A Symbolic Physics Solver for FSAE

About

Everything Aero is a symbolic aerodynamics and vehicle-dynamics calculator designed for Formula Student / Formula SAE Rules Quiz preparation and concept checking. It is open source and can easily be expanded to other fields/domains.

If you want to contribute missing equations, please add an issue or PR on GitHub!

Instead of hard-coded calculators, this tool uses a large library of physics-based equations (aero forces, load transfer, tyre models, drag, moments, etc.) and automatically assembles and solves the relevant system based on the inputs you provide.

How to use it:

• Each symbol needs to be appended with a numeric index (e.g. rho0, v0, A0, C_L0). All available symbols and formulas are listed below the calculator.
• Enter known values or constraints as equations (e.g. v0 = 30, C_L0 = -3.2)
• Optionally add assumptions that might lead to an inconsistent system (e.g. "assume no lateral weight transfer" can be added as "W_lat0=0")
• Specify which variables you want to solve for. This is used when dropping equations for over-constrained systems and when formatting the output.
• There are many possible aerodynamic F_z available. To do vehicle dynamics, tell the solver which normal force to use for the generic F_z e.g. "F_z0=F_z_df0" to use only the downforce. For e.g. Reynolds number scaling calculations, add "Re0=Re1".

The solver determines which equations apply, builds a consistent system, and returns the solution along with the equations actually used.

The calculator itself runs a full python environment in the browser. It therefore needs a few seconds to start and install packages with pip (in the browser environment only!). Furthermore sympy is loaded after the first solve button press. So the first press will take a few seconds.

All known symbols and equations are listed below.

If the tool fails to load, reload the page and check the console (press F12 key).

Calculator

Usage

The calculator offers three input fields: Equations, Assumptions, and Looking for. Each field accepts one input per line. After entering your inputs, click Solve to run the symbolic solver. An example input is shown in fig. 1 below.

Conceptual Overview

Everything Aero does not follow a fixed calculation order. Instead, it collects all known symbols and equations, automatically activates all solvable equation templates, solves the resulting system symbolically, and relaxes assumptions if the system is over-constrained. You only need to provide enough information — not a specific workflow.

Indexed Variables

All symbols must be indexed. Each index represents a separate operating point. Typical use cases include \(v_0\), \(v_1\) for two different speeds, \(\rho_0\), \(\rho_1\) for different atmospheric conditions, or \(F_{z0}\), \(F_{z1}\) for before and after a setup change. If your problem involves only one condition, use index 0 everywhere.

Note: if any symbol is missing an index, the solver will not run.

v0
rho0
F_z_total1

Input Fields

Equations

The Equations field is used for known numerical values, known relationships from the problem statement, and custom equations not already built into the solver. All equations entered here are treated as hard constraints — if the system is inconsistent, no solution will be found.

v0 = 30
rho0 = 1.225
Re0 = Re1

Assumptions

Assumptions are soft constraints that may override physics. This field is optional. If assumptions conflict with physical equations, the solver will try to preserve equations related to variables in the Looking for field and relax assumption equations if necessary. This mirrors how assumptions are treated in real engineering problems.

W_lat0 = 0

Looking for

This field is optional but strongly recommended. It serves two purposes: it highlights variables in the LaTeX solution output, and it tells the solver which equations to prioritize when resolving conflicts.

F_z_total_cornering_fl
F_z_total_cornering_fr

Solve

Once inputs are entered, click Solve. Scalar-only problems solve instantly; symbolic functions may take a few seconds. If no solution is found, check indexing first.

Custom Equations

Custom equations can be added directly in the Equations field, and new symbols are allowed. The example below solves FS-Quiz Question 366 and correctly finds \(v_1 = 86.6666\).

Re0 = Re1
nu0 = nu1
rho0 = rho1

Troubleshooting

If nothing happens after pressing Solve, the most likely cause is that not every symbol has an index correctly appended to it. Add the correct index and solve again.

How It Works

Everything Aero is built around a generic symbolic equation manager. Instead of hard-coding calculation steps, the solver dynamically assembles and solves a system of equations based on a large library of equation templates and which variables are present or provided.

High-Level Flow

When you submit inputs, the solver extracts all symbols and creates indexed variables, then activates any equation templates that are solvable given the known symbols. A symbolic system is assembled and solved using SymPy. If the system is over-constrained, assumptions are relaxed iteratively until a solution is found. The result may include functions, not just scalar values.

