A frequency domain decomposition method is formulated to extract modal information of multi-degree-of-freedom systems. Previously, state-variable modal decomposition (SVMD), smooth orthogonal decomposition (SOD), and also proper orthogonal decomposition (POD), have used time domain signals for modal decomposition. Analysis shows that similar formulas can be obtained in the frequency domain, with eigenvalues as the resonance frequencies, and the eigenvectors as the inverse of the linear normal modes (LNMs). The method can be expressed in symmetric or non-symmetric eigenvalue problem format. The symmetric format resembles SOD in the frequency domain, while the non-symmetric method is similar to SVMD. One of the advantages of a frequency-domain method is that signals can be divided into frequency segments to find out the existence of natural frequencies and normal modes within a limited frequency range, such that the noise influence can be reduced. Furthermore, the obtained eigenvectors can be used to check if they actually are normal modes or noises. The method was examined by a six degree-of-freedom mass-spring system under free vibration and random excitation conditions.

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