# Known Equations

## Bernoulli Equation (two points)

$$
\begin{aligned}
g_{i} h_{i} \rho_{i} + p_{i} + \frac{\rho_{i} v_{i}^{2}}{2} &= g_{j} h_{j} \rho_{j} + p_{j} + \frac{\rho_{j} v_{j}^{2}}{2}
\end{aligned}
$$


## Continuity Equation for Incompressible Flow

$$
\begin{aligned}
\frac{v_{i}}{v_{j}} &= \frac{A_{j}}{A_{i}}
\end{aligned}
$$


## Reynolds Number

$$
\begin{aligned}
Re_{i} &= \frac{L_{i} \rho_{i} v_{i}}{\eta_{i}}
\end{aligned}
$$


## Kinematic Viscosity Definition

$$
\begin{aligned}
\nu_{i} &= \frac{\eta_{i}}{\rho_{i}}
\end{aligned}
$$


## Aerodynamic Drag Force

$$
\begin{aligned}
F_{d i} &= \frac{A_{ref i} C_{d i} \rho_{i} v_{i}^{2}}{2}
\end{aligned}
$$


## Aerodynamic Lateral Force

$$
\begin{aligned}
F_{y aero i} &= \frac{A_{ref i} C_{y i} \rho_{i} v_{i}^{2}}{2}
\end{aligned}
$$


## Aerodynamic Yawing Moment

$$
\begin{aligned}
M_{yaw aero i} &= F_{y aero i} x_{rel aero i}
\end{aligned}
$$


## Vehicle Speed from Velocity Components

$$
\begin{aligned}
v_{i} &= \sqrt{v_{x i}^{2} + v_{y i}^{2}}
\end{aligned}
$$


## Vehicle Slip Angle at CG

$$
\begin{aligned}
\beta_{sideslip i} &= \operatorname{atan}{\left(\frac{v_{y i}}{v_{x i}} \right)}
\end{aligned}
$$


## Longitudinal Force Equilibrium

$$
\begin{aligned}
- F_{d i} - F_{rr i} + F_{x drive i} - F_{x inertia i} &= 0
\end{aligned}
$$


## Rolling Resistance Force

$$
\begin{aligned}
F_{rr i} &= C_{rr i} F_{z i}
\end{aligned}
$$


## Longitudinal Power from Force

$$
\begin{aligned}
P_{x i} &= F_{x drive i} v_{i}
\end{aligned}
$$


## Yaw Moment Equation of Motion

$$
\begin{aligned}
M_{z i} &= I_{z i} r_{dot i}
\end{aligned}
$$


## Maximum Longitudinal Tyre Force

$$
\begin{aligned}
F_{x max i} &= F_{z i} \mu_{long i}
\end{aligned}
$$


## Axle Normal Force from Wheel Loads

$$
\begin{aligned}
F_{z f i} &= F_{z fl i} + F_{z fr i} \\
F_{z r i} &= F_{z rl i} + F_{z rr i}
\end{aligned}
$$


## Ideal Gas Law

$$
\begin{aligned}
p_{i} &= R_{i} T_{i} \rho_{i}
\end{aligned}
$$


## ISA Temperature Lapse Rate (Assumption)

$$
\begin{aligned}
T_{i} &= - L_{ISA i} h_{i} + T_{0 i}
\end{aligned}
$$


## Coast-Down Time (Drag Only)

$$
\begin{aligned}
t_{coast i} &= \frac{2.0 m_{i} \log{\left(\frac{v_{i}}{v_{final i}} \right)}}{A_{ref i} C_{d i} \rho_{i} v_{i}}
\end{aligned}
$$


## Coast-Down Distance (Drag Only)

$$
\begin{aligned}
s_{coast i} &= \frac{2.0 m_{i} \left(- v_{final i} + v_{i}\right)}{A_{ref i} C_{d i} \rho_{i}}
\end{aligned}
$$


## Required Thrust vs Speed

$$
\begin{aligned}
F_{thrust req i} &= F_{d i} + F_{x inertia i}
\end{aligned}
$$


