Advances in Mechanical Engineering 2017, Vol. 9(7) 1–13
Ó The Author(s) 2017 DOI: 10.1177/1687814017707649 journals.sagepub.com/home/ade
Identifying the stiffness parameters of
a structure using a subspace approach
and the Gram–Schmidt process in a
wavelet domain
Wei-Chih Su
1, Chiung-Shiann Huang
2, Ho-Cheng Lien
1and
Quang-Tuyen Le
2Abstract
This article presents a procedure to improve the accuracy of calculated stiffness matrix of a structure based on the identified modal parameters from its measured responses. First, a continuous wavelet transform is applied to the mea-sured responses of a structure, and the state–space model can be reconstructed by the wavelet coefficients of accelera-tion that can be obtained from the measured noisy responses. The modal parameters are identified using the subspace approach. Second, the identified mode shapes are corrected via Gram–Schmidt process. Finally, the identified natural fre-quencies and the corrected mode shapes in previous steps are utilized to build the stiffness matrix of structure. The accuracy of the proposed approach is numerically confirmed, and the noise effects on the ability to precisely identify the stiffness matrix are also investigated. The measured data of two eight-story steel frames in a shaking table test are ana-lyzed to demonstrate the applicability of the procedure to real structures.
Keywords
Damage assessment, subspace approach, Gram–Schmidt process, modal identification, wavelet transform
Date received: 14 June 2016; accepted: 6 April 2017 Academic Editor: Stephen D Prior
Introduction
Damage to a structure is caused by many sources, espe-cially intense loading during a strong earthquake and the degradation of structural material. Identifying the location and level of structural damage is critical to the investigation of the serviceability and safety of struc-tures. Early evaluation of damage or structural degra-dation is essential for preventing disastrous accidents. Studies on structural health monitoring have been receiving increasing attentions in the field of civil engineering.
The modal parameters of a structure can be used to figure out the stiffness characteristics of a structure. For this reason, the damaged structures can be detected by the change of the modal parameters. In recent
decades, the modal identification methods that are implemented in time domain, frequency domain, or time–frequency domain are simple and extensively adopted.1–3 Various formulations in time–frequency domain are performed to process the noisy data from various field tests.4,5 The proposed procedure is a
1
National Center for High-Performance Computing, National Applied Research Laboratories, Hsinchu, Taiwan
2
Department of Civil Engineering, National Chiao Tung University, Hsinchu, Taiwan
Corresponding author:
Wei-Chih Su, National Center for High-Performance Computing, National Applied Research Laboratories, No. 7, R&D 6th Rd., Hsinchu Science Park, Hsinchu 30076, Taiwan.
Email: [email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
typical example of the cooperation between a state– space model and a subspace approach6,7 in time– frequency domain for the modal identification. However, the accuracy of the identified modal para-meters is always reduced by the imperfections of the measured responses, which usually comprise certain noise from unknown input or system resolutions.
Due to the strong capability of data de-noising, con-tinuous wavelet transforms have been discussed in some studies that aim to identify the modal parameters of a linear system.8–10Huang and Su11utilized continu-ous wavelet transform for the earthquake responses of a structure using various wavelet functions. They pro-posed modal identification procedures that are based on a time series model to determine structural modal parameters of the system using the coefficient of a wavelet transform.
As mentioned previously, the identified modal para-meters could be used to observe whether the structure is damaged or not. But, the damaged location is difficult to be found from the results of modal identification. A stiffness matrix–based approach7,12that employs identi-fied modal parameters and given mass properties is a vibration-based damage detection approach for finding the damaged location. The stiffness parameters have been identified for controlling the flexible members in robotic manipulator or marine riser system.13–15In the ideal experimental condition, the clear and noise-free responses would be measured. However, the noisy vibration is always obtained in the on-site test; for this reason, the de-noising procedure needs to apply on the modal identification process.
