## 4 Application  
### Model Setup  
To further test the effectiveness of OSP in SHM for complex structures, we conducted an experiment on a three-story aluminum frame structure, illustrated in Figure 4a with a 50 cm ruler for scale, as an attempt to mimic offshore jacket structures. The structure was composed of four square frames and four slanted legs. The sides of the square structure had dimensions of 0.5, 0.45, 0.4, and 0.35 m from bottom to top. The spacing of each layer from bottom to top was 0.5 m; thus, the height of the structure was 1.5 m. The material used was aluminum, and all joints were connected with bolts. The frame was constructed with an angle section profile (\(25\times 25\times 2\) mm\({}^{3}\)). The bottom layer was fixed on an optical table to prevent horizontal motion and vibration. The mechanical properties of aluminum are as follows: [PERSON]'s modulus = 68.9 GPa, Poisson's ratio = 0.33, and density = 2850 kg/m\({}^{3}\). A corresponding finite-element model was developed for OSP, as illustrated in Figure 4b,c. The model had 232 nodes, 244 beam elements, and 1176 DOFs. The same angle section profile was applied on all elements. Modal analysis was performed with fixed constraints at the four bottom corners. Because the structure was vertically symmetrical, its natural frequencies were expected to be the same in horizontal directions. At the beginning of modal analysis, by using the IIRS technique, only 70 translational DOFs (along the x-direction and y-direction) were considered for OSP. The results of modal analysis were fed to an optimization process, which output the eight most effective locations for sensors. MOP optimization was conducted for the same configuration as that of Canton Tower to determine the optimal locations for sensors. In addition, we used the traditional forward sequential sensor placement (FSSP) algorithm [7] to discover different optimization solutions. The FSSP algorithm \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & \(\mathbf{f_{1}}\) & \(\mathbf{f_{2}}\) & \(\mathbf{f_{3}}\) & \(\mu_{1}\) & \(\mu_{2}\) & \(\mu_{3}\) & **CD** \\ \hline MOP-No. 28 & 3.1799 & 6.7150 & 1.9181 & 0.9934 & 0.0001 & 0.9624 & 1.4087 \\ Reference [27] & 6.8546 & 6.1141 & 3.1888 & - & - & - & - \\ Reference [6] (0.1) & 4.7892 & 7.6955 & 2.7343 & - & - & - & - \\ Reference [6] (0.2) & 2.1766 & 6.0536 & 1.9181 & 1.0000 & 0.0247 & 0.6191 & 1.1764 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the solutions obtained with different methods when using the proposed objective function and closeness. minimizes the sum of the least squares of the off-diagonal elements of a MAC matrix (i.e., the objective is MinMAC). The third method involved random placement by excluding the nodes selected by the FSSP algorithm. A vibration test was then performed on the physical model. The structure was instrumented with eight single-axis accelerometers. The OSP locations were obtained using the MOP, FSSP, and random placement methods. Four sensors were oriented in the x-direction, and the other four were oriented in the y-direction. A random impulse signal was applied at the upper corner of the frame (Node 29) on the structure along the y-direction. The sample rate of data acquisition was 25,600 Hz, and data were recorded for 10 s. The modal properties of the structure were identified using frequency domain decomposition on the basis of all the nodal measurements. Then, vibration signals were processed to obtain the power spectral density (PSD) matrix. The vibration signals were also compared with modal frequencies from FEM analysis. Moreover, the relative displacements between sensors were computed at the first six modal frequencies to obtain a mode shape matrix for calculating objective functions. Because we did not measure all the nodes for potential sensor locations, \(f_{3}\) could not be calculated.