Full-Scale Dynamic Testing And Modal Identification Of A Coupled floor Slab System

In order to determine the damping ratios at vibration levels comparable to the target serviceability limits of milli-g level, e.g., ISO 2631-2[14], forced vibration ker tests were perfor

Full-scale dynamic testing and modal identification of a coupled floor slab system S.K Au⇑, Y.C Ni, F.L Zhang, H.F Lam

Department of Building and Construction, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

a r t i c l e i n f o

Article history:

Received 7 March 2011

Revised 9 December 2011

Accepted 12 December 2011

Available online 2 February 2012

Keywords:

Ambient vibration test

Bayesian FFT

Forced vibration test

Modal identification

a b s t r a c t

This paper presents work on full-scale vibration testing of the 2nd and 3rd floor slabs of the Tin Shui Wai Indoor Recreation Center The slabs are supported by one-way long span steel trusses, which are con-nected by diagonal members and vertical columns to form a mega-truss On the 2nd floor are a large multi-function room and children play area, while the 3rd floor hosts two basketball courts Based on their expected usage, significant cultural vibrations with possible rhythmic activities can be expected

To determine the dynamic characteristics of the constructed slab system, ambient and forced vibration tests were performed Thirty-five setups were carried out in the ambient test to determine the mode shapes using six triaxial accelerometers A recently developed Fast Bayesian FFT Method is used to iden-tify the modal properties using the ambient data in individual setups The mode shapes from the individ-ual setups are assembled by a least square fitting procedure Forced vibration tests were performed by loading the slabs at resonance with a long-stroke electromagnetic shaker, resulting in vibration ampli-tudes in the order of a few milli-g A steady-state frequency sweep was carried out and the modal prop-erties were identified by least square fitting of the measured steady-state amplitude spectra with a linear dynamic model The dynamic properties identified from the ambient and forced vibration tests, as well as their posterior uncertainty and setup-to-setup variability, will be compared and discussed The field tests provide an opportunity to apply the Fast Bayesian FFT Method in a practical context

Ó 2011 Elsevier Ltd All rights reserved

1 Introduction

The Tin Shui Wai (TSW) public library cum Indoor Recreation

Center is an ex-Provisional Regional Council project to meet the

demand for library and recreational facilities of the Tin Shui Wai

district in the New Territories of Hong Kong It is a three-storied

concrete building with a height of approximately 40 m Fig 1

shows the exterior view of the building at the time of

instrumen-tation The slabs on the 2nd floor (2/F) and 3rd floor (3/F) span over

a 35  35 m area and are supported by one-way long span steel

trusses The two floor slabs are connected by six vertical columns

and diagonal members at about quarter spans, forming a combined

system where the slab dynamics are likely to be coupled On the 2/

F are a large multi-function room and a children playground The 3/

F hosts two basketball courts Based on their expected usage,

sig-nificant cultural vibrations with possible rhythmic activities are

expected At the design stage, a finite element model was created

to estimate the dynamic properties of the slab system, revealing

natural frequencies of 5.4 and 6.6 Hz for the first two vertical

modes Realizing the limitations in the model and the absence of

the damping ratios[1], it was of interest to both the building

own-er and design engineown-er to expown-erimentally detown-ermine the modal

properties in order to assess the likely vibration level under service loading to a higher confidence than was possible from the informa-tion available at the design stage A series of field vibrainforma-tion tests were performed with these objectives in mind They include ambi-ent vibration test, forced vibration (shaker) test and service load (jumping) test

Full-scale testing provides an important means for acquiring in-situ knowledge of a constructed facility[2–5] Ambient vibration tests can be performed without artificial loading and hence require less equipment[6–9] They were performed first to obtain a first-hand estimate of the natural frequencies, damping ratios and mode shapes A number of setups were performed to determine the mode shapes using six triaxial accelerometers Due to the nature

of ambient loading, the modal properties are applicable only for low vibration levels (e.g., up to 100lg) This qualification is espe-cially relevant for the damping ratios, which are well-known to be amplitude dependent[10–13] In order to determine the damping ratios at vibration levels comparable to the target serviceability limits (of milli-g level, e.g., ISO 2631-2[14]), forced vibration (ker) tests were performed with a long-stroke electrodynamic sha-ker, where the mode shapes along a critical line of the slab were also identified Service load tests with a large number of partici-pants jumping to cause resonance were finally performed to obtain the likely vibration in some design scenario This paper presents the field instrumentation and modal identification of the coupled 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved.

