## 5 Concluding Remarks  
A novel approach that integrates the subspace method with the SWPT was developed for identifying the modal parameters in structures, encompassing both linear and nonlinear behaviors. The introduction of a Gaussian weighting function, coupled with a moving window technique, enhances the approach's accuracy in identifying time-varying modal parameters. Leveraging the Meyer wavelet as a band-pass filter and its corresponding scale function as a low-pass filter constructs the basis functions for SWPT subspaces. The fine filtering properties inherent in the SWPT not only facilitate mode discrimination but also enable the identification of numerous modes from a limited number of measured degrees of freedom. The validation of the proposed approach commenced with the processing of numerically simulated acceleration responses from six-story shear buildings, exhibiting constant, periodically time-varying, or sharply time-varying stiffness and modal damping ratios. The results underscore the effectiveness and robustness of the method in accurately determining time-varying modal parameters, even when base excitation and the responses of only two floors, with a 10% NSR, were utilized. Compared with true values, the identified frequencies and modal damping ratios for all six modes in the time-invariant shear building exhibited maximum differences of 0.9% and 5%, respectively. For the periodically time-varying shear building, the maximum differences in frequency were less than 1% for all six modes, with maximum modal damping ratio differences less than 15% for the first five modes and 20% for the sixth mode. In the case of the sharply time-varying shear building, the identified frequencies for the first four modes deviated by less than 1%, while the  
Figure 16: Identified time-varying modal frequencies and damping ratios for the steel frame. differences were less than 2% for the fifth and sixth modes. Correspondingly, the identified modal damping ratios differed from the true values by less than 15%. In extending the application of the proposed approach, it was employed to process measured linear transverse velocity responses and support acceleration excitations from a two-span cable-stayed bridge during an earthquake, as well as nonlinear acceleration responses from the two floors of a five-story steel frame under a shaking table test. The identified modal parameters for the first eight transverse modes of the bridge demonstrated admirable agreement with those obtained from finite element analysis and ambient vibration measurements. Moreover, the identified time-varying modal parameters for the steel frame exhibited favorable agreement with published findings obtained using a time-varying autoregressive model with exogenous input with the continuous Cauchy wavelet transform. Significant changes in the natural frequencies and damping ratios indicated earthquake-induced damage to the frame. Overall, the proposed approach show-cased its versatility and efficacy in accurately identifying modal parameters for structures undergoing both linear and nonlinear dynamic conditions. Conceptualization, W.-C.S. and C.-S.H. ; methodology, W.-C.S. and C.-S.H. ; software, W.-C.S. ; validation, W.-C.S., L.-J.C. and C.-S.H. ; formal analysis, L.-J.C. and C.-S.H. ; investigation, W.-C.S., L.-J.C. and C.-S.H. ; resources, C.-S.H. ; data curation, L.-J.C.; writing--original draft preparation, C.-S.H. ; writing--review and editing, C.-S.H. ; supervision, C.-S.H. ; project administration, C.-S.H. ; funding acquisition, C.-S.H. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Science and Technology Council, Taiwan, through research grant number MOST 111-2221-E-A49-015-MY2. Not applicable. Data available statementData is contained within the article. The work reported herein was supported by the National Science and Technology Council, Taiwan, through research grant number MOST 111-2221-E-A49-015-MY2. This support is gratefully acknowledged.