# Operational Modal Analysis and Safety Assessment of a Historical Masonry Bell Tower
## 3 Dynamical Identification  
An Ambient Vibration Testing program was carried out on the tower to investigate the dynamical behavior under environmental sources. This approach is largely used to investigate the dynamical behavior of masonry bell towers [19; 20]. The experimental setup consisted of 4 triaxial and 2 biaxial force-balanced accelerometers, with 2.5 V/g sensitivity and bandwidth 0-200 Hz, range \(\pm\)2g, damping 0.707. The triaxial sensors were kept in the same position in all stages of the test: two sensors (T2, T3) were placed at 8.75 m height and the remaining two (T1 and T4) were placed at 12.25 m height (Figure 6). The biaxial sensors (B5, B6) were alternately placed at 5.25 m (setup I) and 8.75 m (setup II), and 15.00 and 17.80 m, respectively (setup III). This multi-setup testing program allowed for the estimation of the modal shapes along the height of the tower, according to the procedure shown in the following and implemented with in-house software developed at the University of Campania \"[PERSON]\". Each setup acquisition lasted 900 s; the first setup was taken at 200 Hz sampling frequency, while setups II and III were applied at 100 Hz to reduce the amount of data. All signals were then corrected to zero average and decimation was applied up to a Nyquist frequency equal to 25 Hz (double decimation for setup I, single decimation for setup II and III). The Fast Fourier Transforms (FFTs) of the signals (Figure 7) consistently show two peaks in two orthogonal directions at about 2.6 Hz and 3.1 Hz. Another lower peak is also recognized for some signals at about 5.9 Hz, while strong power density is also observed for accelerometer B6 in setup III for many frequencies between 6 and 12 Hz. Since this sensor was placed on the top of a wall not significantly constrained out of its plane (Figure 6), it seems clear that these frequencies are related to local modes of the top floor walls. Frequencies and mode shapes were obtained by means of Frequency Domain Decomposition (FDD) [21], applying the procedure proposed by [PERSON] and [PERSON] [22] to merge multiple setups having reference and roving sensors. In Figure 8, the first singular value (SVD 1) of the merged signals is plotted against frequencies. Three clear peaks are recognizable at 2.59 Hz, 3.08 Hz and 4.15 Hz. Some help in recognizing real modes when using FDD is offered by the analysis of the Modal Complexity Factor (MCF) for a given mode shape \(\psi_{r}\), defined as in [23]  
\[MCF=1-\frac{\left(S_{xx}-S_{yy}\right)^{2}+4S_{xy}^{2}}{\left(S_{xx}+S_{yy} \right)^{2}} \tag{1}\]  
Figure 5: Cracking phenomenon observed on the east facade: (**a**) photo of the external wall and inspection with thermal imager from ground (**b**) and with drone (**c**). The vertical crack is highlighted in white. where  
\[S_{xx}=Re\{\psi_{r}\}^{T}Re\{\psi_{r}\},S_{yy}=Im\{\psi_{r}\}^{T}Im\{\psi_{r}\},S_ {xy}=Re\{\psi_{r}\}^{T}Im\{\psi_{r}\} \tag{2}\]  
The MCF is a measure of how complex a mode is, and ranges from 0 (real mode) to 1 (imaginary mode). Since at each frequency \(f\) of the SVD plot it is possible to estimate a corresponding modal shape \(\psi_{r}(f)\), \(MCF(\psi_{r})\) can be seen as a function of \(f\). In Figure 8, the value 1-\(MCF(f)\) is plotted at each frequency level: it is possible to see that the maximum values of 1-\(MCF\), i.e., real modes, are observed corresponding to the SVD peaks. Real modes are also observed at two additional frequencies equal to 5.81 Hz and 6.15 Hz where PSD peaks are not evident. Thanks to this approach, these additional modal shapes and frequencies can be identified as modes of the structure. The five identified modes are depicted in Figure 9. While the first two modes are translational modes, mainly along the two principal directions, with little effect of the interaction with the church, the fourth mode rather clearly appears as a torsional mode.