⇑ Corresponding author Tel.: +852 3442 2769; fax: +852 2788 7612.

E-mail address: siukuiau@cityu.edu.hk (S.K Au).

Contents lists available atSciVerse ScienceDirect

Engineering Structures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / e n g s t r u c t

slab system, focusing on the ambient tests and shaker tests The

field tests are described in detail with particular reference to their

implications on the identified modal properties The paper also

contributes to the application of established Bayesian modal

iden-tification theory and discussion of practical issues encountered

The posterior uncertainty and setup-to-setup variability of modal

properties shall also be discussed from a Bayesian and frequentist

point of view, respectively

2 Ambient vibration test

2.1 Instrumentation

In the ambient tests, six force-balance triaxial accelerometers,

Guralp CMG5T, were used to obtain acceleration time histories

synchronously in each setup The analogue signals were

transmit-ted through cable and acquired digitally by a 24 bit data logger

The overall channel noise is about 0:1lg= ffiffiffiffiffiffi

Hz p

in the frequency band above 1 Hz Acceleration data of 6  3 = 18 channels from

the six triaxial accelerometers were acquired at a sampling rate

of 2048 Hz (the lowest allowed by hardware) and later decimated

by 8 to an effective sampling rate of 256 Hz for analysis

2.1.1 Sensor location

For the purpose of identifying mode shapes, the slabs were

divided into segments by grid lines, whose intersections defined

the sensor locations Setting out was performed by the building

contractor, with precision adequate for field testing In order to

cov-er all the target degrees of freedom (DOFs) with a limited numbcov-er of

sensors (six only), a number of setups were planned The measured

DOFs in different setups were designed to share a common set of

DOFs so that their mode shape information covering different parts

of the structure can be assembled (or ‘glued’) together

Figs 2 and 3show the overall setup plans for 2/F and 3/F,

respec-tively The instrumented area on each floor measures 30 m long by

20 m wide A total of 9  7  2 = 126 locations were planned to be

measured triaxially, giving 126  3 = 378 DOFs In these figures, the

number in the rectangular box shows the location number Typical

locations are filled yellow Next to the box shows the setup number

underlined The location numbers are assigned with the following

convention that facilitates field implementation: the first number

indicates the floor; the second number indicates the number of

the row; the third and fourth number indicate the column number

For example, ‘2101’ refers to the first row and the first column on 2/

F This nomenclature allows easy recalling of position on site It also

allows additional sensor locations to be added without disturbing

the existing location numbers

2.1.2 Reference sensors

To allow for the assembling of mode shape information on the two floors from different setups, two reference sensors, one on each floor, were placed and remained recording in all setups Loca-tions 2404 and 3404 have been chosen to be the reference, which are filled light brown inFigs 2 and 3, respectively Both theoretical and practical considerations have been taken into account in the choice of these reference locations On the theoretical side, they should have significant response in the modes of interest At the planning stage an attempt was made to avoid nodal locations based on intuitive guess of the mode shape On the practical side, limited cable lengths (max 45 m in this case) required that the ref-erence locations be near the central area of the slab, although this was complicated by the blocking of the partition walls surrounding the multi-function room on 2/F (see column lines 2 and D inFig 2)

A hole, indicated by ‘H’ inFig 3, was drilled on 3/F to allow the pas-sage of signal cable between 2/F and 3/F Without this hole, one would have resorted to run the cable through the staircase near C-3 inFig 2, which would require much longer cable and create additional safety issues on site Drilling of this hole could be facil-itated as the internal servicing of the building was still in progress 2.1.3 Roving setups

