# Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing
## 3 Brake Calipers
### Vehicle Dynamics  
Three are the most applied models for the vehicle dynamics calculations that characterize its ride quality, such as vehicle's transmissibility, suspension travel, and tire deflection together with the tire-ground adhesion: the quarter car model, the 2-DOF half-car model, and the 4-DOF half-car model. These models are based on the following assumptions: rigid unsprung and sprung masses, linear suspension spring, and viscous damping. In addition, the tire in these models is represented as a combination of spring and damper. However, the tire damping is omitted in the analysis of all models [30]. The used models in this research work were the quarter car model and the 4-DOF half-car model.  
The quarter-car model represents one-fourth of a vehicle. Its degrees of freedom are the translational displacement of both the sprung and unsprung mass, making it a 2-DOF model. Figure 2 illustrates a quarter-car model [29].  
Here, the \(m_{s}\) and the \(m_{u}\) represent the quarter sprung and unsprung mass of a car, respectively. In addition, \(k_{s}\), \(c_{s}\) represents the spring stiffness and damper coefficient of the shock absorber. Finally, \(k_{u}\) and \(c_{u}\) are the spring-damper effect of the tire. Both the sprung and the unsprung masses can be calculated by the following equations:  
\[\text{Sprung mass}:\ m_{\tilde{s}}\ddot{x}_{s}=-k_{s}(x_{s}-x_{u})-c_{s}\big{(} \dot{x}_{s}-\dot{x}_{u}\big{)} \tag{7}\]  
\[\text{Unsprung mass}:\ m_{u}\ddot{x}_{u}=k_{s}(x_{s}-x_{u})-c_{s}\big{(}\dot{x }_{s}-\dot{x}_{u}\big{)}-k_{u}(x_{u}-y)-c_{u}\big{(}\dot{x}_{u}-\dot{y}\big{)} \tag{8}\]  
Translated into matrix form, we get the following equation of motion:  
\[m\ddot{x}+c\dot{x}+kx=F \tag{9}\]  
Figure 2: The quarter-car model, adapted from [PERSON] [29].  
where  
\[x=\left[\begin{array}{c}x_{s}\\ x_{u}\end{array}\right]\text{, }m=\left[\begin{array}{cc}m_{s}&0\\ 0&m_{u}\end{array}\right]\text{, }c=\left[\begin{array}{cc}c_{s}&-c_{s}\\ -c_{s}&c_{s}+c_{u}\end{array}\right]k=\left[\begin{array}{cc}k_{s}&-k_{s}\\ -k_{s}&k_{s}+k_{u}\end{array}\right]\text{, }F=\left[\begin{array}{c}0\\ k_{u}y+c_{u}\dot{y}\end{array}\right]\]  
This model was utilized for the calculation of the suspension vibration and helped the authors to identify the effect of a potential unsprung mass reduction in the case study of the racecar. However, this model takes into account only the bounce motion of the vehicle. For this reason, the 4-DOF half-car model was used in addition to the quarter-car model.  
The 4-DOF half-car model was exploited for the pitch angle estimations, given the dynamic load distribution between the front and the rear axle and with regards to the front and rear braking load. At the 4-DOF half-car model depicted in Figure 3, the half body (\(m\)) of the car as well as one front (\(m_{1}\)) and one rear wheel (\(m_{2}\)) are presented. The \(k_{t1}\) and \(k_{t2}\) are the spring stiffness of the tires. Furthermore, \(c_{1}\) and \(c_{2}\) are the damping coefficients, and \(k_{1}\) and \(k_{2}\) the stiffness of the shock absorbers.