Architecture

The calculator is split into three independent parts: the equation engine (generic and reusable), equation templates (domain-specific physics), and the web UI built with ngapp. This separation makes it straightforward to add new equations, create calculators for other domains, or extend the physics without touching the solver. Deployment and documentation are handled automatically via GitHub Actions.

Equation Templates

All physics equations are defined as templates and registered with the equation manager. A Reynolds number template looks like this:

aero_eq_manager.add_equation_template(
    equations_to_add=["Re{{i}}=L{{i}} * rho{{i}} * v{{i}} / eta{{i}}"],
    relevant_vars=[
        ("Re",  "Reynolds number [-]"),
        ("L",   "Characteristic length [m]"),
        ("rho", "Air density [kg/m³]"),
        ("v",   "Velocity [m/s]"),
        ("eta", "Dynamic viscosity [Pa·s]")
    ],
    vars_to_check=[(["Re", "L"], 1)],
    name="Reynolds Number"
)

equations_to_add is a list of strings in lhs = rhs format. Symbols use {{i}} and {{j}} as index placeholders, and multi-index equations are expanded automatically.

relevant_vars is a list of (symbol, description) tuples used for documentation and UI clarity. Descriptions should be precise to avoid input errors.

vars_to_check is a list of ([symbols], n) tuples that gate template activation: a template is only added to the system if at least n of the listed symbols already exist. This prevents unsolvable equations from being included and keeps the solver fast.

Assumption Handling

If the system is over-constrained, equations involving assumption symbols are selectively removed in an iterative process. Variables listed in Looking for are preserved where possible, and relaxation continues until a solution is found or all options are exhausted.

Output

Solutions are symbolic and displayed in LaTeX, with variables from the Looking for field highlighted. This allows further analysis such as differentiation or evaluation at specific parameter values.

FS Rules Quiz Quick Start

This page shows how to use Everything Aero to solve typical FS Rules Quiz problems quickly and reliably. If the problem can be solved with equations, this tool can usually handle it with minimal setup.

Mental Model

Everything Aero is not a step-by-step calculator. You do not choose formulas manually, decide calculation order, or worry about solving intermediate variables. You only write down what the problem gives you, add any explicit assumptions, and tell the solver what you want to find. The solver figures out the rest.

Every Symbol Must Have an Index

Every variable must have an index appended. If the problem describes only one condition, use index 0 throughout. The following is correct:

v0 = 30
rho0 = 1.225

Writing v = 30 or rho = 1.225 without an index will prevent the solver from running.

Typical FS-Quiz Workflow

Step 1: Write down given values

Copy everything the problem gives you numerically into the Equations field.

v0 = 30
rho0 = 1.225
C_L0 = -3.0
A0 = 1.2
m0 = 280

Step 2: Add stated assumptions

If the problem states things like "assume no lateral load transfer", "neglect drag", or "ignore aerodynamic moments", input them in Assumptions. Assumptions are treated as soft constraints and can be relaxed if needed.

W_lat0 = 0

Step 3: Ask the question

Put exactly what the question asks for into Looking for. This tells the solver what to highlight in the output and what to prioritize if equations conflict. This field is optional but strongly recommended.

Re0
F_z_total_cornering_fl
v1

Example 1: Reynolds Number

A car drives at 30 m/s with a characteristic length of 0.3 m, air density 1.225 kg/m³, and dynamic viscosity 1.8×10⁻⁵ Pa·s. We want the Reynolds number.

Equations:

v0 = 30
L0 = 0.3
rho0 = 1.225
eta0 = 1.8e-5

Looking for:

Re0

The solver returns \(Re_0 = 612500.0\).

Example 2: Wind Tunnel Scaling

A 50% scale model must match the Reynolds number of the full car. Air properties are identical and full-scale speed is 30 m/s. We want the required tunnel speed.

Equations:

Re0 = Re1
L0 = 1.0
L1 = 0.5
v0 = 30
rho0 = rho1
eta0 = eta1

Looking for:

v1

The solver returns \(v_1 = 60.0\).

Final Advice

When in doubt: put everything the problem gives you in Equations, put stated assumptions in Assumptions, and put the question in Looking for. Let the solver do the rest.