Using the six triaxial accelerometers, the 126 measurement locations inFigs 2 and 3were covered in 35 setups, with 16 setups

on 2/F and 19 setups on 3/F The setups on 2/F and 3/F were per-formed separately on two consecutive days, from 8am to 6pm In all setups two sensors were always placed at the reference loca-tions 2404 and 3404 The remaining four sensors were roved in dif-ferent setups to cover the other locations InFigs 2 and 3, the color

of the number in rectangular box distinguishes the particular sen-sor placed, e.g., blue for TM54 and red for TM55 As the setups pro-ceeded, the sensors typically marched from the figure North to South, moved to the right column and then North to South again The last two setups inFig 2 were exceptions in order to cover the right slab boundary

Ambient test of 3/F, which was done one day after 2/F, followed

a similar plan in the early setups until Setup 8, where the channels associated with TM54 failed due to faulty cable Subsequent setups were revised immediately on site and resorted to proceed with the remaining three roving sensors As a result, three setups were added to cover all the remaining locations, leading to 19 setups The plan shown inFig 3is the one actually used

During the test, one person was responsible for a particular sen-sor When transiting between setups, each roving sensor was moved to the next corresponding location Including taking pic-tures and leveling, the transition typically could be finished in

5 min Vibration data in each setup was recorded for 15 min Exceptions were Setups 17–19 on 3/F, where only 10 min of data were collected due to time limitation and in view of their boundary nature (relatively unimportant) Nevertheless these exceptions have little effect on data quality for the purpose of modal identifi-cation All sensors were oriented with their North aligning with North direction of the figure

As a note, it rained on the day when the setups on 3/F were per-formed As the roof was not completely covered nor sealed, rain water pooled on the 3/F slab Upon inspection of data on site the rain was found to have insignificant effect on data quality The pooling of rain water might have increased the dead weight and damping of the slab but the effect was insignificant, as evidenced from the identification results (see later)

2.2 Ambient modal identification Using the data in each setup, the modal properties of the struc-ture are identified following a Bayesian FFT approach The original Fig 1 The TSW Indoor Recreation Center.

formulation is due to[15] A recently developed fast algorithm[16]

allows practical implementation The method makes use of the Fast

Fourier Transform (FFT) of measured ambient data on a selected

fre-quency band for modal identification The basic idea is that, for a

structure under broad-banded excitation, the real and imaginary

part of the FFT of the measured response follows a

multi-dimen-sional Gaussian distribution which can be characterized analytically

in terms of the modal parameters By maximizing the posterior

probability density function (PDF) of the modal parameters given

the FFT data, or equivalently minimizing the negative log-likelihood

function, the most probable modal properties can be determined

Posterior uncertainty of the modal parameters in the context of

Bayesian inference can also be calculated from the Hessian of the

negative log-likelihood function The theory is outlined below

The measured acceleration data f€yjg is assumed to consist of the

structural ambient vibration signal and prediction error:

€

where €xj2 Rn and ej2 Rnðj ¼ 1; 2;    ; NÞ are the acceleration

re-sponse of the structure and prediction error, respectively; N is the

number of samples per channel; n is the number of measured DOFs

in a given setup The prediction error accounts for the discrepancy

between the measured response and the (theoretical) model

re-sponse for given modal parameters, which may arise due to

model-ing error and/or measurement noise The FFT of f€yjg is defined as:

Fk¼

ffiffiffiffiffiffiffiffi

2Dt

N

r

XN

j¼1

€

yjexp i2pðk  1Þðj  1Þ

N

ð2Þ

whereDt is the sampling interval; k = 1, , Nq, Nq= int(N/2) + 1 is the

index corresponding to the Nyquist frequency; int(.) denotes the

integral part of its argument

Let Zk¼ ½ReFk; ImFk 2 R2nbe an augmented vector of the real and imaginary part of Fk In practice, only the FFT data confined

to a selected frequency band dominated by the target mode(s) is used for identification Let such collection of FFT data be denoted

by {Zk} Using Bayes’ Theorem and assuming no prior information, the posterior PDF of the set of modal parameters h (say) given {Zk}

is proportional to the likelihood function, i.e.,

The ‘most probable value’ (MPV) of h is the one that maximizes p(h|{Zk}), and hence p({Zk}|h) It is convenient to write in terms of the ‘negative log-likelihood function’ (NLLF)

such that p(h|{Zk}) / exp [L(h)] The MPV of h is then the one that minimizes the NLLF

