# Mathematical Simulations and Analyses of Proportional Electro-Hydraulic Brakes and Anti-Lock Braking Systems in Motorcycles

^{*}

## Abstract

**:**

## 1. Introduction

_{v}is the vehicle speed, and V

_{w}is the wheel speed.

## 2. Proportional Pressure Control Valve

#### 2.1. Proportional Valve Body

_{m}) equation is as follows:

_{em}is the proportional solenoid force; F

_{cal}is the caliper force; P

_{m}is the master cylinder pressure; P

_{cal}is the caliper pressure; A

_{sp}is the spool valve cross-sectional area; and A

_{ol}is the needle valve hole cross-sectional area.

#### 2.2. Proportional Electromagnet

## 3. Mathematical Model and Controller

#### 3.1. Mathematical Model of Motorcycle Motion

#### 3.2. Wheel Braking Model

#### 3.3. Tire and Ground Model

#### 3.4. PEHB Mathematic Model Analysis

#### 3.5. Traditional Discrete Switch Control

#### 3.6. Proportional–Integral–Derivative Controller

## 4. Simulation and Results Analysis

#### 4.1. PEHB System Simulations and Analyses

#### 4.2. Motorcycle ABS Simulation Model with an EHB

#### 4.3. Motorcycle ABS Simulation Model for New PEHB

_{p}= −40, K

_{i}= −5, and K

_{d}= −0.5.

#### 4.4. Analyses of Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The novel control circuit for a proportional electro-hydraulic brake (PEHB) actuator of a motorcycle.

**Figure 14.**The proportional valve relief simulation in 100 bars: (

**a**) step wave command; (

**b**) iron core stroke; and (

**c**) caliper pressure.

**Figure 15.**The proportional valve relief simulation in 100 bars: (

**a**) triangular wave command; (

**b**) iron core stroke; and (

**c**) caliper pressure.

**Figure 18.**EHB performing anti-lock braking on a dry road surface: (

**a**) vehicle speed and wheel speed; (

**b**) slip; and (

**c**) brake pressure.

**Figure 21.**The difference results with and without the bang-bang controller in the PID controller: (

**a**) front wheel; (

**b**) rear wheel.

**Figure 22.**PEHB performing anti-lock braking on a dry road surface: (

**a**) vehicle speed and wheel speed; (

**b**) slip; and (

**c**) brake pressure.

**Figure 23.**PEHB performing anti-lock braking on a wet road surface: (

**a**) vehicle speed and wheel speed; (

**b**) slip; and (

**c**) brake pressure.

Flux Path Model | Mean Path Length | $\mathbf{Permeance}({\mathit{P}}_{\mathit{a}\mathit{i}\mathit{r}}$) |
---|---|---|

I | $\mathrm{g}$ | $2\pi {\mu}_{0}(\mathrm{r}+\frac{\mathrm{g}}{2})\mathrm{h}/\mathrm{g}$ |

II | 1.22$\mathrm{g}$ | $3.3{\mu}_{0}(\mathrm{r}+\frac{\mathrm{g}}{2})$ |

III | 1.22$\mathrm{g}$ | $3.3{\mu}_{0}(\mathrm{r}+\frac{\mathrm{g}}{2})$ |

IV | $\sqrt{\mathrm{g}(\mathrm{g}+\mathrm{t})}$ | $4{\mu}_{0}(\mathrm{r}+\sqrt{\mathrm{g}(\mathrm{g}+\mathrm{t})})\mathrm{ln}(\frac{\mathrm{g}+\mathrm{t}}{\mathrm{g}})$ |

V | $\sqrt{\mathrm{g}(\mathrm{g}+\mathrm{r})}$ | $4{\mu}_{0}(\mathrm{r}+\mathrm{g}-\sqrt{\mathrm{g}(\mathrm{g}+\mathrm{r})})\mathrm{ln}(\frac{\mathrm{g}+\mathrm{r}}{\mathrm{g}})$ |

VI | $\mathrm{z}$ | $\frac{{\mu}_{0}\pi}{z}{(\mathrm{r}+\mathrm{g}-2\mathrm{z}/\pi )}^{2}$ |

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

${M}_{v}$ | total mass of the rider and the motorcycle | 220 Kgf |

${H}_{G}$ | height of the center of the mass of the vehicle | 0.6 m |

${H}_{w}$ | average height of the wind force acting on the motorcycle | 0.7 m |

$L$ | wheelbase between the front and rear wheels of the motorcycle | 1.2 m |

${L}_{a}$ | distances between the front wheels and the center of mass of the vehicle | 0.7 m |

${t}_{f}$ | time-delay constant of the front shock absorber | 0.2 |

${t}_{r}$ | time-delay constant of the rear shock absorber | 0.1 |

${\rho}_{w}$ | air density | 1.18 kg/m^{3} |

${C}_{w}$ | coefficient of air resistance | 0.48 |

${A}_{w}$ | frontal area of the vehicle | 0.55 m^{2} |

${R}_{w}$ | tire radius | 0.21 m |

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

${M}_{s}$ | shuttle shaft and iron core mass | 0.1 Kgf |

$B$ | damping coefficient | 0.20 Kgf-s/mm |

$K$ | spring constant | 0.18 Kgf/mm |

${X}_{c}$ | spring initial compression value | 0.3 mm |

$d$ | spool valve hole diameter | 1.2 mm |

$\theta $ | coning angle of the needle valve | 25 |

$\beta $ | bulk modulus of the liquid | 190 Kgf/mm^{2} |

${P}_{cal}/{V}_{c}$ | Caliper pressure per unit volume | 0.06 Kgf/mm^{5} |

Control Module and Mode | Road State | Braking Time (s) | Stopping Distance (m) | Slip (Steady State) |
---|---|---|---|---|

PEHB + PID | Dry | 2.29 | 20.88 | 0.2 ± 0.06 |

PEHB + PID | Wet | 2.97 | 26.42 | 0.2 ± 0.03 |

EHB + Bang-Bang | Dry | 2.40 | 21.73 | 0.2 ± 0.1 |

Without ABS | Dry | >3 | >34.35 | 1 |

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**MDPI and ACS Style**

Chen, C.-P.; Chiang, M.-H.
Mathematical Simulations and Analyses of Proportional Electro-Hydraulic Brakes and Anti-Lock Braking Systems in Motorcycles. *Actuators* **2018**, *7*, 34.
https://doi.org/10.3390/act7030034

**AMA Style**

Chen C-P, Chiang M-H.
Mathematical Simulations and Analyses of Proportional Electro-Hydraulic Brakes and Anti-Lock Braking Systems in Motorcycles. *Actuators*. 2018; 7(3):34.
https://doi.org/10.3390/act7030034

**Chicago/Turabian Style**

Chen, Che-Pin, and Mao-Hsiung Chiang.
2018. "Mathematical Simulations and Analyses of Proportional Electro-Hydraulic Brakes and Anti-Lock Braking Systems in Motorcycles" *Actuators* 7, no. 3: 34.
https://doi.org/10.3390/act7030034