# Modal Identification of Train Passenger Seats Based on Dynamic Tests and Output-Only Techniques
## 2 Modal Identification Methodologies
### Enhanced Frequency Domain Decomposition (EFDD)  
The EFDD method is an upgrade of the Frequency Domain Decomposition (FDD) method, proposed by [PERSON] et al. [44], and allows an accurate estimation not only of the natural frequencies, but also mode shapes and damping coefficients. This method is based on the Singular Value Decomposition (SVD) of the Power Spectral Density (PSD) matrix, \(S_{yy}(\omega)\), which can be obtained by the Fourier transform of the time-domain autocorrelation functions. Thus, the main diagonal of the PSD matrix is composed by the auto-spectral density functions, and the out-of-diagonal terms represent the cross-spectral density functions. The SVD decomposition of the PSD matrix can be summarised by the following equation:  
\[S_{yy}(\omega)=\mathrm{U}\Sigma\mathrm{V}^{H} \tag{2}\]  
where \(\Sigma\) is a diagonal matrix containing the singular values, and U and V are similar matrices containing the singular vectors. [PERSON] et al. [44] demonstrated that, for structures with low values of damping and subject to white noise excitations, the matrix of singular values corresponds to the power spectral density functions of a set of one degree-of-freedom (DOF) oscillators, whose singular values and vectors coincide with those of the real structure. If the white noise excitation assumption is not fulfilled, the method can still be applied, and the obtained results are approximated. Those results are considered reliable and enhanced compared to those obtained on the classical peak-picking method [44; 45]. Then, the SVD allows for isolating the contribution of each of the mode vibrations as a 1-DOF system. The main steps (1 to 4) of the EFDD method are summarised  
Figure 1: Typical transmissibility curve on conventional train seats (red dot represents the resonant frequency and blue dot represents the antiresonance frequency) (adapted from [42]). in Figure 2 and involve mathematical operations first in the time domain and then in the frequency domain. In the case of well-separated modes, only the first singular value will have significant information. However, in the case of closely spaced modes, one will be identified in the first singular value curve, while the others will be identified in the second- or higher-order singular values curves. The natural frequencies and damping coefficients are derived from the time domain autocorrelation function, which is calculated by the Inverse Fourier Transform (IFFT) of the singular values around each resonance peak. The comparison between vectors can be performed employing a scalar index denominated as the Model Assurance Criterion (MAC), which investigates the correlation between two modal configurations [44; 45]. By averaging the singular vectors of the neighbourhood points of the resonance frequencies with MAC higher than 0.8, and weighting by the value of the corresponding singular values, the estimation of modal configurations associated with each natural frequency is calculated. This way, the selection of the SVD points is based on the similarity between their singular vectors calculated through the MAC parameter. Respecting the assumptions of the method, these functions will be proportional to the free vibration decay of the 1-DOF oscillator. Thus, it is possible to estimate the natural frequency based on a dedicated counting of vibration cycles and damping coefficient, by applying the logarithm decrement method, which involves adjusting the exponential term (\(e^{-\zeta\omega_{n}t}\)) to the free vibration decay movement [44; 46; 47]. Regarding the present work, pairing comfort and standard seat modes can be conducted through the MAC parameter, which, as stated, measures the linear correlation between modal vectors through their mean square deviation [48]. The MAC parameter is a scalar, varying between 0 and 1, whose unit value indicates that the modes are coincident or linearly dependent [49; 50].