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

Engineering Structures 37 (2012) 167–178

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

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 connected 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 identify the modal properties using the ambient data in individual setups. The mode shapes from the individual 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 amplitudes in the order of a few milli-g. A steady-state frequency sweep was carried out and the modal properties 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 instrumentation. The slabs on the 2nd floor (2/F) and 3rd floor (3/F) span over
a 35 Â 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, significant 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 owner and design engineer to experimentally determine the modal
⇑ Corresponding author. Tel.: +852 3442 2769; fax: +852 2788 7612.
E-mail address: (S.K. Au).
0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engstruct.2011.12.024

properties in order to assess the likely vibration level under service
loading to a higher confidence than was possible from the information available at the design stage. A series of field vibration tests
were performed with these objectives in mind. They include ambient vibration test, forced vibration (shaker) test and service load
(jumping) test.
Full-scale testing provides an important means for acquiring insitu 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 firsthand 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 100 lg). This qualification is especially 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 (shaker) tests were performed with a long-stroke electrodynamic shaker, where the mode shapes along a critical line of the slab were
also identified. Service load tests with a large number of participants 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


168

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

Fig. 1. The TSW Indoor Recreation Center.

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 identification 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 transmitted through cable and acquired digitally bypaffiffiffiffiffiffi24 bit data logger.
The overall channel noise is about 0:1 lg= Hz in the frequency
band above 1 Hz. Acceleration data of 6 Â 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 cover all the target degrees of freedom (DOFs) with a limited number 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 3 show the overall setup plans for 2/F and 3/F, respectively. The instrumented area on each floor measures 30 m long by
20 m wide. A total of 9 Â 7 Â 2 = 126 locations were planned to be
measured triaxially, giving 126 Â 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. Locations 2404 and 3404 have been chosen to be the reference, which
are filled light brown in Figs. 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 reference 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 in Fig. 2).
A hole, indicated by ‘H’ in Fig. 3, was drilled on 3/F to allow the passage 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 in Fig. 2, which would require much longer cable and create
additional safety issues on site. Drilling of this hole could be facilitated 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 in Figs. 2 and 3 were 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 performed separately on two consecutive days, from 8am to 6pm. In
all setups two sensors were always placed at the reference locations 2404 and 3404. The remaining four sensors were roved in different setups to cover the other locations. In Figs. 2 and 3, the color
of the number in rectangular box distinguishes the particular sensor placed, e.g., blue for TM54 and red for TM55. As the setups proceeded, 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 in Fig. 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 in Fig. 3 is the one actually used.
During the test, one person was responsible for a particular sensor. When transiting between setups, each roving sensor was
moved to the next corresponding location. Including taking pictures 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 identification. 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 performed. 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 structure are identified following a Bayesian FFT approach. The original


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S.K. Au et al. / Engineering Structures 37 (2012) 167–178

TM54
2101

1
2201

2
2301

3
2401

4
2501

5
2601

6
2701

7

TM56

TM55
2103

2102

1

8

2203

2202

2

9

2702

14

2306

3

2604

6
7

14

7

2308

14

2209

15
2309

15

10

3

2408

2409

11 15,16

4

2508

2509

12

5
2607

13
2706

15

9

2507

12
2606

6
2705

2208

2

2407

11
2506

5
2605

13
2704

10

2109

8

2307

2406

2505

2108

1
2207

9

2405

12

2603

2703

2206

2305

2504

2107

8

4

5

13

2106

2

2404

2503

12
2602

2205

4

11
2502

2204

10,11

2403

2402

1

2304

3

10

2105

8

9

2303

2302

2104

TM57

2608

16
2609

13

6
2707

2708

7

16
2709

14

16

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

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 frequency 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-dimensional 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.
€ j g is assumed to consist of the
The measured acceleration data fy
structural ambient vibration signal and prediction error:

€j ¼ x
€ j þ ej
y

ð1Þ

€ j 2 Rn and ej 2 Rn ðj ¼ 1; 2; Á Á Á ; NÞ are the acceleration rewhere x
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 response for given modal parameters, which may arise due to model€ j g is defined as:
ing error and/or measurement noise. The FFT of fy

Fk ¼

rffiffiffiffiffiffiffiffi N



2 Dt X
ðk À 1Þðj À 1Þ
€j exp Ài2p
y
N j¼1
N

ð2Þ

where Dt 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 Zk ¼ ½ReF k ; ImF k Š 2 R2n be an augmented vector of the real
and imaginary part of F k . 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.,

pðhjfZk gÞ / pðfZk gjhÞ

ð3Þ

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)