Despite the de-noising procedure reducing the influ-ence of noise in the frequency identification, it often removes certain parts’ signal energy and causes identifi-cation error for mode shapes in higher modes. The orthogonality property, which cooperates with the mass and stiffness matrices of the mode shapes, can be used to correct the identified mode shapes. In linear algebra, many orthogonalization algorithms and approaches have been proposed and widely used for computing an orthogonal basis, such as Householder transformations, Givens rotations, and Gram–Schmidt process.16There are two basic computational variants for executing the Gram–Schmidt process: the classical Gram–Schmidt algorithm and the modified Gram–Schmidt algo-rithm.17 For the rounding error, the classical Gram– Schmidt algorithm may produce a set of vectors which is lost orthogonality.18 The modified Gram–Schmidt algorithm has better numerical properties stable than the classical Gram–Schmidt algorithm.19
This study proposes a comprehensive procedure to accurately obtain a stiffness matrix using the identified modal parameters based on Gram–Schmidt process.
The identified modal parameters are calculated from a subspace approach in Morlet wavelet domain. The iden-tified mode shapes are corrected by applying the given mass properties and Gram–Schmidt process. The structural stiffness parameters can be calculated via the previous identified natural frequencies and the corrected mode shapes. The applicability of this approach was verified in a numerical analysis using simulated earth-quake acceleration responses of a six-story shear build-ing. The measured noisy responses and input with 10% variance of the noise-to-signal ratio (NSR) are pro-cessed. The proposed procedure was successfully applied to the measured acceleration responses of two eight-story steel frames in a shaking table test and proved the feasibility of this procedure in practical cases. The col-umns in the first and third story of the steel frame were constructed of steel plates with cutoff.
Methodology
Subspace approach in Morlet wavelet domain
In the case of a linear structure, the equation of motion for the dynamic responses can be expressed as follows
M€x + C _x + Kx¼ f ð1Þ where M, C, and K represent the mass matrix, damping matrix, and stiffness matrices, respectively, of the sys-tem; €x is the measured acceleration response vector; _x is the measured velocity response vector; x is the displa-cement response vectors; and f represents for the input force vectors. Consider that the observed degrees of freedoms apply to the case of incomplete observation and only acceleration or velocity responses are mea-sured. Here, the expression y¼ Lx is used to describe the observed responses, in which L is a matrix with components equal to 0 or 1 according to the non-observed or non-observed conditions.
The state–space model is generally considered in the following formulation
zk + 1 ¼ Azk+ Bfk yk¼ Ezk+ Dfk
ð2Þ where A, B, E, and D are the system matrices that are related to M, C, and K; z¼ x T _xTTis a state vari-able. As shown in equation (2), the following recursive formula can be obtained
yk + s¼ EAsz
k+ Dfk + s+ Xs
i¼1
EAi1Bfk + s + i1 ð3Þ Applying the continuous Morlet wavelet transform to equation (3) and treating y and f as vector functions yield the following
W csyk + s ¼EAsW cszk+ DW csfk + s +X s i¼1 EAi1BW csfk + s + i1 ð4Þ
where the definition of the continuous wavelet trans-form for a function fðtÞ can be written as follows
