# Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing
## 4 Methodology  
In this research work, the development of front and rear brake calipers for a student racecar is presented. The in-house calipers were based on the commercial ISR calipers: the ISR 22-048 for the front caliper and the ISR-049 for the rear. Both are fixed calipers originally designed for 125 cc bikes and Formula SAE racecars. However, the ISR 22-048 (front) uses four pistons while the ISR-049 (rear) two. The assemblies of the ISR calipers are illustrated in Figure 4.  
At the beginning of this research, the identification of mass reduction possibilities was implemented. For this reason, an analysis of the vehicle's braking behavior, together with the ISR calipers, was conducted. Then, the calipers were optimized using a three-level optimization procedure in order to reduce their weight, and thus the total unsprung mass of the racecar. A reduction of the vehicle's unsprung mass could improve its performance. The implemented methodology is depicted in Figure 5.  
Figure 4: The ISR calipers in SolidWorks: (**a**) Front caliper and (**b**) Rear caliper. Figure 5: The optimization procedure. The analysis of the vehicle and the ISR calipers comprises two main tasks: the lap time simulation of the vehicle in Matlab and the physical testing of the commercial ISR calipers using an in-house test jig. The calipers were a part of the disc brakes of the racecar, which in turn were connected to a kinetic energy recovery system (KERS). The utilized KERS had limited torque output at the motor's top rpm. Thus, the vehicle's top speed was used as the dimension load. As has already been mentioned in Section 2, it is a good engineering practice to use the car tires as the limiting factor for the brake system. The chosen tires for the racecar were the Continental C19. A dynamic tire model was developed in MATLAB in order to conduct a lap time simulation. This model was based on the Continental's technical specifications and the Pacejka M5.2 tire model [31] and is given by the following tire formula:  
\[y=D*\sin\Bigl{\{}C*\tan^{-1}\Bigl{[}B_{x}-E\Bigl{(}B_{x}-\tan^{-1}(B_{x}) \Bigr{)}\Bigr{]}\Bigr{\}} \tag{10}\]  
where  
\(y\): force or moment resulting from a slip parameter  
\(x\): slip parameter  
\(B\): stiffness factor  
\(C\): shape factor  
\(D\): peak factor  
\(E\): curvature factor  
Five lap time simulations were implemented for five different torques, range 300-700 Nm with a 100 Nm step. These simulations contributed to the calculation of the brake system's load case using the above tire model, the vehicle load transfer, as well as the aerodynamic- and motor representations. Furthermore, the minimum braking force that did not compromise the lap times was identified. Finally, the temperature development on the discs was estimated based on empirical data from previous autocross runs. These data were collected by an INFKL 800 \({}^{\circ}\)C IR brake temperature sensor placed into the hydraulic system of the disc brakes. Figure 6 depicts the results from the lap simulations of the calculated braking distance and velocity of the car and the distribution of the longitudinal forces, in the case of braking, from 120 km/h to 0 km/h. In addition, a temperature-time diagram is presented based on the sensor's data. Figure 6: Braking from 120 km/h to 0 km/h: (**a**) Distance-time graph for different torques, (**b**) Velocity-time graph for different torques, (**c**) Longitudinal forces-time graph, and (**d**) Temperature-time graph. The conducted simulations in Matlab showed a convergence of maximum brake force in cohesion to the tire model. Torque values above 600 Nm gave slower brake time as the brake force exceeds the tire's grip limit. In addition, from the longitudinal forces-time graph (graph c), it is observed that it is required a 0.8 brake balance between the front and the rear calipers. Finally, the empirical data of the temperature on the disc showed that a temperature up to 450 \({}^{\circ}\)C was expected on the disc brakes.