LðhÞ ¼ À ln pðfZk gjhÞ

ð4Þ

such that p(h|{Zk}) / exp [ÀL(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 instantaneously 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 identified 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 density (PSD) of modal force S, PSD of prediction error Se, and mode
shape U e Rn. From first principle for sufficiently small Dt and long


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S.K. Au et al. / Engineering Structures 37 (2012) 167–178

TM54
3102

3101

1

3103

3202

2

3402

3401

4

3403

3502

3501

6

3603

3702

3703

16

7

3604

3704

13

6
3705

14

3607

7

3707

17
3609

13
3708

7

14

18,19
3509

12
3608

6

3706

17
3409

3508

5

3606

3309

10

11

3507

12

18,19

3408

4

3506

H

3605

13

7

5

3407

11

3209

9
3308

3

3406

3505

12

6

15

3701

3504

5

3602

3601

3405

17

3208

3307

10

3109

8

2

3306

4

3503

16

5

3404

4

15

3207

9

3

3108

1

3206

3305

10,11

3107

8

2

3304

3

3106

3205

9

3303

16

TM57

1

3204

2

3302

3

3105

8

3203

15

3301

3104

1

8

3201

TM56

TM55

18
3709

14

19

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

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

LðhÞ ¼

1X
1 X T À1
ln det Ck þ
Z C Zk
2 k
2 k k k

ð5Þ

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

"
SDk UUT
Ck ¼
2
0

0

#
þ

UUT

Se
I2n
2

ð6Þ

is the theoretical covariance matrix of Zk;

Dk ¼

½ðb2k

2

2 À1

À 1Þ þ ð2fbk Þ Š

ð7Þ
2nÂ2n

and bk = f/fk; where fk is the FFT frequency abscissa I2n 2 R
is the
identity matrix.

The inverse on Ck in Eq. (5) renders the dependence of the NLLF
on h highly non-trivial. Nevertheless, reformulating using eigenspace decomposition, it can be shown that the NLLF can be rewritten in the following canonical form:

LðhÞ ¼ ÀnN f ln 2 þ ðn À 1ÞN f ln Se þ

X

lnðSDk þ Se Þ

k
2
T
þ SÀ1
e ðd À U AU=kUk Þ

ð8Þ

where Nf is the number of frequency ordinates in the selected band;
kUk ¼ ðUT UÞ1=2 is the Euclidean norm of U; and

X

ð1 þ Se =SDk ÞÀ1 Dk

ð9Þ

Dk ¼ ReF k ReF Tk þ ImF k ImF Tk
X
ReF Tk ReF k þ ImF Tk ImF k
d¼

ð10Þ

A¼

k

k

ð11Þ

The significance of Eq. (8) is that it is explicitly in terms of U as a
quadratic form and the inverse in Eq. (5) has been resolved. It follows from standard results of linear algebra that the MPV of U is
simply the eigenvector of A with the largest eigenvalue. Consequently, only four parameters, ff ; f; S; Se g, need to be optimized
numerically. The computational process is significantly shortened
with no dependence on the number of measured DOFs n. For moderate 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 sufficient 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 order Taylor series of the NLLF about the MPV, it can be shown that
the posterior covariance matrix is equal to the inverse of the Hessian of the NLLF. Analytical expressions have been derived for calculating 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 variation = posterior standard deviation/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 random mode shape following the posterior distribution and satisfying the norm constraint can be represented as [20]
0

U ¼

1þ

n
X
j¼1

!À1=2
d2j Z 2j

^0 þ
U

n
X
j¼1

!
d j Z j uj

ð12Þ


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S.K. Au et al. / Engineering Structures 37 (2012) 167–178

^ 0 2 Rn is the most probable mode shape (normalized to
where U

unity); fd2j : j ¼ 1; . . . ; ng and fuj 2 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; fZ j : j ¼ 1; . . . ; ng are independent
and identically distributed standard Gaussian random variables.
The Modal Assurance Criteria (MAC) between the uncertain
^ 0 indicates the deviation in direction
mode shape U0 and its MPV U
and hence the uncertainty of U0 . In the current context the MAC is
a random variable given by