Determining the MPV of the modal parameters h requires numerically minimizing the NLLF The computational time grows drastically with the dimension of h, which is proportional to the number of measured DOFs n in a given setup This renders direct solution based on the original formulation impractical in real applications In view of this, fast algorithms have been developed recently which allow the MPV to be obtained almost instanta-neously in the case of well separated modes [16] or in general (i.e., closely-spaced modes) [17,18] The modes studied in this work can be considered well-separated and so they can be identi-fied using the FFT on separate frequency bands (i.e., m = 1) For a single mode in the selected frequency band, h consists of only one set of natural frequency f, damping ratio f, power spectral den-sity (PSD) of modal force S, PSD of prediction error Se, and mode shapeUeRn From first principle for sufficiently smallDt and long

3

12

2301

2401

2501

2601

2701

2302

2402

2502

2602

2702

2303

2403

2503

2603

2703

2304

2404

2504

2604

2704

2305

2405

2505

2605

2705

2306

2406

2506

2606

2706

2307

2407

2507

2607

2707

2308

2408

2508

2608

2708

2309

2409

2509

2609

2709

1

2

3

4

5 6 7

9

10

11

12 13 14

2

4

5 6 7

9

10,11

12 13 14

2

3

4

5 6 7

9

10

11

13 14

2

3

4

5 6 7

9

10

11

12 13 14

15

15

15,16

16

16

16

Fig 2 Ambient test setup (2/F); solid circles-columns; mega truss near numbered lines.

duration of data (often met in practice) the NLLF can be shown to

be

LðhÞ ¼1

2

X

k

ln det Ckþ1

2 X

k

ZTC1

where the sum is over all frequencies in the selected band;

Ck¼SDk

2

UUT 0

0 UUT

þSe

is the theoretical covariance matrix of Zk;

Dk¼ ½ðb2k 1Þ2þ ð2fbkÞ21 ð7Þ

and bk= f/fk; where fkis the FFT frequency abscissa I2n2 R2n2nis the

identity matrix

The inverse on Ckin Eq.(5)renders the dependence of the NLLF

on h highly non-trivial Nevertheless, reformulating using

eigen-space decomposition, it can be shown that the NLLF can be

rewrit-ten in the following canonical form:

LðhÞ ¼ nNfln 2 þ ðn  1ÞNfln SeþX

k

lnðSDkþ SeÞ

þ S1e ðd  UTAU=kUk2

where Nfis the number of frequency ordinates in the selected band;

kUk ¼ ðUTUÞ1=2is the Euclidean norm of U; and

A ¼X

k

d ¼X

The significance of Eq.(8)is that it is explicitly in terms ofUas a quadratic form and the inverse in Eq.(5)has been resolved It fol-lows from standard results of linear algebra that the MPV ofUis simply the eigenvector of A with the largest eigenvalue Conse-quently, only four parameters, ff ; f; S; Seg, need to be optimized numerically The computational process is significantly shortened with no dependence on the number of measured DOFs n For mod-erate to a large number of DOFs, say, 30, this typically requires a few seconds only

2.2.1 Posterior uncertainties The posterior uncertainty of modal parameters is associated with the spreading of the posterior PDF about the MPV With suf-ficient data, the modal parameters are asymptotically jointly Gaussian, and so their uncertainty can be fully characterized by the covariance matrix of the posterior PDF[19] Using a second or-der Taylor series of the NLLF about the MPV, it can be shown that the posterior covariance matrix is equal to the inverse of the Hes-sian of the NLLF Analytical expressions have been derived for cal-culating the Hessian without resorting to finite difference[16] The uncertainty of f ; f; S and Se, which are scalar quantities, can

be conveniently assessed by their posterior COV (coefficient of var-iation = posterior standard devvar-iation/most probable value) On the other hand, special care should be exercised for the mode shape because of its vectorial nature and the fact that its components are subject to unit norm constraint It has been shown that a ran-dom mode shape following the posterior distribution and satisfy-ing the norm constraint can be represented as[20]

U0¼ 1 þXn

d2jZ2j

!1=2

^

U0þXn

djZjuj

!

