# Dynamic Handling Characterization and Set-Up Optimization for a Formula SAE Race Car via Multi-Body Simulation

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## Abstract

**:**

## 1. Introduction

## 2. Basis of Multi-Body Dynamics

**q**is the vector of the system generalized coordinates. It is assumed that the number of equations is equal to the number of unknown variables. By taking first derivative of Equation (1), the velocity kinematic constraint equations are obtained as

**q**, and $\dot{q}$ the vector of generalized velocities. By differentiating Equation (2), one obtains the acceleration kinematic equations:

**M**being the mass matrix,

**Q**the vector containing the external forces and $\mathit{\lambda}$ the vector of Lagrangian multipliers. By adding the m Equations (3), Equation (4), which is a system of n equations (n: number of dofs) in n + m variables, can be solved as

## 3. Full-Vehicle Modeling via Multi-Body Approach

## 4. Dynamic Handling Characterization and Baseline Vehicle Set-Up

- ${\delta}_{p}$ is the front tire steer angle;
- $l$ is the vehicle wheelbase;
- $R$ is the cornering radius;
- ${a}_{y}$ is the lateral acceleration;
- $g$ is the gravity acceleration.

## 5. Set-Up Optimization

#### 5.1. Tilt Test

#### 5.2. Brake Test

- Clutch—the driver inserts the clutch on and increases the engine rev:
- ⚬
- Throttle: step-like control, with a duration of $3\mathrm{s}$, up to $100\%$;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: constant control, unitary value;
- ⚬
- Conditions: stop when $rp{m}_{engine}\gg 6450\mathrm{rpm}$.

- Release—the driver releases the clutch at the right engine rev:
- ⚬
- Throttle: constant control, relative null value;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: step-like control, duration $0.1\mathrm{s}$, up to $0\%$;
- ⚬
- Conditions: terminate when $time=0.1\mathrm{s}$.

- Acceleration_1Gear—the driver releases the clutch and accelerates:
- ⚬
- Throttle: relative step like control, duration of 0.5 s, up to 100%;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: constant control, null value;
- ⚬
- Conditions: terminate when $velocity>65.7\mathrm{km}/\mathrm{h}$.

- Braking—the driver performs emergency braking:
- ⚬
- Throttle: step-like control, duration of 0.5 s, up to 0%;
- ⚬
- Braking: step-like control, starts at $0.5\mathrm{s}$, duration of $0.2\mathrm{s}$, up to $100\%$;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: step-like control, duration of $0.5\mathrm{s}$, up to unitary value;
- ⚬
- Conditions: terminate when $velocity<0.36\mathrm{km}/\mathrm{h}$.

- Increasing inflation pressure to reduce the tire contact patch;
- Increase the camber angle to reduce the tire contact patch;
- Increase brake balance to compensate for weight shift during braking.

- Front/rear inflation pressure: $\mathrm{100,000}\mathrm{Pa}$
- Front/rear camber angle: $-0.043\mathrm{rad}\left(-2.5\xb0\right)$
- Front braking balance: $74\%$

#### 5.3. Acceleration Event

- Clutch—the driver inserts the clutch on and increases the engine rev:
- ⚬
- Throttle: step-like control, duration $3\mathrm{s}$, up to $100\%$;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: constant control, unitary value;
- ⚬
- Conditions: terminate when $rp{m}_{engine}\gg 6450\mathrm{rpm}$.

- Release—the driver releases the clutch:
- ⚬
- Throttle: constant control, relative null value;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: step-like control, duration $0.1\mathrm{s}$, up to $0\%$;
- ⚬
- Conditions: terminate when $time=0.1\mathrm{s}$.

- Acceleration_1Gear—the driver accelerates up to 100%:
- ⚬
- Throttle: relative step-like control, duration of $0.5\mathrm{s}$, up to 100%;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: constant control, first gear;
- ⚬
- Clutch: constant control, null value;
- ⚬
- Conditions: terminate when $rp{m}_{engine}\gg 4000\mathrm{rpm}$.

- Acceleration—the driver keeps accelerating:
- ⚬
- Throttle: constant control, up to $100\%$;
- ⚬
- Braking: constant control, null value;
- ⚬
- Gear: machine-type control;
- ⚬
- Clutch: machine-type control;
- ⚬
- Conditions: terminate when $distance\gg 75\mathrm{m}$.

- Inflation pressure higher for the front tires to reduce the contact patch;
- Camber angle larger for the front tires to reduce the contact patch;
- Softer anti-roll spring stiffness to the rear to increase the elastic-type weight shift on the rear drive tires.

