SYSTEM IDENTIFICATION OF STRUCTURES FROM SEISMIC RESPONSE DATA VIA WAVELET PACKET METHOD

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13 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No. 2272 SYSEM IDENIFICAION OF SRCRES FROM SEISMIC RESPONSE DAA VIA WAVELE PACKE MEHOD C. S. Huang 1, S. L.

1 13 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No SYSEM IDENIFICAION OF SRCRES FROM SEISMIC RESPONSE DAA VIA WAVELE PACKE MEHOD C. S. Huang 1, S. L. Hung 2 and W. C. Su 3 SMMARY his paper presents a procedure of estabishing the discrete equations of motion for a structure via waveet packet method from its measured responses and inputs. hen, the natura frequencies, damping ratios and mode shapes of the structure can be directy determined. he proposed procedure is appied to process the acceeration responses of a five-story stee frame, subected to 8%, 2%, 4%, 52% and 6% of the strength of the Kobe earthquake, in shaking tabe tests. he stee frame responded nonineary when it subected to 6% Kobe earthquake, whie it responded ineary when it subected to the other strengths of the Kobe earthquake. his work aso investigates the differences in moda parameters identified from the responses to the inputs with different strengths of the Kobe earthquake. he accuracy of the present resuts is confirmed by comparing the present resuts with the pubished obtained from the responses to the 8% Kobe earthquake input. INRODCION Monitoring of an existing structure is increasingy important in current efforts to evauate the safety of the structure when its materia deteriorates or when the structure has been subected to severe oading ike a strong earthquake. It is desirabe to use the measured responses of a structure to conform the construction quaity, to vaidate or improve anaytica finite eement structura modes, or to determine whether a structure is damaged and further, the nature of any such damage. he dynamic characteristics of a structure are frequenty extracted from its measured seismic responses, and the damage of a structure is conventionay assessed from observed dynamic responses by detecting changes in the moda parameters of the structure. he concept underying such an approach is that damage 1 Professor, Dept of Civi Engineering, Nationa Chiao ung niversity, 11 a-hsueh Rd., Hsinchu, aiwan. Emai: cshuang@mai.nctu.edu.tw 2 Professor, Dept of Civi Engineering, Nationa Chiao ung niversity, 11 a-hsueh Rd., Hsinchu, aiwan. 3 Graduate student, Dept of Civi Engineering, Nationa Chiao ung niversity, 11 a-hsueh Rd., Hsinchu, aiwan.