Known Symbols

A	Cross-sectional area [m²]
A_ref	Aerodynamic reference area [m²]
C_L	Lift coefficient [-]
C_d	Drag coefficient [-]
C_rr	Rolling resistance coefficient [-]
C_y	Side force coefficient [-]
Delta_F_z	Change in tyre normal load [N]
Delta_Fz_drag_f	Front axle normal load change from drag moment [N]
Delta_Fz_drag_fl	Front-left normal load change from drag moment [N]
Delta_Fz_drag_fr	Front-right normal load change from drag moment [N]
Delta_Fz_drag_r	Rear axle normal load change from drag moment [N]
Delta_Fz_drag_rl	Rear-left normal load change from drag moment [N]
Delta_Fz_drag_rr	Rear-right normal load change from drag moment [N]
Delta_Fz_lat	Total lateral normal load transfer [N]
Delta_Fz_lat_f	Front axle lateral load transfer [N]
Delta_Fz_lat_fl	Front-left load change from lateral effects [N]
Delta_Fz_lat_fr	Front-right load change from lateral effects [N]
Delta_Fz_lat_r	Rear axle lateral load transfer [N]
Delta_Fz_lat_rl	Rear-left load change from lateral effects [N]
Delta_Fz_lat_rr	Rear-right load change from lateral effects [N]
Delta_Fz_long	Total longitudinal load transfer [N]
Delta_Fz_long_f	Front axle normal load change from longitudinal effects [N]
Delta_Fz_long_fl	Front-left load change from longitudinal effects [N]
Delta_Fz_long_fr	Front-right load change from longitudinal effects [N]
Delta_Fz_long_geom	Geometric load transfer component [N]
Delta_Fz_long_r	Rear axle normal load change from longitudinal effects [N]
Delta_Fz_long_rl	Rear-left load change from longitudinal effects [N]
Delta_Fz_long_rr	Rear-right load change from longitudinal effects [N]
Delta_Fz_long_total	Net longitudinal load transfer [N]
Delta_delta_z_tire	Change in tyre deflection [m]
F_bump	Bump stop force [N] (active only if delta_z_bump > 0)
F_centripetal	Required centripetal force [N]
F_d	Aerodynamic drag force [N]
F_f_max	Maximum available friction force magnitude [N]
F_rr	Rolling resistance force [N]
F_slope	Longitudinal force due to road slope [N]
F_spring	Suspension spring force [N]
F_tan	Resultant tangential force magnitude [N]
F_thrust_req	Required thrust force [N]
F_x_drive	Total driven longitudinal force at wheels [N]
F_x_inertia	Longitudinal inertial force (vehicle frame) [N]
F_x_max	Maximum longitudinal tyre force [N]
F_x_tire	Longitudinal tractive force [N]
F_x_tire_max	Maximum available longitudinal tyre force [N]
F_y_aero	Aerodynamic side force [N]
F_y_available	Available lateral force [N]
F_y_inertia	Lateral inertial force (vehicle frame) [N]
F_y_max	Maximum lateral force [N]
F_z	Generic normal force on tyre [N]
F_z_arb_left	ARB vertical force at left wheel [N]
F_z_arb_right	ARB vertical force at right wheel [N]
F_z_df	Aerodynamic downforce [N]
F_z_df_f	Front axle aerodynamic downforce [N]
F_z_df_fl	Front-left aerodynamic downforce [N]
F_z_df_fr	Front-right aerodynamic downforce [N]
F_z_df_r	Rear axle aerodynamic downforce [N]
F_z_df_rl	Rear-left aerodynamic downforce [N]
F_z_df_rr	Rear-right aerodynamic downforce [N]
F_z_f	Front axle normal force [N]
F_z_fl	Front-left wheel normal force [N]
F_z_fr	Front-right wheel normal force [N]
F_z_r	Rear axle normal force [N]
F_z_rl	Rear-left wheel normal force [N]
F_z_rr	Rear-right wheel normal force [N]
F_z_static	Static normal force [N]
F_z_static_df_drag_fl	Front-left normal force: static + downforce + drag moment [N]
F_z_static_df_drag_fr	Front-right normal force: static + downforce + drag moment [N]
F_z_static_df_drag_rl	Rear-left normal force: static + downforce + drag moment [N]
F_z_static_df_drag_rr	Rear-right normal force: static + downforce + drag moment [N]
F_z_static_df_f	Front axle normal force: static + downforce [N]
F_z_static_df_fl	Front-left normal force: static + downforce [N]
F_z_static_df_fr	Front-right normal force: static + downforce [N]
F_z_static_df_r	Rear axle normal force: static + downforce [N]
F_z_static_df_rl	Rear-left normal force: static + downforce [N]
F_z_static_df_rr	Rear-right normal force: static + downforce [N]
F_z_static_f	Static front axle normal force [N]
F_z_static_fl	Static front-left normal force [N]
F_z_static_fr	Static front-right normal force [N]
F_z_static_r	Static rear axle normal force [N]
