## 4 Application
### Model Setup
We slightly modified Equation (7) by setting the AMKE obtained from the MOP method as the reference value, which was divided by the AMKE obtained from other methods, as per Equation (16). If the value obtained after the aforementioned division was greater than 1, the results of this sensor placement method were worse than those of the MOP method for \(f_{3}\). \[f_{3}=\frac{\sum_{i=1}^{s}{AMKE_{i}}^{MOP}}{\sum_{i=1}^{s}{AMKE_{i}}}, \tag{16}\]  
Figure 4: Three-story two-bay experimental frame structure: (**a**) Physical model, (**b**) finite-element model with sectional profile rendering, and (**c**) finite-element mesh with nodes (red dots) and fixed boundary conditions (white triangles). Subsequently, the FEM results were input into OSP algorithms, which output the eight best locations, as listed in Table 5. The sensor positions obtained using three methods are illustrated in Figure 6, where Node 29 is the excitation node, green arrows represent x-direction sensors, and magenta arrows represent y-direction sensors. Node 25 in the MOP method was collocated by two sensors, which were not limited by the MOP algorithm. These sensors were placed on the selected nodes of the physical model according to Table 5, and we performed vibration tests for each selection. Figure 7 illustrates the acceleration PSDs of each sensor when using the MOP method. This figure spans a frequency range from 0 to 130 Hz, which covers the first 10 modes of the frame structure. Note that the PSD of No. 8 sensor at Node 34 shows the greatest amplitude and cleanest signals among the eight deployed sensors. The primary reason is attributed to the nature of the operational modal analysis method in the present study when performing singular value decomposition for identifying the PSDs. Either the sensor location near the free ends or close to excitation node may also affect the signal amplitudes, but is of secondary importance. Nevertheless, only peak frequencies without the need of signal amplitude of the PSDs are of interest for determining the modal frequencies. The peak frequencies are obtained (See examples indicated by the pink-dashed vertical lines in the PSDs of No. 2 and 6 sensors) and listed in Table 4, and compared with the values predicted using the FEM. The symmetry of modes was also resolved; however, the frequencies were on average 14% lower than those obtained through the FEM. The signals from different sensors did not indicate significant differences in modal frequencies. In terms of the characteristics of frequency response, the modes along the x-direction and the torsional modes were well excited. Distinctive peaks were observed, which proved that rich structural information was included in the measurements at the OSP locations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **X-Direction** & **Y-Direction** & \(\mathbf{f_{1}}\) & \(\mathbf{f_{2}}\) & \(\mathbf{f_{3}}\) \\ \hline MOP & 5, 16, 25, 31 & 6, 15, 25, 34 & 0.2633 & 1.8006 & 1 \\ FSSP & 7, 12, 26, 28 & 4, 6, 18, 22 & 0.1475 & 1.9335 & 2.2136 \\ Random & 11, 19, 23, 31 & 5, 9, 17, 21 & 0.4197 & 2.7686 & 0.9078 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Sensor placement and performance evaluation for the three OSP methods. Figure 6: Three different OSP results along the x- and y-directions: The (**a**) multiobjective optimization problem (MOP); (**b**) forward sequential sensor placement (FSSP); and (**c**) random placement methods. The signals were further processed to evaluate the mode shape matrices of sensing DOFs through singular value decomposition. Three objective functions were computed with their corresponding matrices (Table 5). The best result for \(f_{1}\) was obtained with the FSSP method because MinMAC was the only objective in FSSP. For the other two objectives, the MOP method outperformed the FSSP method. Surprisingly, the random placement method achieved optimal performance for \(f_{3}\); however, for \(f_{1}\) and \(f_{2}\), its performance levels were notably inferior to those of the other two methods.