Study Of Data-driven Operational Modal Analysis Methods

Experimental modal analysis is of great value to mechanical engineering and has made great success in many fields such as structural dynamics analysis, fault diagnosis, online health monitoring, etc. One of the most remarkable field is operational modal analysis (OMA), in which the modal parameters of system can be identified only by using system’s response under ambient excitation. OMA has been applied in modal analysis of many large engineering structures because it has advantages that it is fast, cheap and can be conducted without isolating the structures. Blind source separation (BSS) is a technique that recovers the sources by utilizing the mixtures captured from the sensors without any knowledge of the sources and the mixing procedure. BSS tech-niques have achieved great development in radar and communication fields since1990s. The procedure of modal response separation coincides with the problem description of BSS, so BSS techniques can be used for modal parameter identification. For under-determined BSS problems in which the number of sensors is less than the number of sources, the sparsity of the signal in some transformed domain can be utilized to recover the sources. This paper studied the conventional modal identification method-stochastic subspace identification method, in the meantime, introduced the BSS techniques and sparse component analysis methods to OMA field. Then a single source point based sparse component analysis method is proposed to carry out the modal identification problem, simulations and experiments verified the validity of the proposed method. The details are as follows:1. On one hand, the modal theory of single degree of freedom (DOF) system, multi-degree of freedom system, undamped system, proportional viscous damped sys-tem and general damped system is reviewed, the methods to achieve the modal response under these different conditions are analyzed and the solutions are ob-tained; on the other hand, the most conventional modal identification method-stochastic subspace identification (SSI) method is introduced, including the basic theory and the solving procedure. These contents provide a theoretical founda-tion and a comparative benchmark to the modal parameter identification methods proposed in the following parts.2. Blind source separation (BSS) technique and its application in modal identifica-tion are studied in details. Two of the most important algorithms of BSS, indepen-dent component analysis (ICA) and second order blind identification (SOBI) are analyzed in terms of model and assumption, existence and uncertainty of the solu-tion, and the solving procedure. In addition, the reasons why BSS technique can be used to achieve modal identification are analyzed. Some modal identification examples using ICA and SOBI are presented.3. The theory and algorithm of sparse component analysis are introduced to solve the underdetermined BSS (UBSS) problem, in which the number of sensors is less than the number of sources and general BSS techniques cannot achieve good result. First, the sparsity of the signal in time domain or some other transformed domain is introduced. Then, typical algorithms of mixing matrix estimation and modal response recovery are presented under the two-step SCA framework. Two types of mixing matrix estimation are introduced, one is based on the hyperplanes formed by the coefficient vectors in the transformed domain, the other is based on the hyperlines. Representative algorithms of both types are presented, that is, mixing matrix estimation algorithm based on hierarchical Hough transform and robust K-hyperline clustering algorithm. Three different l1-norm minimization algorithms, basis pursuit (BP), basis pursuit denoising (BPDN) and Lasso are presented to recover the sparse sources. The optimizing objective function of these three methods are different and can be used in different conditions. BP algorithm performs well in noiseless case but cannot obtain good result in noisy case. BPDN and Lasso can achieve better performance than BP in noisy case by relaxing the restriction conditions defined in BP. Then the concept of single source point (SSP) is introduced and a method is proposed to improve the estimation accuracy of the mixing matrix by detecting all the SSPs in the data space.4. An algorithm named SSP-based-SCA is proposed to conduct modal identifica-tion, the algorithm flow chart is presented. Numerical simulation and experi-ment results verified the effectiveness of the proposed algorithm. A3-dof spring mass system is used to conduct the simulation. Well-separated (WS) modes and closely-spaced (CS) modes are obtained by modifying the mass matrix, stiffness matrix and the damping matrix. Under WS and CS conditions, the mixing matrix can be estimated for both determined case and underdetermined case. The natural frequencies, mode shapes and damping ratios are obtained by using-norm min-imization and S-DOF modal parameter fitting method. The high modal assurance criterion (MAC) values and comparability of the estimated modal parameters and the theoretical ones validate the effectiveness of the proposed algorithm. The ro-bustness of the algorithm is also studied in the presence of noise. The experiment is carried out to identifying the modal parameters of a steel cantilever beam. The vibration displacements of different markers which sticked to the beam are cap-tured by a high-speed camera and extracted by using image processing techniques. Then the natural frequencies and damping ratios are estimated by using the pro-posed SSP-based-SCA algorithm. The results are compared with the theoretical ones and the results of SSI method. The proposed algorithm obtained comparable results with the SSI method, although SSP-based-SCA using data from only two sensors and SSI using all the21sensors, which indicates the superiority of the proposed algorithm in modal identification.