# Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions will be mitigated by 8–11 g/km. There are several ways to reduce car weight, such as the use of lighter materials in manufacturing, the downsizing of the car, and the removal of unwanted material from car components [2]. One of the most implemented material-removal methods is topology optimization (TO).

## 2. The General Structural Optimization Problem and the SIMP Approach

_{e}; 1 for required material, and 0 for void. Hence, the nested mathematical formulation, based on the homogenization theory developed by Bendsøe [18], is the following:

_{e}), ρ

_{e}) is the objective function, the ρ

_{e}, in the state function u(ρ

_{e}), is the density of each element in the design domain Ω. G

_{j}(u(ρ

_{e}), ρ

_{e}) are the constraints, and V is the total volume of the structure. The most implemented objective function is the compliance of a structure. Compliance is the reciprocal of the stiffness, so in other words, by minimizing the compliance, the stiffness of the structure is increased.

_{e}. Furthermore, a minimum density ρ

_{min}≠ 0 was used as a lower bound of density in order to avoid a calculation of a zero structure’s elasticity. The “penalization” of the intermediate finite elements; elements with density ρ

_{min}< ρ

_{e}< 1 is conducted by the penalty factor, p. In other words, the SIMP method prevents the formation of the intermediate elements by increasing the structure’s density to an exponent equal to p. According to Sigmund (2001), the ideal value of the penalty factor is three. Hence, the introduction of this factor reduces the elasticity, and in turn, the global stiffness is reduced.

## 3. Brake Calipers

#### 3.1. Brake System Kinetics

_{kinetic}, of the vehicle is converted to thermal energy, E

_{thermal}, which in turn is absorbed by the brake system. A simplified formula describing the relation between a vehicle’s mass with a given velocity and the difference in temperature of the brake system is the following [24]:

_{tire}. In addition to the wheel torque, other parameters such as the aerodynamic drag, the drivetrain losses, the gear meshing, the oil viscosity, and the rolling resistance can contribute to the braking force [24].

#### 3.2. Vehicle Dynamics

_{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:

_{1}) and one rear wheel (m

_{2}) are presented. The k

_{t}

_{1}and k

_{t}

_{2}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. Finally, the illustrated rigid bar in the model represents half of the car’s mass, m, with a lateral moment of inertia, I

_{y}.

## 4. Methodology

^{3}) compared to the other two available options, the Steel MS1 (8.1 g/cm

^{3}) and the Ti6Al4V (4.4 g/cm

^{3}). Concerning the used case study in this paper, every gram counts in the development of calipers for a racecar. However, the calipers of a racecar are exposed to high braking forces and temperatures. Hence, the Ti6Al4V with an exceptional yield strength (1147 MPa), even at 500 °C (890 MPa), was considered an excellent choice [33].

_{XY}= 120 GPa and E

_{Z}= 110 GPa, Poisson’s ratio ν = 0.31, and density ρ = 4.41 g/cm

^{3}. The material properties were taken from the EOS data sheet, which was the used material of the 3D printed housings.

_{b}= 600 Nm. On the other hand, the validation of the rear caliper was set with a pretension load F = 17 KN, a piston pressure P = 8 MPa, and a brake disc torque M

_{b}= 300 Nm. Finally, the same boundary conditions were used in both cases, as they are shown in Figure 8.

## 5. Results

## 6. Topology Optimization for Manufacturing

## 7. Conclusions

## 8. Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The quarter-car model, adapted from Jazar [29].

**Figure 3.**The 4-DOF half-car model adapted from Jazar [29].

**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.

**Figure 7.**The design space, the loads, and the boundary conditions of the housings: (

**a**) Housing of the front caliper, (

**b**) Housing of the rear caliper.

**Figure 8.**The used assemblies for the validation studies with their loads and boundary conditions: (

**a**) Assembly of the front caliper, (

**b**) Assembly of the rear caliper.

**Figure 9.**The assemblies of the caliper design concepts after the first optimization level in SolidWorks: (

**a**) Front, (

**b**) Rear.

**Figure 11.**The results of the validation studies in ABAQUS for the front caliper: (

**a**) Stress plot of the front housing, (

**b**) Stress plot of the front caliper, (

**c**) Total displacement of the front housing, (

**d**) Total displacement of the front caliper.

**Figure 12.**The results of the validation studies in ABAQUS for the rear caliper: (

**a**) Stress plot of the rear housing, (

**b**) Stress plot of the rear caliper, (

**c**) Total displacement of the rear housing, and (

**d**) Total displacement of the rear caliper.

**Figure 14.**A geometry comparison between the CAD models and the 3D printed parts of the housings: (

**a**) The front housing and (

**b**) The rear housing.

Diameter [mm] | Depth [mm] | Pad Distance [mm] | |
---|---|---|---|

CAD of Front housing | 25 | 59 | 10 |

3D printed Front housing | 24.87 | 59.01 | 9.93 |

CAD of Rear housing | 17 | 49.5 | 10 |

3D printed Rear housing | 16.93 | 49.58 | 9.95 |

Average Deviation | −0.47% | 0.09% | −0.60% |

Component/Assembly | Weight (g) | Max Stress (MPa) | Total Displacement (mm) | Displacement in Y (mm) | ||
---|---|---|---|---|---|---|

FRONT CALIPER | Housing | ISR 22-048 | 320 | 425 | 0.663 | 0.516 |

Optimized (raw model) | 185.75 | 434 | 0.47 | 0.39 | ||

Optimized after redesign | 189.64 | 400 | 0.27 | 0.25 | ||

Optimized after 3D printing preparation | 228.58 | 350 | 0.33 | 0.29 | ||

3D printed | 230.6 | - | - | - | ||

Caliper | ISR 22-048 assembly | 483 | - | - | 0.5 | |

Design space assembly | 2285 | 73 | 0.041 | 0.037 | ||

Optimized assembly | 304.7 | 400 | 0.27 | 0.25 | ||

REAR CALIPER | Housing | ISR 22-049 | 210 | 465 | 0.543 | 0.461 |

Optimized (raw model) | 74.10 | 312 | 0.47 | 0.36 | ||

Optimized after redesign | 83.37 | 447 | 0.45 | 0.33 | ||

Optimized after 3D printing preparation | 112.4 | 405 | 0.37 | 0.36 | ||

3D printed | 119.2 | - | - | - | ||

Caliper | ISR 22-049 Assembly | 320.8 | - | - | 0.4 | |

Design space assembly | 1254 | 44 | 0.012 | 0.009 | ||

Optimized Assembly | 165.11 | 447 | 0.45 | 0.33 |

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**MDPI and ACS Style**

Tyflopoulos, E.; Lien, M.; Steinert, M.
Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing. *Appl. Sci.* **2021**, *11*, 1437.
https://doi.org/10.3390/app11041437

**AMA Style**

Tyflopoulos E, Lien M, Steinert M.
Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing. *Applied Sciences*. 2021; 11(4):1437.
https://doi.org/10.3390/app11041437

**Chicago/Turabian Style**

Tyflopoulos, Evangelos, Mathias Lien, and Martin Steinert.
2021. "Optimization of Brake Calipers Using Topology Optimization for Additive Manufacturing" *Applied Sciences* 11, no. 4: 1437.
https://doi.org/10.3390/app11041437