144 M. Tarpø et al. 13.2 Theory 13.2.1 Operational Modal Analysis and the Frequency Domain Decomposition Operational Modal Analysis is an experimental technique that only use the system response for modal identification. We assume that the system is linear and time invariant and the excitation of the system has a high level of white Gaussian noise characteristics [13]. For this paper, we will use the Frequency Domain Decomposition as identification technique [14]. The spectral density matrix is defined as [13] Gyy.!/ DF E y.t/yT.t C / (13.1) where F denotes the Fourier transformation, E is the expectation operator, y.t/ 2 N 1 is the system response and N denotes the number degrees-of-freedom. The Modal Decomposition states that the response of the linear system, y.t/, is a linear combination of the mode shape, B 2 N M, and the modal coordinates, q.t/ 2 M 1. Here Mis the number of modes. y.t/ DBq.t/ (13.2) This is also called a linear transformation. We insert the modal decomposition into the spectral density matrix. Gyy.!/ DBGqq.!/BH (13.3) where Gqq.!/ 2 M M is the spectral density matrix of the modal coordinates, q.t/. Next, we preform a singular value decomposition of the spectral density matrix. Gyy.!/ DU.!/S.!/U.!/ H (13.4) We see a similarity between Eqs. (13.3) and (13.4). The Frequency Domain Decomposition is based on the approximation that the singular vectors are analogous to the mode shapes and the singular values are similar to the spectral density matrix for the modal coordinates. However the technique is a approximation [13]. For one thing, we assume that the modal coordinates are uncorrelated, which is only the case for a linear system. 13.2.2 Strain Estimation In this section, we will show how to estimate the strain response in any point of a structure using the Modal Expansion technique. We want to expand the experimental mode shapes so we create a linear relationship between the modal matrices using the Local Corresponding principle [15]. OADBaP (13.5) whereAis the experimental modal matrix, OAis the smoothen experimental modal matrix, Ba is a reduced modal matrix from a highly correlated finite element model and Pis the transformation matrix. We expand the experimental modal matrix by replacing the reduced finite element modal matrix with the full modal matrix, B. OAfull DBP (13.6) where OAfull is the expanded experimental modal matrix. Next, we want to find the response of the entire structure so we estimate the modal coordinates from Eq. (13.2). Oq.t/ DACy.t/ (13.7)
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