^ 0T U0
U

q ¼ ^0
¼ 1þ
kU kkU0 k

n
X

!À1=2
d2j Z 2j

ð13Þ

j¼1

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

E½qŠ $

1þ

n
X

!À1=2
d2j

ð14Þ

j¼1

Thus, the closer the E½qŠ 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
in Fig. 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 observed in the frequency bands 2–4 and 6–10 Hz. The former corresponds 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 frequency 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 1 shows 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 results for the mode shape U shall 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. 5 shows the
histogram of the MPV of f and f among all setups. As expected,
the identified values vary among setups. As seen in Table 1, the

-3

10

-4

[g/sqrt(Hz)]

10

-5

10

-6

10

-7

10

0

1

2

3

4

5

6

7

8

9

10

Frequency (Hz)

(a) PSD spectrum
-5

x 10
12

Vertical modes
of slab

[g/sqrt(Hz)]

10
8

Mode 3
Mode 1

6

Mode 2

Lateral modes
of whole building

4

2
0

1

2

3

4

5

6

7

8

Frequency (Hz)

(b) Singular value spectrum
Fig. 4. PSD and singular value spectrum in ambient test, Setup 204.

9

10


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S.K. Au et al. / Engineering Structures 37 (2012) 167–178

Table 1
Identification results (MPV) of all setups in ambient tests.
Setup

201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
301
302
303
304
305
306
307

308
309
310
311
312
313
314
315
316
317
318
319
Mean
COV
*

Mode 1

Mode 2
*

f (Hz)

f
(%)

S

6.214
6.237

6.219
6.210
6.229
6.218
6.215
6.219
6.220
6.221
6.211
6.217
6.202
6.190
6.167
6.213
6.235
6.230
6.231
6.220
6.223
6.216
6.224
6.199
6.192
6.180
6.191
6.216
6.195
6.199
6.202
6.210

6.207
6.213
6.221
6.212
0.3%

0.97
1.23
0.91
1.07
0.86
0.99
1.20
0.87
0.99
1.22
1.19
1.12
0.87
1.21
1.24
1.17
1.03
1.14
1.05
1.06
1.08
0.78
1.17
1.27

1.44
0.98
1.22
1.08
1.31
1.20
1.11
1.18
0.95
1.38
1.06
1.10
14%

6.3
10.8
4.0
2.6
2.7
6.3
9.8
7.3
5.3
9.3
12.2
6.7
6.6
6.2
12.3
6.1

9.6
4.7
5.3
5.5
3.6
5.2
11.8
7.3
21.9
16.6
14.4
21.0
18.5
19.4
18.6
14.7
12.2
10.6
6.7
9.8
55.6%

Mode 3
*

Se*

Modal RMS at
2404 (lg)

f (Hz)

f
(%)

S

27.1
33.0
34.3
13.3
17.3
17.2
33.2
27.1
23.8
29.7
44.8
19.4
48.0
29.5
30.9
26.9
37.5
54.2
50.3
34.8
34.0
42.1
20.0

25.1
74.8
93.4
85.0
85.1
58.3
52.7
124.2
48.1
22.6
19.2
13.4
40.9
62%

56
65
46
35
39
56
63
64
51
61
71
54
61
50
69

51
67
45
50
50
41
57
70
53
86
91
76
97
83
89
90
78
79
61
56
63
25%

7.764
7.706
7.770
7.765
7.805
7.766
7.747

7.735
7.742
7.734
7.706
7.750
7.816
7.751
7.831
7.774
7.752
7.769
7.758
7.754
7.745
7.747
7.680
7.729
7.676
7.694
7.738
7.671
7.653
7.644
7.648
7.618
7.755
7.769
7.810
7.736
0.7%

1.65
2.36
1.85
1.59
1.97
2.45
2.88
2.33
2.32
2.16
2.24
1.93
2.20
2.25
2.50
2.28
2.06
1.88
2.17
2.10
1.95
1.94
2.19
1.61
1.72
1.87
2.11
1.73
1.92

1.78
1.44
2.09
2.08
2.29
2.37
2.06
15%

3.6
16.3
3.8
2.7
3.0
5.8
8.6
6.2
5.6
6.7
9.9
5.5
7.2
6.2
8.2
6.3
9.8
8.3
7.7
7.5
4.4

4.8
12.3
4.7
11.8
8.5
9.8
12.1
13.7
10.2
9.4
8.2
6.8
7.0
7.1
7.7
40.0%

Se

*

11.5
72.6
29.9
20.4
17.3
18.3
16.3
18.8
29.9

53.9
69.8
23.7
45.4
23.5
17.5
15.4
44.1
54.9
57.0
58.9
21.9
19.9
25.8
18.3
72.7
84.1
118.5
87.3
51.8
30.9
84.0
45.7
14.6
9.7
9.1
39.8
69%

