EXAMPLES OF MODAL IDENTIFICATION OF STRUCTURES IN JAPAN BY FDD AND MRD TECHNIQUES

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1 EXAMPLES OF MODAL IDENTIFICATION OF STRUCTURES IN JAPAN BY FDD AND MRD TECHNIQUES Yukio Tamura, Tokyo Polytechnic University, Japan Akihito Yoshida, Tokyo Polytechnic University, Japan Lingmi Zhang Nanjing University of Aeronautics and Astronautics China Takayoshi Ito Tokyo Electric Power Services Japan Shinji Nakata Asahikasei Corporation Japan Kazuhiro Sato MHS Planners, Architects & Engineers Japan Abstract The feasibility and efficiency of two simple and user-friendly but accurate damping evaluation techniques are discussed. One is the Frequency Domain Decomposition (FDD) technique, which uses Singular Value Decomposition (SVD) of the cross-spectral density matrix and the other is the Multi-mode Random Decrement (MRD) technique. Both techniques can be applied to ambient excitations such as wind, turbulence, traffic, and/or micro-seismic tremors, enabling easy handling of closely-spaced and even repeated modes. Good correspondence is shown between the with vibration characteristics obtained by these techniques, and various important points to note on the traditional damping evaluation techniques are also discussed. Introduction In order to evaluate responses of buildings, their dynamic characteristics such as natural frequencies, vibration modes and damping ratios should be accurately known. It is well known that a dynamic structure can be damped by mechanisms with different internal and external characteristics: friction between atomic/molecular or different parts, impacts, air/fluid resistance, and so on. Combination of different phenomena results in various types of damping. Generally, their mathematical descriptions are quite complicated, and not suitable for vibration analysis for complicated structures. In structural dynamics, damping is described by viscous, hysteretic, coulomb or velocity-squared models. Viscous damping occurs when the damping force is proportional to the velocity, and is mathematically convenient because it results in a linear second order differential equation for engineering structures. Transient decay of a viscously under-damped system will decay exponentially. From a practical point of view, equivalent viscous damping, which models the overall damped behaviour of structural systems as being viscous, is often adopted in structural dynamics.

2 Clearly, structural damping is the most important, but most uncertain parameter affecting dynamic responses of buildings. This uncertainty significantly reduces the reliability of structural design for dynamic effects. Thus, accurate determination of structural damping is very important, not only for evaluating structural responses, but also for designing active and passive auxiliary damping devices to be installed in buildings and structures. However, there is no theoretical method for estimating damping in buildings. It is very difficult, if not impossible, even for viscous damping. Thus, the design damping ratio has been estimated on the basis of actual measurements, which are widely dispersed for various reasons. Many full-scale data have estimated damping ratios larger than the actual values because of inappropriate use of damping evaluation techniques. This paper examines two simple and userfriendly but accurate damping evaluation techniques. One is the Frequency Domain Decomposition (FDD) technique (Brincker et al., 2 []) using Singular Value Decomposition (SVD) of the cross-spectral density matrix and the other is the Multi-mode Random Decrement (MRD) technique (Tamura et al., 22 [2]). Both techniques can be applied to ambient excitations such as wind, turbulence, traffic, and/or micro-seismic tremors, enabling easy handling of closely-spaced and even repeated modes. Full-scale measurements were carried out on various types of buildings and structures. Good correspondence was shown between the vibration characteristics obtained by these techniques, and various important points to note on the traditional damping evaluation techniques are discussed. 2 Comparisons of Damping Evaluation Techniques Including Multi-mode Random Decrement (MRD) Technique and Frequency Domain Decomposition Technique (FDD) 3cm 22cm 22cm 22cm 22cm y Centroid x Accelerometers 2. Primitive laboratory investigation of z y dynamic properties of a building model 3cm x To investigate the features of the various damping evaluation techniques, ambient Figure 4-story building model response measurements of a 4-story building model were conducted using fifteen servo-type accelerometers. Figure shows an elevation and plan of the tested 4-story model, which can oscillate with x, y and θ components. Three accelerometers were installed at each level for the x and y directions to be translated into equivalent motions (x, y and θ components) at the centroid. It is assumed that the floor was subject to lateral rigid body motion. The sampling rate was set at Hz, with a Nyquist frequency of 5Hz. The sampling length of the ambient vibration record was set at 2 hours. 2.2 Damping evaluation techniques used for comparisons Two different evaluation techniques in time domain and five techniques in frequency domain were used for evaluation of damping based on ambient vibration records as follows: Techniques in Time Domain:

