# Optimization and Experimental Verification of Lower
## 2 Optimization Model for Lower Articulation
### Structure and Fatigue Life Evaluation of Optimized Lower Articulation  
#### 2.3.1 Structural Design Using Meta-Model  
The objective function of the optimal design of the lower articulation of the low-floor tram is to minimize the weight while ensuring that the stress remains within the allowable stress range. The constraints are defined based on the stresses derived from the tensile and compressive load conditions as follows:  
\[\text{Minimize W} =\text{W}\left(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\right) \text{Subject to}\] \[\text{g} =\text{Allowable stress}\leq 650\text{ MPa}\] \[\text{Design variables}\] \[\text{Lower}\leqq x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8} \leqq\text{Upper}\]  
Here, g represents the stress for each design variable, and W denotes the weight of the articulation. The initial analysis results indicated that the stress of the lower articulation did not exceed the allowable stress limit of 650 MPa. Consequently, the Incomplete Small Composite Design-II (ISCD-II) method was employed to ensure that the stress remained below the allowable limit while minimizing the weight. The commercial software Easy Design [20] was used to create the analysis conditions. Table 3 presents the experimental design matrix and analysis results for the eight factors using the ISCD-II technique. A total of 16 analyses were conducted to account for the variations in stress and weight due to changes in the eight factors. Table 3 also includes the maximum stress and weight values. The Kriging model was utilized in conjunction with the optimization algorithm to add experimental points and determine the optimal values. The optimization results, achieved by adding three experimental points, were identified as SAO #1-3. For example, Figure 7 shows the stress distribution results for SAO #3. The trend in the results is consistent with the initial model analysis. The model that meets the allowable stress of the design and minimizes the weight is identified as SAO #3. This model achieved a 10% reduction in weight from 143.7 kg in the initial model to 128.7 kg, fulfilling the goal of lightweight optimization. Additionally, the optimized lower articulation demonstrated appropriate stress levels within the allowable stress range, thereby achieving the optimization objectives.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Case** & \(\mathbf{x_{1}}\) & \(\mathbf{x_{2}}\) & \(\mathbf{x_{3}}\) & \(\mathbf{x_{4}}\) & \(\mathbf{x_{5}}\) & \(\mathbf{x_{6}}\) & \(\mathbf{x_{7}}\) & \(\mathbf{x_{8}}\) & \
\begin{tabular}{c} **Stress** \\ **[MPa]** \\ \end{tabular} &
\begin{tabular}{c} **Mass** \\ **[kg]** \\ \end{tabular} \\ \hline
1 & 15.0 & 70.0 & 45.0 & 54.0 & 140 & 25.5 & 39.5 & 58.0 & 1046.6 & 162.5 \\
2 & 5.0 & 90.0 & 45.0 & 64.0 & 120 & 25.5 & 39.5 & 58.0 & 886.1 & 144.4 \\
3 & 15.0 & 90.0 & 25.0 & 64.0 & 140 & 31.5 & 33.5 & 58.0 & 1300.0 & 136.2 \\
4 & 5.0 & 70.0 & 45.0 & 64.0 & 140 & 31.5 & 39.5 & 66.0 & 1321.7 & 156.2 \\
5 & 5.0 & 90.0 & 25.0 & 54.0 & 120 & 31.5 & 39.5 & 66.0 & 1393.4 & 129.2 \\
6 & 5.0 & 70.0 & 25.0 & 64.0 & 140 & 25.5 & 33.5 & 66.0 & 702.7 & 134.2 \\
7 & 15.0 & 70.0 & 25.0 & 64.0 & 120 & 31.5 & 39.5 & 58.0 & 1302.1 & 128.9 \\
8 & 15.0 & 70.0 & 45.0 & 54.0 & 120 & 31.5 & 33.5 & 66.0 & 904.9 & 151.0 \\
9 & 15.0 & 90.0 & 45.0 & 64.0 & 120 & 25.5 & 33.5 & 58.0 & 795.0 & 147.8 \\
10 & 5.0 & 90.0 & 45.0 & 54.0 & 140 & 31.5 & 33.5 & 58.0 & 673.5 & 159.8 \\
11 & 15.0 & 90.0 & 25.0 & 54.0 & 140 & 25.5 & 39.5 & 66.0 & 533.2 & 139.6 \\
12 & 5.0 & 70.0 & 25.0 & 54.0 & 120 & 25.5 & 33.5 & 58.0 & 666.1 & 130.5 \\
13 & 10.0 & 80.0 & 35.0 & 59.0 & 130 & 28.5 & 36.5 & 62.0 & 529.7 & 143.7 \\ SAO \#1 & 13.9 & 89.6 & 25 & 108.7 & 134.5 & 26.3 & 51.5 & 66 & 649.5 & 134.4 \\ SAO \#2 & 5.1 & 70.4 & 25.3 & 107.3 & 120 & 25.5 & 46.5 & 59.5 & 650.1 & 128.7 \\ SAO \#3 & 5.1 & 70.4 & 25.4 & 107.2 & 120 & 25.5 & 46.5 & 59.5 & 648.5 & 128.7 \\ \hline \hline \end{tabular}
\end{table}
Table 3: ISCD-II of design matrix.