2 to a structure reduces its natura frequencies, increases the moda damping, and changes the moda shapes. Conventiona techniques in the time domain, such as the time series approach associated with the ARX or ARMAX mode [1-3] and the subspace approach [4, 5], have been often appied to determining the dynamic characteristics of a structure from its seismic responses. Hearn and esta [6] appied a perturbation method to process measured dynamic responses of a stee frame in a aboratory, and found that changes in natura frequencies and moda damping are good indices for damage. However, from dynamic tests on bridges, Aampai and Fu [7] and Saawu and Wiiams [8] concuded that the change in natura frequencies is not sufficienty sensitive to detect oca damage in the structure. In recent years, a new and powerfu mathematica too, caed waveet transformation, has been deveoped, the history of whose deveopment can be found in introductory papers [9, 1] and books [11, 12]. nike a Fourier transform, which expresses a signa in terms of frequency components, the waveet transform decomposes a signa into frequency components that are functions of time. he advantages of the waveet transformation over the Fourier transformation have been addressed in severa papers and books [1-12]. he waveet transformation has been successfuy appied in mathematics, physics and engineering, especiay for signa processing and soving noninear probems. Waveet transformations have aso caught the attention of researchers in the fied of system identification, for use in determining the dynamic characteristics of a time invariant inear system. Schoenwad [13] identified the parameters in the equation of motion for a system with singe degree of freedom by appying a continuous waveet transform to the equation of motion. Ruzzene et a. [14] appied a discrete waveet transform and the Hibert transform technique to determine the natura frequencies and damping of a structure system from its free vibration responses. Robertson et a. [15, 16] deveoped a procedure for extracting impuse response data from the dynamic responses of a structure and used an eigensystem reaization agorithm to identify the dynamic characteristics of the structure. Gouttebroze and Lardies [17] deveoped a waveet identification approach in the time-frequency domain for eucidating the natura frequencies and damping of a structure from free vibration responses. heir approach cannot directy determine the mode shapes. Lardies and Gouttebroze [18] further appied their waveet identification technique [17] to process the measured ambient vibrations of a V tower, by first extracting a free vibration signa from the measured ambient vibration responses, using the conventiona random decrement technique. hese existing methodoogies, invoving waveet transformation for identifying the dynamic characteristics of a structure, use the waveet transform either to extract impuse response functions or to determine natura frequencies and damping from free vibration responses. his work proposes a new procedure for appying a waveet packet transformation to the earthquake responses of a structure, to determine its natura frequencies, damping ratios and mode shapes. his procedure appies a waveet packet transformation to discrete equations of motion of a structure. hen, the parameters of the equations of motion are determined by a east-squares approach, and are directy used to determine the dynamic characteristics of the structure. his procedure is appied to the measured dynamic responses of a fivestory stee frame in a shaking tabe test to demonstrate the feasibiity of appying the proposed procedure to rea data. he moda parameters of the frame subected to 8%, 2%, 4%, 52% and 6% of the strength of the Kobe earthquake are identified from the measured acceeration responses. he stee frame responded nonineary when it subected to 6% Kobe earthquake. Hence, this work aso investigates the differences in moda parameters identified from the responses to the inputs with different strengths of the Kobe earthquake. he accuracy of the present resuts is confirmed by comparing the present resuts with the pubished [5] obtained from the responses to the 8% Kobe earthquake input.

3 MEHODOLOGY Waveet packet As the natura extension of the waveet transform theory, Coifman, Meyer and Wickerhauser [19-21] first proposed their generaization resuts as waveet packets to design efficient schemes for representing and compressing signas and images. he idea is to construct a ibrary of orthonorma bases for L 2 ( R ). Assume a measured signa, s(t), beongs to functiona space +1. In the waveet packet anaysis, the space is orthogonay decomposed into as many subspaces as needed (see Fig. 1), and is expressed as, + 1 = where each space ( m) (2m) + 1 = (1) = ( m) 1 1 (1) 1 (2) 1 (3) 1 = L = k 1 (1) k 1 + is decomposed into two orthogona subspaces L ( 2m) k + 1 (2 2) k 1 k + 1 (2 1) k 1 (1) ( 2 + 1) and m. hat is (2m+ 1) (m). Space is spanned by (, m) waveet packet that is a set of orthonorma functions defined as / 2 µ ( t) = 2 (2 t k), k Z. (2) { } m,, k µ m he waveet packets can be easiy obtained using MALAB waveet toobox and choosing a wanted mother waveet function. ( ) +1 ( ) () 1 ( ) 1 () 1 1 ( 2) 1 ( 3) 1 ( ) 2 () 1 2 ( 6) 2 ( 7) 2 ( ) () 1 k k ( 2 2 ) k k k ( 2 1) k ( ) k 1 () 1 k 1 k 1 ( 2 2) + k 1 k 1 ( 2 1) + k 1 Fig. 1. Decomposition of functiona space in waveet packet method he functions µ satisfy the foowing orthonorma conditions: m,, k 1 when k = < µ m,, k, µ m,, >= and < µ m,, k, µ n,, >= when m n, (3) when k

