# A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model
## 3 Simulated Data Analysis
### Lattice Tower Example  
The second simulated example used to validate the ML approach discussed in Section 2.2 consists of a finite element model which is illustrated in Figure 11. The example consists of a lattice tower structure constituted by two segments of equal height and by variable equilateral triangular sections. The lower section is scaled with regard to the upper one by a factor of 2. This model is composed by beam elements with 6 DOFs per node. The nodes of the foundations are clamped and the others have 3 DOFs: two translations in \(x\) and \(y\) and one rotation around \(z\)-axis; and the remaining DOFs are set equal to zero. These settings result in a FE model with a total of \(N_{m}=18\) DOFs. The symmetry of the structure was slightly broken by defining different geometry and materials to the column elements in order to simulate the behaviour of tower-like structures, which normally present very closely spaced of bending modes. The real mode shapes of the tower structure are shown in Table 6. The first 6 modal configurations and the corresponding modal parameters of the FE model of the lattice tower are presented in Figure 12 and Table 7, respectively. The structural damping, on the other hand, is modelled as the special case of proportional damping by setting the damping coefficients of all modes equal to 1%. The structure was excited, independently, at node 8 in \(x\) and \(y\)-direction and at node 9 in \(y\)-direction with white noise inputs. The simulated responses were measured in acceleration at the 6 DOFs indicated in Figure 13. By assuming that the two triangular sections behave as rigid diaphragms, the measured outputs are enough to yield the modal configurations of the structure. The FRF matrix with 6 rows and 3 columns was calculated in the frequency range of 0-10 Hz with a resolution of 10 mHz. Afterwards, a colored noise was introduced in the FRF matrix with a standard deviation of 5%. Figure 11: Finite element model of the tower example used in the simulated analysis. Figure 12: First six mode shapes of the tower structure: 1 st bending mode in Y direction (BY1) (**a**), 1 st bending mode in X direction (BX1) (**b**), 1 st torsional mode (T1) (**c**), 2 nd bending mode in Y direction (BY2) (**d**), 2 st bending mode in X direction (BX2) (**e**) and 2 nd torsional mode (T2) (**f**). The noise was added to the FRFs following the same procedure described in the previous example. The exact and noisy element(1,1) of the FRF matrix, the corresponding exact standard deviation of the noise as well as the exact natural frequencies are shown in Figure 14. A set of 1000 FRFs with 1000 frequency lines contaminated with noise was generated to perform Monte Carlo simulations in order to assess the efficiency of the proposed MLE-PMM when dealing with closely spaced modes. In the first step of the identification of each dataset, a stabilisation diagram was constructed with the LSCF technique from the measured FRFs and the approach based on Hierarchical Clustering described in References [29; 30] was subsequently used to automatically identify the physical modal properties. A typical stabilization diagram constructed by identifying models with order ranging from 30 to 60 is represented in Figure 15. Next, the FRF matrix was synthesized from the estimates obtained with the LSCF technique was then used as starting guess by the MLE-PMM. Finally, a total of 20 iterations were performed, both to optimize the estimates obtained with LSCF in the first step and to compute confidence intervals of the optimized estimates. Differently from the previous application example, a higher number of iterations was used in these simulations to secure the convergence of the cost function \(l(\Theta)\). This difference is explained by the fact that the system being optimized in this particular simulation example comprises vibration modes with closely spaced natural frequencies and, therefore,