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

Engineering Structures 37 (2012) 167–178 Contents lists available at SciVerse ScienceDirect 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 March 2011 Revised December 2011 Accepted 12 December 2011 Available online 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 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 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  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: siukuiau@cityu.edu.hk (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 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 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 Hz Acceleration data of  = 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 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 and 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   = 126 locations were planned to be measured triaxially, giving 126  = 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 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 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 and 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 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 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 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 Vibration data in each setup was recorded for 15 Exceptions were Setups 17–19 on 3/F, where only 10 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 169 S.K Au et al / Engineering Structures 37 (2012) 167–178 TM54 2101 2201 2301 2401 2501 2601 2701 TM56 TM55 2103 2102 2203 2202 2702 14 2306 2604 14 2308 14 2209 15 2309 15 10 2408 2409 11 15,16 2508 2509 12 2607 13 2706 15 2507 12 2606 2705 2208 2407 11 2506 2605 13 2704 10 2109 2307 2406 2505 2108 2207 2405 12 2603 2703 2206 2305 2504 2107 13 2106 2404 2503 12 2602 2205 11 2502 2204 10,11 2403 2402 2304 10 2105 2303 2302 2104 TM57 2608 16 2609 13 2707 2708 16 2709 14 16 Fig 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 Rn and ej 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   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) + is the index corresponding to the Nyquist frequency; int(.) denotes the integral part of its argument Let Zk ¼ ½ReF k ; ImF k Š 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 170 S.K Au et al / Engineering Structures 37 (2012) 167–178 TM54 3102 3101 3103 3202 3402 3401 3403 3502 3501 3603 3702 3703 16 3604 3704 13 3705 14 3607 3707 17 3609 13 3708 14 18,19 3509 12 3608 3706 17 3409 3508 3606 3309 10 11 3507 12 18,19 3408 3506 H 3605 13 3407 11 3209 3308 3406 3505 12 15 3701 3504 3602 3601 3405 17 3208 3307 10 3109 3306 3503 16 3404 15 3207 3108 3206 3305 10,11 3107 3304 3106 3205 3303 16 TM57 3204 3302 3105 3203 15 3301 3104 3201 TM56 TM55 18 3709 14 19 Fig 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 X T À1 ln det Ck þ Z C Zk k k k k ð5Þ where the sum is over all frequencies in the selected band; " SDk UUT Ck ¼ 0 # þ UUT Se I2n ð6Þ is the theoretical covariance matrix of Zk; Dk ¼ ½ðb2k 2 À1 À 1Þ þ ð2fbk Þ Š ð7Þ 2nÂ2n and bk = f/fk; where fk is the FFT frequency abscissa I2n 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 þ ðn À 1ÞN f ln Se þ X lnðSDk þ Se Þ k 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] U ¼ 1þ n X j¼1 !À1=2 d2j Z 2j ^0 þ U n X j¼1 ! d j Z j uj ð12Þ 171 S.K Au et al / Engineering Structures 37 (2012) 167–178 ^ Rn is the most probable mode shape (normalized to where U unity); fd2j : j ¼ 1; ; ng and fuj 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 ^ 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 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 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 10 Frequency (Hz) (a) PSD spectrum -5 x 10 12 Vertical modes of slab [g/sqrt(Hz)] 10 Mode Mode Mode Lateral modes of whole building 2 Frequency (Hz) (b) Singular value spectrum Fig PSD and singular value spectrum in ambient test, Setup 204 10 172 S.K Au et al / Engineering Structures 37 (2012) 167–178 Table 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 Mode * 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 * 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 ([...]... International Organization for Standardization; 2003 [15] Yuen KV, Katafygiotis LS Bayesian Fast Fourier Transform Approach for modal updating using ambient data Adv Struct Eng 2003;6(2):81–95 [16] Au SK Fast Bayesian FFT method for ambient modal identification with separated modes J Eng Mech 2011;137(3):214–26 [17] Au SK Fast Bayesian ambient modal identification in the frequency domain, part I: posterior... graduate student at the University of Hong Kong, participated in the field tests References [1] Satake N, Suda KI, Arakawa T, Sasaki A, Tamura Y Damping evaluation using full-scale data of buildings in Japan J Struct Eng 2003;129(4):470–7 [2] Papadimitriou C, Haralampidis G, Sobczyk K Optimal experimental design in stochastic structural dynamics Prob Eng Mech 2005;20:67–78 [3] James GH, Carne TG, Lauffer... 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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... 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... stochastic loading model The experiments are essentially repeatable ð17Þ where D¼ X a2 k 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. .. 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. .. [9] Bayoglu Flener E, Karoumi R Dynamic testing of a soil_steel composite railway bridge Eng Struct 2009;31:2803–11 [10] Davenport AG, Hill-Carroll P Damping in tall buildings: its variability and treatment in design In proceedings of ASCE Spring Convention, Seattle, USA, Building Motion in Wind, 1986; p 42–57 [11] Kareem A, Gurley K Damping in structures: its evaluation and treatment of uncertainty ... 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... 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... Yip, graduate student at the University of Hong Kong, participated in the field tests References [1] Satake N, Suda KI, Arakawa T, Sasaki A, Tamura Y Damping evaluation using full-scale data of buildings