## Required Power vs Speed

$$
\begin{aligned}
P_{required i} &= F_{thrust req i} v_{i}
\end{aligned}
$$


## Continuity Equation (Mass Conservation)

$$
\begin{aligned}
A_{i} \rho_{i} v_{i} &= A_{j} \rho_{j} v_{j}
\end{aligned}
$$


## CG Distance Definitions (Front / Rear)

$$
\begin{aligned}
a_{CG i} &= x_{CG i} \\
b_{CG i} &= l_{wb i} - x_{CG i}
\end{aligned}
$$


## Tyre Vertical Stiffness Definition

$$
\begin{aligned}
k_{tire i} &= \frac{F_{z i}}{\delta_{z tire i}}
\end{aligned}
$$


## Tyre Deflection Change from Load Change

$$
\begin{aligned}
\Delta_{\delta z tire i} &= \frac{\Delta_{F z i}}{k_{tire i}}
\end{aligned}
$$


## Wheel Vertical Position from Tyre Deflection

$$
\begin{aligned}
z_{wheel i} &= - \delta_{z tire i} + z_{chassis i}
\end{aligned}
$$


## Suspension Motion Ratio (Displacement)

$$
\begin{aligned}
\delta_{z spring i} &= MR_{i} \delta_{z wheel i}
\end{aligned}
$$


## Suspension Spring Force

$$
\begin{aligned}
F_{spring i} &= \delta_{z spring i} k_{spring i}
\end{aligned}
$$


## Suspension Force at Wheel

$$
\begin{aligned}
F_{z susp i} &= \frac{F_{spring i}}{MR_{i}}
\end{aligned}
$$


## Wheel Vertical Displacement Decomposition

$$
\begin{aligned}
\delta_{z wheel i} &= \delta_{z susp i} + \delta_{z tire i}
\end{aligned}
$$


## Suspension Deflection from Wheel Load

$$
\begin{aligned}
\delta_{z susp i} &= \frac{F_{z i} MR_{i}}{k_{spring i}}
\end{aligned}
$$


## Effective Wheel Rate

$$
\begin{aligned}
k_{wheel i} &= \frac{k_{spring i}}{MR_{i}^{2}}
\end{aligned}
$$


## Anti-Roll Bar Torque from Wheel Displacement

$$
\begin{aligned}
T_{arb i} &= k_{arb i} \left(\delta_{z wheel left i} - \delta_{z wheel right i}\right)
\end{aligned}
$$


## Anti-Roll Bar Vertical Wheel Forces

$$
\begin{aligned}
F_{z arb left i} &= \frac{T_{arb i}}{t_{track i}} \\
F_{z arb right i} &= - \frac{T_{arb i}}{t_{track i}}
\end{aligned}
$$


## Wheel Displacement from Heave and Pitch

$$
\begin{aligned}
\delta_{z wheel f i} &= a_{CG i} \theta_{pitch i} + \delta_{z heave i} \\
\delta_{z wheel r i} &= - b_{CG i} \theta_{pitch i} + \delta_{z heave i}
\end{aligned}
$$


## Bump Stop Engagement Deflection

$$
\begin{aligned}
\delta_{z bump i} &= - \delta_{z clearance i} + \delta_{z susp i}
\end{aligned}
$$


## Bump Stop Force (Linear, Engaged Only)

$$
\begin{aligned}
F_{bump i} &= \delta_{z bump i} k_{bump i}
\end{aligned}
$$


## Total Suspension Force Including Bump Stop

$$
\begin{aligned}
F_{z susp total i} &= F_{bump i} + F_{z susp i}
\end{aligned}
$$


## Geometric Longitudinal Load Transfer (Anti-Dive / Anti-Squat)

$$
\begin{aligned}
\Delta_{Fz long geom i} &= F_{x tire i} \tan{\left(\theta_{anti i} \right)}
\end{aligned}
$$


## Net Longitudinal Load Transfer Including Geometry

$$
\begin{aligned}
\Delta_{Fz long total i} &= - \Delta_{Fz long geom i} + \Delta_{Fz long i}
\end{aligned}
$$


## Effective Vertical Stiffness (Tyre + Suspension)

$$
\begin{aligned}
\frac{1}{k_{vert i}} &= \frac{1}{k_{wheel i}} + \frac{1}{k_{tire i}}
\end{aligned}
$$


