Automatic Parameter Setting of Random Decrement Technique for the Estimation of Building Modal Parameters

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1 Automatic Parameter Setting o Random Decrement Techniue or the Estimation o Building Modal Parameters Fatima Nasser, Zhongyang Li, Nadine Martin and Philippe Gueguen Gipsa-lab, Departement Images Signal BP F-384 Saint Martin d Heres, France {atima.nasser, zhong-yang.li, nadine.martin}@gipsa-lab.grenoble-inp.r ISTerre, Univ. Grenoble /CNRS/IFSTTAR BP 53, 384 Grenoble cedex 9, France Pilippe.gueguen@obs.uj-grenoble.r Abstract This paper examines the use o the Random Decrement Techniue (RDT) or the estimation o building modal parameters, in particular the natural reuency and the damping ratio. It ocuses on establishing the processing parameters one must speciy to use the techniue to its best advantage. This work is motivated by the high inluence o these parameters on the RDT uality. Despite the widespread use o the RDT and the enormous eort or understanding the choice o its parameters, no work has been done on how to automate this selection. From this point, this paper comes out with a proposition o an automatic procedure to guide up the choice o these parameters. The results o this paper, on simulated and real-world ambient vibrations, indicate that the proposed automatic setting procedure o the dierent parameters o the RDT has opened the way to more reliable results and much credible estimation o the modal parameters. Introduction The RDT was introduced by Cole during the late 96s [], or the analysis o response measurements only. The principle o this techniue is to estimate Random Decrement Signatures (RDSs) by averaging time segments o the measured structural responses. These segments are selected under certain conditions known as triggering conditions. The modal parameters o the structure can be then extracted rom the RDSs using methods developed to extract these parameters rom ree decays. Since its development, the theory behind this techniue has been widely investigated [, 3, 4, 5], with a signiicant concentration on the parameters contributing to the perormance o the RDT. For example, studying the iltering eect [6], exploring the diiculties o choosing the triggering levels or a given triggering condition [3], and discussing the number o segments needed to yield reliable estimates [7], however none o them had addressed a way o setting these parameters automatically. This paper responds to this deiciency by proposing an automatic procedure or setting the RDT related parameters. The natural starting point is the deinition and the estimation o RDS which is shown in section. The contribution o this work is discussed in section 3. Sections 4 and 5 explore respectively an overview o the proposed algorithm and a detailed explanation o its steps. The obtained results are examined in section 6 over simulated and real-world ambient vibration signals. Section 7 draws the conclusions o this paper. RDS deinition and estimation The RDSs D ( ) are deined as the mean value.. T at an actual time t, Y (t ) Y D Y ( ) E Y( t ) T Y ( t ) E o a process Y(t) given some triggering conditions, () For a time series, the RDSs can be estimated unbiased as an empirical mean N Dˆ Y ( ) y( ti ) Ty ( ti), () N i

