETH)Zurich. Switzerland

Benedikt Weber, born in 1953. receivedhis civil engineering degree from the ETHin 1977 and his \lasters degree fromStanford Univ. in 1980. After five years inprofessional practice. he returned to theETH and received his doctoral degreethere in 1994. His main research interest isin structural dynamics and earthquakeanalysis of arch dams.

4g. Peer-reviewed by internationalexperts and accepted h' the IABSEPublications Committee

Theoretical Background

Two-Degrees-of-Freedom System

The simplest model for a structurewith a tuned vibration absorber is atwo-degrees-of-freedom system. It isused here to develop the theoreticalbackground and will be discussedsubsequently in connection with thedesign of tuned vibration absorbers.Fig. 1 shows the general set-up of atwo-degrees-of-freedom system.

10

00.70 0.80 0.90 1.00 1.10 1.20

frequency ratio

0.90 1.00 1.10frequency ratio

Fig. 3: Contours of naxinuan aniplifica—(ions

1.00

0.98

0.96

0.94

0.92

0.90

________________________

0.00 0.01 0.02 0.03 0.04mass ratio m.J./rnH

Fig. 4: Optina! absorber frequency

this surface for the previous example.The minimal amplification is 11.6 atthe pointf7 If = 0.99 and 1= 6%.

The sensitivity of the tuning can be ap-preciated by considering the contourlevel for an amplification of 15. To ob-tain a maximum amplification belowthat value, the tuning of the frequencyhas to be in the range cL = 0.96—1.02.i.e.. within ±3%. On the other hand.the damping value can he in the rangeof 1 = 3%—ll%. which is betweenhalf and almost double the optimalvalue. Clearly the optimal point is rela-

tivelv insensitive to the damping, butthe exact frequency tuning is crucial.By calculating numerically the optimalfrequency for different mass ratios andfor different structural damping val-ues. the curves given in Fig. 4 havebeen obtained. The curve for , = 0%corresponds to the formula given inEqs. (I) and (2). For higher structuraldamping values and high mass ratios.the optimal frequency of the absorberis slightly less than the one given bythe formula. However, since tuned vi-bration absorbers are only effectivefor lightly damped structures, the for-mulas in Eqs. (1) and (2) are sufficient-ly accurate for any practical applica-tion.

Design Procedure

Excitation

The excitation force due to various hu-man activities is described in [1]. It canbe expressed as a Fourier series as:

F(t)=G+iG1 sin(2Jrft)+ 1G7 sin(2 . 2Jrft — -, ) +... (3)

As a simplified procedure. each har-monic is considered separately. Themaximum steady-state response is ob-tained if the excitation frequency f =if,, equals the frequency of the modeconsidered. Because the activity ratelies within a certain range, each modecan usually only be excited by a certainharmonic. Depending on the eigenfre-quencies of the structure, the first, sec-ond or third term is important. Themost severe loading is typically due tojumping. with a force amplitude ofAG = 1.8G. Assuming the weight ofthe person jumping to be G = 800 N.this leads to an amplitude of the first

0.05 Fourier term of AG1 = 1440 N.

Equivalent Single-Degree-of-Freedom System

A first step in designing an absorber isto determine the dynamic parametersof the structure. i.e.. the mass and thestiffness of an equivalent single-de-gree-of-freedom system correspondingto a certain mode of vibration. Forpractical reasons. the stiffness is usual-ly calculated from the mass and theeigenfrequencv. The eigenfrequencycan be accurately determined based onexperimental data. A reliable value isessential to obtain an optimal frequen-cy tuning. The modal mass can be esti-mated assuming a mode shape and cal-culating the corresponding kinetic en-

ergv (Rayleighs quotient). To obtainan equivalent single-degree-of-free-dom system with the same displace-ment as the structure. the assumedmode shape has to be scaled to a valueof 1 at the location and in the directionof the absorber. Similar considerationshave to be made in a finite elementanalysis.

Absorber Parameters

\\'hen designing an absorber, a majordesign decision is the choice of themass ratio mT/in/I. The mass ratio di-rectlv influences the response of thestructure and the relative movementof the absorber mass. The mass ratiocan be chosen with the aid of the de-sign curves given in Figs. 5 and 6.