## Roll Stiffness Distribution (Front Fraction)

$$
\begin{aligned}
\phi_{roll f i} &= \frac{k_{roll f i}}{k_{roll f i} + k_{roll r i}}
\end{aligned}
$$


## Ackermann Inner Wheel Steering Angle

$$
\begin{aligned}
\delta_{inner i} &= \operatorname{atan}{\left(\frac{l_{wb i}}{R_{turn i} - \frac{t_{track i}}{2}} \right)}
\end{aligned}
$$


## Ackermann Outer Wheel Steering Angle

$$
\begin{aligned}
\delta_{outer i} &= \operatorname{atan}{\left(\frac{l_{wb i}}{R_{turn i} + \frac{t_{track i}}{2}} \right)}
\end{aligned}
$$


## Average Steering Angle (Bicycle Equivalent)

$$
\begin{aligned}
\delta_{avg i} &= \frac{\delta_{inner i}}{2} + \frac{\delta_{outer i}}{2}
\end{aligned}
$$


## Turn Radius from Steering Angle

$$
\begin{aligned}
R_{turn i} &= \frac{l_{wb i}}{\tan{\left(\delta_{avg i} \right)}}
\end{aligned}
$$


## Available Lateral Force on Banked Surface (for flat roads, theta_bank=0)

$$
\begin{aligned}
F_{y available i} &= F_{z i} \left(\mu_{i} \cos{\left(\theta_{bank i} \right)} + \sin{\left(\theta_{bank i} \right)}\right)
\end{aligned}
$$


## Banked Turn Lateral Force Limit

$$
\begin{aligned}
F_{centripetal i} &= F_{y available i}
\end{aligned}
$$


## Aerodynamic Downforce Axle Split

$$
\begin{aligned}
F_{z df f i} &= F_{z df i} balance_{df i} \\
F_{z df r i} &= F_{z df i} \left(1 - balance_{df i}\right)
\end{aligned}
$$


## Aerodynamic Efficiency (L/D)

$$
\begin{aligned}
\eta_{aero i} &= \frac{C_{L i}}{C_{d i}}
\end{aligned}
$$


## Barometric Formula

$$
\begin{aligned}
p_{i} &= p_{0 i} e^{- \frac{g_{i} h_{i} \rho_{0 i}}{p_{0 i}}}
\end{aligned}
$$


## Density Variation with Altitude

$$
\begin{aligned}
\rho_{i} &= \rho_{0 i} e^{- \frac{g_{i} h_{i} \rho_{0 i}}{p_{0 i}}}
\end{aligned}
$$


## Aerodynamic Lever Arms relative to CG

$$
\begin{aligned}
x_{rel aero i} &= - x_{CG i} + x_{COP i} \\
z_{rel aero i} &= - z_{CG i} + z_{COP i}
\end{aligned}
$$


## Aerodynamic Pitching Moment about CG

$$
\begin{aligned}
M_{pitch aero i} &= F_{d i} z_{rel aero i} - F_{z df i} x_{rel aero i}
\end{aligned}
$$


## Aerodynamic Balance from COP Location

$$
\begin{aligned}
balance_{df i} &= \frac{l_{wb i} - x_{COP i}}{l_{wb i}}
\end{aligned}
$$


## Pitching Moment from Downforce Distribution

$$
\begin{aligned}
M_{pitch df i} &= F_{z df i} \left(x_{CG i} - x_{COP i}\right)
\end{aligned}
$$


## Pitching Moment from Drag Force

$$
\begin{aligned}
M_{pitch drag i} &= F_{d i} \left(z_{CG i} - z_{COP i}\right)
\end{aligned}
$$


## Longitudinal Inertial Force

$$
\begin{aligned}
F_{x inertia i} &= a_{long i} m_{i}
\end{aligned}
$$


## Total Longitudinal Load Transfer (mass based)

$$
\begin{aligned}
\Delta_{Fz long i} &= \frac{F_{x inertia i} z_{CG i}}{l_{wb i}}
\end{aligned}
$$


## Longitudinal Load Transfer Axle Split

$$
\begin{aligned}
\Delta_{Fz long f i} &= - \Delta_{Fz long i} \\
\Delta_{Fz long r i} &= \Delta_{Fz long i}
\end{aligned}
$$