2 where D ˆ ( ) is the estimated RDS, Y y(t) is the measurement and N is the number o the triggering points. The triggering condition presents the basic reuirement or the initial condition o the time segments in the averaging process at the time lag. All the dierent triggering conditions used can be considered as speciic ormulation o the applied general triggering condition, where a, a,b and b are the triggering levels. T Y ( t ) T Y ( t ) a Y t) a, b Y( t) ( b. (3) The most reuently used triggering conditions are level crossing, T L Y t a Y t) ( ) a Y( t a, local extrema, a Y t) a, Y( t) T P Y ( t ) ) TY E ( t) ( (, positive point,, and zero-up crossing, T Z ( ) Y( t) a, Y Y t ( t). The number o triggering points controls the estimation time and the accuracy o the estimates [4]. 3 The contribution o this work When irstly introduced by Cole [], RDT was intended to extract one vibration mode out o the narrow-band iltered response measurement, called the RDS. Accordingly, a iltering process is reuired as a irst step in a multi-mode context. Since iltering is the irst step to condition the data or analysis in the RDT; thereore, it should be carried out with high precision. To achieve that and to initialize this process, we propose to set the ilter parameters rom a spectrum o the process Y(t) as described in Section 5. The second step consists o segmenting the ilter output o each mode in order then to average the segments and to retrieve an approximated system response with a correct modal parameters. For this step, we compare the dierent triggering conditions proposed in the literature, such as [5]. Ater averaging, the method provides a signature o a single-degree-o-reedom system or each mode. From these signatures, the modal parameters o each mode can be estimated. To evaluate the reliability o the estimation, we tested our proposed method over real-world and simulated ambient vibration signals. This paper has greatly shown that when the irst step o the RDT is well done, i.e. the iltering process is very well perormed; the other parameters, like the segment length and the dierent types o triggering conditions, will have less eect on the RDS estimation and the modal parameter extraction. 4 Overview o the proposed algorithm In the proposed algorithm, the reuency bands o the signal components were detected using Welch s spectral estimator. The data-driven interpretation o the spectrum is carried out over each reuency sample to determine whether the reuency corresponds to a mode or noise only. The noise spectrum, which has to be estimated as the spectral content corresponding to the additive noise, is extracted using a multi-pass iltering method [8] [9]. Based on the result o the noise spectrum estimation, a peak detection method is applied with a given alse alarm probability to distinguish the spectral content o interest rom the noise spectrum []. In this paper, the spectral peaks are detected by a Neyman-Pearson hypothesis test that reuires the choice o a alse alarm probability and the estimation o the noise spectrum. The peak detection method estimates Q peaks that we associate with the modes o the signal. For setting the ilter bandwidth o each peak we propose to estimate it rom the bandwidth o these detected peaks. The method is simply based on localizing the minima o the Q peaks, [ L (let minimum), R (right minimum)]. Followed by a iltering process, the detected peaks and their related minima are connected in seuence to orm the input o the ilter. Accordingly, a iltered signal will be in use to estimate the RDS o each mode, which will be in turn used or modal parameters extraction. In this way, the iltering process is automatically achieved with paying high attention to its parameters, yielding a good RDS and modal parameter estimations.

3 5 Algorithm details In this section, we describe the main steps o the algorithm proposed. Figure shows the entire estimation process that can be summed up in 4 steps. Concerning the need to recognize the modal parameters o all components o the signal, the algorithm starts by calculating the spectrum y t using a Hamming window o L t points where L t is chosen as S y o signal Fs L t b3 db *. (4) res where Fsis the sampling reuency, b 3 db and res are the -3 db bandwidth o the main lobe o the spectral window ( b 3dB. 3 or Hamming) in reuency bins and Hz respectively. Instead o ixing the window length, we chose to ix the bandwidth res, regarded as the reuency resolution o the analysis, according to a priori physical knowledge o the signal. Each maximum o the spectrum corresponds to two categories, either the noise t e or the signal o interest s t. As in [9], we use a peak detection method to classiy all o the maxima according to the two categories o spectral contents. The principle is to build a hypothesis test based on these two hypotheses. The algorithm reuires the ollowing parameters: Q is the assumed number o the detected peaks; L is t the time window length o the Welch estimator; L is the reuency window length o the multipass ilter; PFA is the alse alarm probability needed or the noise spectrum estimation; P is the number o passes; PFA is the alse alarm probability. These parameters and steps o the algorithm are detailed in Section 5. d The great advantage o the proposed algorithm is that the peak detection strategy is adjusted to the data and the statistical properties o the spectrum estimator. Thereore, heavy manual conigurations are avoided. With the noise spectrum ˆ estimated rom the local spectrum itsel using a multipass iltering method, the calculation o the detection threshold depends on the alse alarm probability chosen by the user. The spectral contents o interest hereby detected are reerred to as peaks, which are essential to the bandwidth estimation, and thus iltering procedure which is an essential step or the RDT. 3

