# Multiobjective Optimization Design for a MR Damper Based on EBFNN and MOPSO
## 4 Optimization Design
### Design of Experiments (DoE)  
The design optimization of MR devices was analyzed in the FEA for magnetic analysis. This leads to an increase in resources and computation time as each optimization process requires individual analysis of an FEA model. The DoE method was proposed to solve these problems. As an alternative methodology, a series of well-chosen design points in the design space were selected for specific analysis. The objective of the DoE is to identify the key design factors from the geometric parameters, systematically analyze the final response and relationship between variables and objectives, and provide the response data for the subsequent establishment of approximate functions in the early stage of the multiobjective optimization design. In this work, the script commands were batched in the ISIGHT software for calling the ANSYS software, and the FEA models of design points based on DoE were developed to achieve our objective function.  
Based on Equations (2) and (3), the basic geometric parameters (effect of the magnetic flux density on the yield stress force) affect the output damping force of the MR damper, including the shear effective length \(L_{1}\) and \(L_{2}\), the radius of individual stage gap \(r\), non-magnetic length \(t_{a}\), coil slot depth \(H_{c}\), gap thickness \(g\), ring thickness \(t_{c}\), and piston radius \(r_{p}\). Since the radius of the damper remained constant in the optimization, the parameters related to the radial size of the damper, such as piston radius, coil slot depth, and external radius of core, were also kept as a constant. In addition, the gap thickness was the same in this work because its reduction would cause a blockage of the MR fluid; thus, it is difficult to ensure the machining accuracy. Four design variables were selected as the shear effective length \(L_{1}\) and \(L_{2}\), nonmagnetic length \(t_{a}\), and ring thickness \(t_{c}\), and their upper and lower limits are presented in Table 2. The Opt LHD method of DoE can achieve the investigation of more samples for each design variables, and these samples are spread evenly. In addition, considering the ability of fitting nonlinear response of the Opt LHD design, this method is used to conduct sampling analysis of MR damper and provide original data for the construction of an approximate model. The sample number is randomly selected as long as it is more than the design variables. Based on the upper and lower limits (Table 2) of design variables, this work generates 80 groups of samples spread evenly in the design space. The corresponding response values for each group were obtained through the joint ANSYS.  
Figure 9 shows the main effects of the four design variables on the output damping force and dynamic range. Here, the correlations between the affecting factors (except the ring thickness) and the output damping force are proportionally monotonic. The relationship between the dynamic range and the first shear effective length \(L_{1}\), the second shear effective length \(L_{2}\), and the nonmagnetic length \(t_{a}\) are all nonlinear. The effect of the ring thickness on the damping force of the MR damper is insignificant. The negative correlation between the dynamic range and the ring thickness is almost linear. However, the previous value of the ring thickness was the lowest. Thus, as one of the damping properties, the ring thickness has an insignificant effect on the design parameter optimization.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Design Variable/mm** & **Lower Limit** & **Upper Limit** & **Initial** \\ \hline First shear effective length \(L_{1}\) & 4 & 12 & 8 \\ Second shear effective length \(L_{2}\) & 4 & 12 & 7 \\ Nonmagnetic length \(t_{a}\) & 2 & 19 & 13 \\ Ring thickness \(t_{c}\) & 3 & 6 & 3 \\ \hline \end{tabular}
\end{table}
Table 2: Upper and lower limits of design variables of the damper structure parameters.