32 Science and Technology Structural Engineering International 1/95

VIIIti CT

F(t)Y1tH

kH ITCH

Fig. 1: Two-degrees-offreedon system

50S. 40

30

20

C,

TopIf1=0.98f!no absorber;

Fig. 2: Ainplificatioiz curves

0.20I-.

0.15

0.10

0.05

0.000.80 1.20

Si_2'b0

Notations

Structure parameters

m11 = massc11 = dashpot constantk11 = spring constantii,, = displacement

11H0 = quasi-static displacementUHO = AGI/kH

f/I = eigenfrequency

f1, = k,j/in /(2)= damping ratio

ci! = c[//(2lJkH/I?I/I)

Absorber parameters

HIT = massCT = dashpot constant

kT = spring constant

11T = displacementfr = eigenfrequency

fr 'kT/fllT/(2Jr)cr = damping ratio

T= CT/(2\kT/FflT)

Excitation force

F(i) = force applied to structureG = weight of active personAG1 = force amplitude of

ith harmonic= activity rate= phase lag of ith harmonic

relative to first harmonic= time

f = frequency of excitation

Fig. 5: Design curve for structure displacement

0.05

Fig. 5 gives the maximum dynamic am-plification of the structure equippedwith an absorber. For design purposes.this value is multiplied by the quasi-static displacement uH() = ziG,/k,, ofthe structure to get the dynamic dis-placement amplitude. Often the designcriteria is based on a maximum accel-eration (typically 0.5 mIs). rather thana displacement. The acceleration canbe calculated by multiplying the dis-placement by (2,rf)2. The mass ratiomT/niH = 0 corresponds to the casewithout an absorber and the amplifica-tion has the value 1I(2ç11). As can beseen from Fig. 5. the absorber is mostefficient for lightly damped structures.For structures with a damping ratio of

= 5 %, the absorber has almost noeffect. Also, enlarging the absorbermass beyond a mass ratio of 1/50 = 0.02has not much effect, even for lightlydamped structures.

The second consideration is the rela-tive displacement of the absorbermass, which can be determined bymultiplying UHO h the maxinium am-plification given in Fig. 6. This valuehas to be considered for the design ofthe absorber itself. Increasing the ab-sorber mass can reduce the absorbermass displacement even if the structur-al displacement is not much reduced.Having decided on the value of themass ratio, the absorber parameterscan readily be calculated by Eqs. (1)and (2). For practical reasons, the ab-sorber usually cannot be built withmass and stiffness values exactly as cal-culated. The stiffness is realised by asuitable combination of availablesprings, and the mass is adjusted ac-cordingly.

Higher-Order 1'! odes

An absorber can only be tuned to oneparticular frequency. normallc to the

frequency of the fundamental mode ofthe structure. If other modes also haveto be damped. more than one absorberis necessary. Typically, if only the fun-damental mode is damped by an ab-sorber. higher-order modes doniinatethe acceleration in a steady-stateanalysis. However, from a practicalpoint of view. these modes are not im-portant. Although it is possible to ex-cite modes with higher frequenciesthrough the higher Fourier terms ofthe excitation force, it is generallyquite difficult to obtain the steady-state response. which requires main-taining the exact excitation frequencyover a long period of time.

Experimental Considerations

Before Installation

To be able to determine the optimalabsorber parameters. the frequencyand the damping of the relevant modeof the structure have to be establishedfirst. This is most precisely accom-plished h testing. A useful test isdropping a sandbag. which excites thestructure in a wide frequency band.The eigenfrequencies show up as sharppeaks in the Fourier spectrum.

Tuning the absorber in the laboratoryis a valuable, even necessary part ofthe installation. Frequency and damp-ing can be determined by measuringthe free vibration response and inter-preting the resulting decay curve. Prac-tically. free vibration is achieved by ex-citing the absorber manually and thenremoving the force, or by moving theabsorber with a lifting device and thensuddenly stopping the absorber's mo-tion. Forced vibration is only valid ifthe excitation is well defined. For ex-

ample, measuring the vibrations dur-ing manual excitation does not yield aunique eigenfrequency because of thehigh damping (typically 10%). Oneproblem which has to be considered isthat the damping devices themselvesare not purely viscous, and also con-tribute to the stiffness. This means thatthe damping and the frequency cannotbe adjusted independently. As can beseen from Fig. 3. the damping does notneed to be very accurate. whereas ex-act frequency tuning is crucial.