Modal RMS at

2404 (lg)

f (Hz)

f
(%)

S*

Se*

Modal RMS at
2404 (lg)

36
65
36
32
31
38
43
40
38
43
52
42
45
41
45
41

54
52
47
47
37
39
58
42
64
52
53
65
66
59
63
48
45
43
43
47
21%

9.048
9.173
9.178
9.157
9.110
9.053
8.963
9.051

9.166
9.182
9.145
9.097
9.027
8.965
9.016
8.993
9.165
9.150
9.181
9.188
9.165
9.099
9.069
9.098
9.095
9.150
9.138
9.101
9.079
9.064
9.172
9.215
9.010
9.037
9.033
9.101
0.8%

1.95
2.29
2.39
2.15
2.47
2.00
4.24
2.32
1.79
2.26
2.28
2.15
2.56
2.87
3.39
2.71
2.76
2.29
2.23
1.85
2.55
2.08
3.44
1.76
2.51
2.84
2.11
2.26
2.33
2.48

2.16
1.88
2.33
2.25
2.60
2.42
21%

1.4
2.7
1.6
1.1
1.5
1.2
6.4
1.3
0.7
1.0
1.2
1.3
3.5
4.3
6.7
3.4
3.4
2.1
1.7
1.2
1.5
1.9

4.2
1.3
11.9
12.3
8.2
10.0
15.5
12.1
8.1
5.6
17.2
12.5
15.3
5.3
94.1%

106.1
513.3
229.2
123.1
131.1
119.8
71.3
114.9
242.0
294.3
307.2
174.1
217.5
97.3

114.9
83.6
192.9
178.4
199.1
206.1
138.2
101.1
62.5
149.7
441.9
494.1
480.2
571.7
636.5
194.0
176.4
151.7
80.6
73.0
76.4
215.6
73%

22
29
22
19
21
21

33
20
17
18
20
21
31
33
37
30
30
26
23
22
20
25
30
23
58
56
53
56
69
59
52
46
72
63
65
35

49%

Unit is 10À12 g2/Hz.

MPV of natural frequency f, regardless of mode, show a small variability (<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%).
In Table 1, the column titled with ‘RMS’ shows the modal contribution of root mean square value of acceleration response. As
a reference for comparison, the values presented are for the vertical DOF at the 2/F reference location (2404). The RMS values are
calculated using
the standard
formula from random vibration thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ory [22], i.e., ( pSf =4f), where the identified modal parameter values have been used. The RMS values are presented so that the
damping ratios could be viewed in an amplitude-dependent perspective. Roughly speaking, the identified damping ratios are applicable for modal vibration level up to 100 lg.
It should be noted that the values of S presented in Table 1 all
correspond to mode shape normalized with the vertical DOF at
the 2/F reference location (2404) equal to unity. Table 2 on posterior 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 Table 1 primarily reflect the change in environmental stochastic loading intensity. With this in mind, one observes that the values of S in
Setups 309–318 in Table 1 are significantly higher than those in the
remaining setups, suggesting stronger stochastic environmental
loading. This is consistent with the fact that these setups were performed 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. 6 shows 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 appears to be more rotational restraint at the left end. This is believed
to be attributed to the partition walls (see Fig. 2), which can impose 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. 7 shows 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 in Fig. 8 has 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.


173

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

Mode 1

Mode 1

10

10

5

5

0
6.15

6.2
Mode 2

0
0.5

6.25

1.5

1

2
Mode 3

3

0

2
4

Damping ratio (%)

6

10
Number

Number

20

1
Mode 2

10
0
7.6

7.7
7.8
Mode 3

5
0

7.9

10

20

5

10

0
8.8

9
9.2
Frequency (Hz)

0

9.4

Fig. 5. Histogram of identified parameters in ambient tests.