W csf p; að Þ ¼ 1 ffiffiffi a p ð + ‘ ‘ f tð Þcs t p a ð5Þ
Variable a is the dilation of the scale parameter, variable p is the translation parameter, and csðtÞ is the mother function. The superscript * denotes the complex conjugate. Morlet wavelets are utilized to form equa-tion (5). Equaequa-tion (6) is used to define the standard Morlet mother wavelet with order s
csð Þ ¼ ut sp1=4et 2=2 eist es2=2 ð6Þ where us¼ ð1 + es 2 2e3s2=4 Þ1=2 is the normaliza-tion constant. Using equanormaliza-tion (4), one can further con-struct the following
W csyk;s¼ YaW cszk+ FaW csfk;s ð7Þ where W csyk;s¼ W csy T k W csy T k + 1 W csy T k + s1 T Ya¼ ET ðEAÞT EAs1 T h iT Fa¼ D 0 0 0 EB D 0 0 EAB EB D 0 .. . EAs1B EAs2B D 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 W csfk;s¼ W csf T k W csf T k + 1 W csf T k + s1 T ð8Þ where the two sets W csyk;sand W csyk;sare the
coef-ficients of Morlet wavelet transform of yk and fk, respectively. Equation (7) can be re-written as follows
^ YN ¼ YaZ^N+ FaF^N ð9Þ where ^ Yk;N ¼ W csyk;s W csyk + 1;s W csyk + N1;s ^ Zk;N ¼ W cszk W cszk + 1 W cszk + N1 ^ Fk;N ¼ W csfk;s W csfk + 1;s W csfk + N1;s 8 > < > : ð10Þ
A brief summary of the subspace approach proce-dure, which was organized by Huang et al.7and in more detail by Huang and Lin,6is described as follows
1. Define an orthogonal projection matrix *?f ¼ I ^FTk;Nð^Fk;NF^
T k;NÞ
1^
Fk;N onto the null space of ^
Fk;N;
2. Introduce the instrumental variables P¼ ½ ^FTp;N Y^Tp;NT;
3. Calculate the weighting matrices Wr¼ I and Wc¼ ðPP?f P
T
=NÞ1=2;
4. Define the matrix H¼ WrY^k;N* ?
f PWc=N ; 5. Apply singular value decomposition on Hffi
QnDnVTn; 6. Let Ga ¼ GaTn, where Ga¼ W1r QnD 1=2 n ; Tn¼ D1=2n Vnð ^Zk;N* ? f PWc=NÞ 1=2 ; 7. Rewrite equation (7) as W csyk;s¼ Gazk+ FaW csfk;s, where zk ¼ TnW cszk; 8. Ga can be established by Ga¼ ET ðE AÞT ðE As1ÞT h iT ; 9. Define Ga1¼ ½ ET ðE AÞT ðE As2ÞTT and Ga2¼ ½ ðEAÞT ðE A2ÞT ðE As1ÞTT; 10. Calculate A by solving the linear equation
Ga2¼ Ga1A.
Identification of modal parameters
The modal parameters of a system are generally identi-fied using the eigenvalues and eigenvectors of the sys-tem matrix A in equation (2). The terms of the space– state variable are used to express the equation of motion. However, matrix A cannot be directly deter-mined from the preceding derivation. To identify the modal parameters of a structure, employ matrix A, which is characterized as follows
lj¼ eljDt; ϕj¼ Tnϕj ð11Þ where lj andϕjare the jth eigenvalue of A and the jth eigenvector of A, respectively, and ljand ϕjare the jth eigenvalue of A and the jth eigenvector of A, respec-tively. In equation (11),ϕjcannot be estimated because Tnis unknown. To overcome this challenge, the modal shape ϕj;y, which corresponds to the observed degrees of freedom, needs to be determined first.
Based on the scheme in section ‘‘Subspace approach in Morlet wavelet domain,’’ a new relationship can be determined
ϕj;y¼ Eϕj ð12Þ The eigenvalues of A and A are complex numbers and can be expressed as follows
lj¼ aj+ ibj
lj¼ aj+ ibj ð13Þ From the relationship between the eigenvalues of A and A in equation (11), one has
aj¼ ln a2 j+ b2j ð Þ 2Dt bj¼an 1 b j=aj ð Þ t=Dt ð14Þ
The modal parameters of the structural system can be determined as follows vj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 j + b2j q jj¼ aj vj ð15Þ
where vj is the jth pseudo-undamped circular natural frequency, and jjis the jth modal damping ratio of the structural system.
