# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 3 Simulated Data Analysis
### Five-DOF System  
The first example used to validate the ML-based approach discussed in Section 2.2 is illustrated in Figure 2. This system was used in Reference [28] to compare different modal parameter estimation techniques in terms of their sensitivity to statistical errors. It is composed by 5 masses supported by cantilever beams and connected among themselves by arch springs. The exact natural frequencies, damping ratios and modal masses of the system are given in Table 2, whereas the real modes are shown in Table 3. These properties were used to generate the FRFs used in simulated EMA. The system was excited by a white Gaussian noise at masses 1 and 2 and the responses were measured at all DOFs, resulting in FRF matrix with two columns and five rows. These FRFs were calculated in the frequency range of 0-80 Hz with a resolution of 0.1 Hz. Afterwards, a colored noise was introduced in the FRF matrix with a standard deviation of 10%. The noise was added to the real and imaginary parts independently and was calculated as a percentage of the absolute value of the FRF at each frequency line. This was achieved by adding a complex random number to the FRF at each frequency line. This number was computed so that its amplitude is a random number of a normal distribution and its phase is an uniform random number between 0 and 2\(\pi\)[13]. The exact and noisy element(1,1) of the FRF matrix and the corresponding \"exact\" standard deviation of the noise are shown in Figure 3a. A set of 5000 FRFs with 800 frequency lines contaminated with noise was generated to perform Monte Carlo simulations in order to assess the efficiency of the proposed MLE-PMM. The modal parameters of each dataset were identified with the LSCF and LSFD estimators and then used as starting values to be optimized by the MLE-PMM algorithm. The identification of each dataset was performed using the full frequency band, that is, with no upper and lower residual terms. Though 1000 Monte Carlo simulations would be more than enough to assess the robustness of the MLE-PMM, a higher number of realizations were performed, in this particular \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & \(f_{n}\) **[Hz]** & \(\xi_{n}\) **[\%]** & \(m_{t}\) **[Kg]** \\ \hline
1 & 26.06 & 2 & 2.52 \\
2 & 36.84 & 2 & 2.97 \\
3 & 51.47 & 2 & 0.90 \\
4 & 56.21 & 2 & 1.09 \\
5 & 62.60 & 2 & 1.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Eigenfrequencies, damping ratios and modal masses of the five-DOF system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**DOF/Mode** & **1** & **2** & **3** & **4** & **5** \\ \hline
1 & 0.7147 & 1.0000 & \(-0.0911\) & \(-0.9230\) & \(-0.6083\) \\
2 & 0.7166 & 0.9999 & \(-0.1493\) & \(1.0000\) & \(-0.1937\) \\
3 & 0.7981 & 0.2257 & 0.1554 & \(-0.1518\) & \(1.0000\) \\
4 & 0.8518 & \(-0.5166\) & 1.0000 & 0.1231 & \(-0.3936\) \\
5 & 1.0000 & \(-0.8590\) & \(-0.5860\) & 0.0196 & \(-0.2041\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Real modes of the five-DOF system. Figure 2: Five-DOF system connected with arch springs [28]. simulation example, to also verify its efficiency in predicting the 95% elliptical confidence regions. A typical stabilization diagram constructed with the LSCF method from the noisy FRF is shown in Figure 3b. As a final step of the identification process, 10 iterations of the MLE-PMM were performed to optimize the modal parameters of each Monte Carlo dataset and estimate their sample standard deviations. The variation of the LSCF and MLE-PMM estimates for the 3 rd and 5 th natural frequencies and damping ratios over the Monte Carlo simulations are shown in Figures 4 and 5, respectively. Figure 4: Monte Carlo simulation results for the 3 rd identified natural frequency (**a**) and damping ratio (**b**): exact quantity (solid black line), LSCF estimates (blue dots) and MLE-PMM estimates after 10 iterations (red dots).