## 4 Optimization
### Updating Parameter Limits and Objective Function  
As an important part of the optimization process, the limits of the updating parameters need to be carefully defined [24]. The present work includes up to seven updating parameters, which can be divided into three groups: support stiffness, material properties, and structure dimensions. The material of the plate was aluminum-magnesium (Al-Mg) alloy, and the material properties, including the Young's modulus, Poisson's ratio, and density, were selected as updating parameters. Normally, these properties are set as constants; however, differences in the material composition and its manufacturing process can induce certain variations. The limits for these parameters were determined by an analogue approach. The material properties of several typical Al-Mg alloys are listed in Table 7. The maximum and minimum values of each material parameter were used as the upper and lower limits of the updating parameters, respectively. The two actual dimensions of the plate, i.e., length and thickness, deviated from the design dimensions, and there was also a deviation in the dimensions measured at different positions. Therefore, the measured dimensions cannot be used directly to build \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \(\mathbf{N}\) & \(\mathbf{\omega}\) & \(c_{1}\) & \(c_{2}\) & \(\mathbf{t_{\rm max}}\) \\ \hline Value & 20, 30, 50 & 0.8 & 2 & 2 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameter setting in the PSO algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Type** & **Young’s Modulus (GPa)** & **Poisson’s Ratio** & **Density (kg/m\({}^{3}\))** \\ \hline
5042 & 68.9 & 0.33 & 2720 \\
5052 & 72 & 0.34 & 2680 \\
5052-H34 & 70 & 0.33 & 2680 \\
5083-H116 & 70 & 0.3 & 2700 \\
5182 & 76 & 0.34 & 2720 \\
5182-O & 70.6 & 0.34 & 2890 \\ \hline Upper limit & 76 & 0.34 & 2890 \\ Lower limit & 68.9 & 0.3 & 2680 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Material properties of several typical Al-Mg alloys [25]. the FE model. In order to determine the limits of the dimensions, it was assumed that the dimensions measured at different positions conform to the normal distribution, and a 95.45% confidence level was chosen, as described in the following relationship:  
\[P(\mu-2\sigma\leq X\leq\mu+2\sigma)\approx 0.9545 \tag{4}\]  
where \(X\) is a random variable, \(\mu\) is the mean value, and \(\sigma\) is the standard deviation. In this work, \(\mu\) and \(\sigma\) were calculated from the measurements. The limits for the length and thickness of the plate were set to be the mean value of the measurements with a deviation of \(+2\sigma\) for the upper limits and \(-2\sigma\) for the lower limits. The optimal dimensions have a 95.45% chance of being inside the given interval. Finally, the limits of the stiffness of the springs used to simulate the support stiffness in the horizontal (\(k_{\rm H}\)) and vertical (\(k_{\rm V}\)) directions were defined. Before determining the limits, the sources of the support stiffness in both directions were analyzed. In the vertical direction (\(k_{\rm V}\)), the support stiffness of the plate mainly comes from the extrusion between the components, which is in accordance with [PERSON]'s law:  
\[F=EA\epsilon \tag{5}\]  
where \(F\) is the support force, \(E\) is the Young's modulus of the plate, \(A\) is the contact area between plate and exciter, \(\epsilon\) is the strain, and \(EA\) is the stiffness of the system. When \(A\) takes the maximum value of 250.5 mm\({}^{2}\), \(EA\) can be used as the upper limit of \(k_{\rm V}\). Considering that the elastic deformation is not the only reason for the support stiffness in this direction, this upper limit can be taken as a higher value. In the horizontal direction (\(k_{\rm H}\)), the support stiffness mainly comes from the friction between the components.