Construction of stiffness matrix
Consider the equation of motion of a structural dynamic response, as shown in equation (1), and assume the damping properties in structure, the modal mode shapes perform the orthogonality property with the mass and stiffness matrix20as follows
MD¼ϕTMϕ; KD¼ϕTKϕ ð16Þ where MD is the diagonal modal mass matrix, and KD is the diagonal modal stiffness matrix of the structure. The square of the modal frequencies is described as an equation of matrix L as follows
L¼ MT=2D KDMD1=2¼ MT=2D ϕTKϕM1=2
D ð17Þ
Assuming that the mass matrix was given, the stiff-ness matrix can be obtained as follows
K¼ϕTMT=2D LM1=2D ϕ1 ð18Þ However, the error in the identified higher modal is larger than the errors in other identified modals because the de-noising procedure removes certain parts’ signal energy. For this reason, a correction procedure of iden-tified mode shapes based on the Gram–Schmidt process is proposed. First, one can obtain the identical equation as follows ϕTϕ ¼ I ð19Þ where ϕ ¼ M1=2ϕM1=2 D ð20Þ
As shown in equation (12), ϕ is a unitary matrix. The column vectors of ϕ are implied to be an orthogo-nal set. However, the ^ϕ constructed by the identified mode shapes during the de-noising procedure is not a unitary matrix. In this situation, the column vectors of
^
ϕ comprise a linear independent set.
Gram–Schmidt orthogonalization, which is also referred to as the Gram–Schmidt process, is a proce-dure that takes a nonorthogonal set of linearly inde-pendent functions and constructs an orthogonal basis over an arbitrary interval with respect to an arbitrary weighting function. The complete procedure is described as follows:
1. Define the projection operator
Pϕ~i ϕ^j ¼ ~ ϕT iϕ^j ~ ϕT iϕ~i ~ ϕi ð21Þ
2. Calculate orthogonal vectors
~
ϕi¼ ^ϕi Xi1
j¼1
Pϕ~jð Þϕ^i ð22Þ 3. Normalize the orthogonal vectors ~ϕi
ϕi¼ ~ ϕi ~ ϕi k k ð23Þ
Based on the given mass properties, the Gram– Schmidt process is applied to the identified model shapes. The identified higher modal is corrected, and the stiffness matrix can be constructed.
Numerical verification
To prove that the proposed procedure is sensitive and realizable in the numerical analysis, a six-story shear building (Figure 1), which is simulated by Runge– Kutta method, was subjected to the 1999 Chi-Chi earth-quake at the base. The modal damping ratio is set to constant value of 5% in the simulation process. Table 1 lists the theoretical modal parameters of the six-story frame. Figure 1 describes the acceleration responses of the shear building and the input excitation at t = 5–45 s with Dt = 0.005 s, which were employed in the identifi-cation process. The frequency response function, which is well-known modal identification approach, was usu-ally used to describe the resonance frequencies of sys-tem. Figure 2 reveals the rough natural frequencies of the structure in 0.85, 2.3, 3.6, 4.8, 5.7, and 6.2 Hz, but the sixth modal is not very obvious.
The modal assurance criterion (MAC) proposed by Allemang and Brown21 was used to check agreement between the identified mode shapes and the theoretical mode shapes
MAC ϕk;i;ϕk;a ¼ ϕ T k;iϕk;a 2 ϕT
k;iϕk;iϕTk;aϕk;a
ð24Þ
where ϕk,i and ϕk,a are the identified ith mode shape and the corresponding analytical mode shape, respec-tively. The MAC values move in the range of [0,1]. The MAC values are close to one, which indicates that the two mode shapes are similar. The MAC values are close
to zero, which indicates that the two mode shapes are orthogonal to each other.
In the numerical example, the ‘‘accurate’’ results are should be defined as follows: the relative errors between the identified frequencies and the theoretical frequencies are within 2%, the relative errors between the identified damping ratio and the theoretical damp-ing ratio are within 20%, and the MAC values exceed 0.95.
Figure 1. Schematic representation of a six-story frame and its simulation responses.
Table 1. Theoretical modal parameters of the six-story frame.