F_z_static_rl	Static rear-left normal force [N]
F_z_static_rr	Static rear-right normal force [N]
F_z_susp	Vertical force transmitted by suspension at wheel [N]
F_z_susp_total	Total suspension vertical force at wheel [N]
I_z	Yaw moment of inertia [kg·m²]
L	Characteristic length [m]
L_ISA	ISA temperature lapse rate [K/m]
MR	Suspension motion ratio [-]
M_pitch_aero	Total aerodynamic pitching moment about CG [N·m]
M_pitch_df	Pitching moment caused by downforce distribution [N·m]
M_pitch_drag	Pitching moment caused by drag force [N·m]
M_yaw_aero	Aerodynamic yawing moment about CG [N·m]
M_z	Sum of yaw moments about CG [N·m]
P_required	Required propulsion power [W]
P_trac	Tractive power at the wheels [W]
P_x	Longitudinal power at wheels [W]
R	Specific gas constant for air [J/(kg·K)]
R_turn	Turn radius to CG [m]
Re	Reynolds number [-]
T	Ambient temperature [K]
T_0	Sea level temperature [K]
T_arb	Anti-roll bar torque equivalent [N·m]
W	Vehicle weight (gravitational force) [N]
a_CG	CG distance to front axle [m]
a_brake_aero	Maximum braking deceleration with aerodynamic downforce [m/s²]
a_lat	Lateral acceleration [m/s²]
a_lat_max	Maximum lateral acceleration (friction-limited) [m/s²]
a_long	Longitudinal acceleration (positive forward) [m/s²]
a_long_max	Maximum longitudinal acceleration (friction-limited) [m/s²]
a_x_max	Maximum traction-limited longitudinal acceleration [m/s²]
b_CG	CG distance to rear axle [m]
balance_df	Aerodynamic balance (front fraction) [-]
balance_lat	Lateral load transfer balance (front fraction) [-]
beta_sideslip	Vehicle sideslip angle at CG [rad]
delta_avg	Average front steering angle [rad]
delta_inner	Inner wheel steering angle [rad]
delta_outer	Outer wheel steering angle [rad]
delta_z_bump	Bump stop compression [m]
delta_z_clearance	Bump stop clearance [m]
delta_z_heave	Chassis heave displacement [m]
delta_z_spring	Spring compression [m]
delta_z_susp	Suspension deflection [m]
delta_z_tire	Tyre vertical deflection [m]
delta_z_wheel	Wheel vertical displacement [m]
delta_z_wheel_f	Front axle wheel vertical displacement [m]
delta_z_wheel_left	Left wheel vertical displacement [m]
delta_z_wheel_r	Rear axle wheel vertical displacement [m]
delta_z_wheel_right	Right wheel vertical displacement [m]
eta	Dynamic viscosity [Pa·s]
eta_aero	Aerodynamic efficiency (lift-to-drag ratio) [-]
f_load	Dimensionless load influence factor [-]
friction_util	Friction utilisation ratio (≤1 = within grip) [-]
g	Gravitational acceleration [m/s²]
h	Geometric height of point [m]
k_arb	Anti-roll bar stiffness [N/m]
k_bump	Bump stop stiffness [N/m]
k_roll_f	Front axle roll stiffness [N·m/rad]
k_roll_r	Rear axle roll stiffness [N·m/rad]
k_spring	Suspension spring stiffness [N/m]
k_tire	Tyre vertical stiffness [N/m]
k_vert	Effective vertical stiffness at wheel [N/m]
k_wheel	Effective wheel rate [N/m]
l_wb	Wheelbase [m]
m	Vehicle mass [kg]
mu	Tyre-road friction coefficient [-]
mu_0	Reference friction coefficient [-]
mu_load_dep	Load-dependent friction coefficient [-]
mu_long	Longitudinal friction coefficient [-]
nu	Kinematic viscosity [m²/s]
p	Static pressure at point [Pa]
p_0	Sea level standard atmospheric pressure [Pa]
phi_roll_f	Front roll stiffness distribution [-]
r_dot	Yaw acceleration [rad/s²]
rho	Density at altitude h [kg/m³]
rho_0	Sea level standard density [kg/m³]
s_coast	Coast-down distance [m]
t_coast	Time to coast from v to v_final [s]
t_track	Track width [m]
theta_anti	Anti-dive / anti-squat angle [rad]
theta_bank	Bank angle [rad]
theta_pitch	Pitch angle (small-angle) [rad]
theta_slope	Road slope angle (positive uphill) [rad]
v	Flow velocity [m/s]
v_final	Final speed [m/s]
v_max_corner	Maximum cornering speed (friction-limited) [m/s]
v_x	Longitudinal velocity component [m/s]
v_y	Lateral velocity [m/s]
x_CG	Center of gravity position [m]
x_COP	Center of pressure position from front axle [m]
x_rel_aero	Longitudinal lever arm from CG to aerodynamic center [m]
z_CG	Center of gravity height [m]
z_COP	Aerodynamic center height [m]
z_chassis	Chassis reference vertical position [m]
z_rel_aero	Vertical lever arm from CG to COP [m]
z_wheel	Wheel center vertical position [m]