4 Amplitude Power distribution (db), the algorithm provides the number o peaks Q, the bandwidth and the segment length estimations, using the parameters L, PFA, PFA, d MSE, MSE, and the iltered signal env Figure : Flow chart o the global algorithm. Given the input signal y t 5. method The method must irstly be computed. The estimation o the power spectrum is carried out by dividing the time signal into successive blocks, calculating the periodogram or each block, and averaging the periodograms o the signal blocks. Let y m denotes the Then the PSD estimate, S y Y t th m block o the signal y, and let M, is given by S y M DFT ( ym) Y m( w ) m m M denote the number o blocks. (5) M Where the notation. m is the time averaging across blocks o data indexed by m..3 Signal Points Freuency (Hz) Figure : A simulated ambient vibration signal o 6 points, sampled at Hz (let), and its PSD estimated by Welch s method with a hamming window o 867 points (right) The PSD estimated by the Welch s method is illustrated in Figure (right). On the calculated spectrum, the Neyman-Pearson hypothesis test described in Section 5. is applied. The analysis based on such a test consists o two major steps. The irst step is to estimate the noise spectrum. The estimation is done by a multi-pass iltering techniue, as described in Section 5.. The second step is to detect the peaks based on the noise spectrum, using a data-driven peak detection method, as described in Section 5.3. Both o these steps are based on the Neyman-Pearson hypothesis test as described in 5.. The detected peaks are inally connected by the bandwidth estimation method proposed in section Hypothesis test or peak detection and removal The spectrum o a general noisy observation y,...,yn S y can be expressed as S S (6) where S s and S e are respectively the local spectrum o the pure signal s,...,sn noise e,...,en s and o the additive, with N being the length o the signal. We deine here two types o spectral contents as two hypothesis respectively, e 4

5 H : Sy H : Sy Se, S S s e (7) When H is not known, as is the case in this paper, we can apply a Neyman-Pearson lemma [], with PFA Prob( T( ) H). (8) where is the test threshold, PFA is the alse-alarm probability. The test result is given by H T ( ). (9) H In this test, the threshold is calculated rom an a-priori chosen PFA and using the statistical properties o the spectrum estimator. For Welch s method without overlapping between the blocks, the random variable S y T( ) can be regarded as having a M S e Gaussian. T ( ) M Combining (8) and (), the detection threshold can be determined by distribution [],o degree o liberty M i the noise e n is () PFA p x) dx p ( x) dx. T ( ) H ( M () Then the test hypothesis (8) becomes H T( ) Se. () H From (), the test depends on the a-priori choice o the alse-alarm probability PFA and the estimation o the unknown noise spectrum S e. In the proposed algorithm, the test is used twice: in the noise spectrum estimation and in the peak detection. For the noise spectrum estimation described in Section 5., it is necessary to remove peaks using this hypothesis test with a alse-alarm probability reerred to as PFA. For the peak detection in Section 5.3, the test is applied using another value, reerred to PFA. d 5.3 Estimation o the noise spectrum We propose to approximate the unknown noise spectrum ˆ through an iterative process; namely, the multipass iltering that was developed in [8] [9]. The entire procedure o multipass iltering can be summarized by Fig. 3. The irst pass in a L -point median ilter, S e as ˆ FILTER S ' median, with ( L ) ( ) ' L,...,, (3) where ˆ is the estimation result o the irst pass. L is an odd number speciying the length o the sliding ilter window. Normally this length is 5

6 Fs Fs 3bmain L 4bmain (4) res res with b main the normalized bandwidth between the two zeros o the main lobe o the chosen window unction (For Hamming window b main 4 ). The estimated noise spectrum is reined in the ollowing P passes. Each pass has two steps, the peaks corresponding to spectrum test described in the previous section, with a alse-alarm probability S p. The second step smooths the remaining part spectrum ˆ p. All peaks veriying S y are removed by applying the hypothesis PFA and the previously estimated noise H are removed rom the local spectrum to yield its peak-ree part S p by an average ilter with a gliding window o M points. The estimated noise spectrum is upgraded by the ilter output. The inal noise spectrum ˆ estimation is the output ater P passes, S e ˆ ˆ. (5) P Figure 3: Decomposition o the block noise specterum estimation in Figure, done by multipass iltering with P passes. Based on the local spectrum ˆ using a sliding window 5.4 Peak detection S y, the noise spectrum is inally estimated as L reuency samples and the alse alarm probability PFA. Based on the estimated noise spectrum, another Neyman-Pearson test is applied to identiy the peaks veriying H. For each reuency index, a hypothesis test is carried out using the test statistic, T y S, (6) ˆ ( ) M 6

