*Energy and mechanical study – Modeling and optimization of the brake disc*

# INTRODUCTION

In an energy transition policy common to different countries, it has a real need to change our modes of travel and reduce our air emissions. So the bike then appears as an economical and ecological way to move daily. However, some inconveniences related to this mode of transport can lead users to leave this solution aside. The appearance of electric bikes has given a new vision of bicycle mobility. Thus, the energy to be provided to move is partly compensated by an electrical system integrated into the bike.

The aim of this project will be to study an e-bike in order to better understand how it works and to explore avenues for possible topological improvement. We will first make the energy approach of this system then the mechanical study to finally try to optimize certain parameters of the e-bike.

# SPECIFICATIONS

## Composition of the VAE

This project focuses on the study of an electrically assisted bicycle composed of:

- From a motor that helps the cyclist, it is placed at the level of the rear wheel
- A motor battery
- An ECU that controls the engine based on sensor data and cycliste control
- Sensors: they allow to know the value of different parameters (speed of the bike, braking, pedaling frequency …) in real time which allows the ECU to correctly control the engine

## Data and assumptions

The cyclist is characterizedby a power supply of 60W with a pedaling frequency of 1Hz, we consider these constant values in permanent motion. The electric motor then ensures the optimal help: either the bike travels at 25km / h, or the motor develops its maximum power of 150W. We are studying an electrically assisted bike with a power of 250W available at start-up.

It is assumed that the cyclist pedals at constant torque for any life situation and if the pedaling frequency decreases, the power provided by the cyclist also decreases in the same proportions and the same goes for the assisting power developed by the electric motor.

By default the cyclist is helped by the electric machine which behaves like a motor but he can also decide not to be helped in which case the machine behaves like a generator. On the descent the cyclist blocks the pedals and the rear wheeland drives thegenerator. The minimum speed of the bike is 9km/h. The rear wheel has a diameter of 0.7m.

The electric machine can be used in its motor or generator operation.

Assumptions:

- We consider in our case a weight of 80 kg for the cyclist and 23 kg for the bike
**m****= 104kg.** - The loss of transmission power is estimated to be 2.5% of the power lost aerodynamically and through wheel/bitumen friction.
- The drag coefficient is S. CX = 0.4 m².
- We consider a coefficient of friction corresponding to a wheel/road contact such that μc = 0.0055

# Energy study

## E-bike reference speed

The reference speed of the e-bike is being determined. For this we apply the kinetic energy theorem to the system {bike + cyclist}.

### Weight

### Friction on the ground

### Air friction

### Mechanical transmission losses

### Kinetic energy theorem

### Digital application

### E-bike use envelope

## Starting the e-bike

We assume that the power evolves according to the form: P=f(V(t)) according to the following graph

### Formal calculation software

### Numeric method with Excel

In our spreadsheet we use an incremental method as below:

Speed1 (km/h) | Speed (m/s) | Ec(tn) (J) | Pmot (W) | Pfrott (W) | Paero (W) | Ec(t+0,1s) (J) |

V(t) | V(t)/3.6 | 0.5*M*( V(t)/3.6)² | =f(V(t)) | =0.0055*M*g* V(t)/3.6 | =0.5*1.27*0.4*( V(t))**3 | =Ec(t)+(Pmot(t)-Pfrott(t)-Paero(t)) |

=Root(2*Ec(t+Δt)/M)*3.6 | V( t+Δt) | 0.5*M*( V(t+Δt)/3.6)² | =f(V(t+Δt)) | =0.0055*M*g* V(t+Δt)/3.6 | =0.5*1.27*0.4*( V(t+ Δt))**3 | =Ec(t+Δt)+(Pmot(t+Δt)-Pfrott(t+Δt)-Paero(t+Δt)) |

This allows us to trace the evolution of the speed of the bike as a function of time :

Finally we find that the speed of **25km / h is reached for a time of 31.6s.**

## Descent and energy recovery

We try to determine the angle of the slope from which the motor of the bike, placed in a situation of descent without pedaling or assistance, will pass into generative operation.

When we are downhill, the Fundamental Principle of Statics gives us:

However the cyclist does not pedal so we have Pcyclist= 0. We look for the slope from which the motor passes into generating operation, that is to say the tipping point Passistance=0.

We now need to know this speed of descent. It corresponds to the speed for which the motor will rotate at its synchronism speed and therefore switch to generative operation.

We looked for the characteristics of a motor similar to that of the case in our study and found one with a synchronism speed of 3000rpm and a reduction ratio of 15 when transmitting speed to the wheel. We must not forget to also take into account the performance of the system.

We must therefore reach a speed of 29.32 km/h. By injecting this value into the equation from the Fundamental Principle of Statics we have: sin(α)=0.0247 ** α=1.42°**

## Calculation of mechanical action components on the brake disc

In order to size and optimize the brake disc, we need to know the mechanical action components that apply to it.

### Flat braking in maximum speed conditions (25km/h)

We place ourselves in a flat braking situation in maximum speed conditions (25km/h) or on a slope of inclination α.

According to the standard, at a speed of **25km/h it takes a total stop in 5 meters.** A constant deceleration equal to **4.82 m/s² is assumed.**

We make the assumption of a rolling without slipping, so there is no friction due to the ground.

There is only the ground reaction Rsol,perpendicular to the contact, which creates a tangential force **Fsol=μ*Rsol** (Coulomb’s law).

The kinetic theorem is then applied to determine the necessary braking torque:

We find the total downtime, **t _{f}= 1.44s** (found by laying Newton’s 1st law).

## Downhill braking

We simplify by V and we integrate on the braking time. We find the value of the braking torque according to the angle of the slope:

α (°) |
C (N/m) |

0 | 247,7 |

5 | 292,5 |

10 | 336,9 |

15 | 380,7 |

20 | 423,6 |

25 | 465,1 |