ð12Þ

16

H

Fig 3 Ambient test setup (3/F); hollow circles-columns; mega truss near numbered lines.

where ^U02 Rn is the most probable mode shape (normalized to

unity); fd2j :j ¼ 1; ; ng and fuj2 Rn:j ¼ 1; ; ng are respectively

the eigenvalues and eigenvectors of the posterior covariance matrix

of U, the latter obtained from the corresponding partition of the full

posterior covariance matrix of h; fZj:j ¼ 1; ; ng are independent

and identically distributed standard Gaussian random variables

The Modal Assurance Criteria (MAC) between the uncertain

mode shapeU0and its MPV ^U0indicates the deviation in direction

and hence the uncertainty ofU0 In the current context the MAC is

a random variable given by

q¼ U^

0TU0

k ^U0kkU0k¼ 1 þ

Xn j¼1

d2jZ2j

!1=2

ð13Þ

It can be shown that asymptotically for small dj or large n,

E½q  1 þXn

j¼1

d2j

!1=2

ð14Þ

Thus, the closer the E½q is to 1, the smaller the uncertainty of

the mode shape Eq.(14)can thus be used as a convenient measure

for the posterior mode shape uncertainty

2.2.2 Mode shape assembly

Using the Fast Bayesian FFT Method, the natural frequencies,

damping ratios and mode shapes can be obtained in each setup

separately It remains to assemble the mode shapes in different

setups to form the overall mode shape containing all the measured

DOFs A recently developed least-square method[21]is used to for

this purpose, whose theory is omitted here due to space limitation

2.3 Modal identification results The ambient modal identification results shall be presented in this section As a starting task the PSD spectra of the acquired data shall be examined first, as they roughly indicate the modes present and guide the choice of frequency bands

2.3.1 PSD spectra

Fig 4(a) shows the PSD calculated using a typical set of data (Setup 204) The corresponding singular value spectrum is shown

inFig 4(b) The channels for sensor East (x), North (y) and Vertical (z) are plotted with dashed, dotted and solid line, respectively Clear resonance peaks characteristic of structural modes can be ob-served in the frequency bands 2–4 and 6–10 Hz The former corre-sponds to lateral modes of the whole building while the latter to vertical modes of the slab Using the data in each setup, for each mode a frequency band and an initial guess for the natural fre-quency are hand-picked from the singular value spectrum The FFT data within the frequency band are used for identifying the mode

2.3.2 Natural frequency and damping

Table 1shows the identified modal parameters in term of their MPV in different setups, including the natural frequency f, damping ratio f, PSD of modal force S and PSD of prediction error Se The re-sults for the mode shapeUshall be presented graphically later The setup-to-setup sample statistics of the identified values among the setups are shown at the bottom of the table.Fig 5shows the histogram of the MPV of f and f among all setups As expected, the identified values vary among setups As seen inTable 1, the

10-7

10-6

10-5

10-4

10-3

Frequency (Hz)

(a) PSD spectrum

2 4 6 8 10 12

x 10-5

Frequency (Hz)

Lateral modes

of whole building

Mode 1

Mode 3

Mode 2

Vertical modes

of slab

(b) Singular value spectrum

MPV of natural frequency f, regardless of mode, show a small

var-iability (<1%) that can be ignored for practical purposes Significant

variability is seen in the damping ratio f (10–20%), and even more

so in the PSD of modal force S and prediction error Se(over 50%)