4 * where the inner product operation is defined as < µ, µ >= µ ( t) µ ( t dt and * denotes the conugate operation. m n m n ) R System identification he dynamicbehaviors of a inear structure are described by the equations of motion, [ M ]{& x } + [ C]{ x& } + [ K]{ x} = { f}, (4) where [M], [C] and [K] are the mass, damping and stiffness matrices of the structure system, respectivey; { x& & }, { x& } and {x} are the acceeration, veocity and dispacement response vectors of the system, and {f} is the input force vector. suay, not a degrees of freedom of the system are measured in a fied experiment for economic reasons. Ony some parts of { x& & } or { x& } are measured. Consequenty, the measured response vector {y}, which can be veocity or acceeration responses, satisfies the foowing discrete equation [22]. I J { y ( t)} = [ Φ] i{ y( t i)} + [ θ] { f ( t )}, (5) i= 1 = where {y(t-i)} and {f(t-i)}are the measured responses and forces at time t-i, respectivey, [Φ] i and [θ] i are the parameter matrices to be determined. Performing a waveet packet anaysis on {y(t-i)} and {f(t-)} with i=, 1, 2,,I and =, 1, 2,,J at the th kˆ eve of subspaces (i.e. at m ( ) m kˆ + 1 subspaces in Fig.1) yieds { y ( t i)} = { y ( m, kˆ, )} µ ( t), (6a) m= = m m, kˆ, ( ) { f ( t )} = { f ( m, kˆ, )} µ ( t). (6b) m= = m, kˆ, Notaby, m is set equa to 2 k ˆ + 1 in the foowing anayses. For finite measured responses and input forces, depends on m. Substituting Eqs. (6a) and (6b) into Eq. (5); performing the inner product with respect to µ ( t) on both sides of the resuting equation, and appying the orthonorma properties m, iˆ, specified by Eq. (3) yieds I J ( ) { y ( m, iˆ, )} = [ Φ] { y ( m, iˆ, )} + [ θ] { f ( m, iˆ, )}. (7) i= 1 i = Rearranging Eq. (7) for different vaues of m and yieds [ [ Y] ] = [ C] [ F] Y, (8) where [ C] = [ Φ] 1 [ Φ] 2 L [ Φ] I [] θ [] θ 1 L [] θ J ], [ ] (1) [ ] [ ] (2) ( I Y Y [ Y ] L [ Y ] ) =, (1) ( J ) [ ] = [ F ] [ F ] [ F ] F L, [ Y ] = [ y (1, iˆ,1) y (1, iˆ,2) L y ( m,ˆ, i ) ]

5 [ F ] = [ f (1, iˆ1, ) f (1, iˆ,2) L f ( m,ˆ, i ) ] ypicay, Eq. (8) represents a set of over-determined inear agebraic equations. he soution for the matrix of parameters [ C ] is determined by a conventiona east-squares approach: 1 Y [ ] [ ] Y Y C = Y. (9) [ ] [ ] [ ] [ F] [ F] [ F] Equation (5) reveas that the moda parameters of a structure are determined from [ Φ] i with i=1, 2,, I. Foowing the procedure given in [23], one can determine the natura frequencies, damping ratios and mode shapes of the structure from the obtained [ Φ ] i. APPLICAION Shaking tabe tests are often performed in a aboratory to examine the behavior of structures in earthquakes. he Nationa Center for Research in Earthquake Engineering in aiwan undertook a series of shaking tabe tests on a 3 m ong, 2 m wide, and 6.5 m high stee frame [24] (Fig. 2) to generate a set of earthquake response data for this benchmark mode of a five-story stee structure. Lead bocks were pied on each foor such that the mass of each foor was approximatey 3664 kg. he frames were subected to the base excitation of the Kobe earthquake, weakened by various eves. he dispacement, veocity, and acceeration response histories of each foor were recorded during the shaking tabe tests. Additionay, some strain gauges were aso instaed in one of the coumns and near the first foor. he samping rate of the raw data was 1 Hz. Notaby, it was reported [24] that the frame responded ineary when it subected to 8%, 1%, 2%, 4%, and 52% of the strength of the Kobe earthquake. Measured strains and visua inspection reveaed that 6% of the strength of the Kobe earthquake input caused the stee coumns near the first foor to yied. In the foowing, ony the responses and inputs in the ong span direction are discussed. Examination of the frame in the Kobe earthquake with various reduction eves he measured acceeration responses from t=4.5 to 12.5 seconds (see Fig.3) corresponding to the base excitations with various reduction eves of the Kobe earthquake, were processed. abe 1 summarizes the identified dynamic characteristics of the frame by using the proposed procedure with setting I =J =7 in eq.(5) and k ˆ = 3 in eqs. (6a) and (6b). Notaby, using arge I and J eiminates the effects of noise and different mother waveet functions on the identified resuts [25].

