# A Simple Mono-Dimensional Approach for Lap Time Optimisation

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## Abstract

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## 1. Introduction

## 2. Vehicle Modelling and Performance Bounds

- Adherence of tyres to the asphalt (grip);
- Engine power;
- Aerodynamic loads, such as drag force and lift/downforce.

## 3. Optimisation Approach

- Calculate the state of motion at time step $\mathrm{i}$ (or, at the beginning of a simulation, use the initial conditions).
- Calculate all of the future vehicle states of motion assuming that the driver is braking, until $k\left(\lambda \right)$ has a maximum.
- Check whether the speed satisfies $k\left(\lambda \right){v}^{2}\le {a}_{y,max}$ at any future time step j.
- If the condition above is always met, it means that the driver can accelerate at step i, and then the state of motion at step i + 1 can be calculated. Otherwise, go back one step and impose that the driver is braking at step i − 1.

## 4. Results

- Circumferences (radius of 100 or 200 m)
- Ellipse (semi-axes of 100 and 150 m)
- Straight lines and hairpin bend (two straight lines joined by a 180° constant radius bend, with a radius of 100 m)

#### 4.1. Circumference

#### 4.2. Ellipse

#### 4.3. Straight Lines and Hairpin Bend

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Friction ellipse for $v=50$ km/h, grip effect only (blue line: acceleration, red line: braking).

**Figure 5.**Circumference, no aerodynamics, R = 100 m: (left) speed and (right) accelerations with respect to $s$.

**Figure 6.**Circumference, aerodynamics, R = 100 m: (left) speed and (right) accelerations with respect to $s$.

**Figure 7.**Circumference, aerodynamics, R = 200 m: (left) speed and (right) accelerations with respect to $s$.

**Figure 11.**Straight lines and hairpin bend, no aerodynamics: (left) speed and (right) accelerations with respect to $s$.

**Figure 12.**Straight lines and hairpin bend, aerodynamics: (left) speed and (right) accelerations with respect to $s$.

Quantity | Symbol (unit) | Value |
---|---|---|

Mass | m (kg) | 620 |

Drag factor | ${k}_{x}$ (Ns^{2}/m^{2}) | 0.72 |

Downforce factor | ${k}_{z}$ (Ns^{2}/m^{2}) | 2.15 |

Power | P (kW) | 550 |

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**MDPI and ACS Style**

Lenzo, B.; Rossi, V.
A Simple Mono-Dimensional Approach for Lap Time Optimisation. *Appl. Sci.* **2020**, *10*, 1498.
https://doi.org/10.3390/app10041498

**AMA Style**

Lenzo B, Rossi V.
A Simple Mono-Dimensional Approach for Lap Time Optimisation. *Applied Sciences*. 2020; 10(4):1498.
https://doi.org/10.3390/app10041498

**Chicago/Turabian Style**

Lenzo, Basilio, and Valerio Rossi.
2020. "A Simple Mono-Dimensional Approach for Lap Time Optimisation" *Applied Sciences* 10, no. 4: 1498.
https://doi.org/10.3390/app10041498