Mode First Second Third Fourth Fifth Sixth
fnðHzÞ 0.846 2.310 3.576 4.831 5.745 6.209 jð%Þ 5.00 5.00 5.00 5.00 5.00 5.00 Mode shapes 1.000 20.951 20.838 0.672 20.443 20.221 0.929 20.450 0.220 20.877 1.000 0.619 0.793 0.288 1.000 20.406 20.815 20.896 0.601 0.874 0.518 1.000 0.025 1.000 0.367 1.000 20.618 0.103 0.783 20.909 0.190 0.663 20.751 20.685 20.753 0.586
Effects of noise
Corrupted noise always exists in the measured responses. To challenge this situation in the numerical process, the simulated responses and input excitation were randomly added by 10% variance of the NSR of noise.
The processing of noisy responses reveals the diffi-culty of obtaining accurate results for each mode at s\ 10. When NSR = 5%, the first to fourth modes attained accurate damping results at s . 35; the fifth mode attained accurate damping result at s . 45; and the sixth mode could not obtain accurate damping result for s \ 50. Figure 3(a) describes the stabilization diagrams of the identified results obtained from the noisy responses without filter.
In the Morlet wavelet domain, the interested fre-quency bands are retained. The continuous wavelet transform typically performs a frequency filter effect. Figure 3(b) illustrates the stabilization diagrams of the identified results in the case of processing noisy data with Morlet wavelet de-noising. The identified results via a filter theorem are as follows: the first to fourth modes attained accurate damping results at s . 15, the fifth mode attained accurate damping result at s . 18, and the sixth mode could not obtain an accurate damp-ing ratio for s \ 22. Figure 3 reveals the more accurate natural frequencies of the structure than the frequency response function in 0.846, 2.310, 3.575, 4.829, 5.747, and 6.207 Hz, including the highest modal. Some inter-ested frequency ranges were kept, which was useful for obtaining accurate damping results.
As mentioned previously, the existing error in the identified higher modal was larger than the errors
obtained for the other identified modals because the de-noising procedure removes certain parts’ signal energy. Figure 3 reveals that the de-noising procedure signifi-cantly altered the identified fifth and sixth mode shapes, and its MAC values are less than 0.98. Figure 4 shows that the MAC values after the Gram–Schmidt process and modified Gram–Schmidt process are applied to the identified mode shapes using various values of s. The identified fifth and sixth mode shapes are corrected and its MAC values exceed 0.99 at (I, J) . 10. The identified mode shapes were corrected by classical Gram–Schmidt algorithm compared a bit poorly with that was cor-rected by modified Gram–Schmidt algorithm, but not obvious. The identified mode shapes are independent and like orthogonal set; for this reason, similar results can be obtained by these two algorithms.
Constructing a stiffness matrix
For the assumption of a given mass matrix, combined with the identified natural frequencies and corrected mode shapes, the stiffness matrix of a structure can be determined by equation (18). Considering the original shear building model as ‘‘frame A,’’ ‘‘frame B’’ was cre-ated by reducing the local stiffness of the first story and third story of ‘‘frame A’’ to 800 and 1200 kN/m, respec-tively. Because both models are shear building, the the-oretical stiffness matrix of both models is a tri-diagonal matrix, and one will focus on the relative errors of tri-diagonal elements in the following numerical study.
Table 2 lists the relative errors of the calculated stiff-ness matrix using equation (18). The modal parameters
Figure 3. Stabilization diagrams of the identified results: (a) without a filter and (b) after de-noising by Morlet wavelet.