After Installation

Final tuning of the absorber frequencyis done after the installation. An accu-rate method consists in exciting thestructure equipped with the absorberby dropping a sandbag. This impulsiveforce covers a wide frequency rangeand the Fourier spectrum of the struc-tural response resembles the amplifi-cation curve of Fig. 2. If the absorber isoptimally tuned. two humps of equalheight will appear. The method onlyworks for smaller structures where theimposed energy is high enough.

Other useful tests are comparisonswith the absorber locked and un-locked. Locking the absorber can beachieved simply by somehow restrain-ing the movement of its mass. The ra-tio of the maximum values with the ab-sorber locked and unlocked are a mea-sure for the effectiveness of the ab-sorber. For these comparisons, it isimportant to apply an identical force inthe corresponding experiments. Ideal-lv. a mechanical shaker would he usedfor the excitation, but froni a practicalpoint of view it is much simpler, al-though not as accurate, to excite thestructure by human activities such asjumping and running using a pace-giv-ing metronome.

Structural Engineering International 1/95 Science and Technology 33

0.00 0.01 0.02 0.03 0.04mass ratio niT /mH

0.00 0.01 0.02 0.03 0.04 0.05mass ratio rn.1. /mH

Fig. 6: Design curve for relative absorber dicplaceineni

2.50m

I

Fig. 7: Girder footbridge Fig. 8: Girder fooihridge: cross section wit/iabsorbers

Practical Examples

Girder Footbridge

The first example is a footbridge withfour spans as shown in Fig. 7. It consistsof two steel girders and a concrete slabmade up of plates of size 2.5 rn x 2.5 m,which are supported by neoprenebearings. The cross section is shown inFig. 8. The bridge showed excessive vi-bration under normal use primarilydue to the low damping of the steelgirders. The fundamental frequencywas determined experimentally to be2.46 Hz and the damping ratio to lie inthe range of çiq = 0.2—0.4%, dependingon the displacement amplitude. Themodal mass was determined usingRayleighs quotient as niH = 30500 kg.The stiffness follows as kH = 7280kN/m.

The design excitation force was takento be \G1 = 1280 N. corresponding toone person jumping with the reso-nance frequency in the center of the

main span. Assuming a damping ratioof H = 0.25 % leads to a dynamic am-plification factor of 200 and a steady-state acceleration amplitude of 8.4rn/s2. An acceptable value of accelera-tion was assumed to be about 0.5 m/s2.Thus the dynamic amplification factorhad to be reduced from 200 down to12. Referring to the design curves inFig. 5. the mass ratio has to be about0.012. The actual absorber mass cho-sen was 350 kg. corresponding to amass ratio of 0.0115. This leads to atheoretical steady-state acceleration of0.053 g. The optimal absorber parame-ters are frn,, = 0.0989 = 2.43 Hz and

= 6.5%. The relative displacenientof the absorber mass, according to thedesign curves in Fig. 6. is ± 15 mm.

The design of the absorber is shown inFig. 9. The main problem was to con-struct an absorber taking into accountthe limited space available. The twoidentical absorbers consist each of amass of 175 kg supported h' foursprings and of two damping devices.

The damping device is a rod sub-merged in a viscous fluid. The frequen-cy can be modulated by a variablemass and the damping can be adjustedby varying the submerged length of therod.The absorber was first tuned in thelaboratory before it was installed onthe bridge. It was fine-tuned on site.using the spectrum of the responsefrom a dropped sandbag (Fig. 10). Theabsorber mass was adjusted until thetwo humps between 2 and 3 Hz wereapproximately equal.

Accelerations due to one person jump-ing are shown in Fig.1l. The high effec-tiveness of the absorber is denionstrat-ed h comparing the values with theabsorber locked and unlocked.

The maximum values of selected testswith one person and with the absorberlocked and unlocked are shown inTable 1. The ratio, which is an indica-tion of the effectiveness of the ab-sorber, is also given. Clearly, with theabsorber locked, the maximum values

1400mm

Fig. 9: Girder footbridge: vibration absorber

300 mmTest Accelerations (iri/s) Ratio

Locked Unlocked

Jumping 9.0 0.48 19.0Walking 1.2 0.24 5.0Running 1.7 0.59 2.9

.0 1 2 3 4 5

frequency [Hz]

Fig. 10: Girder footbridge: spectrum (sand-bag)

34 Science and Technology Structural Engineering International 1/95

16.Om 25.lm

absorber

20.1 m 17.6m

L1C. _.