Known Equations

$$ \begin{aligned} g_{i} h_{i} \rho_{i} + p_{i} + \frac{\rho_{i} v_{i}^{2}}{2} &= g_{j} h_{j} \rho_{j} + p_{j} + \frac{\rho_{j} v_{j}^{2}}{2} \end{aligned} $$ $$ \begin{aligned} \frac{v_{i}}{v_{j}} &= \frac{A_{j}}{A_{i}} \end{aligned} $$ $$ \begin{aligned} Re_{i} &= \frac{L_{i} \rho_{i} v_{i}}{\eta_{i}} \end{aligned} $$ $$ \begin{aligned} \nu_{i} &= \frac{\eta_{i}}{\rho_{i}} \end{aligned} $$ $$ \begin{aligned} F_{d i} &= \frac{A_{ref i} C_{d i} \rho_{i} v_{i}^{2}}{2} \end{aligned} $$ $$ \begin{aligned} F_{y aero i} &= \frac{A_{ref i} C_{y i} \rho_{i} v_{i}^{2}}{2} \end{aligned} $$ $$ \begin{aligned} M_{yaw aero i} &= F_{y aero i} x_{rel aero i} \end{aligned} $$ $$ \begin{aligned} v_{i} &= \sqrt{v_{x i}^{2} + v_{y i}^{2}} \end{aligned} $$ $$ \begin{aligned} \beta_{sideslip i} &= \operatorname{atan}{\left(\frac{v_{y i}}{v_{x i}} \right)} \end{aligned} $$ $$ \begin{aligned} - F_{d i} - F_{rr i} + F_{x drive i} - F_{x inertia i} &= 0 \end{aligned} $$ $$ \begin{aligned} F_{rr i} &= C_{rr i} F_{z i} \end{aligned} $$ $$ \begin{aligned} P_{x i} &= F_{x drive i} v_{i} \end{aligned} $$ $$ \begin{aligned} M_{z i} &= I_{z i} r_{dot i} \end{aligned} $$ $$ \begin{aligned} F_{x max i} &= F_{z i} \mu_{long i} \end{aligned} $$ $$ \begin{aligned} F_{z f i} &= F_{z fl i} + F_{z fr i} \\ F_{z r i} &= F_{z rl i} + F_{z rr i} \end{aligned} $$ $$ \begin{aligned} p_{i} &= R_{i} T_{i} \rho_{i} \end{aligned} $$ $$ \begin{aligned} T_{i} &= - L_{ISA i} h_{i} + T_{0 i} \end{aligned} $$ $$ \begin{aligned} t_{coast i} &= \frac{2.0 m_{i} \log{\left(\frac{v_{i}}{v_{final i}} \right)}}{A_{ref i} C_{d i} \rho_{i} v_{i}} \end{aligned} $$ $$ \begin{aligned} s_{coast i} &= \frac{2.0 m_{i} \left(- v_{final i} + v_{i}\right)}{A_{ref i} C_{d i} \rho_{i}} \end{aligned} $$ $$ \begin{aligned} F_{thrust req i} &= F_{d i} + F_{x inertia i} \end{aligned} $$ $$ \begin{aligned} P_{required i} &= F_{thrust req i} v_{i} \end{aligned} $$ $$ \begin{aligned} A_{i} \rho_{i} v_{i} &= A_{j} \rho_{j} v_{j} \end{aligned} $$ $$ \begin{aligned} a_{CG i} &= x_{CG i} \\ b_{CG i} &= l_{wb i} - x_{CG i} \end{aligned} $$ $$ \begin{aligned} k_{tire i} &= \frac{F_{z i}}{\delta_{z tire i}} \end{aligned} $$ $$ \begin{aligned} \Delta_{\delta z tire i} &= \frac{\Delta_{F z i}}{k_{tire i}} \end{aligned} $$ $$ \begin{aligned} z_{wheel i} &= - \delta_{z tire i} + z_{chassis i} \end{aligned} $$ $$ \begin{aligned} \delta_{z spring i} &= MR_{i} \delta_{z wheel i} \end{aligned} $$ $$ \begin{aligned} F_{spring i} &= \delta_{z spring i} k_{spring i} \end{aligned} $$ $$ \begin{aligned} F_{z susp i} &= \frac{F_{spring i}}{MR_{i}} \end{aligned} $$ $$ \begin{aligned} \delta_{z