3 () Traditional random decrement technique (TRD) [3][4]; (2) Multi-mode random decrement technique (MRD) [2]. Techniques in Frequency Domain: () Half-power method (HP) based on power spectral density function of tip acceleration; (2) / 2 method ( / 2 ) based on transfer function of tip and base accelerations; (3) Phase gradient method (PG) based on transfer function of tip and base accelerations; (4) Curve fit of transfer function (TF) of tip and base accelerations; (5) Frequency domain decomposition (FDD) [] based on the singular value decomposition of cross-spectral density matrix of 2 vibration components of the 4-story model..2.2 Multi-mode Random Decrement Technique (MRD) The TRD technique assuming a SDOF system can efficiently evaluate the damping ratios and the natural frequencies only for well-separated vibration modes. It is very efficient for evaluating the amplitude dependency of the dynamic characteristics of buildings. However, if there are closely located predominant frequency components, a beating phenomenon is observed in the Random Decrement signature (RD signature). In such case, TRD cannot be used for evaluation of damping ratio. In order to evaluate multiple closely located vibration modes, the Multi-mode Random Decrement technique (MRD) has been proposed (Tamura et al., 22 [2]), where the superimposition of the multi SDOF systems with different dynamic characteristics is made. The RD signature with beating phenomenon is approximated by superimposition of different damped free oscillations as follows: xi hiω it = 2 R i ( t) e cos hi ωit φ i 2 h R( t) = n i = R ( t) + m i where R (t) :original RD signature, R i (t) :RD signature for the i-th mode component, x i :initial value of i-th mode component, h i :i-th mode damping ratio, ω i :i-th mode circular frequency, t :time, φ i :phase shift, and m : mean value correction of original RD signature Frequency Domain Decomposition Instead of using PSD directly, as in the classical frequency domain technique, the cross spectrum density (CSD) matrix is decomposed at each frequency line via Singular Value Decomposition (SVD). SVD has a powerful property of separating noisy data from disturbance caused by unmodeled dynamics and measurement noise. For the analysis, the Singular Value (SV) plots, as functions of frequencies, calculated from SVD can be used to determine modal frequencies and mode shapes. It has been proved that the peaks of a singular value plot indicate the existence of structural modes (Brincker et al., 2 []). The singular vector corresponding to the local maximum singular value is unscaled mode shape. This is exactly true if the excitation process in the vicinity of the modal frequency is white noise. One of the major advantages of the FDD technique is that closely-spaced modes, even repeated modes, can be easily dealt with. The only approximation is that mode shape orthogonality is assumed.

