Chapter17 System Identification of Structures with Modal Interference Chang-Sheng Lin Abstract Previous studies show that when a system has repeated modes in frequency domain, the multiple-input and multiple-output (MIMO) concept can be applied to effectively identify the modal parameters. In this study, the MIMO concept is extended to the time domain for system identification of systems with modal interference. When performing a modal analysis of a structural system with close modes or high damping, the results of identification may be poor due to modal interference, and the estimation of system order is important to the effectiveness of modal identification. By introducing the concept of the singular value decomposition (SVD) in Eigensystem Realization Algorithm (ERA), and simplifying the identification process of ERA, the system order will be estimated effectively even for a system having close (even repeated) modes. In addition, the correlation matrix is also used to perform SVD, and then directly identify modal parameters, i.e., omitting the additional process of constructing the generalized Hankel matrix, which provides a more efficient method of identification of modal parameter. Also, in this paper, the SVD algorithm is introduced to the identification process of Ibrahim Time-Domain Method. The order of system matrix is efficiently determined, and the modal identification of a system with close modes through the ITD method can then be well implemented. Keywords Singular value decomposition (SVD) • Eigensystem realization algorithm (ERA) • Ibrahim time-domain method (ITD) 17.1 Introduction Modal estimation techniques have been extensively developed in the past, including the frequency-domain methods and the time-domain methods. The time-domain methods manipulate signals in time domain and sometimes work under the consideration that only output response is available, so that we can usually take advantage of time-domain methods in performing modal estimation form practical response data. The frequency-domain methods often require evaluation of the frequency response functions (FRFs) from the measured input and output data, and then estimate modal parameters of a system [1]. Based on the Prony’s theory, Brown et al. developed the least square complex exponential algorithm (LSCE) [2] using a squared output matrix constructed by multichannel impulse response functions, which is a well-known technique in conventional modal analysis yielding global estimates of residues and poles. In 1982, Vold and Rocklin further proposed poly reference complex exponent method (PRCE) [3] to perform modal identification for the case that one of the modes may not be present in the response data. In 1985, Juang and Pappa [4] proposed the Eigensystem Realization Algorithm (ERA) using the impulse response or the free vibration response of the system to construct the Hankel matrix, which is an augmented matrix containing Markov parameters, for reducing the effect of noise, and making the parameters estimation more accurate. Subsequently, they also proposed a modification of ERA, generally known as ERA/DC (Eigensystem Realization Algorithm with Data Correlation) [5, 6]. ERA/DC uses data correlations to reduce the noise effect in the process of modal-parameter identification. In addition, when performing a modal analysis of a structural system, the results of identification may be poor due to modal interference, which might even introduce the problem of identifiability. Modal interference refers to the phenomenon that vibration energy of each mode of a system may overlap with other modes within certain frequency range. This phenomenon usually results in the difficulty of identification of modal parameters, especially when the identification is performed in the frequency domain. The causes of modal interference may include closely spaced modal frequencies, high damping ratios, and large damping non-proportionality [7]. The influences of modal interference to each cause are difference, and the well-identified modal parameters of a system may not be obtained due to each cause. C.-S. Lin ( ) Department of Vehicle Engineering, National Pingtung University of Science and Technology, Neipu, Taiwan e-mail: changsheng@mail.npust.edu.tw © The Society for Experimental Mechanics, Inc. 2018 A. Baldi et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62899-8_17 115
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