7 The test threshold is deined by a new alse-alarm probability Then the hypothesis o peak detection (9) becomes ( T ( ) H M PFA as d PFAd p x) dx p ( x) dx. (7) S y H ˆ. H (8) Where ˆ is the estimated noise spectrum. An example o the peak detection is shown in Figure 4. The peaks are deined as the vertex o the bell shapes above the detection threshold ˆ. Each peak is represented by its reuency and the amplitude a. Figure 4: Peak detection on local spectra o the signal o Figure, with Detected peaks over the threshold ˆ calculated with 5.5 Bandwidth estimation are treated as noise. 4 PFA 4 and PFA 3. ( ) PFA. Local maxima below the ˆ For each detected peak Q,we propose to localize its let and right minima, denoted as L and R respectively. Then a symmetry test is carried out or all the peaks o neither lowest nor highest reuency ranges. The test consists o accepting the symmetry o either L or other mode is included in the symmetry range. However, or the lowest and the highest reuency ranges peaks, the respectively as the bandwidth range or the iltering process as shown in Table. Fre. range Filter reuency band Neither lowest nor highest re. R under a condition that no R and Lowest re. LPF R Highest re. HPF L BPF min( L, R ), min( L, R ) d L will be used 7

8 Table : Filter bandwidth range o the detected modes with L and respectively, and 5.6 Modal parameter estimation the reuency o the detected peak R the localized let and right minima Given that the RDS, D ˆ Y ( ), decays as e n, estimates o the damping ratios,, are obtained by minimising the Least Suare Error o the envelope itting using an exponential unction. The method takes care o the smallest error between the envelope and the itted unction. The euation or this method is LSE N N i D ( ) n i Y i e (9) where i are time instants corresponding to the i th extrema o the RDS. However, or the estimates o the modal reuencies, the zero crossing reuency estimation is used. This method is widely used because o its simplicity [], it is based on a zero crossing scheme. The reuency o a signal is estimated uite simply as the zero-crossing rate. N N i, i, i () where, i is the i th zero crossing o the RDS. 6 Perormance analysis To test the relevancy o the RDT over the proposed method, many dierent simulated and real-world ambient vibration signals were used, here we present two simulated and one real-world data. In this paper MATLAB programming is used or implementation o proposed algorithm. The Butterworth ilter has been used to ilter the signals under study. It was supplied with the desired passband p, and stopband cuto reuencies s, the allowable passband ripple R p, and the minimum stopband attenuation R s. Then it provides the ilter order and the normalized bandwidth W n o such a ilter. 6. Application on simulated signals The simulated signals are obtained rom a continuous physical modeling o a building deined as an euivalent Timoshenko beam with a simple D linear lumped mass model. At each level o the building, an ambient vibration is generated as the dynamic response o this model, which is excited by a ground solicitation assumed to be white and Gaussian [3]. Two signals o dierent conigurations were simulated as is shown in Table. Signal Sig Sig N mod e 3 3 (Hz) [, 5, ] [.8,, 4] (%) [, 3, 4] [.5,,.5] F s (Hz) signal length (points) 6 8 N loor 5 8

9 Power distribution (db) Power distribution (db) Table : Two simulated signals ( Sig, Sig ), with dierent conigurations o N mod e being the number o modes, and the modal reuencies and the damping ratios o each mode, F the sampling reuency and N loor the number o loors o the building at which the signal is simulated. The spectrum using Welch s method, the detected peaks and the localized minima o each mode o the two signals Sig, and Sig are illustrated in igures 5, and 6 respectively. s Welch Noise spectrum Detected peaks Let min. Right min Welch Noise spectrum Detected peaks Let min. Right min Freuency (Hz) Freuency (Hz) Figure 5: The spectrum estimated by Welch s method, the detected peaks at.95, 5.7 & 9.96 Hz, and the localized minima o each peak o Sig (let), a zoom around the three modes o interest (right) The results o the proposed method over the two simulated signals Sig, and Sig are summarized in Tables 3, and 4 respectively. Mode Filter band (Hz) T Y ( t ) Level Crossing Local Extrema Positive Point Zero-Up Crossing Period st [-3.9] nd [4.-6.5] 3 rd [8.-.7] ˆ ˆ ˆ ˆ ˆ ˆ Table 3: The estimated ilter band, natural reuency ˆ, and damping ratio ˆ o Sig using all the our triggering conditions T Y ( t ) 9