- Front inflation pressure: $\mathrm{100,000}\mathrm{Pa}$
- Front camber angle: $-0.043\mathrm{rad}\left(-2.5\xb0\right)$
- Rear spring stiffness: $\mathrm{43,800}\mathrm{N}/\mathrm{m}\left(250\mathrm{lb}/\mathrm{in}\right)$

#### 5.4. Skidpad Event

- Skidpad—the driver tracks the optimal trajectory:
- ⚬
- Steering: machine-type control based on a predefined trajectory in the form of a point set or “path map”;
- ⚬
- Throttle/Braking: machine-type control to maintain the initial speed constant;
- ⚬
- Gear: relative constant control, null value;
- ⚬
- Clutch: relative constant control, null value;
- ⚬
- Conditions: terminate when $distance\gg 210\mathrm{m}$

- Front/rear camber angle: $-0.043\mathrm{rad}\left(-2.5\xb0\right)$
- Front anti-roll bar height: $0.065\mathrm{m}$
- Rear anti-roll bar height: $0.035\mathrm{m}$

#### 5.5. Discussion and Lessons Learnt

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Vehicle aggregate mass (kg) and Inertia tensor (kg mm^{2}) | 275.0 | I_{XX}: 5.7183132261 × 10^{7}I _{YY}: 8.607411095 × 10^{8}I _{ZZ}: 8.5355796004 × 10^{8}I _{XY}: 7.5633943553 × 10^{−10}I _{ZX}: 1.0636880734 × 10^{8}I _{YZ}: 0.0 |

Center of aggregate mass location (x, y, z mm) | 1547.5, 0.0, 249.9646 | |

Wheelbase (mm) | 1550 | |

Front suspension | Rear suspension | |

Width (mm) | $1150$ | $1100$ |

Kingpin angle (°) | $6.13$ | $5.68$ |

Scrub Radius (mm) | $28.7$ | $30.6$ |

Spindle length (mm) | $52.4$ | $53.4$ |

Caster angle (°) | $5.75$ | $5.32$ |

Mechanical trail (mm) | $25.2$ | $23.4$ |

Anti-roll bar diameter (mm) and length (mm) | 8 200 | 8 130 |

**Table A2.**Hardpoint locations of the front right suspension expressed in the model global reference system.

Hardpoint | Loc_x | Loc_y | Loc_z |

arb_droplink | 763.5 | 253.807 | 609.29 |

BC_axis | 750 | 245 | 555 |

BC_center | 763.5 | 245 | 555 |

damper_inboard | 763.5 | 58.6 | 600 |

damper_outboard | 763.5 | 245.1 | 592 |

lca_front | 641.5 | 245 | 151 |

lca_outer | 763.5 | 529 | 161 |

lca_rear | 867.5 | 245 | 158 |

prod_inboard | 763.5 | 310 | 617.3 |

prod_outboard | 763.5 | 496.6 | 182.7 |

ride_height | 641.5 | 245 | 151 |

tierod_inner | 772.5 | 250 | 172.5 |

tierod_outer | 689.3 | 551.7 | 180.2 |

uca_front | 578.9 | 245 | 302 |

uca_outer | 778.5 | 513 | 310 |

uca_rear | 863.1 | 245 | 294 |

wheel_center | 772.5 | 575 | 232.45 |

**Table A3.**Hardpoint locations of the rear right suspension expressed in the model global reference system.

Hardpoint | Loc_x (mm) | Loc_y (mm) | Loc_z (mm) |
---|---|---|---|

BC_axis | 2200 | 259.99 | 399 |

BC_center | 2288.73 | 259.99 | 399 |

damper_inboard | 2288.73 | 74 | 443.8 |

damper_outboard | 2288.73 | 262.39 | 437.93 |

drive_shaft_inr | 2322.5 | 150 | 228.6 |

lca_front | 2082.5 | 259.99 | 152 |

lca_outer | 2313.5 | 504 | 155 |

lca_rear | 2392.5 | 260.01 | 131 |

prod_inboard | 2288.73 | 328.39 | 444.22 |

prod_outboard | 2288.73 | 504 | 156.59 |

ride_height | 2392.5 | 260.01 | 131 |

tierod_inner | 2226.97 | 260 | 200 |

tierod_outer | 2257 | 523.02 | 232.45 |

uca_front | 2092.5 | 259.99 | 276 |

uca_outer | 2328.5 | 488 | 316 |

uca_rear | 2402.5 | 260 | 266 |

wheel_center | 2322.5 | 550 | 232.45 |

**Table A4.**Hardpoint locations of the rear/left antiroll bar system expressed in the model global reference frame.

Front Antiroll System | |||

Hardpoint | Loc_x (mm) | Loc_y (mm) | Loc_z (mm) |

droplink/left | 763.5 | −253.807 | 609.29 |

droplink/right | 763.5 | 253.807 | 609.29 |

arb_bend | 763.5 | 0 | 580 |

arb_center | 963.5 | 0 | 580 |

Rear Antiroll System | |||

Hardpoint | Loc_x (mm) | Loc_y (mm) | Loc_z (mm) |

droplink/left | 2288.73 | -259.779 | 419 |

droplink/right | 2288.73 | 259.779 | 419 |

arb_bend | 2288.73 | 0 | 423.797 |

arb_center | 2418.73 | 0 | 423.797 |

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**Figure 4.**Skid-pad virtual test via successive step-steer maneuvers, vehicle output-lateral acceleration.

