James Brandt

This page documents my personal projects, including programming and research. My posts here are summarised versions of projects I have undertaken in my free time. I predominantly use Python on VSC, with plans to use C++ in the future.

7 minute read

Rationale

Dynamic brake proportioning (commonly referred to as brake bias) is essential for optimising F1 lap performance. This precise calibration of the braking force between the front and rear axles is critical for vehicle stability, contributing to time gains. Drivers adjust their brake bias mid-lap to compensate for fluctuating vehicle speed, variations in corner geometry, and tyre conditions.

Determining the optimal brake bias for specific corners and conditions requires quantitative data from real vehicle testing. Thus, the purpose of an optimised mathematical model is to be utilised as a baseline for teams during testing and practise sessions. This project largely explores avoiding the requirement of computational simulation software such as CFD — which has a restricted number of hours of usage per FIA regulations — by simplifying complex concepts into functions and equations. Rudimentary Python code further automates and accelerates these calculations, allowing the user to manually input the conditions of a specific corner to output the calculated optimal brake bias.

Background Information

The Physics of Brake Bias

Brake bias defines the ratio of total braking force distributed between the front and rear axles (for example, 55% brake bias indicates that 55% of the deceleration force is distributed to the front of the vehicle). The requirement for front-axle bias is a necessity due to the following factors:

  • Longitudinal Load Transfer: Upon braking, the normal load on the front tyres swiftly increases while the load on the rear diminishes, due to the inertia generated by the vehicle’s rapid deceleration.

  • Friction Ellipse Constraint: A tyre can generate longitudinal (braking) and lateral (cornering) forces simultaneously, but both cannot be maximised at once. That is, tyres have a certain capacity for forces that must be shared longitudinally and laterally.

  • Stability Constraint: The optimal bias must be precisely constrained to maintain stability. Excessive front bias increases the risks of lock-ups, while excessive rear bias induces rear wheel lock-ups, and increases the chance of the vehicle spinning out. A 90% front to 10% rear bias would violate this constraint, for example.

The optimal bias is the precise point wherein both axles operate at their maximum available friction limit.

Mathematical Model

This mathematical model employs the following equations and simplifications:

  • The vehicle mass $m$, wheelbase $L$, centre-of-gravity (CoG) height $h_{cg}$, and front-wheel CoG distance $a_{cg}$ are fixed.

  • Static normal loads on front and rear axles are given by the simple lever arm of weight distribution:

\[N_{f,static} = m \times g \times \frac{L - a_{cg}}{L}, N_{r,static} = m \times g \times \frac{a_{cg}}{L}\]
  • Under braking deceleration $a_{x}$ the longitudinal transfer is:
\[\Delta N = \frac{m \times a_{x} \times h_{cg}}{L}\]

  • Aero downforce on each axle is approximated as $D = \frac{1}{2} \rho A C_{l} v^{2}$ , separated into front and rear fractions.

  • Lateral acceleration $a_{y}$ is derived from corner radius $R$ and entry speed $v$: $a_{y} = \frac{v^{2}}{R}$. That gives lateral force total $F_{y,total} = m \times a_{y}$.

  • For each axle, the remaining braking (longitudinal) capacity is computed using a simplified friction circle (where $\mu$ is peak tyre friction assumed equal front/rear in this model):

\[F_{f,max} = \sqrt{(\mu \times N)^{2} - (F_{y,used})^{2}}\]

  • The total braking force required: $F_{x,total} = m \times a_{x}$.

  • Therefore, optimum front bias fraction is:

\[bias_{front} = \frac{F_{x,front,max}}{F_{x,front,max} + F_{x,rear,max}}\]
  • The search range (for realism and to ensure vehicular stability) is set to be between 45% and 60% front.

Assumptions, Inclusions and Omissions

  • The following factors are omitted: a variation in friction between the front and rear tyres, tyre compound differences, the effects of weight transfer (pitch or roll), hybrid powertrain regeneration effects, and driver behaviour (whether the driver employs trail braking, or traditionally brakes and then turns).

  • It is assumed that the vehicle is at a constant entry velocity, applying a constant deceleration before cornering. Only a single corner is modelled, though the exact conditions would be modifiable.

  • The model neglects engine braking, temperature changes, tyre wear, path variation and bumpiness.

Methodology

Code Overview

Below are excerpts of some important Python functions used in the project.