10 Power distribution (db) Power distribution (db) Welch Noise spectrum Detected peaks Let min. Right min Welch Noise spectrum Detected peaks Let min. Right min Freuency (Hz) Freuency (Hz) Figure 6: The spectrum estimated by Welch s method, the detected peaks at.78,.95, & 4. Hz, and the localized minima o each mode o Sig (let), a zoom around the three modes o interest (right) Mode Filter band (Hz) T Y ( t ) Level Crossing Local Extrema Positive Point Zero-Up Crossing Period st [-.78] nd [.87-.7] 3 rd [.7-.9] ˆ ˆ ˆ ˆ ˆ ˆ Table 4: The estimated ilter band, natural reuency ˆ, and damping ratio ˆ o Sig using all the our triggering conditions The two simulated signals Sig and Sig are composed o three modes. The proposed method has successully detected the reuencies o all the simulated modes as shown in Figures 5 and 6. The detection is achieved thanks to the estimated noise spectrum which helps to separate the modes rom the noise. In the reuency band between [4-] Hz and [.5-7] Hz o Sig and Sig respectively, the noise spectrum is over estimated because it is inluenced by a wide band mode. From Tables 3, and 4, the perormance o the algorithm is relatively robust. The estimated modal parameters are extracted correctly and they show very slight dierences under all parameter settings, like the dierent types o triggering conditions and the dierent number o periods. T Y ( t ) 6. Application on real-world ambient vibration signals The signal o this part is a real-world ambient vibration signal, recorded at the top o the 7 th loor Taipo Tower, the republic o china (Taiwan), Figure, using multiple sensors placed on several stories to measure simultaneously the vibrations in three directions, longitudinal, transverse and vertical. This signal is sampled at Hz. For this signals we study the analysis uniuely over the data in the longitudinal direction. Focusing on the irst three modes represented as, the irst longitudinal, the irst transverse, and the irst torsion mode respectively.

11 Power distribution (db) Figure 7: The Taipo tower at the city center o Taipei (let), and its geographical location (right) The spectrum using Welch s method, the detected peaks and the localized minima o each mode o Taipo signal are illustrated in Figure 8. The estimation o the modal parameters by the proposed algorithm is shown in Table Welch Noise spectrum Detected peaks 35 3 (Zoom around the modes o interest) Freuency (Hz) Freuency (Hz) Figure 8: The spectrum estimated by Welch s method, the detected peaks at.4,.3, &. Hz o the Taipo signal (let), a zoom around the three modes o interest with the localized minima o each mode (right) Mode Filter band (Hz) T Y ( t ) Level Crossing Local Extrema Positive Point Zero-Up Crossing Period st [.9-.7] nd [.7-.55] 3 rd [.6-.4] ˆ ˆ ˆ ˆ ˆ ˆ