Table 2. Estimated stiffness matrix and its relative error (without correcting the identified mode shapes). Stiffness matrix Frame A 1005.679 (0.568%) 21009.295 (0.929%) 21.707 226.118 25.944 21.221 21009.295 (0.929%) 2018.604 (0.930%) 2984.881 (1.512%) 51.134 25.516 1.408 21.707 2984.881 (1.512%) 1922.019 (3.899%) 2997.626 (0.237%) 248.759 5.081 226.118 51.134 2997.626 (0.237%) 1978.508 (1.075%) 2921.725 (7.827%) 290.224 25.944 25.516 248.759 2921.725 (7.827%) 2603.219 (4.129%) 21520.698 (1.380%) 21.221 1.408 5.081 290.224 21520.698 (1.380%) 2924.168 (2.528%) Frame B 1032.500 (3.250%) 21070.332 (7.033%) 52.096 277.855 25.960 270.882 21070.332 (7.033%) 2090.303 (4.515%) 2997.328 (0.267%) 67.212 41.677 165.054 52.096 2997.328 (0.267%) 1915.541 (4.223%) 2996.853 (0.315%) 2157.634 271.003 277.855 67.212 2996.853 (0.315%) 1899.878 (5.549%) 2778.033 (2.746%) 158.965 25.960 41.677 2157.634 2778.033 (2.746%) 2170.846 (5.615%) 21511.971 (0.798%) 270.882 165.054 271.003 158.965 21511.971 (0.798%) 2806.405 (3.941%)
Figure 4. Stabilization diagrams of the identified mode shapes: (a) corrected by the Gram–Schmidt process and (b) corrected by the modified Gram–Schmidt process.
Table 3. The estimated stiffness matrix and its relative error (identified mode shapes corrected by Gram–Schmidt process). Stiffness matrix Frame A 993.009 (0.699%) 2987.187 (1.281%) 210.334 4.971 28.821 10.730 2987.187 (1.281%) 1975.543 (1.223%) 2973.735 (2.626%) 220.714 25.241 225.385 210.334 2973.735 (2.626%) 1961.160 (1.942%) 2962.204 (3.780%) 238.300 32.145 4.971 220.714 2962.204 (3.780%) 1959.807 (2.010%) 2975.316 (2.468%) 211.932 28.821 25.241 238.300 2975.316 (2.468%) 2523.684 (0.947%) 21543.922 (2.928%) 10.730 225.385 32.145 211.932 21543.922 (2.928%) 3066.367 (2.212%) Frame B 1009.182 (0.918%) 21017.580 (1.758%) 17.394 25.771 7.541 24.765 21017.580 (1.758%) 2035.744 (1.787%) 21039.315 (3.932%) 20.523 222.101 16.429 17.394 21039.315 (3.932%) 2035.590 (1.780%) 21022.276 (2.228%) 21.317 219.013 25.771 20.523 21022.276 (2.228%) 1821.097 (1.172%) 2817.916 (2.239%) 14.198 7.541 222.101 21.317 2817.916 (2.239%) 2312.837 (0.558%) 21502.725 (0.182%) 24.765 16.429 219.013 14.198 21502.725 (0.182%) 2692.001 (0.296%)
that would be used in equation (18) were identified from the noisy simulation dynamic responses in the Morlet wavelet domain. For the ‘‘frame A’’ and ‘‘frame B,’’ the maximum relative errors are 7.827% and 7.033%, respectively. Evaluating whether the variant of the local stiffness is less than 5% is difficult based on the identified modal parameters. For this reason, the difference of local stiffness between ‘‘frame A’’ and
‘‘frame B’’ which were created without a mode shape correction procedure is not very clear.
The proposed procedure is used to create the stiff-ness of ‘‘frame A’’ and the stiffstiff-ness of ‘‘frame B’’ from the noisy dynamic responses. Table 3 also lists the rela-tive errors of the calculated stiffness matrix using the Gram–Schmidt process and equation (18). The modal parameter that is employed in the Gram–Schmidt
process and equation (18) was also identified from the noisy simulation dynamic responses in the Morlet wavelet domain. For ‘‘frame A’’ and ‘‘frame B,’’ the maximum relative errors are 3.780% and 3.932%, respectively. Evaluating whether the variant of the local stiffness exceeds 5% is easy based on the identified modal parameters. Therefore, the difference in local stiffness between ‘‘frame A’’ and ‘‘frame B,’’ which were created by the proposed procedure, is very clear.