InTable 1, the column titled with ‘RMS’ shows the modal

con-tribution of root mean square value of acceleration response As

a reference for comparison, the values presented are for the

verti-cal DOF at the 2/F reference location (2404) The RMS values are

calculated using the standard formula from random vibration

the-ory[22], i.e., ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pSf =4f

p

), where the identified modal parameter val-ues have been used The RMS valval-ues are presented so that the

damping ratios could be viewed in an amplitude-dependent

per-spective Roughly speaking, the identified damping ratios are

appli-cable for modal vibration level up to 100lg

It should be noted that the values of S presented inTable 1all

correspond to mode shape normalized with the vertical DOF at

the 2/F reference location (2404) equal to unity.Table 2on

poster-ior COV later suggests that the value of S in each setup is identified

with good accuracy This means that the values of S presented in

Ta-ble 1primarily reflect the change in environmental stochastic

load-ing intensity With this in mind, one observes that the values of S in

Setups 309–318 inTable 1are significantly higher than those in the

remaining setups, suggesting stronger stochastic environmental

loading This is consistent with the fact that these setups were

per-formed when there was a rain storm and the 3/F slab was hit by rain

drops (recalling that the roof was not finished yet)

2.3.3 Mode shape The MPV of the mode shape is next presented graphically Recall that the mode shape for the whole slab system is assembled in a least square sense from those identified in the individual setups

Fig 6shows the mode shape of mode 1 This is the fundamental bending mode along the x-direction It is clear that the 2/F and 3/

F slabs are coupled The maximum deflection occurs at about the midspan along the long direction, although the deflected shape

on the left and right side from the midspan are somewhat different

In particular, there is less deformation on the left side and there ap-pears to be more rotational restraint at the left end This is believed

to be attributed to the partition walls (seeFig 2), which can im-pose local boundary restraints The mode shapes of the slabs on 2/F and 3/F are quite similar, attributed to coupling by the interior steel posts.Fig 7shows the mode shape of mode 2 There is almost one full sine curve in the mode shape along the x-direction with maximum deflection at the quarter-span from each end Similar

to mode 1, the left side from the midspan appears to have less deformation compared to the right side, again attributed to the partition walls The mode shape of mode 3 inFig 8has one and

a half sine curve along the x-direction, with significant deflection

at the left end This is consistent with the fact that the left end is

a cantilever Overall speaking, the mode shapes are reasonable,

as they make good physical sense Apart from analysis algorithms, proper setting out of sensor locations and alignment are vital to achieving this quality

Table 1

Identification results (MPV) of all setups in ambient tests.

f (Hz) f

(%)

S * S e* Modal RMS at

2404 (lg)

f (Hz) f (%)

S * S e* Modal RMS at

2404 (lg)

f (Hz) f (%)

S * S e* Modal RMS at

2404 (lg)

* Unit is 10 12

g 2

/Hz.

6.15 0 6.2 6.25 5

10

Mode 1

0 5 10

Mode 1

0 10 20

Mode 2

0 5 10

Mode 2

0 5 10

Mode 3

Frequency (Hz)

0 10 20

Mode 3

Damping ratio (%) Fig 5 Histogram of identified parameters in ambient tests.

Table 2

Posterior COV of modal parameters in ambient test.

f (%) f (%) S (%) S e (%) a* (10 6 ) f (%) f (%) S (%) S e (%) a* (10 6 ) f (%) f (%) S (%) S e (%) a* (10 6 )

*a q

2.3.4 Posterior uncertainty

The uncertainties associated with the modal parameters are

next discussed, from both a Bayesian and frequentist perspective

Table 2shows the posterior COV of modal parameters in all setups,

equal to the square-root of the posterior variance divided by the

MPV These values reflect the remaining uncertainty associated

with the modal parameter given the data in a particular setup

and assuming the model used It is seen that the posterior COVs

are in the order of 0.1% for the natural frequency f, a few percents

for the damping ratio f and PSD of modal force S; and 1% for the

PSD of prediction error Se These values indicate that the amount

of data used for identification in each setup is adequate, as the

remaining uncertainty is quite small It is not necessary to increase

the data duration to improve identification accuracy

The posterior uncertainty of the mode shape is assessed

through Eq.(14) The results are shown inTable 2under the

col-umn titled ‘a’, which is defined asa¼ 1  E½q, in view of the close proximity of the calculated values of E½q to unity The higher thea, the higher the posterior mode shape uncertainty It is seen that the values ofaare very small, indicating high accuracy in the identified mode shape in each setup A previous study[20]revealed empiri-cally that the expected MAC is of similar order of magnitude as the setup-to-setup variability, the latter assuming that the same experiment were performed repeatedly under the same stochastic environment