6 Fig. 2. A photo and simpe sketch of a five-story stee frame

7 acc.(g) acc.(g) acc.(g) acc.(g) th foor 4th foor 3rd foor 2nd foor acc.(g) acc.(g) st foor input excitation ime(sec) Fig. 3. Response histories for 6% Kobe earthquake input. In abe 1, e is defined as [26] * ( φir aφic ) ( φir aφic ) 1/ 2 e = ( ) (1) * φ ir φir * where the compex constant, a, is obtained by minimizing ( φir aφic ) ( φir aφic ), * denotes the compex conugate, φ ir and φ ic represent the i th compex mode shapes for the reference state and the current state to which it is to be compared, respectivey. When the two moda shapes are highy correated, e is cose to zero. e vaues are more sensitive to the changes in moda shapes than MAC (moda assurance criterion) vaues [27]. Here, the vaues of e designate the correation between the moda shapes for an input of 8% Kobe earthquake and those for inputs with other reduction eves.

8 abe 1. Identified moda parameters for different inputs. Excitation Mode Frequency(Hz Damping(%) e(%) 8% 2% 4% 52% 6% (1.4) 1.48 (1.3) (.35) (4.53).17 (.16) (.76) (8.23).19 (.19) (2.76) (12.39).13 (.13) (.58) (15.99).12 (.1) (.47) Comparing the present resuts with the parenthesized resuts that were obtained by using a subspace method [5] confirms the accuracy of the present resuts. abe 1 reveas that the frequencies for each mode generay decrease as the excitation magnitude increases, but the changes in frequency are not so considerabe. Generay, the moda damping vaues increase with excitation magnitude. he moda damping vaues for the 6% Kobe earthquake are much greater than those for the 8% Kobe. Interestingy, ony the damping for the first mode exceeds 1% whie the damping for the other modes is typicay much ess than 1%. Notaby, e vaues ceary show that the moda shapes of the higher modes (3 rd to 5 th ) for the 6% Kobe earthquake notaby differ from those for the 8% Kobe earthquake, since the corresponding e vaues far exceed 5%. he damping and e vaues truy refect the fact of possibe damage of the frame under the 6% Kobe earthquake input. In the present procedure of identifying the dynamic characteristics of a structure from its dynamic responses, the discrete equations of motion (i.e. eq. (5)) for the measured degrees of freedom were aso estabished. It is worthwhie to investigate the errors of the one step ahead predicted output from the estabished discrete equations. Figures 4 and 5 show the comparisons

9 between the observed responses and predicted ones for the 8% and 6% Kobe earthquake inputs, respectivey. Notaby, the responses in these two figures were normaized according to the maximum responses in each figure, respectivey. Comparing Figs. 4 and 5 finds that the discrepancies between prediction and measurement for the 6% Kobe earthquake input far exceed those for the 8% Kobe earthquake input. his observation is usefu for structura damage assessment. Fig. 4. Comparison between the measured and predicted responses for 8% Kob earthquake Fig. 5. Comparison between the measured and predicted responses for 6 % Kobe earthquake