4 Not only the natural frequency and the mode shape but also the damping ratio can be estimated by the FDD technique. The basic idea of the FDD technique is as follows (Brincker et al., 2 [5]). The singular value in the vicinity of the natural frequency is equivalent to the PSD function of the corresponding mode (as a SDOF system). This PSD function is identified around the peak by comparing the mode shape estimate with the singular vectors for the frequency lines around the peak. As long as a singular vector is found that has a high Modal Amplitude Coherence (MAC) value with the mode shape, the corresponding singular value belongs to the SDOF function. If at a certain line none of the singular values has a singular vector with a MAC value larger than a certain limit value, the search for matching parts of the PSD function is terminated. From the fully or partially identified SDOF spectral density function, the natural frequency and the damping ratio can be estimated by taking the PSD function back to the time domain by inverse FFT as a correlation function of the SDOF system. From the free decay function, the natural frequency and the damping are found by the logarithmic decrement technique. 2.3 Damping ratio from various techniques Figure 2 shows the RD signature using the x component of the tip accelerations of the 4-story building model. The initial amplitude of the acceleration to get the RD signature was set at the standard deviation, σ acc. The lowest natural frequency of the x component of the translational vibration mode of this model and that of the y component are closely located. Therefore, the RD signature with a beating phenomenon was observed as shown in Fig. 2, although it was not clear. As described above, the TRD technique should not be used for the evaluation of the dynamic characteristics. The MRD technique is an appropriate approach in such a case to identify the two different dynamic characteristics. However, when a beating phenomenon is not clear as in this case, it is difficult to identify the two different dynamic characteristics by the MRD technique. Thus, coordinate transformation is needed to make the acceleration records, which have almost the same energy for the acceleration in the x and y directions, as shown in Fig. 3. Figure 4 shows the RD signature using the tip acceleration in the x direction. It can be seen that a beating phenomenon is much clearer than that using the tip acceleration in the x direction as already shown in Fig. 2. The natural frequency and the damping ratio evaluated by MRD for the RD signature using the tip acceleration in the x direction are 3.6 Hz and.24% for the st mode, and 3.7 Hz and.37% for the 2nd mode. Figure 5 shows the variations of the damping ratio with the number of data points used for Discrete Fourier Transform (DFT) calculation. In the frequency domain approaches, the power spectral density functions or the transfer function is calculated via DFT. It is well known that leakage error in PSD estimation always takes place due to data truncation of DFT. Leakage is a kind of bias error, which cannot be eliminated by windowing, e.g. by applying a Hanning window, and is harmful to the damping estimation accuracy, which relies on the PSD measurements. The bias error caused by leakage is proportional to the square of the frequency resolution (Bendat & Piersol, 986 [6]). Therefore, increasing frequency resolution is a very effective way to reduce leakage error. As shown in Fig. 5, the damping ratio evaluated by frequency domain approaches decreases with increasing number of data points used for DFT calculation and converge to a constant value. It is noted that the number of data points used for DFT calculation should be larger than 6,384 in this case, which is almost 6 times the natural frequency of the model, to accurately identify the damping ratio. This means that the sample should have enough length. An ensemble averaging operation with a sufficient number of samples is also necessary to obtain a correct power spectral density function. Therefore, a very long record of ambient vibration is necessary to accurately evaluate the damping ratio by using a sufficient number of samples with enough length.

5 Acceleration(cm/s 2 ) f =3.Hz h =9.76% f 2 =3.6Hz h 2 =.25% Time (s) y y θ x' = x cosθ y sinθ x y' = x sinθ + y cosθ x' Figure 3 Coordinates transformation Acceleration(cm/s 2 ) Figure 2 RD signature of tip acceleration (x dir.) f =3.6Hz h =.24% f 2 =3.7Hz h 2 =.37% Time (s) Figure 4 RD signature of tip acceleration (x dir.) Number of data points used for DFT calculation Figure 5 Variations of damping ratios with DFT data points 3 Dynamic Characteristics of a 5-story Office Building 3. Tested 5-story building A series of field measurements were made of ambient vibrations of a middle-rise 5-story office building, and its dynamic characteristics were evaluated (Tamura et al., 22 [7], and Miwa et al., 22 [8]). The building extends from 6.m underground to 59.5m above basement level, as shown in Fig. 6. The columns are concrete-filled tube and the beams are wide-flange steel. The floor comprises a concrete slab and steel deck. The exterior walls of the first floor are of pre-cast concrete. The walls from the second floor to the top are of autoclaved lightweight concrete. The plan of a standard story is 22.2m long by 3.8m wide, and the floor-to-floor height is 3.8m. The concrete strengths were 24N/mm 2 underground, 42N/mm 2 for the column filling, and 2N/mm 2 (lightweight concrete) above ground. 25.9m 3.8m Damping ratio (%) RM MRD FDD Accelerometers / 2 HP PG PG HP RM MRD FDD 59.5m 3.8m y 25.9m x (a) Northeast face (b) Northwest (c) General plan Figure 6 Elevations of a 5-story office building