Application to a shake table test
To extend the applicability of the proposed approach when using practical measurements, the responses of two eight-story steel frame were measured in shaking table tests. The tests were performed in the laboratory of the National Center for Research on Earthquake Engineering in Taipei City, Taiwan (Figure 5). The first frame, which is structurally regular, was denoted as ‘‘standard.’’ The length, width, and height of the frame were 1.8, 1.2, and 8.5 m (Figure 5), respectively. Lead plates were piled on each floor such that the total mass of the steel frame was approximately 4.519 tons. The
second frame, which is structurally irregular with respect to stiffness and denoted as ‘‘cutoff’’ was identi-cal to ‘‘standard,’’ with the exception that the columns of the first and third stories were constructed of steel plates with cutoff, as shown in Figure 6, and others are constructed with intact steel plates.
Both frames were subjected to base excitations of the 1999 Chi-Chi earthquake, which occurred on 21 September 21 1999 in Chi-Chi, Nantou County, Taiwan, with a 100 gal (1 m/s2) reduced level. Data were recorded at a sampling rate of 200 Hz. The acceleration responses of the base and all floors at t = 5–35 s were employed in the evaluation of modal parameters of the frame. Figure 5 displayed the base excitations and accel-eration responses of all floors in the long-span direction of the ‘‘standard’’ frame, which was subjected to a 100 gal loading of the 1999 Chi-Chi earthquake.
Table 4 summarizes the identified modal parameters of both frames, and Figure 7 shows the identified mode shapes that were corrected by the Gram–Schmidt pro-cess. Figure 6 implies that the columns of the first and third stories, which were constructed by steel plates with cutoff, did not significantly alter the mode shapes. As expectation, the identified frequencies of the ‘‘stan-dard’’ frame are slightly larger than the identified fre-quencies ‘‘cutoff’’ frame. Even in the higher modes, the identified modal damping ratios for different frames were smaller than 2%. A 2% of modal damping is typi-cally adopted in the dynamic analysis of a steel struc-ture in the design process.
Table 5 shows the final identified stiffness matrix of both frames. Variables K(i)(i 2 1) in the stiffness matrix represent the local stiffness of the story between the ith and i 2 1th floors. Table 6 lists the relative errors between the ‘‘standard’’ and the ‘‘cutoff’’ frames. K1gin Table 6 indicates the stiffness values of the structural elements between the first floor and the ground floor, where P8i¼1K1i+ K1g¼ 0, thus, K1gcan be calculated based on K1i(i = 1–8) for each frame.
For the condition when both the ‘‘standard’’ and the ‘‘cutoff’’ frames were subjected to a 100-gal Chi-Chi earthquake loading, the relative errors of the local stiff-ness between ‘‘standard’’ and ‘‘cutoff’’ frames are listed in Table 6. Compared with the ‘‘standard’’ frame, the stiffness values for the second, fourth, fifth, sixth, seventh, and eighth floors of ‘‘cutoff’’ frame do not
Table 4. Identified natural frequencies and damping ratios of both the ‘‘standard’’ and ‘‘cutoff’’ frames.
Frame Mode First Second Third Fourth Fifth Sixth Seventh Eighth
Standard fnðHzÞ 1.052 3.112 5.179 7.123 8.921 10.469 11.725 12.512
jð%Þ 0.636 0.251 0.213 0.176 0.175 0.172 0.179 0.195
Cutoff fnðHzÞ 1.007 3.057 5.089 6.981 8.825 10.407 11.572 12.441
jð%Þ 0.693 0.379 0.301 0.287 0.219 0.216 0.185 0.170
significantly change, but the stiffness values of the first and third floors decrease to 8.864% and 8.008%,
respectively. These values reveal that the first and third stories of the ‘‘cutoff’’ frame were slightly damaged.
Conclusion
This study presented an orthogonalization-based approach for correcting the identified mode shapes that were evaluated from structural dynamic responses in Morlet wavelet domain. The corrected mode shapes and given mass properties will be used to create an accurate stiffness matrix of a structure. This procedure is divided into the following three stages: (1) identifying the modal parameters of a structure from measured seismic responses, (2) assuming that the mass properties were given and correcting the identified mode shapes via the Gram–Schmidt process, and (3) constructing the stiffness from the identified natural frequencies and correcting the mode shapes.