**Figure 8.**Tilt test—best set-up; til/tir: left/right tire. Please note that since the test is performed at a constant tilt rate of 0.017 rad/s (1°/s), the labels along the x-axis correspond to the platform tilt angle in degree units.

**Figure 12.**Acceleration Event-Driver input. (

**a**): the y-axis shows the steering demand (°), the gear ratio, and the clutch demand that varies in the window [0 1]. (

**b**): brake and throttle demand ranging between 0 and 100%.

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

$l$ | $1.55\mathrm{m}$ |

${\delta}_{v}$ | $0.170\mathrm{rad}$ |

${\delta}_{p}$ | $0.031\mathrm{rad}$ |

${\tau}_{s}$ | $0.182$ |

Baseline Set-Up | Front | Rear |
---|---|---|

Tire pressure | $\mathrm{70,000}\mathrm{Pa}$ | $\mathrm{70,000}\mathrm{Pa}$ |

Camber angle | $0\mathrm{rad}$ | $0\mathrm{rad}$ |

Toe-in | $0\mathrm{rad}$ | $0\mathrm{rad}$ |

Anti-roll rocker height | $0.050\mathrm{m}$ | $0.050\mathrm{m}$ |

Suspension spring stiffness | $\mathrm{61,294.4}\mathrm{N}/\mathrm{m}\left(350\mathrm{lb}/\mathrm{in}\right)$ | $\mathrm{61,294.4}\mathrm{N}/\mathrm{m}\left(350\mathrm{lb}/\mathrm{in}\right)$ |

Suspension damper configuration | $C13R13\left(0-5/0-5\right)$ | $C13R13\left(0-5/0-5\right)$ |

Brake balance | $62\%$ | $38\%$ |

Baseline Set-Up | Front | Rear |

Tire pressure | $\mathrm{70,000}\mathrm{Pa}$ | $\mathrm{70,000}\mathrm{Pa}$ |

Camber angle | $0\mathrm{rad}$ | $0\mathrm{rad}$ |

Toe-in | $0\mathrm{rad}$ | $0\mathrm{rad}$ |

Anti-roll rocker height | $0.050\mathrm{m}$ | $0.050\mathrm{m}$ |

Spring stiffness | $\mathrm{61,294.4}\mathrm{N}/\mathrm{m}\left(350\mathrm{lb}/\mathrm{in}\right)$ | $\mathrm{61,294.4}\mathrm{N}/\mathrm{m}\left(350\mathrm{lb}/\mathrm{in}\right)$ |

Damper configuration | $\mathrm{C}13\mathrm{R}13\left(0-5/0-5\right)$ | $\mathrm{C}13\mathrm{R}13\left(0-5/0-5\right)$ |

Brake balance | $62\%$ | $38\%$ |

Set-Up Tilt Test | Front | Rear |

Anti-roll rocker height | $0.060\mathrm{m}$ | $0.050\mathrm{m}$ |

Set-Up Brake Test | Front | Rear |

Tire pressure | $\mathrm{100,000}\mathrm{Pa}$ | $\mathrm{100,000}\mathrm{Pa}$ |

Camber angle | $-0.043\mathrm{rad}\left(-2.5\xb0\right)$ | $-0.043\mathrm{rad}\left(-2.5\xb0\right)$ |

Brake balance | $74\%$ | $26\%$ |

Set-Up Acceleration Event | Front | Rear |

Tire pressure | $\mathrm{100,000}\mathrm{Pa}$ | $\mathrm{70,000}\mathrm{Pa}$ |

Camber angle | $-0.043\mathrm{rad}\left(-2.5\xb0\right)$ | $0\mathrm{rad}$ |

Spring stiffness | $\mathrm{61,294.4}\mathrm{N}/\mathrm{m}\left(350\mathrm{lb}/\mathrm{in}\right)$ | $\mathrm{43,800}\mathrm{N}/\mathrm{m}\left(250\mathrm{lb}/\mathrm{in}\right)$ |

Set-Up Skidpad Event | Front | Rear |

Camber angle | $-0.043\mathrm{rad}\left(-2.5\xb0\right)$ | $-0.043\mathrm{rad}\left(-2.5\xb0\right)$ |

Anti-roll rocker height | $0.065\mathrm{m}$ | $0.035\mathrm{m}$ |

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**MDPI and ACS Style**

Balena, M.; Mantriota, G.; Reina, G.
Dynamic Handling Characterization and Set-Up Optimization for a Formula SAE Race Car via Multi-Body Simulation. *Machines* **2021**, *9*, 126.
https://doi.org/10.3390/machines9060126

**AMA Style**

Balena M, Mantriota G, Reina G.
Dynamic Handling Characterization and Set-Up Optimization for a Formula SAE Race Car via Multi-Body Simulation. *Machines*. 2021; 9(6):126.
https://doi.org/10.3390/machines9060126

**Chicago/Turabian Style**

Balena, Matteo, Giacomo Mantriota, and Giulio Reina.
2021. "Dynamic Handling Characterization and Set-Up Optimization for a Formula SAE Race Car via Multi-Body Simulation" *Machines* 9, no. 6: 126.
https://doi.org/10.3390/machines9060126