12 Table 5: The estimated ilter band, natural reuency ˆ, and damping ratio ˆ o the Taipo signal using all the our triggering conditions In this ambient vibration signal, the three modes under study are very closely spaced in reuency. The analysis o each mode is thereore carried out over a very narrow reuency band. Figure 8 shows the result o the proposed method on detecting the three modes o interest. We have particular interest o these modes because their physical interpretation has been established by the physicians. Despite the reuency closeness o peaks, the noise spectrum estimation and the peak detection methods proposed here are able to detect correctly the three modes at.4,.3, and. Hz. To urther show the applicability o the proposed method, Table 5 summarizes the result o modal parameters estimations. For all the three modes, regardless o the type o triggering condition and the number o period, the undamental reuencies were extracted correctly as compared to the values o the a priori inormation rom the physicians. On the other hand, the damping ratio estimations show very slight variance. 7 Conclusions In this paper, we propose an automatic method or setting the RDT processing parameters. The proposed method guides up the automatic choice o the parameters or the iltering process, which is the irst step to condition the applicability o the RDT over multicomponent signals. For the iltering process an automatic bandwidth measurements are achieved by irst apply the Welch s method to have a spectral representation o the signal. At each local spectrum a peak detection method is applied to extract peaks rom the noise spectrum, which is irst estimated by a multipass iltering. For each detected peak, the corresponding minima are localized with a symmetric test. The test consists o accepting the symmetry o either the let minimum o the right one o the detected peak with a condition that no other mode is included in the symmetry range. The automatic iltering o each mode is then ollowed by the RDT, namely, segmenting the ilter output in order then to average the segments and to retrieve an approximate system response with good modal parameter estimation. The perormance analysis on both the simulated and real-world data show that the proposed method works correctly despite the dierent types o the triggering conditions and the number o periods that is set with a low range in order not to increase the variance o the estimated RDS. The modal parameters are then estimated with a relative robustness and the modes detected correctly and automatically. This proves that when the iltering process is carried out very careully, the RDT perormance is enhanced and thus the modal parameter estimations. The proposed method is carried out by adding original techniues in many steps o the RDT with much better adaptability reserving all the important characteristics o the RDT, like the simplicity and the speed, thus allows working on the multi modes o the signal simultaneously with ew manual conigurations. Possible extension o the present work includes generalizing the current selection method to the cases when the excitations are seismic. Acknowledgments This work has been supported by French Research National Agency (ANR) through RISKNAT program (project URBASIS ANR-9-RISK-9). Reerences [] H. A. Cole, On-the-line Analysis o Random Vibrations, AIAA: ASME Structures, Structural Dynamics and Materials Conerence, Palm Springs (968). [] J. K. Vandiver, A. B Dunwoody, R. B Campbell, and M. F. Cook, A Mathematical Basis o the Random Decrement Vibration Signature Analysis Techniue, J. o Mechanical Design, Vol. 4, pp , (98). T Y ( t )

13 [3] J. Asmuussen, Modal Analysis Based on the Random Decrement Techniue: Application to Civil Engineering Structures, PhD Dissertation, U. o Aalborg, D. (997). [4] S. R. Ibrahim, Random Decrement Techniue or Modal Identiication o Structures, Journal o Spaccecrat and Rockets, vol. 4, No., pp (997). [5] J. Antoni and M. El Badaoui, The Discrete-Time Random Decrement Techniue: Closed-orm Solutions or the Blind Identiication o SIMO Systems, International Conerence on SSI (). [6] A. Zubaydi, The Filtering Eect o Random Responses o Stiened Plates on Their Random Decrement Signatures and Natural Freuencies, Majalah IPTEK, Vol. 6, No., (5). [7] J.C.S Yang, N. Dagalakis, and M. Hirt, Application o the Random Decrement Techniue in the Detection o an Induces Crack on an Oshore Platorm Model, Computer Methods or Oshore Structures. ASME, 65-, pp , (98). [8] M. Durnerin, A strategy or interpretation in spectral analysis, Ph.D. dissertation, Institut National Polytechniue de Grenoble, France, 99. [9] C. Mailhes, N. Martin, K. Sahli, and G. Lejeune, A Spectral Identiy Card, in EUropean SIgnal Processing Conerence, EUSIPCO 6,, Florence, Italy, 6. [] N. Martin, Spectral Analysis, Parametric, Non-parametric and Advanced Methods, F. CASTANIE, Ed. WILEY-VCH,. [] F. Millioz and N. Martin, Time-Freuency Segmentation or Engine Speed Monitoring, in ICSV3, Vienna, Austria, July [] S. M. Kay and R. Sudhaker, "A Zero Crossing Based Spectrum Analyzer," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-34, pp. 96-4, February 986. [3] C. Michel, S. Hans, P. Gueguen, and C. Boutin, In Situ Experiment and Modelling o RC-Structure Using Ambient Vibration and Timoshenko Beam, First European Conerence on Earthuake Engineering and Seismology, Switzerland 6. 3