Table 2
Posterior COV of modal parameters in ambient test.
Setup

201
202
203
204
205
206
207
208

209
210
211
212
213
214
215
216
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
*

Mode 1

Mode 2

Mode 3

f (%)

f (%)

S (%)

Se (%)

a* (10À6)

f (%)

f (%)

S (%)

Se (%)

a* (10À6)

f (%)

f (%)

S (%)

Se (%)

a* (10À6)

0.06
0.06
0.05
0.06
0.05
0.06
0.06
0.05
0.06
0.06
0.06
0.06
0.05
0.06
0.07
0.06
0.06
0.06
0.06
0.06
0.06
0.05
0.06
0.07
0.07

0.06
0.06
0.06
0.07
0.07
0.06
0.06
0.07
0.09
0.07

6.3
5.8
6.4
6.1
6.5
6.2
5.9
6.5
6.2
5.8
5.8
6.0
6.5
5.9
5.8
5.9
6.1
6.0
6.1

6.1
6.1
6.8
5.9
5.7
5.5
6.3
5.8
6.0
5.7
6.0
6.2
5.9
7.8
7.2
7.5

3.4
3.5
3.4
3.4
3.3
3.4
3.5
3.3
3.4
3.5
3.5
3.4
3.4

3.5
3.5
3.5
3.4
3.5
3.4
3.4
3.5
3.3
3.4
3.5
3.6
3.4
3.5
3.4
3.6
3.6
3.6
3.5
4.1
4.6
4.2

0.70
0.69
0.70
0.70
0.70
0.70
0.70

0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.70
0.77
0.77
0.77
0.77
0.77
0.79
0.78
0.77
0.94
0.98
0.94

12

8
13
9
12
8
15
8
9
7
8
7
18
21
13
21
10
24
14
11
19
16
7
11
10
9
12
9
11
12
29

15
11
18
14

0.07
0.10
0.08
0.07
0.09
0.10
0.12
0.10
0.10
0.09
0.09
0.08
0.10
0.10
0.11
0.10
0.09
0.08
0.09
0.09
0.08
0.08
0.09
0.07
0.08

0.08
0.09
0.08
0.08
0.08
0.07
0.09
0.11
0.12
0.13

5.7
5.8
5.7
5.8
5.7
5.9
6.1
5.8
5.8
5.8
5.8
5.7
5.8
5.8
5.9
5.8
5.8
5.7
5.8

5.8
5.7
5.7
5.8
5.7
5.7
5.7
5.8
5.8
5.7
5.9
5.9
5.8
7.1
7.5
7.1

5.2
6.3
5.5
5.1
5.8
6.5
7.3
6.3
6.3
6.0
6.1
5.6
6.2

6.2
6.7
6.2
5.8
5.6
6.0
5.9
5.7
5.6
6.0
5.1
5.3
5.5
5.9
5.3
5.6
5.5
5.0
5.9
7.2
8.1
7.9

0.81
0.81
0.81
0.81
0.81
0.81
0.82

0.82
0.82
0.81
0.81
0.81
0.81
0.81
0.81
0.81
0.81
0.82
0.81
0.81
0.81
0.81
0.81
0.82
0.90
0.90
0.90
0.90
0.89
0.92
0.91
0.90
1.10
1.2
1.10

11

18
20
16
20
19
17
15
21
24
23
15
34
25
17
17
18
19
20
22
16
17
13
11
11
16
24
16
12
13
16

17
20
16
14

0.09
0.10
0.10
0.09
0.10
0.09
0.24
0.10
0.08
0.09
0.09
0.09
0.11
0.14
0.16
0.13
0.11
0.09
0.09
0.08
0.10
0.09
0.15
0.08
0.11

0.12
0.09
0.10
0.10
0.11
0.09
0.08
0.13
0.13
0.14

5.9
6.2
6.1
6.0
6.2
6.0
8.2
6.2
5.9
6.1
6.1
6.0
6.4
6.6
7.2
6.5
6.5
6.1
6.0

5.8
6.3
6.0
7.2
5.9
6.4
6.6
6.1
6.3
6.4
6.6
6.1
5.9
7.5
8.0
7.8

6.4
7.2
7.3
6.7
7.4
6.5
12.4
7.3
6.2
7.1
7.1
6.8
7.8

8.7
9.9
8.2
8.2
7.0
6.9
6.2
7.6
6.6
9.9
6.0
7.7
8.5
6.8
7.4
7.6
8.0
6.9
6.3
8.8
9.3
9.5

0.8
0.8
0.8
0.8
0.8
0.8
0.8

0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
1.1
1.2
1.1

84

92
35
26
48
83
74
117
88
67
76
61
98
81
78
78
77
39
28
28
45
51
62
82
113
88
98
189
226
143
24

25
62
80
79

a = 1 À E(q).