# Static normal loads
def static_normal_loads(mass, gravity, load, a_cg):
    Nf = mass * gravity * (load - a_cg) / load
    Nr = mass * gravity * (a_cg) / load
    return Nf, Nr

# Longitudinal load transfer
def longitudinal_load_transfer(mass, ax_brake, h_cg, load):
    return mass * ax_brake * h_cg / load

# Aero downforce functions
def Df_front(velocity):
    return 0.5 * rho * area * Cl_front * velocity**2

def Df_rear(velocity):
    return 0.5 * rho * area * Cl_rear  * velocity**2

# Friction circle remaining braking force
def friction_circle_remaining_longitudinal(mu, N_axle, Fy_used):
    term = (mu * N_axle)**2 - Fy_used**2
    if term < 0:
        return 0.0
    return math.sqrt(term)

# Compute brake bias
def compute_brake_bias_from_radius(v_entry, R_corner, ax_brake):
    ay_target = v_entry**2 / R_corner
    return compute_brake_bias(v_entry = v_entry, ay_target = ay_target, ax_brake = ax_brake)

Full Process and Experimentation

  1. Define car parameters (mass, geometry, aero coefficients, tyre friction coefficient, brake bias search range).
  2. Build the model functions for load transfer, aero load, lateral force split, braking capacity per axle.
  3. For each trial input (entry speed $v$, corner radius $R$, deceleration $a_{x}$) compute optimum front bias fraction.
  4. Compare experimental brake bias values to predicted values.

Results & Analysis

Experimental Trials

Three distinct trials were tested to evaluate the accuracy and precision of the model:

  1. Trial A: Heavy braking zone from high speed into tight corner. $v_{entry}$ = 90m/s, $R_{corner}$ = 60m, $a_{x,brake}$ = 4.5g.
  2. Trial B: Medium braking zone, moderate speed and radius. $v_{entry}$ = 70m/s, $R_{corner}$ = 120m, $a_{x,brake}$ = 3.5g.
  3. Trial C: Light braking into flowing corner. $v_{entry}$ = 60m/s, $R_{corner}$ = 200m, $a_{x,brake}$ = 2.5g.
Trial Experimental Bias (front %) Expected Bias (front %) % Difference
A 50.00 % 55.0 % −9.09 %
B 62.01 % 60.0 % +3.35 %
C 59.73 % 60.0 % −0.45 %

Table 1: Table of results of the three test trials, including the percentage difference between the expected bias values and the experimental bias values. It is important to note that the expected values are somewhat contextual, and affected by factors that have been omitted from this model. They are a general indication of where the experimental values should lie.

Interpretation

  • The results show that the model produces different optimal biases for the different corner scenarios, proving a degree of reliability though not indicative of its accuracy.
  • Trial A being at 50% frontal brake bias highlights a scenario with relatively lesser deceleration, causing the front bias requirement to decrease. This bias is somewhat lower than the expected5 55-60% for heavy braking. Although, this value is not a strictly inaccurate, as 50% bias may be employed under certain conditions.
  • Trial B’s 62% frontal brake bias may demonstrate that the input values represent an unusually heavy braking zone, or that the model possesses a degree of inaccuracy under higher braking conditions.
  • Trial C’s 59.73% frontal brake bias lies within the typical 60% front range expected for this turn. It is an indication of accuracy for this specific trial.

Real-World Comparison

Because there are no publicly published ideal bias percentages for specific track corners, the absolute accuracy of the model cannot be validated against real telemetry. Instead, the expected values utilised from published simulation data intends to highlight whether the simulation data is within an appropriate range, the variations in brake bias is consistent, and the trend is accurate.

Evaluation

Strengths

  • The results are reasonable, mostly accurate, and vary with input corner severity, indicating a successful correlation between the input variables and the brake bias ratio.

  • The model captures key physics concepts in Formula One: deceleration, weight shift, tyre load, and friction.

Weaknesses

  • The tyre model is very simplified through its uniform front and rear friction and its lack in tyre variation.

  • Braking via hybrid energy recovery (which affect rear braking load in modern Formula One cars) is omitted. The rear braking in Formula One is partly handled by engine braking and the ERS system, so the simple rear braking force assumption is an approximation when the hybrid powertrain is considered.

  • The scenario inputs were not tied to publicly-verified data with known bias values. Without actual bias values for those corners, the model cannot be truly validated.

Conclusion

This mini-project has shown that by utilising the fundamental physics of braking, weight transfer, lateral versus longitudinal forces, and aerodynamic load, one can program a moderately accurate model to predict optimal brake bias for a Formula One car in a cornering-braking scenario. Although the model cannot be validated against published corner-specific bias values (which are not publicly available), the results demonstrate credibility: they sit within the expected range (50% to 60% front), and they illustrate the expected correlation between the input variables and the output brake bias.

References

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