It is instructive to compare the posterior COVs inTable 2with the setup-to-setup sample COVs at the bottom ofTable 1 Taking the first mode for example, the sample COV is 0.3% for the natural frequency, 14% for the damping ratio, 56% for the PSD of modal force and 62% for the PSD of prediction error For the natural fre-quency and damping ratio, the posterior and sample COV are of similar order of magnitude Both type of COV bring out a similar

-5 0 10

25 35 -5

0 10 20 -5 0 5 10

X (m) Y(m)

-5 0 5 10 15 20 -5

0 5 10

Y(m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10

X (m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10 15 20

X (m)

Fig 6 Ambient mode shape (Mode 1).

-5 0 10 25

35 -5

0 10 20 -5 0 5 10

X (m) Y(m)

-5 0 5 10 15 20 -5

0 5 10

Y(m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10

X (m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10 15 20

X (m)

Fig 7 Ambient mode shape (Mode 2).

conclusion regarding the posterior uncertainty (given data in a

sin-gle setup) or setup-to-setup variability For the PSD of modal force

and prediction error, however, the sample COV values are

signifi-cantly larger than the corresponding posterior COV values On

sec-ond thoughts, this difference is understandable because for these

parameters the sample COV reflects the change in environmental

conditions, which cannot be captured by the posterior COV that

is only based on the data in a given setup

3 Forced vibration test

A series of shaker tests were performed one day after the

ambi-ent tests This time span allowed the team to re-organize

equip-ment and update testing plan taking into account the

information from the ambient tests At the same time, this time

window was sufficiently short and fell on a Saturday so that there

was essentially no change in the structural properties This ensures

that the results from the ambient tests and shaker tests refer to the

same structure The primary focus of the shaker test is to obtain the

damping ratio at vibration level comparable to serviceability levels,

as it may be amplitude-dependent The modal mass can also be

determined, which is not possible in ambient tests

3.1 Instrumentation

Using a single long-stroke shaker (APS 113, APS Dynamics),

fre-quency sweeps (steady-state) for the first two modes were

per-formed and their modal properties identified accordingly A

frequency sweep was not performed for the third mode as it was

found on site that the resonance amplitude was not significantly

above ambient vibration level Four setups, namely, 201, 202,

301 and 302, were performed Here, the first index in the setup

number indicates the floor where the shaker and most sensors

are placed The last index indicates the mode around which

fre-quency sweep is performed For example, Setup 302 corresponds

to frequency sweep around the second mode where the shaker

and most sensors are placed on 3/F

For Setups 201 and 202, the vertical acceleration at nine

loca-tions along a line on 2/F from 2401 to 2409 (seeFig 2) were

mea-sured Analogously, the locations from 3401 to 3409 (seeFig 3)

were measured for Setups 301 and 302 The shaker location was

decided primarily to achieve a good modal participation, based

on mode shape information from the ambient tests On the floor being excited, the vertical accelerations at ten locations were mea-sured In all setups, the two reference locations 2404 and 3404 were always measured so that the mode shape on the two floors from different setups could be assembled to give an overall mode shape The structural acceleration at the shaker location was al-ways measured so that the modal mass could be identified In addi-tion to the Guralp sensors used in the ambient tests, four Kistler K8330 uniaxial accelerometers with a noise floor of about l mi-cro-g were used to make up the measurement array Kistler 8776 uniaxial modal accelerometer was used to measure the accelera-tion of the shaker mass to calculate input force

For a given mode, frequency sweep always started with the nat-ural frequency that was identified from ambient vibration tests This was intended to create resonance and give first-hand informa-tion on the maximum vibrainforma-tion level that could be produced by the shaker Subsequent frequencies were determined in an ad-hoc manner to obtain a set of amplitudes that distributed more or less evenly over the half-power bandwidth Frequencies that were far-ther from resonance were spaced wider as they were less important

For each shaker frequency, acceleration data was recorded from before the shaker was turned on to long after it was turned off Each recorded time history typically consists of an initial phase, a growing phase, a steady-state phase of about 30 s, and a decaying (free vibration) phase of about 60 s A total of 120 s is recorded for each setup A 20 s segment of steady-state time history is extracted from the record, from which the steady-state amplitude can be determined and later used for modal identification