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S.K. Au et al. / Engineering Structures 37 (2012) 167–178

10

5

5
Z(m)

Z(m)

10

0
-5
20
10

Y(m)

0-5

-5 0

10

35

25

0
-5
-5

X (m)

0

5

10

15

20

Y(m)

10

20
15

Y(m)

Z(m)

5
0

10
5
0

-5
-5

0

-5
-5

5 10 15 20 25 30 35

0

5 10 15 20 25 30 35

X (m)

X (m)

Fig. 6. Ambient mode shape (Mode 1).

10

5

Z(m)

Z(m)

10

0
-5
20

10

Y(m)

0-5

-5 0

10

25

35

5
0
-5
-5

0

X (m)

5

10

15

20

Y(m)

10

20

Y(m)

Z(m)

15

5
0

10
5
0

-5
-5

0

5 10 15 20 25 30 35

-5
-5

0

X (m)

5 10 15 20 25 30 35

X (m)

Fig. 7. Ambient mode shape (Mode 2).

2.3.4. Posterior uncertainty
The uncertainties associated with the modal parameters are
next discussed, from both a Bayesian and frequentist perspective.

Table 2 shows 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 in Table 2 under the col-

umn titled ‘a’, which is defined as a ¼ 1 À E½qŠ, in view of the close
proximity of the calculated values of E½qŠ to unity. The higher the a,
the higher the posterior mode shape uncertainty. It is seen that the
values of a are very small, indicating high accuracy in the identified
mode shape in each setup. A previous study [20] revealed empirically 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 in Table 2 with
the setup-to-setup sample COVs at the bottom of Table 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 frequency and damping ratio, the posterior and sample COV are of
similar order of magnitude. Both type of COV bring out a similar

175

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

10

5

Z(m)

Z(m)

10

0
-5
20

10

Y(m)

0-5

-5 0

10

25

5
0

-5

35

-5

X (m)

0

5

10

15

20

Y(m)

10

20

Y(m)

Z(m)

15
5
0

10
5
0

-5
-5

0

5 10 15 20 25 30 35

-5
-5

0

X (m)

5 10 15 20 25 30 35

X (m)

Fig. 8. Ambient mode shape (Mode 3).

conclusion regarding the posterior uncertainty (given data in a single setup) or setup-to-setup variability. For the PSD of modal force

and prediction error, however, the sample COV values are significantly larger than the corresponding posterior COV values. On second 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 ambient tests. This time span allowed the team to re-organize equipment 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), frequency sweeps (steady-state) for the first two modes were performed 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 frequency 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 locations along a line on 2/F from 2401 to 2409 (see Fig. 2) were measured. Analogously, the locations from 3401 to 3409 (see Fig. 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 measured. 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 always measured so that the modal mass could be identified. In addition to the Guralp sensors used in the ambient tests, four Kistler
K8330 uniaxial accelerometers with a noise floor of about l micro-g were used to make up the measurement array. Kistler 8776
uniaxial modal accelerometer was used to measure the acceleration of the shaker mass to calculate input force.
For a given mode, frequency sweep always started with the natural frequency that was identified from ambient vibration tests.
This was intended to create resonance and give first-hand information on the maximum vibration 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 farther 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 method is used for identifying the modal parameters using the steadystate amplitudes at different excitation frequencies. The theory is
outlined below.
Assuming the shaker is operating at frequency f k near resonance and the mode shape is normalized to unity at the load


176

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

DOF of the shaker, the theoretical steady-state amplitudes at the
measured DOFs, collected in an n-by-l vector, is given by:

mak
Dk U
M

Mode

Setup

Natural
frequency f (Hz)

Damping
ratio f (%)

Modal RMS
at 2404 (lg)

Modal
mass (ton)

1

201
301

202
302

6.248
6.251
7.641
7.662

1.1
1.2
1.7
1.8

1124
1171
833
897

597
585
441
439

ð15Þ

-4
-3
2401
x 10
x 10

4
1
2
0.5
0
0
6
6.2
6.4
6.6
6
-3
-3
2403
x 10
x 10
2
2
1
1
0
0
6
6.5
6
-3
-3
2405
x 10
x 10

4
4
2
2
0
0
6
6.5
6
-3
-3
2407
x 10
x 10
2
1
1
0.5
0
0
6
6.5
6
-4
-3
2409
x 10
x 10
2
2

1
1
0
0
6
6.2
6.4
6.6
6
Excitation frequency(Hz)

2

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

Jðf ; f; U; rÞ ¼

-3

2402

6.5
2404

6.5
2406

6.5
2408

6.5
3404

6.5

-4

0

Amplitude(g)