0.50

0.40

0.30

0.20

Table 1: Girder footbridge: mnaxinlumn verti-cal accelerations

Fig. 11: Girder footbridge: accelerations (jumping)

z7z_/

absorber

30 ni

Fig. /2: Cable-staved footbridge

-C 45 m

Fig. 13: Cable-staved footbridge cross sec-(ion

depend on the kind of loading and itsduration. The steady-state case is onlyobtained for jumping. For walking andrunning, the time needed to cross thebridge is too short to reach steady-state conditions. The values with theabsorber unlocked all lie around theacceptable value of 0.5 m/s2.

Cable-Stayed Footbridge

The second example is a cable-stayedfootbridge with a steel girder (Fig. 12).This case is different from the othertwo in the respect that the installationof an absorber was planned before thebridge was built. Therefore, the firstanalyses were performed by finite ele-ment calculations. The fundamentalfrequency was determined to be fH=1.98 Hz. the modal stiffness k1, =3500 N/mm and the modal mass mH =23000 kg. The damping ratio was as-sumed to be ,, = 0.5 %. The locationfor the absorber was chosen becausethis is where the fundamental modeshape has its largest amplitude. Usingan excitation force of G1 = 1.8 x800 N acting at the location of theabsorber yields an acceleration of6.3 mIs. which is about ten times theacceptable value. With a planned ab-sorber mass of 800 kg this value woulddecrease to 0.46 m/s2. This absorbermass was chosen somewhat larger thanabsolutely necessary, because spacewas no problem in this case. Theplacement of the absorber is shown inFig. 13.

The design of the absorber is shown inFig. 1-I. Because the absorber parame-ters were based on a finite elementanalysis rather than on measurements,the absorber was designed to be ad-justable for a wide range of frequen-cies. In addition to variation of themass. there were also three sets ofsprings with different stiffnesses avail-able. The actual absorber mass afterthe frequency tuning was 1000 kg, cor-responding to a mass ratio of 0.044.leading to a theoretical maximum ac-celeration of 0.44 mi's and a maximum

relative movement of the absorbermass of± 14mm.

After the bridge was built. the eigen-frequencies were determined experi-mentally h' analyzing the free vibra-tions produced by a dropped sandbagand by a person jumping. The funda-mental frequency was fH = 2.12 Hz.which is close to the calculated valueof 1.98 Hz. Higher modes were alsoquite close to the calculated values.The damping ratio was about 0.5 % forlow acceleration amplitudes in therange of 0.1—0.2 m/s2. However, forlarger amplitudes the damping wasmuch more than anticipated. Up to anamplitude of 1 m/s2 it was about 1%,and for higher amplitudes it was evenlarger. Because the fundamental fre-quency was close to the predicted val-ue, the different sets of springs werenot actually needed. The tuning wasperformed in the laboratory by vary-ing the mass only.

Various tests with one or two personsjumping or running were performed.Fig. /5 shows the comparison of the ac-celeration response due to one person

jumping, both with the absorberlocked and unlocked. The absorber isnot as effective as in the first example.One reason is that the locked caseused as a reference already has a lowvalue because the structure itself hasconsiderable damping. especially forhigher displacement amplitudes. Theother reason is that the influence ofhigher modes is no longer negligible,as can be seen from the time history.Maximum values for one person jump-ing and running are shown in Table 2.Although the ratios between thelocked and the unlocked cases arerather low, the absorber performs wellin that it reduces the response from anexcessive to an acceptable value.

Test Accelerations (mis2) Ratio

Locked Unlocked

Jumping 1.1 0.45 2.4Running 0.45 0.3 1.5

Table 2: Cable-stayed footbridge: ,naximurnvertical accelerations

Structural Engineering International 1/95 Science and Technology 35

10mm

—- -.700 mm

2500 mm 700 mni

Fig. 14: Cable-staved footbridge: vibration absorber

0

0 5 10 15

Fig. 15: Cable-stayed footbridge: accelerations (jumping)

time [s]20

Diving Platform

The last example is a reinforced con-crete diving platform (Fig. 16). It con-sists of a Y-shaped shaft with two slabsat the 3 m and 5 m levels. The twobranches are connected by a tie-beamat the level of the lower slab. The plat-form could he excited excessively byshaking it horizontally on the railingparallel to the pool. Some cracks in theshaft were observed.