10 Identifying dynamic characteristics from windowed responses Since 6% of the strength of the Kobe earthquake input caused the stee coumns near the first foor to yied, the frame shoud be considered as a time varying system. Consequenty, it is interesting to study the changes in moda parameters obtained from different portions of response records. he responses from t=4 to 14 seconds were divided into nine segments, each of 2 seconds, each of which overays 1 second of the previous one. abe 2 presents the moda parameters, obtained from different segments of data for the 8% and 6% Kobe earthquakes. Notaby, the e vaues for the first segment were computed with reference to the moda shapes obtained using the responses between t=4.5 and 12.5 seconds, whie the e vaues for other segments refer to the moda shapes for the first segment. abe 2. Identified moda parameters from windowed responses. Segment Excitation Parameters Mode Mean(µ) SD(σ) σ/µ Frequency(Hz) % Damping(%) e(%) Frequency(Hz) % Damping(%) e(%)

11 abe 2 shows that using different segments of data eads to no significant variation in the identified frequencies. However, the ratios of the mean vaues to standard deviations of the identified frequencies from different segments are much arger for the 6% Kobe earthquake input than those for the 8% Kobe earthquake input. he damping vaues do differ somewhat with different segments of data. he moda shapes obtained from different segments for the 8% Kobe earthquake input show no significant differences. Interestingy, considerabe differences in higher moda shapes (e 5%) are observed for ony two segments in the 6% Kobe earthquake input. CONCLDING REMARKS his paper presented a waveet-based approach for identifying the dynamic characteristics of a structure from its seismic responses. A waveet packet method was appied to process the measured responses. he discrete equations of motion corresponding to the measured degrees of freedom were reconstructed using the orthonorma properties of waveet packets. he coefficients in the discrete equations of motion were determined by a conventiona east-squares approach. hen, the natura frequencies, damping ratios and mode shapes of the structure were cacuated from these coefficients. he proposed procedure was appied to process the acceeration responses of a five-story stee frame, subected to 8%, 2%, 4%, 52% and 6% of the strength of the Kobe earthquake, in shaking tabe tests. he proposed method of estimating the moda parameters was verified by exceent agreement between the present resuts and those obtained by a subspace method, for the frame subected to the 8% Kobe earthquake. he reported noninear responses to the 6% Kobe earthquake input were found to change significanty moda shapes from those for the frame subected to the other strengths of the Kobe earthquake. Furthermore, the prediction errors of the estabished discrete equations of motion for the responses to 6% Kobe earthquake input were consideraby arger than those for the 8% Kobe earthquake input. ACKNOWLEDGEMENS he research reported herein was sponsored by the Centra Weather Bureau, Ministry of ransportation and Communications (MOC-CWB-92-E-11), which is gratefuy acknowedged. he appreciation is aso extended to the Nationa Center for Research on Earthquake Engineering for providing shaking tabe test data. REFERENCES 1. Safak E, Ceebi M. Seismic response of ransamerica buiding, II: system identification. Journa of Structura Engineering, ASCE 1991; 117: Loh CH, Lin HM. Appication of off-ine and on-ine identification techniques to buiding seismic response data. Earthquake Engineering and Structura Dynamics 1996; 25: Satio, Yokota, H. Evauation of dynamic characteristics of high-rise buidings using system identification. Journa of Wind Engineering and Industria Aerodynamics 1996; 59:299-37

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13 ROC, 23 (in Chinese). 26. rifunac D. Comparisons between ambient and forces vibration experiments. Earthquake Engineering and Structura Dynamics 1972; 1: Aemang RL, Brown DL. A correation coefficient for moda vector anaysis. Proceeding of the first Internationa Moda Anaysis Conference, Bethe, Connecticut, SA. 1983;

2330. A study on Gabor frame for estimating instantaneous dynamic characteristics of structures Wei-Chih Su 1, Chiung-Shiann Huang 2 1

2330. A study on Gabor frame for estimating instantaneous dynamic characteristics of structures Wei-Chih Su 1, Chiung-Shiann Huang 2 1 National Center for High-Performance Computing, National Applied Research