6 3.2 Field measurement Fourteen servo-type accelerometers were used for one setup with two accelerometers at the 5th floor as references. It is assumed that the floor was subject to lateral rigid body motion. The measured vibration was translated into equivalent motions at the desired corners. Accelerometers excluding reference accelerometers were used as roving sensors for the st, 2nd, 3rd and 4th setups. Three accelerometers were typically placed in the southeast (x direction) and northeast (x and y directions) corners from the 7th to 5th floors as well as in the roof. Six accelerometers were placed at the 2nd, 4th and 6th floors. Fifty-three components were measured in total. The FDD technique was applied to evaluate dynamic characteristics of the building based on the measured ambient vibrations. Cross spectral densities were estimated using the full data of fiftythree components. The Hanning window was applied with 66.7 % overlap to compensate for the ensemble averaging effects. 3.3 Dynamic characteristics estimated by FDD Figure 7 shows the SV plots of the office building. There were many peaks of less than 5Hz corresponding to the natural frequencies, and it was possible to obtain up to the 9th mode below 5Hz. Figure 8 depicts the corresponding nine mode shapes. Figure 9 shows a typical bell of the SDOF system the st and 2nd modes of the 5-story building. From the identified SDOF spectral density function, the modal frequency and the damping can be estimated by taking the PSD function back to time domain by inverse FFT as auto-correlation function of the SDOF system. From the free decay function, the modal frequency and the damping are found. Singular Values Hz.86Hz.6Hz 2.37Hz 2.6Hz 4.37Hz 2.9Hz 4.Hz 4.6Hz Frequency (Hz) Figure 7 SV plots of a 5-story office building f =.76Hz f 2 =.85Hz f 3 =.Hz f 4 =2.23Hz f 5 =2.46Hz f 6 =2.94Hz f 7 =3.85Hz f 8 =4.25Hz f 9 =4.49Hz Figure 8 Mode shapes of a 5-story office building

7 In order to show the influence of the frequency resolution on the damping estimation accuracy, 256, 52, 24, 248 and 496 data points were used to calculate the CSD functions. The corresponding frequency resolutions were.783,.392,.95,.977 and.488 Hz, respectively. Figure presents the variations of the damping ratios with the number of data points used for DFT calculation. It is very interesting to observe that, as predicted by the theory of random data procession, the damping ratios of all modes decrease, while the number of data points, or frequency resolution, increases. It appears that damping estimates converge when the number of data points is large enough (up to 496 or 892). Table shows the dynamic characteristics of 5-story building obtained by the FDD technique, where the number of data points used for DFT calculation is 496. Corresponding natural frequencies obtained by FEM analysis are also shown in Table. In the original FEM analysis, the building s stiffness was estimated only for the members of the main structure, and the natural frequencies by FEM analysis were evaluated slightly smaller than the full-scale values. The stiffness of the building s exterior walls was therefore added so that the first-mode natural frequency was identical to the full-scale value. As a result, satisfactory agreement was obtained up to the 9th vibration mode as shown in Table. Mode Normalized Singular Values 2 db Natural Frequency (Hz) Damping Ratio (%) FEM FDD FDD Hz.85Hz Frequency (Hz) Figure 9 Close-up view of SV plots of a 5-story building (%) Table Dynamic characteristics of 3 st mode 2nd mode a 5-story office building 2 3rd mode Damping Ratio (a) st - 3rd mode 2.5 (%) Damping Ratio (b) 4th - 6th mode 2.5 (%) Damping Ratio 2 6th mode 5th mode 4th mode 7th mode.5 9th mode.5 8th mode (c) 7th - 9th mode Number of FFT DFT data points Figure Variations of damping ratios with DFT data points

8 4 Dynamic Characteristics of a Tall Chimney 4. Field Measurement Set-up Ambient response measurements of a 23mhigh chimney were conducted, and its dynamic characteristics were evaluated (Tamura et al., 22 [2], Yoshida and Tamura, 24 [9]). Figure shows an elevation and plan of the tested chimney, which consisted of steel trusses and a concrete funnel. The chimney has an octagonal cross section. Servo-type accelerometers were installed on three different levels, as shown in Fig.. Two horizontal components (x, y) and one vertical component (z) were measured at each level. A sonic anemometer was also installed at the top of the chimney. The sampling rate of the acceleration records was set at Hz, and the ambient responses were measured for 9 minutes in total. 4.2 System Identification by MRD and FDD Figure 2 shows the power spectra of accelerations at three different heights of the chimney. Peaks corresponding to several natural frequencies are clearly shown. The TRD technique assuming a SDOF system was applied for system identification using the ambient accelerations of the y direction at the top: GL+22m. By processing with a numerical band-pass filter with a frequency range of.6hz -.Hz, only the frequency components around the lowest peak near.4hz depicted in Fig.2 were extracted. The initial amplitude of the acceleration to get the RD signature was set at the standard deviation, σ acc. From the clear peak near.4hz in the power spectra shown in Fig. 2, it was believed that the dynamic property of the first mode would be easily obtained with appropriate accuracy, but this was not correct. Figure 3 shows the obtained RD signature, PSD of acceleration S acc (f) G.L.+22m G.L.+48m G.L.+76m G.L. : Accelerometers : Sonic anemometer 23m Figure A 23m-high chimney Width of band pass filter.4hz.4hz. 4 Frequency (Hz) Figure 2 Power spectra of accelerations (y-dir.) of a 23m-high chimney h =.8% h 2 =.3%.5 f =.4Hz f 2 =.4Hz Acceleration (cm/s 2 ) -.5 y 22m 48m 76m Time (s) Figure 3 RD signature of tip acceleration (y-dir.) of a 23m-high chimney 4 db.4hz 2 2.7Hz.4Hz.52Hz 2.38Hz.47Hz Normalized Singular Values Frequency (Hz) Figure 4 SV plots of a 23m-high chimney x