In a first stage. the structure was thor-oughly studied, both by finite elementcalculations and experimental modalanalysis. The analysis showed that themain vibrations were in the first modeas depicted in Fig. 16. Measured valueswere f11 = 2.6—2.7 Hz for the funda-mental frequency and , = 1.5—2.0%for the damping ratio. A special prob-1cm in this example as the fact thatthe movement of the structure at theabsorber location is oblique, with alarger component in the horizontaldirection, as indicated. An ideal ab-sorber would have to move in the samedirection, but such a device is difficultto construct. Instead it was decided todesign a horizontally moving absorber.The classical solution of a pendulumwas not feasible in this case becausethe length of the pendulum would onlybe about 3 cm. The novel design finallyused is shown in Fig. 17. It consists of amass hanging on steel plates acting assprings. It moves like a pendulum butthe stiffness is provided by the springs.Besides the mass, the stiffness can alsobe varied h adjusting the length of thesprings. The damping device is again arod submerged in a viscous fluid. Therod. however, moves horizontally in-stead of vertically.

The modal mass of the platform wasdetermined by the finite element mod-el with the appropriate scaling of themode shape as rn11 = 12000 kg. For thedesign of the absorber a reduction fac-tor for the accelerations of about 5 waspostulated. Assuming a damping ratioof 1.5%. the dynamic amplifica-

tion has to be reduced from 33 down to7. According to the design curves inFig. 5, the corresponding mass ratiois 0.03. The mass actually chosen was335 kg corresponding to a mass ratio of0.028.

First testing of the platform showed anamplitude-dependent eigenfrequencywhich was somewhat higher than theone determined by the experimentalmodal analysis, because in the mean-time the cracks in the shaft had beenrepaired by epoxy injection. The fre-quencv was 2.9 Hz for accelerationamplitudes of 0.05 m/s2 and 2.7 Hz foramplitudes of 2 mIs2. The absorberwas tuned for an expected frequencyof 2.8 Hz.

Tests with the absorber locked and un-locked were performed exciting the

structure by jumping or by shaking iton the railing. Fig. 18 shows the com-parison of the accelerations inducedby two persons shaking with the ab-sorber locked and unlocked. Maxi-mum values of accelerations due toshaking by two persons and due tojumping by one person are comparedin Table 3. As can he observed, the ab-sorber is very effective. The desired re-duction factor of 5 was more or lessobtained in the tests shown here and inother tests.

Test Accelerat ions (mis2) Ratio

Locked Unlocked

Jumping 1.3 0.3 4.3Shaking 3.2 0.5 6.4

Table 3: Diving platform: ,naxi,nuni lion—

coma! accelerations

ConcIusion

Vibration absorbers can be designedbased on simple dynamics. The two de-sign charts in Figs. 5 and 6. and the for-mulas from Den Hartog given in Eqs.(1) and (2) are sufficient to determinethe absorber's parameters.

Recommended values for the mass ra-tio are 0.03—0.04 if space for the tunedvibration absorber is not restricted.Lower values, down to 0.01. still leadto effective absorbers for lightlydamped structures. This choice, how-ever, leads to large relative move-ments of the absorber mass.

The exact tuning of the absorber fre-quencv is crucial, whereas the dampingneed only be adjusted approximately.For long-term operation of tuned vi-bration absorbers it is important thatthe device is accessible for mainte-nance and re-tuning in case the dv-namic properties of the structurechange.

References

[1] BACII\IANN. H.. A\1\IANN. \V. etal. Vibration Problems in Structures — Prac—

uail Guidelines. Birkhäuser Verlag. Basel.40 Boston. Berlin. 1994.

[2] DEN HARTOG. J. P. Mechanical Vi-brations. \lcGraw-Hill. New York. 1940.

36 Science and Technology Structural Engineering International 1/95

absorber

£

Fig. /6: Diving platform

C

610mm

Fig. 17: Diving platform: vibration absorber

0 10 20 30

Fig. 18: Diving platformmz: accelerations shaking)

time [sJ