# The Hybrid Brake Model and Its Validation

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^{†}

## Abstract

**:**

## 1. Introduction

- Very low power density compared to friction brakes;
- Braking to a standstill is not possible.

## 2. Concept of the Hybrid Brake

## 3. Design Method

## 4. System Model

## 5. Electromagnetic Model

- The model is a reluctance network;
- The model is a quasi static model;
- The magnetic circuit is modelled two-dimensionally in a pole cross-section at the mean effective radius in order to calculate the mean eddy current across all pin columns in the radial direction.

^{−1}and n = 8000 min

^{−1}.

## 6. Spring Model

## 7. Spring Parameter Analysis

## 8. FEM Analyses

^{−2}, but a spring steel with a tensile strength of σ

_{all}= 1800 N mm

^{−2}is used. In order to achieve the desired spring characteristics, even with the modified design, a parameter study was performed using the FEM model. Figure 13a shows an example of the relative mean square deviation of the spring characteristic from the required one, which was obtained for different contour radii ${r}_{\mathrm{c}}$. Figure 13b shows the spring characteristic curve of the final spring design calculated with the FEM. The relative mean square error of this spring curve is less than 3% compared to the desired one.

## 9. Experimental Setup and Methods

#### 9.1. Measurement Setup

#### 9.2. Measurement of Friction Torque

^{−1}and an excitation current of I

_{ex}= 50 A. It can be seen that the signal was most likely disturbed by temperature effects.

^{−1}, the error between the torque calculated with the measured magnetic fluxes and the directly measured torque was less than 10%. The large difference at speeds above n = 6500 min

^{−1}was most likely due to deformation of the rotor due to centrifugal forces (see discussion).

## 10. Results

#### 10.1. Stationary Electromagnetic Validation

#### 10.2. Stationary Validation Membrane Spring

#### 10.3. Measurement Results and Validation of the Overall System

^{−1}to n = 7500 min

^{−1}. Additionally, in the lower-speed region at excitation currents of ${I}_{\mathrm{ex}}=30$ A to ${I}_{\mathrm{ex}}=50$ A, there are relatively large errors. The reason for the errors in the low-torque region are most likely a result of frictional torques in the test bed. At all other measured points, it is assumed that the errors are mostly a result of the fact that the air gap is not homogenous in the radial direction, and the air gap is only measured at a radius of $r=0.1$ m when the outer radius of the poles is ${r}_{\mathrm{o}}=0.125$ m.

## 11. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Theoretical torques of an eddy current brake with a linearly decreasing speed over time and a friction brake for low speeds.

**Figure 2.**Exemplary CAD model of a hybrid brake (

**a**) and superordinate operating principle of the hybrid brake (

**b**).

**Figure 5.**CAD model of the rotational axial flux ECB, where detail a shows one half pole cross-section in the tangential–axial plane at the middle effective radius corresponding to Figure 5b (

**a**) and the geometrical parameters of one pole cross-section (

**b**).

**Figure 6.**Two-dimensional reluctance network. Air reluctances are presented in white, steel reluctances are grey, and air gap reluctances are light grey because they include the pin reluctances next to the air gap corresponding to ${h}_{\mathrm{gl}}$ in Figure 5b.

**Figure 7.**Eddy current torque ${M}_{\mathrm{ec}}$ (

**a**) and magnetic normal force ${F}_{\mathrm{n}\mu}$ (

**b**) as a function of the air gap and excitation current for different speeds (n = 1000 min

^{−1}and n = 8000 min

^{−1}).

**Figure 8.**Schematic three-dimensional geometry of the spring mechanism with a nondeflected spring with a spring wavenumber of ${N}_{\mathrm{sw}}=6$ (

**a**). Quarter geometry of the middle with the force F deformed and a spring sheet with the limiting contour (

**b**).

**Figure 9.**Results of the spring parameter analyses. ${t}_{\mathrm{con}}$: time until contact, ${v}_{\mathrm{imp}}$: impact velocity, ${\gamma}_{\mathrm{ap}}$: aperiodic factor. The coloured points (designs 1 to 3) are related to the spring curves in Figure 10 and the state space trajectories in Figure 11.

**Figure 10.**Spring curves for spring parameters resulting in the quality criteria marked with the red, green, and blue points in Figure 9 and the corresponding time-dependent magnetic normal forces projected in the spring curve plane.

**Figure 11.**State space trajectories for the axial movement of the rotor for different spring parameters resulting in different impact velocities, times to impact, and aperiodic factors.

**Figure 12.**Torques over time for the spring designs with the spring curves shown in Figure 10.

**Figure 13.**Optimisation of the contour of the spring mechanism in the FEM (

**a**) and the desired spring curve vs. the final spring curve as a result of the FEM calculation (

**b**).

**Figure 16.**Positions of the flux probes (fp) in the material structure (

**a**) flux probes at the pole core and pole plate (

**b**).

**Figure 17.**Positions of the laser distance sensors (

**a**) and the laser distance sensors at the hybrid brake (

**b**).

**Figure 19.**Measured torques and torques calculated with the measured flux in the material structure at different speeds and excitation currents.

**Figure 20.**Measured and predicted flux densities in the pole core ${B}_{\mathrm{pc}}$ at ${\delta}_{\mu}=0.5\mathrm{mm}$ for different excitation currents (

**a**) and the measured and predicted flux densities in the pole core and in the steel pins at ${I}_{\mathrm{ex}}=90\mathrm{A}$ for different air gaps (

**b**).

**Figure 21.**Force–displacement curves of the membrane spring as a result of the FEM analysis and experiment (

**a**) and deformed spring mechanism as a result of a FEM analysis (

**b**).

**Figure 22.**Map of the experimentally evaluated proportion of torque due to eddy currents in the space of torque and speed.

**Figure 23.**Map of the relative errors in the predicted and experimentally evaluated torques due to eddy currents in the space of torque and speed for different excitation currents.

**Figure 24.**Map of the simulated normal force with the measured magnetic air gap and speed for different excitation currents.

Parameter | Symbol | Value | Unit |
---|---|---|---|

outer active diameter | ${d}_{\mathrm{o}}$ | $0.25$ | $\mathrm{m}$ |

active length | ${l}_{\mathrm{ecb}}$ | $0.05$ | $\mathrm{m}$ |

active length one rotor | ${l}_{\mathrm{r}}$ | $0.0167$ | $\mathrm{m}$ |

active length stator | ${l}_{\mathrm{s}}$ | $0.02$ | $\mathrm{m}$ |

maximum magnetic air gap | ${\delta}_{\mu \mathrm{max}}$ | $0.0013$ | $\mathrm{m}$ |

maximum mechanical air gap | ${\delta}_{\mathrm{max}}$ | $0.0007$ | $\mathrm{m}$ |

predeformation | ${s}_{0}$ | $0.00065$ | $\mathrm{m}$ |

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## Share and Cite

**MDPI and ACS Style**

Holtmann, C.; Köhler, C.; Weber, C.; Rinderknecht, F.
The Hybrid Brake Model and Its Validation. *Electronics* **2023**, *12*, 2632.
https://doi.org/10.3390/electronics12122632

**AMA Style**

Holtmann C, Köhler C, Weber C, Rinderknecht F.
The Hybrid Brake Model and Its Validation. *Electronics*. 2023; 12(12):2632.
https://doi.org/10.3390/electronics12122632

**Chicago/Turabian Style**

Holtmann, Christoph, Christoph Köhler, Christian Weber, and Frank Rinderknecht.
2023. "The Hybrid Brake Model and Its Validation" *Electronics* 12, no. 12: 2632.
https://doi.org/10.3390/electronics12122632