2
1
0
4
2
0
2
1
0

x 10

3401

6
-3
x 10

3403

6
-3
x 10

3405

6
-3
x 10

3407

6
-4
x 10

3409

-3

6.5

6.5

6.5

6.5

2
1
0
6
6.2
6.4
6.6
Excitation frequency(Hz)

1
0.5
0
2
1
0
4
2
0
1
0.5
0
2
1
0

3402

6
-3
x 10

3404

6
-3
x 10

3406

6
-3
x 10

3408

6
-3
x 10

2404

(c) Setup 301

ð16Þ

2402

7.6 7.8
2404

8

7.6 7.8
2406

8

7.6 7.8
2408

8

7.6 7.8
3404

8

7.6 7.8

8

(b) Setup 202

x 10

6

-3

2401

x 10
x 10
1
2
0.5
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-3
-3
2403
x 10
x 10
2
2
1
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-4
-3
2405
x 10
x 10
4
2

2
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-3
-3
2407
x 10
x 10
2
1
1
0.5
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-4
-3
2409
x 10
x 10
2
2
1
1
0
0

7.2 7.4 7.6 7.8 8
7.2 7.4
Excitation frequency(Hz)

(a) Setup 201
5

2
X

~ k

Ak ðf ; f; U; rÞ À A

k

Amplitude(g)

Amplitude(g)

where m and ak are the shaker mass and its acceleration amplitude
respectively; M is the modal mass; and Dk is given by Eq. (7) but
with bk ¼ f =f k . Note that the sign of each entry in Ak follows the corresponding entry in U. In the shaker tests, m = 13.2 kg and ak can be
determined from the recorded acceleration time history of the shaker mass. The set of modal parameters to be identified consists of
f, f, U and M; or equivalently, f, f, U and r, where r = m/M is the ratio of the shaker mass to the modal mass.
~ k 2 Rn collect the measured steady-state
On the other hand, let A
amplitudes at the measured DOFs. The modal parameters are
identified as the values that minimize a measure of fit function.

-3

6.5

6.5

6.5

6.5

6.5

Amplitude(g)

Ak ¼

Table 3
Summary of results from shaker tests.

-3

3401
x 10
x 10
1
2
0.5
1
0
0

7.2 7.4 7.6 7.8 8
7.2 7.4
-3
-3
3403
x 10
x 10
2
2
1
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-4
-3
3405
x 10
x 10
4
2
2
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-3
-3

3407
x 10
x 10
2
1
1
0.5
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
-4
-3
3409
x 10
x 10
4
2
2
1
0
0
7.2 7.4 7.6 7.8 8
7.2 7.4
Excitation frequency(Hz)

(d) Setup 302

Fig. 9. Steady-state amplitude versus excitation frequency.

3402

7.6 7.8
3404

8

7.6 7.8
3406

8

7.6 7.8
3408

8

7.6 7.8
2404

8

7.6 7.8

8


177

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

Determining the best set of modal parameters originally involves solving a numerical optimization problem whose dimension
grows with the number of measured DOFs. The dimension of the
problem, however, can be significantly reduced to two by noting
that the optimal U and r can be found analytically in terms of f
and f, taking advantage of the fact that the objective function is a
quadratic form in U. As a result, the best f and f can be obtained
by minimizing the following objective function (along which U
and r have been optimized):

Jðf ; fÞ ¼ ÀBT DB

identification results are practically unaffected, since the contribution from these DOFs to the overall measure of fit function is small.
Table 3 shows the identified natural frequencies and damping
ratios in the shaker tests. The identified values in different setups
are quite consistent with each other. The RMS values in the shaker
tests are about twenty times larger than their ambient counterparts in Table 4. The last column in Table 3 shows the identified
modal mass. Note that the modal mass depends on the scaling of
mode shape. The results presented here assume that the vertical
DOF at the 2/F reference (2404) is unity, so that they can be compared. It is seen that for both modes the values identified from different setups are quite consistent with each other. The percentage
difference is only 2% and 0.5% for mode 1 and mode 2, respectively.
Fig. 10 shows the mode shape identified from the shaker tests
for mode 1 and mode 2 (solid line), respectively. For comparison,
the results from the ambient tests are also plotted with dashed
lines. It is seen that the mode shapes identified from the two types
of tests are quite close to each other. In fact, their MAC values are
calculated to be 0.9966 and 0.9905 for the first and second mode,
respectively.
Finally, Table 4 summarizes the natural frequencies and damping ratios identified in different tests. The values presented here
are sample average over the setups performed. The identified values are quite consistent across tests. For the first mode, the natural