In the modal identification stage, a continuous Morlet wavelet transform was applied to the measured responses; therefore, the state-variable model was recon-structed in the wavelet domain. The modal parameters of the structure were calculated from subspace approach. To reduce the numerical error of the identified mode shapes during the de-noising procedure, the mass proper-ties and Gram–Schmidt process were used to correct the identified mode shapes. The identified natural frequen-cies and corrected mode shapes were employed to con-struct the stiffness matrix of the con-structure.
To validate the proposed procedure, the numerically simulated responses of a six-story shear building model subjected to earthquake input were employed in the iden-tification process. The noise effects on the ability to pre-cisely identify the stiffness matrix were also investigated. The proposed procedure determined stiffness parameters that were more accurate than the stiffness parameters of the procedure without mode shape correction.
The real measured data of two eight-story steel frames, with a length, width, and height of 1.8, 1.2, and 8.5 m, respectively, from a shaking table test are ana-lyzed to demonstrate the applicability of the proposed procedure to real structures. The proposed method suc-cessfully proved that the stiffness values of these two frames differed between the first story and third story. This comprehensive procedure, which was applied to the experimental responses, demonstrates its practical applicability to a real symmetrical building.
Acknowledgements
The authors are grateful for the help provided by The Ministry of Science and Technology of the Republic of China, Taiwan. They also appreciate both the National Center for Research on Earthquake Engineering and J. J. Cloud Corp. for providing shaking table test data and com-munication of the measured vibration, respectively.
Table 5. Identified stiffness matrix of both the ‘‘standard’’ and ‘‘cutoff’’ frames. Stiffness matrix (kN/m) Standard 780.2342 2837.836 43.98267 22.97081 3.552926 2.809886 0.511029 6.375347 2837.836 1722.024 2917.276 53.47203 25.93369 20.04274 0.175858 2.253992 43.98267 2917.276 1722.219 2912.034 56.94597 23.09068 20.37461 4.029386 22.97081 53.47203 2912.034 1719.429 2908.316 53.59368 23.69292 4.611722 3.552926 25.93369 56.94597 2908.316 1700.953 2901.942 56.85761 1.024018 2.809886 20.04274 23.09068 53.59368 2901.942 1706.639 2911.585 59.29386 0.511029 0.175858 20.37461 23.69292 56.85761 2911.585 1700.403 2896.087 6.375347 2.253992 4.029386 4.611722 1.024018 59.29386 2896.087 1693.801 Cutoff 772.7837 2830.678 47.54845 22.44991 2.546618 2.228547 2.19979 1.452748 2830.678 1707.279 2917.123 53.07994 23.74075 0.63243 21.50896 20.2842 47.54845 2917.123 1723.348 2912.251 55.49978 22.69367 0.541615 20.81442 22.44991 53.07994 2912.251 1721.041 2909.132 53.36249 22.59622 20.40708 2.546618 23.74075 55.49978 2909.132 1703.184 2898.932 53.25404 24.66255 2.228547 0.63243 22.69367 53.36249 2898.932 1635.668 2838.581 50.38481 2.19979 21.50896 0.541615 22.59622 53.25404 2838.581 1631.751 2892.115 1.452748 20.2842 20.81442 20.40708 24.66255 50.38481 2892.115 1590.285
Note: Because of shear building, the bold values are represented to results focused on bi-diagonal elements of matrix.
Table 6. Relative errors of identified local stiffness parameters between the ‘‘standard’’ and ‘‘cutoff’’ frames.
K87 K76 K65 K54 K43 K32 K21 K1g
Relative error (%) 0.854 0.017 0.024 0.090 0.334 8.008 0.443 8.864
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is financially supported by The Ministry of Science and Technology of the Republic of China, Taiwan, contract no. MOST 105-2622-M-492-001-CC2.
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