9 where a beating phenomenon is observed, suggesting two closely located dominant frequency components. By a careful examination of the close-up view of the peak near.4hz, it is seen that there are actually two peaks at.4hz and.4hz. These peaks are named f and f 2, respectively, in this paper. Then, the MRD technique explained in 2.2. was applied, and the damping ratio and the natural frequency of the chimney were estimated at.8% and.4hz for the st mode, and.3% and.4hz for the 2nd mode. The dynamic characteristics of the 3rd and 4th modes were also estimated by the MRD technique. The FDD technique was also applied to the six horizontal components of the acceleration responses at the three different heights to evaluate the chimney s dynamic characteristics in detail. Figure 4 shows the frequency distribution of the SV plots obtained by the FDD technique. Figure 5 shows the variations of the damping ratios of the lowest two modes with the number of data points used for PSD calculation. The damping ratios converge to certain values with increase in the number of data points. Table 2 shows the dynamic characteristics of a 23m-high chimney obtained by MRD and FDD. Damping ratio (%) Data points used for PSD calculation Figure 5 Variations of damping ratios of a 23m-high chimney with DFT data points Table 2 Dynamic characteristics of a 23m-high chimney Mode Natural Frequency (Hz) st mode 2nd mode Damping Ratio (%) RD FDD RD FDD Dynamic Characteristics of a Long-span Roof 5. Field measurement setup Ambient response measurements were 49m performed on a long-span roof 42.8m high, 8m 8m wide, and 49m deep in cantilever form, as shown in Photo. Servo-type accelerometers were installed on roof beams to measure the vertical accelerations. Three 42.8m accelerometers were fixed at the reference points and twelve accelerometers were moved and set at various points. Ambient response measurements were carried out for four setups, Photo A long-span roof and a total of fifty-one measurement points were obtained. The sampling rate was set at Hz, and measurements made for hour each setup.

10 5.2 System identification by FDD It was possible to identify the natural frequencies up to 6th mode below 6 Hz. The mode shapes up to the 5th mode obtained by the FDD method are shown in Fig. 6. Mode shapes could be estimated very clearly even from very small ambient vibrations. Table 3 compares the natural frequencies obtained by the FDD and FEM analyses up to the 5th mode. It was found that the natural frequencies obtained from the field measurements were more than % higher than those obtained from the FEM analysis. This is considered to be due to the contribution of the stiffness of the secondary members, which is not estimated by the FEM model. (a)f =.3Hz (b) f 2 =.9Hz (c) f 3 =.3Hz (d) f 4 =.93Hz (e) f 5 =2.58Hz (f) f 6 =2.74Hz (g) f 7 =2.88Hz (h) f 8 =2.97Hz (i) f 9 =3.3Hz (j) f =3.9Hz (k) f =3.94 (l) f 2 =4.58Hz (m) f 3 =4.86Hz (n) f 4 =5.38Hz (o) f 5 =5.57Hz Figure 6 Mode shapes and natural frequencies of a long-span roof obtained by FDD Table 3 Natural frequencies of a long-span roof Mode Natural Frequency Difference (Hz) (%) FEM FDD Dynamic Characteristics of transmission towers 6. Field measurement setup Ambient response measurements and human excited response measurements were performed on ten transmission towers, whose heights were from 2m to 8m as shown in Fig. 7. Three accelerometers were installed on top of each tower to measure the acceleration of X, Y, θ components. The sampling rate was set at Hz, and measurements were made for 3 minutes. 6.2 System identification by RDT Table 4 compares the natural frequencies and the damping ratios obtained by RDT. Figure 8 shows the changes of natural frequency with tower height. As shown in Table 4 and Figure 8, the natural frequency decreases with increasing tower height. Since most of these tower had a square or