frequencies are all near 6.2 Hz, differing by only 0.6%. The damping
ratios identified by the ambient and forced vibration test are practically the same, differing by only 4.3%. For the second mode, the
difference in frequency is 1.3%. The difference in the damping ratio
is larger, being 20%. Despite the difference in the order of magnitude of vibration RMS, the damping ratios show little difference,
i.e., little degree of amplitude dependence. In general, the identified modal properties from the ambient and shaker test are quite
consistent with each other and sufficiently accurate to represent
the dynamic properties of subject structure at the time of instrumentation. This consistency supports assumptions on linear
dynamics, classical damping mechanism and stochastic loading
model. The experiments are essentially repeatable.

ð17Þ

where

D¼

X

a2k D2k In

ð18Þ

~k
ak Dk A

ð19Þ

k

B¼

X
k

and In 2 Rn is the identity matrix. The optimal value of r, i.e., ^r, is
equal to the entry at the load (shaker) DOF of the vector c = DÀ1B
^ ¼ ^r À1 c. Note that this
and the optimal mode shape is given by U
mode shape is normalized to unity at the load DOF.
3.3. Modal Identification results
For each shaker frequency, the steady-state amplitudes are first
determined by least square fitting of the measured time histories
[23]. The acceleration amplitude of the shaker mass in different
setups is roughly constant at about 1.1g, leading to a harmonic load
amplitude of 13.2 Â 1.1 Â 9.81 = 142N. Fig. 9 shows the steadystate amplitude versus excitation frequency for different setups.
Since the force amplitude is approximately constant with frequency, the amplitude plots have similar shape as the frequency
response function (FRF). The dots show the measured steady-state
~ k ). The theoretical steady-state amplitudes correamplitudes (A
sponding to the best fit model are shown with solid lines. It is seen
that the fit is generally good. Exceptions are location 2405 in Setup
202 and location 3405 in Setup 302. It is because the shaker-induced second mode responses at these locations are not significantly larger than the ambient response. In this case, the
measured amplitudes do not reflect correctly the steady-state
amplitudes due to the shaker. Despite, the overall fitting and modal

4. Conclusion
The dynamic characteristics of the coupled slab system on the
2/F and 3/F of the Tin Shui Wai Indoor Recreation Center have been

Table 4
Comparison of results from ambient and shaker test.

Mode

1
2
3

Natural frequency f (Hz)

Damping ratio f (%)

Modal RMS at 2404 (lg)

Ambient

Shaker

Ambient

Shaker

Ambient

Shaker

6.21
7.74
9.10

6.25
7.65

–

1.1
2.1
2.4

1.2
1.7
–

63
47
35

1148
865
–

10

10

5

5

0

0

-5

0

5

10

15
X(m)

(a) Mode 1

20

25

30

-5

0

5

10

15
X(m)

20

(b) Mode 2

Fig. 10. Identified mode shape from shaker tests.

25

30


178

S.K. Au et al. / Engineering Structures 37 (2012) 167–178

identified by means of ambient and forced vibration test. Three
vertical modes have been presented. Global mode shapes have
been obtained comprising the two floors using a limited number
of sensors with a large number of setups. The resulting mode
shapes are intuitively sound. The modal properties identified from
the ambient and shaker tests are consistent with each other. Despite the difference in the vibration level under testing, the identified damping ratios from the ambient and forced tests are similar.
One possibility was that at the time of instrumentation there was
not much internal servicing. The uncertainties associated with the
modal parameters have been investigated empirically from both a
Bayesian and frequentist point of view. For the natural frequencies
and damping ratio, the posterior COV (Bayesian) is found to be of
similar order of magnitude as the sample (setup-to-setup) COV.
The same is not true for the spectral density of modal force or
the prediction error, as their sample COV merely reflects the
change in environmental conditions.

Acknowledgments
The authors would like to record their special thanks to the
Director of Architectural Services for her kind permission for using
the data in the project in publishing the paper. Thanks are also due
to the following persons at the Architectural Services Department
for design information of the slab system and kind assistance in
the project: Ir C. T. Wong, Chief Structural Engineer; Ir M. K. Leung,
Senior Structural Engineer; and Ir C.Y. Kan, Structural Engineer of
the project. Ir L.H. Wan, Site Agent of the construction project, provided excellent logistics support for the field tests. Mr. H.L. Yip,
graduate student at the University of Hong Kong, participated in
the field tests.
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