Underdetermined blind modal identification of structures by earthquake and ambient vibration measurements via sparse component analysis

Sparse component analysis (SCA) approach was adopted to handle underdetermined blind modal identification of structures, where the number of sensors is less than the number of active modes. To exploit the sparsity of structural responses in time–frequency domain, Short Time Fourier Transform (STFT) was used in this study. The proposed SCA-based approach has two main stages: modal matrix estimation and modal displacement estimation. In the first stage, hierarchical clustering algorithm was used to estimate the modal matrix. The clustering algorithm was preceded by a preprocessing step to select the points in time–frequency domain that only one mode makes contribution in the responses. These points were fed to the clustering algorithm as an input. Performing this analysis enhanced the modal matrix estimation accuracy and reduced the computational cost while conducting clustering analysis. Having estimated mixing matrix, the complex-valued modal responses in the transformed domain were recovered via Smoothed zero-norm (SL-0) algorithm. In a broad sense, using the SL-0 algorithm permits researchers to use any kind of transform in seeking sparsity, regardless of obtaining real-valued or complex-valued signals in transformed domain. Natural frequencies and damping ratios were extracted from the recovered modal responses. Performance of the proposed method was investigated using a synthetic example and a benchmark structure with earthquake and ambient excitation, respectively.