# Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model
## 3 Results and Discussion  
### Case 1: Frequency Separation Optimization  
In brake rotor design, an important issue is to avoid the coupling of ND and T modes, i.e., to prevent T modes from growing too close to ND modes in frequency. Due to the manufacturing process, two brake rotors with the same design specification can yield a variation in natural frequencies as high as 7%, especially at high frequencies [27]. Other factors, e.g., wear, may also deviate the natural frequencies of a rotor from its design values. Therefore, separating T modes from ND modes as far as possible is a prudent decision. The modal analysis of the baseline model showed that, in the first 40 modes, 1T mode was the only T mode, and it was sandwiched by 6 ND and 7 ND modes. Therefore, in this optimization case, 1T mode was forced to be separated as far as possible from 6 ND and 7 ND modes by optimizing a few geometric parameters. The frequency differences between 6 ND and 1T modes and 1T and 7 ND modes were combined as one objective function and then maximized. The mathematical formulation to define the frequency separation optimization problem in this research can be stated as  
\[\text{Maximize }f_{obj}=f_{sep1}+f_{sep2} \tag{2}\]  
\[\text{subjected to }L_{DV}\leq DV\leq U_{DV} \tag{3}\]  
\[M_{a}-M_{base}\leq\varepsilon_{M} \tag{4}\]  
\[f_{sep1}=f_{1T}-f_{6 ND}>0 \tag{5}\]  
\[f_{sep2}=f_{7 ND}-f_{1T}>0 \tag{6}\]  
\[\text{and }DV= \begin{bmatrix}&DI_{1}&T_{F1}&T_{F2}&T_{RIB}\end{bmatrix} ^{T} \tag{7}\]  
where \(f_{obj}\) is the objective function of this optimization problem; \(f_{sep1}\) and \(f_{sep2}\) are the separation frequencies between the 6 ND (\(f_{6 ND}\)) and 1T (\(f_{1T}\)) modes and 1T and 7 ND (\(f_{7 ND}\)) modes, respectively; \(DV\) contains the design variables and \(L_{DV}\) and \(U_{DV}\) denote the lower and upper bounds, respectively, of the design variables; \(M_{base}\) is the total mass of the baseline model, \(M_{a}\) represents the mass of current model; \(\varepsilon_{M}\) is a small positive value. Equation (4) ensures that the mass of the optimized model will not exceed the baseline model by more than a small amount. In this study, \(\varepsilon_{M}\) was set to 1% of the baseline mass. The weight issue is always an important concern when designing automotive parts. Four design variables \(DI_{1}\), \(T_{F1}\), \(T_{F2}\), and \(T_{RIB}\) were chosen based on engineering judgment and sensitivity analysis that revealed their high sensitivities. The sensitivity of a frequency \(f_{j}\) to a design variable \(DV_{i}\) is defined as  
\[\text{s}_{ij}=\frac{\partial f_{j}}{\partial DV_{i}}\cong\frac{f_{j}(DV_{i}+ \Delta DV_{i})-f_{j}(DV_{i})}{\Delta DV_{i}}. \tag{8}\]  
by assigning the baseline value and setting a 1% increment to one design variable at a time, and then repeating the finite element analysis, the sensitivities of 1T mode to \(DI_{1}\), \(T_{F1}\), \(T_{F2}\), and \(T_{RIB}\), for example, were found to be \(-11.96\), \(66.00\), \(-13.33\) and \(-75.6\) Hz/mm, respectively. The upper and the lower bounds were set as \(\pm 5\%\) from the baseline value for \(DI_{1}\), and \(\pm 20\%\) for \(T_{F1}\), \(T_{F2}\), and \(T_{RIB}\). Over relaxed upper and/or lower bounds might cause numerical difficulties during finite element analysis in some situations. Applying Equations (2)-(7) with an APDL code using the subproblem approximation method (SAM) [28], which is an advanced, zero-order optimization procedure using approximations to all dependent variables and can be applied efficiently to a broad spectrum of engineering problems, convergence was achieved in 12 iterations. Table 4 lists the optimization results for Case 1. Although \(f_{sep1}\) decreased, \(f_{sep2}\) increased by a large amount after optimization, resulting in a combined improvement of 181.4 Hz in the objective function. Figure 10 shows the 6 ND, 1T, and 7 ND mode shapes and frequencies for the optimized model. Notably, the mode orders of these three modes all changed.