wheel i} &= \delta_{z susp i} + \delta_{z tire i} \end{aligned} $$ $$ \begin{aligned} \delta_{z susp i} &= \frac{F_{z i} MR_{i}}{k_{spring i}} \end{aligned} $$ $$ \begin{aligned} k_{wheel i} &= \frac{k_{spring i}}{MR_{i}^{2}} \end{aligned} $$ $$ \begin{aligned} T_{arb i} &= k_{arb i} \left(\delta_{z wheel left i} - \delta_{z wheel right i}\right) \end{aligned} $$ $$ \begin{aligned} F_{z arb left i} &= \frac{T_{arb i}}{t_{track i}} \\ F_{z arb right i} &= - \frac{T_{arb i}}{t_{track i}} \end{aligned} $$ $$ \begin{aligned} \delta_{z wheel f i} &= a_{CG i} \theta_{pitch i} + \delta_{z heave i} \\ \delta_{z wheel r i} &= - b_{CG i} \theta_{pitch i} + \delta_{z heave i} \end{aligned} $$ $$ \begin{aligned} \delta_{z bump i} &= - \delta_{z clearance i} + \delta_{z susp i} \end{aligned} $$ $$ \begin{aligned} F_{bump i} &= \delta_{z bump i} k_{bump i} \end{aligned} $$ $$ \begin{aligned} F_{z susp total i} &= F_{bump i} + F_{z susp i} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz long geom i} &= F_{x tire i} \tan{\left(\theta_{anti i} \right)} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz long total i} &= - \Delta_{Fz long geom i} + \Delta_{Fz long i} \end{aligned} $$ $$ \begin{aligned} \frac{1}{k_{vert i}} &= \frac{1}{k_{wheel i}} + \frac{1}{k_{tire i}} \end{aligned} $$ $$ \begin{aligned} \phi_{roll f i} &= \frac{k_{roll f i}}{k_{roll f i} + k_{roll r i}} \end{aligned} $$ $$ \begin{aligned} \delta_{inner i} &= \operatorname{atan}{\left(\frac{l_{wb i}}{R_{turn i} - \frac{t_{track i}}{2}} \right)} \end{aligned} $$ $$ \begin{aligned} \delta_{outer i} &= \operatorname{atan}{\left(\frac{l_{wb i}}{R_{turn i} + \frac{t_{track i}}{2}} \right)} \end{aligned} $$ $$ \begin{aligned} \delta_{avg i} &= \frac{\delta_{inner i}}{2} + \frac{\delta_{outer i}}{2} \end{aligned} $$ $$ \begin{aligned} R_{turn i} &= \frac{l_{wb i}}{\tan{\left(\delta_{avg i} \right)}} \end{aligned} $$ $$ \begin{aligned} F_{y available i} &= F_{z i} \left(\mu_{i} \cos{\left(\theta_{bank i} \right)} + \sin{\left(\theta_{bank i} \right)}\right) \end{aligned} $$ $$ \begin{aligned} F_{centripetal i} &= F_{y available i} \end{aligned} $$ $$ \begin{aligned} F_{z df f i} &= F_{z df i} balance_{df i} \\ F_{z df r i} &= F_{z df i} \left(1 - balance_{df i}\right) \end{aligned} $$ $$ \begin{aligned} \eta_{aero i} &= \frac{C_{L i}}{C_{d i}} \end{aligned} $$ $$ \begin{aligned} p_{i} &= p_{0 i} e^{- \frac{g_{i} h_{i} \rho_{0 i}}{p_{0 i}}} \end{aligned} $$ $$ \begin{aligned} \rho_{i} &= \rho_{0 i} e^{- \frac{g_{i} h_{i} \rho_{0 i}}{p_{0 i}}} \end{aligned} $$ $$ \begin{aligned} x_{rel aero i} &= - x_{CG i} + x_{COP i} \\ z_{rel aero i} &= - z_{CG i} + z_{COP