11 circular section, the natural frequencies in the X and Y directions show almost the same values. Figure 9 shows the relationship between the natural periods obtained by the FEM analysis and RDT. The natural periods obtained by RDT were almost the same as those obtained by FEM analysis. 8m 7m 6m 5m 4m Ntower Stower Atower Ktower Itower Gtower Dtower Otower Ttower 3m Mtower 2m m Fig. 7 Elevation of transmission towers Natural frequency (Hz) Tower height (m) Figure. 8 Changes of natural frequency of transmission tower with height Natural period by FEM (s) Natural period by RDT (s) Figure. 9 Relationship between natural periods obtained by FEM analysis and RDT Table 4 Dynamic characteristics of transmission towers obtained by RDT Name Height Natural frequency (Hz) Damping ratio (%) (m) X dir. Y dir. θ dir. X dir. Y dir. θ dir. N tower S tower A tower K tower I tower G tower D tower O tower T tower M tower

12 7 Dynamic Characteristics of Other Structures by FDD Table 5 shows the dynamic characteristics of other structures obtained by FDD. For H tower, the acceleration of only two translation components is measured, and the dynamic characteristics for the torsional component couldn t be identified. For both control towers, the Tuned Liquid Damper (TLD) is installed near the top. Table 5 Dynamic characteristics of other structures Height Natural frequency (Hz) Damping ratio (%) Name (m) X dir. Y dir. θ dir. X dir. Y dir. θ dir. H tower N flight control tower (With TLD) H flight control tower (With TLD) UR tower Concluding Remarks Various damping evaluation techniques were discussed, and the efficiency and feasibility of the Frequency Domain Decomposition (FDD) technique and the Multi-mode Random Decrement (MRD) technique were demonstrated. Both techniques can be applied for ambient excitations, enabling easy handling of closely-spaced and even repeated modes. Fairly good correspondence was shown between the vibration characteristics obtained by the MRD and the FDD. Various important points to note regarding traditional damping evaluation techniques were also discussed, and it was emphasized that a sufficient number of data points for DFT is necessary for spectral damping evaluation techniques. In addition, the dynamic characteristics of various structures were investigated. 9 References [] Brincker, R., Zhang, L.M. and Anderson, P., 2, Modal identification from ambient response using frequency domain decomposition, Proceedings of the 8th International Modal Analysis Conference (IMAC) [2] Tamura Y., Zhang L.-M., Yoshida A., Nakata S. and Itoh T., 22, Ambient vibration tests and modal identification of structures by FDD and 2DOF-RD technique, Structural Engineers World Congress (SEWC), Yokohama, Japan, October 9-2, T--a-, pp.8 [3] Cole, H. A., 973, On-line failure detection and damping measurement of aerospace structures by the random decrement signatures, NASA CR-225 [4] Jeary, A., 986, Damping in buildings - a mechanism and a predictor, Journal of Earthquake Engineering and Structural Dynamics, 4, pp [5] Brincker, R., Ventura, C.E. and Andersen, P., 2, Damping estimation by frequency domain decomposition, Proceedings of the 9th International Modal Analysis Conference (IMAC), pp [6] Bendat, J. and Piersol, A., Random Data, 986, Analysis and Measurement Procedures, John Wiley & Son, New York, USA, 986 [7] Tamura, Y., Zhang, L., Yoshida, A., Cho, K., Nakata, S., and Naito, S., 22, Ambient vibration testing & modal identification of an office building with CFT columns, Proceedings of the 2th International Modal Analysis Conference (IMAC), pp.4-46 [8] Miwa, M., Nakata, S., Tamura, Y., Fukushima, Y. and Otsuki, T., 22, Modal identification by FEM analysis of a building with CFT columns, Proceedings of the 2th International Modal Analysis Conference (IMAC) [9] Yoshida A. and Tamura Y., 24, System identification of structure for wind-induced response, Proceedings of the 5th International Colloquium on Bluff Body Aerodynamics and Applications (BBAA), Ottawa, Canada, July -5, pp