3.2 Modal identification by least square fitting of steady-state amplitudes

The identification of modal parameters using the data in the shaker tests corresponds to one where both the input excitation and the output response are measured A least square fitting

meth-od is used for identifying the mmeth-odal parameters using the steady-state amplitudes at different excitation frequencies The theory is outlined below

Assuming the shaker is operating at frequency fk near reso-nance and the mode shape is normalized to unity at the load

-5 0 10 25

35 -5

0 10 20 -5 0 5 10

X (m) Y(m)

-5 0 5 10 15 20 -5

0 5 10

Y(m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10

X (m)

-5 0 5 10 15 20 25 30 35 -5

0 5 10 15 20

X (m)

Fig 8 Ambient mode shape (Mode 3).

DOF of the shaker, the theoretical steady-state amplitudes at the

measured DOFs, collected in an n-by-l vector, is given by:

Ak¼mak

where m and akare the shaker mass and its acceleration amplitude

respectively; M is the modal mass; and Dkis given by Eq.(7)but

with bk¼ f =fk Note that the sign of each entry in Akfollows the

cor-responding entry inU In the shaker tests, m = 13.2 kg and akcan be

determined from the recorded acceleration time history of the

sha-ker mass The set of modal parameters to be identified consists of

f, f,Uand M; or equivalently, f, f,Uand r, where r = m/M is the

ra-tio of the shaker mass to the modal mass

On the other hand, let ~Ak2 Rncollect the measured steady-state

amplitudes at the measured DOFs The modal parameters are

identified as the values that minimize a measure of fit function

The latter is defined to account for the discrepancy between the theoretical and measured steady-state amplitudes over all excita-tion frequencies:

Jðf ; f; U; rÞ ¼X

k

Akðf ; f; U; rÞ  ~Ak



6 6.2 6.4 6.6 0

2

4x 10 -4 2401

0 0.5

1x 10 -3 2402

0 1

2x 10 -3 2403

0 1

2x 10 -3 2404

0 2

4x 10 -3

2405

0 2

4x 10 -3 2406

0 1

2x 10 -3 2407

0 0.5

1x 10 -3 2408

6 6.2 6.4 6.6 0

1

2x 10 -4

Excitation frequency(Hz)

2409

0 1

2x 10 -3 3404

7.2 7.4 7.6 7.8 8 0

0.5

1x 10 -3 2401

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2402

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2403

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2404

7.2 7.4 7.6 7.8 8 0

2

4x 10 -4

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2406

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2407

7.2 7.4 7.6 7.8 8 0

0.5

1x 10 -3 2408

7.2 7.4 7.6 7.8 8 0

1

2x 10 -4

Excitation frequency(Hz)

2409

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3404

(a) Setup 201 (b) Setup 202

0

5x 10 -4 3401

0 0.5

1x 10 -3 3402

0 1

2x 10 -3 3403

0 1

2x 10 -3 3404

0 2

4x 10 -3

0 2

4x 10 -3 3406

0 1

2x 10 -3 3407

0 0.5

1x 10 -3 3408

6 6.2 6.4 6.6 0

1

2x 10 -4

Excitation frequency(Hz)

3409

0 1

2x 10 -3 2404

7.2 7.4 7.6 7.8 8 0

0.5

1x 10 -3 3401

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3402

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3403

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3404

7.2 7.4 7.6 7.8 8 0

2

4x 10 -4

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3406

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 3407

7.2 7.4 7.6 7.8 8 0

0.5

1x 10 -3 3408

7.2 7.4 7.6 7.8 8 0

2

4x 10 -4

Excitation frequency(Hz)

3409

7.2 7.4 7.6 7.8 8 0

1

2x 10 -3 2404

(c) Setup 301 (d) Setup 302

Table 3 Summary of results from shaker tests.

Mode Setup Natural

frequency f (Hz)

Damping ratio f (%)

Modal RMS

at 2404 (lg)

Modal mass (ton)