i} \end{aligned} $$ $$ \begin{aligned} M_{pitch aero i} &= F_{d i} z_{rel aero i} - F_{z df i} x_{rel aero i} \end{aligned} $$ $$ \begin{aligned} balance_{df i} &= \frac{l_{wb i} - x_{COP i}}{l_{wb i}} \end{aligned} $$ $$ \begin{aligned} M_{pitch df i} &= F_{z df i} \left(x_{CG i} - x_{COP i}\right) \end{aligned} $$ $$ \begin{aligned} M_{pitch drag i} &= F_{d i} \left(z_{CG i} - z_{COP i}\right) \end{aligned} $$ $$ \begin{aligned} F_{x inertia i} &= a_{long i} m_{i} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz long i} &= \frac{F_{x inertia i} z_{CG i}}{l_{wb i}} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz long f i} &= - \Delta_{Fz long i} \\ \Delta_{Fz long r i} &= \Delta_{Fz long i} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz long fl i} &= \frac{\Delta_{Fz long f i}}{2} \\ \Delta_{Fz long fr i} &= \frac{\Delta_{Fz long f i}}{2} \\ \Delta_{Fz long rl i} &= \frac{\Delta_{Fz long r i}}{2} \\ \Delta_{Fz long rr i} &= \frac{\Delta_{Fz long r i}}{2} \end{aligned} $$ $$ \begin{aligned} F_{y inertia i} &= a_{lat i} m_{i} \end{aligned} $$ $$ \begin{aligned} a_{lat i} &= \frac{v_{i}^{2}}{R_{turn i}} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz lat i} &= \frac{F_{y inertia i} z_{CG i}}{t_{track i}} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz lat f i} &= \Delta_{Fz lat i} balance_{lat i} \\ \Delta_{Fz lat r i} &= \Delta_{Fz lat i} \left(1 - balance_{lat i}\right) \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz lat fl i} &= \frac{\Delta_{Fz lat f i}}{2} \\ \Delta_{Fz lat fr i} &= - \frac{\Delta_{Fz lat f i}}{2} \\ \Delta_{Fz lat rl i} &= \frac{\Delta_{Fz lat r i}}{2} \\ \Delta_{Fz lat rr i} &= - \frac{\Delta_{Fz lat r i}}{2} \end{aligned} $$ $$ \begin{aligned} W_{i} &= g_{i} m_{i} \end{aligned} $$ $$ \begin{aligned} F_{z static i} &= W_{i} \end{aligned} $$ $$ \begin{aligned} F_{z static fl i} &= \frac{F_{z static f i}}{2} \\ F_{z static fr i} &= \frac{F_{z static f i}}{2} \\ F_{z static rl i} &= \frac{F_{z static r i}}{2} \\ F_{z static rr i} &= \frac{F_{z static r i}}{2} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz drag fl i} &= \frac{\Delta_{Fz drag f i}}{2} \\ \Delta_{Fz drag fr i} &= \frac{\Delta_{Fz drag f i}}{2} \\ \Delta_{Fz drag rl i} &= \frac{\Delta_{Fz drag r i}}{2} \\ \Delta_{Fz drag rr i} &= \frac{\Delta_{Fz drag r i}}{2} \end{aligned} $$ $$ \begin{aligned} F_{z static df f i} &= F_{z df f i} + F_{z static f i} \\ F_{z static df r i} &= F_{z df r i} + F_{z static r i} \end{aligned} $$ $$ \begin{aligned} F_{z static df fl i} &= F_{z df fl i} + F_{z static fl i} \\ F_{z static df fr i} &= F_{z df fr i} + F_{z static fr i} \\ F_{z static df rl i} &= F_{z df