# Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model
## 2 Research Methods
### The Fully Parameterized Brake Rotor Model  
The geometry of the vented brake rotor under investigation was fully parameterized by 22 geometrical input parameters, excluding the bolt holes on top of the rotor hub, and these parameters can be seen in Figure 5 (shown in a quarter-sectioned model) and Figure 6. The parameter \(\theta_{RIB}\), shown in Figure 6, also determined the number of ribs sandwiched between the upper and lower discs. For example, \(\theta_{RIB}=10^{\circ}\) leads to \(360^{\circ}/10^{\circ}=36\) ribs and \(\theta_{RIB}\) must be set in such a way that the division results in an integer. \(S_{CH1}\), \(S_{CH2}\), and \(S_{CH3}\) represent the depths (or widths) of the respective \(45^{\circ}\) chamfers. These 22 parameters completely and uniquely defined the geometry of the brake rotor. With the help of ANSYS Parametric Design Language, assigning the values adopted from a commercial brake rotor to the input parameters (values given in Table 1) rendered a solid model, as previously shown in Figure 2, which was defined as the baseline model. APDL is a programing language capable of parametrically creating solid models, meshing, building, and solving finite element models, manipulating FEA results, and even performing optimization. A finite element grid of this baseline model can be seen in Figure 7. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(D_{\text{CEN}}\) & \(D\mathbf{I_{1}}\) & \(D\mathbf{I_{2}}\) & \(D\mathbf{I_{3}}\) & \(DO_{1}\) & \(DO_{2}\) & \(DO_{3}\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(S_{CH1}\) \\ \hline
61 & 123 & 142.6 & 147.6 & 135 & 230.6 & 222.6 & 2 & 2 & 1 \\ \hline \(S_{CH2}\) & \(S_{CH3}\) & \(S_{CHX}\) & \(S_{CHY}\) & \(T_{C}\) & \(T_{FI}\) & \(T_{F2}\) & \(T_{RIB}\) & \(W_{RIB}\) & \(Y_{1}\) & \(\theta_{RIB}\) \\ \hline
3.5 & 2 & 6 & 2 & 6 & 6 & 6 & 5 & 5.4 & 27.5 & \(10^{\circ}\) \\ \hline \hline \end{tabular} \({}^{\lx@sectionsign}\) All length units are in millimeters.  
\end{table} 
Table 1: Initial values of the 22 geometrical parameters for the baseline model \({}^{\lx@sectionsign}\). Figure 4: In-plane modes: (**a**) radial mode; (**b**,**c**) tangential modes including (**b**) shear mode and (**c**) compression mode. The brake rotor was made of grey cast iron, the material properties of which are listed in Table 2. Based on the results of a preliminary study, modal analyses of the rotor model under a free-free boundary condition with 40 flexural modes were enough for the current study. Among the first 40 flexural modes, the ND modes and T modes were particularly of interest in brake rotor design. Analysis results showed that one nodal diameter (1 ND mode) did not exist for the free-free rotor model. However, 2 ND through 7 ND modes and  
Figure 5: Geometrical parameters of the vented brake rotor. Figure 6: Geometrical parameters defining the ribs of the rotor. Figure 7: Finite element grid of the baseline model. 1T mode of the baseline model were among the 40 flexural modes. Figure 8 shows these ND and T modes with their mode shapes, mode orders, and frequencies. In order to properly show their mode shapes, two different coordinate systems are used in Figure 8. For ND modes, vibration amplitudes in the y-direction (perpendicular to the rotor disc plane) are much more prominent, and therefore, a Cartesian coordinate system is used. For T modes, oscillations are mainly confined on the rotor disc surface and behave similarly to torsion motions, and therefore, a cylindrical coordinate system with its z-axis perpendicular to the disc surface is employed. The deep red and deep blue color contours on the mode shapes in Figure 8 represent two regions with amplitudes of opposite directions. For a nearly axially symmetric structure such as the brake rotor under investigation, ND modes are expected to appear in nearly orthogonal pairs with two consecutive modes phase-shifted 90\({}^{\circ}\) apart and very close in frequency, and so are the T modes.