# Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model
## 3 Results and Discussion
### Case 1: Frequency Separation Optimization
The mode orders of the 6 ND, 1T, and 7 ND modes were initially 26 th, 30 th, and 34 th (baseline model), and after optimization, they were 28 th, 32 nd, and 38 th (optimized model), respectively. In fact, during the optimization process, the mode orders of all analyzed modes were constantly changing as the design variables evolved. It is critical that, in each iteration, a new set of ND and T modes were determined based on the automatic identification scheme mentioned in the last section. If the modes of interest were not correctly tracked during optimization, the process would most likely fail to converge to an optimal solution. The heat flux loading for this study can be seen in Figure 12, with the peak heat flux of 1.2 W/mm\({}^{2}\) at 1.0 s and then linearly decreasing to 0 at 5.0 s. The heat flux data were acquired during a previous project in which the authors were involved. The simulation duration lasted 5.5 s, and the measuring time was also set at 5.5 s. Since the sliced model was only a 1/36 model of the complete rotor, a much finer mesh could be used. Figure 13 shows a finite element grid of the sliced baseline model that contains 38,917 elements and 98,100 nodes. Performing a transient thermal analysis and a structural analysis on the sliced baseline model with the material properties listed in Table 2 yielded the baseline results that were compared with those after optimization. Figure 14 shows temperature distributions (\({}^{\circ}\)C) of the sliced baseline model at various time steps, and Figure 15 illustrates deformations in the y-direction (meters) at various times for the same model. Large temperature gradients were observed near the junction of the hub and the outboard disc throughout different time steps, as shown in Figure 14. At time = 5.5 s, \(U_{y1}\) through \(U_{y4}\) were identified to be 0.1122, 0.0481, 0.0924 and 0.0344 mm, respectively, and coming was calculated on the outboard disc as \(U_{y1}-U_{y2}=0.0641\) mm and on the inboard disc as \(U_{y3}-U_{y4}=0.0580\) mm. Figure 11: A sliced rotor model and its boundary conditions. Figure 12: Heat flux loading on the sliced rotor model. Figure 14: Temperature distributions of the sliced baseline model at various time steps. Figure 13: Finite element grid of the sliced rotor model. In optimization Case 2, thermal distortion of the frictional surfaces, i.e., the brake rotor coning, was minimized. The mathematical formulation to define this problem can be stated as  
\[\text{Minimize}\ f_{obj}=(\delta_{1}+\delta_{2})/2 \tag{9}\]  
\[\text{subjected to}\ L_{DV}\leq DV\leq U_{DV} \tag{10}\]  
\[M_{a}-M_{base}\leq\epsilon_{M} \tag{11}\]  
\[\delta_{1}=U_{y1}-U_{y2}\geq 0 \tag{12}\]  
Figure 14.0  
Figure 14.1  
Figure 14.2  
Figure 14.3  
Figure 14.4  
\[\text{and}\ DV=\left[ \begin{array}{cccc}DI_{1}&T_{F1}&T_{F2}&T_{RIB}\end{array}  \right]^{T} \tag{18}\]  
where the objective function of this optimization problem is defined as the average coning calculated on the inboard and the outboard discs; \(\delta_{1}\) is the deformation difference between \(U_{y1}\) and \(U_{y2}\) at time = 5.5 s, and \(\delta_{2}\) is the difference between \(U_{y3}\) and \(U_{y4}\) at the same time. During optimization, we sought to ensure that the structural deformations in friction surfaces achieve a coning effect by forcing \(U_{y1}\) through \(U_{y4}\), with \(\delta_{1}\) and \(\delta_{2}\) all positive, i.e., Equations (12)-(17); nevertheless, we wanted this effect to be as small as possible. By applying Equations (9)-(18) and incorporating them with the SAM optimization scheme in precisely the same way as the previous case in another APDL program, convergence was observed in 15 iterations. In each iteration, the six steps to acquire brake rotor coning at time = 5.5 s, including a transient thermal analysis and a structural analysis, were  
Figure 15: Deformations in y-direction of the sliced baseline model. realized and repeated. Table 5 lists the optimization results for Case 2.