rl i} + F_{z static rl i} \\ F_{z static df rr i} &= F_{z df rr i} + F_{z static rr i} \end{aligned} $$ $$ \begin{aligned} F_{z static df drag fl i} &= \Delta_{Fz drag fl i} + F_{z static df fl i} \\ F_{z static df drag fr i} &= \Delta_{Fz drag fr i} + F_{z static df fr i} \\ F_{z static df drag rl i} &= \Delta_{Fz drag rl i} + F_{z static df rl i} \\ F_{z static df drag rr i} &= \Delta_{Fz drag rr i} + F_{z static df rr i} \end{aligned} $$ $$ \begin{aligned} F_{z static f i} &= \frac{W_{i} \left(l_{wb i} - x_{CG i}\right)}{l_{wb i}} \\ F_{z static r i} &= \frac{W_{i} x_{CG i}}{l_{wb i}} \end{aligned} $$ $$ \begin{aligned} F_{z df i} &= \frac{A_{ref i} C_{L i} \rho_{i} v_{i}^{2}}{2} \end{aligned} $$ $$ \begin{aligned} F_{z df fl i} &= \frac{F_{z df f i}}{2} \\ F_{z df fr i} &= \frac{F_{z df f i}}{2} \\ F_{z df rl i} &= \frac{F_{z df r i}}{2} \\ F_{z df rr i} &= \frac{F_{z df r i}}{2} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz drag f i} &= \frac{M_{pitch drag i}}{l_{wb i}} \\ \Delta_{Fz drag r i} &= - \frac{M_{pitch drag i}}{l_{wb i}} \end{aligned} $$ $$ \begin{aligned} \Delta_{Fz drag fl i} &= \frac{\Delta_{Fz drag f i}}{2} \\ \Delta_{Fz drag fr i} &= \frac{\Delta_{Fz drag f i}}{2} \\ \Delta_{Fz drag rl i} &= \frac{\Delta_{Fz drag r i}}{2} \\ \Delta_{Fz drag rr i} &= \frac{\Delta_{Fz drag r i}}{2} \end{aligned} $$ $$ \begin{aligned} F_{f max i} &= F_{z i} \mu_{i} \end{aligned} $$ $$ \begin{aligned} \mu_{load dep i} &= f_{load i} \mu_{0 i} \end{aligned} $$ $$ \begin{aligned} a_{long max i} &= \frac{F_{z i} \mu_{i}}{m_{i}} \end{aligned} $$ $$ \begin{aligned} a_{lat max i} &= \frac{F_{z i} \mu_{i}}{m_{i}} \end{aligned} $$ $$ \begin{aligned} v_{max corner i} &= \sqrt{\frac{F_{z i} R_{turn i} \mu_{i}}{m_{i}}} \end{aligned} $$ $$ \begin{aligned} F_{tan i} &= \sqrt{F_{x tire i}^{2} + F_{y inertia i}^{2}} \end{aligned} $$ $$ \begin{aligned} friction_{util i} &= \frac{F_{tan i}}{F_{z i} \mu_{i}} \end{aligned} $$ $$ \begin{aligned} \frac{F_{x tire i}^{2}}{F_{x tire max i}^{2}} + \frac{F_{y inertia i}^{2}}{F_{y max i}^{2}} &= 1 \end{aligned} $$ $$ \begin{aligned} P_{trac i} &= F_{x tire i} v_{i} \end{aligned} $$ $$ \begin{aligned} a_{brake aero i} &= \frac{\mu_{i} \left(F_{z df i} + F_{z static i}\right)}{m_{i}} \end{aligned} $$ $$ \begin{aligned} F_{centripetal i} &= \frac{m_{i} v_{i}^{2}}{R_{turn i}} \end{aligned} $$ $$ \begin{aligned} F_{slope i} &= g_{i} m_{i} \sin{\left(\theta_{slope i} \right)} \end{aligned} $$ $$ \begin{aligned} a_{x max i} &= \frac{F_{x tire max i}}{m_{i}} \end{aligned} $$

License

This project (with exceptions) is published under the CC Attribution-ShareAlike 4.0 International License.