## Longitudinal Load Transfer Per Wheel

$$
\begin{aligned}
\Delta_{Fz long fl i} &= \frac{\Delta_{Fz long f i}}{2} \\
\Delta_{Fz long fr i} &= \frac{\Delta_{Fz long f i}}{2} \\
\Delta_{Fz long rl i} &= \frac{\Delta_{Fz long r i}}{2} \\
\Delta_{Fz long rr i} &= \frac{\Delta_{Fz long r i}}{2}
\end{aligned}
$$


## Lateral Inertial Force

$$
\begin{aligned}
F_{y inertia i} &= a_{lat i} m_{i}
\end{aligned}
$$


## Lateral Acceleration in a Turn

$$
\begin{aligned}
a_{lat i} &= \frac{v_{i}^{2}}{R_{turn i}}
\end{aligned}
$$


## Total Lateral Load Transfer

$$
\begin{aligned}
\Delta_{Fz lat i} &= \frac{F_{y inertia i} z_{CG i}}{t_{track i}}
\end{aligned}
$$


## Lateral Load Transfer Axle Distribution

$$
\begin{aligned}
\Delta_{Fz lat f i} &= \Delta_{Fz lat i} balance_{lat i} \\
\Delta_{Fz lat r i} &= \Delta_{Fz lat i} \left(1 - balance_{lat i}\right)
\end{aligned}
$$


## Lateral Load Transfer Per Wheel (clockwise positive turn)

$$
\begin{aligned}
\Delta_{Fz lat fl i} &= \frac{\Delta_{Fz lat f i}}{2} \\
\Delta_{Fz lat fr i} &= - \frac{\Delta_{Fz lat f i}}{2} \\
\Delta_{Fz lat rl i} &= \frac{\Delta_{Fz lat r i}}{2} \\
\Delta_{Fz lat rr i} &= - \frac{\Delta_{Fz lat r i}}{2}
\end{aligned}
$$


## Vehicle Weight Definition

$$
\begin{aligned}
W_{i} &= g_{i} m_{i}
\end{aligned}
$$


## Static Normal Force (Total)

$$
\begin{aligned}
F_{z static i} &= W_{i}
\end{aligned}
$$


## Static Normal Force Per Wheel

$$
\begin{aligned}
F_{z static fl i} &= \frac{F_{z static f i}}{2} \\
F_{z static fr i} &= \frac{F_{z static f i}}{2} \\
F_{z static rl i} &= \frac{F_{z static r i}}{2} \\
F_{z static rr i} &= \frac{F_{z static r i}}{2}
\end{aligned}
$$


## Drag-Induced Normal Load Shift Per Wheel

$$
\begin{aligned}
\Delta_{Fz drag fl i} &= \frac{\Delta_{Fz drag f i}}{2} \\
\Delta_{Fz drag fr i} &= \frac{\Delta_{Fz drag f i}}{2} \\
\Delta_{Fz drag rl i} &= \frac{\Delta_{Fz drag r i}}{2} \\
\Delta_{Fz drag rr i} &= \frac{\Delta_{Fz drag r i}}{2}
\end{aligned}
$$


## Normal Force: Static + Downforce (Axle)

$$
\begin{aligned}
F_{z static df f i} &= F_{z df f i} + F_{z static f i} \\
F_{z static df r i} &= F_{z df r i} + F_{z static r i}
\end{aligned}
$$


## Normal Force: Static + Downforce (Per Wheel)

$$
\begin{aligned}
F_{z static df fl i} &= F_{z df fl i} + F_{z static fl i} \\
F_{z static df fr i} &= F_{z df fr i} + F_{z static fr i} \\
F_{z static df rl i} &= F_{z df rl i} + F_{z static rl i} \\
F_{z static df rr i} &= F_{z df rr i} + F_{z static rr i}
\end{aligned}
$$


## Normal Force: Static + Downforce + Drag Moment (Per Wheel)

$$
\begin{aligned}
F_{z static df drag fl i} &= \Delta_{Fz drag fl i} + F_{z static df fl i} \\
F_{z static df drag fr i} &= \Delta_{Fz drag fr i} + F_{z static df fr i} \\
F_{z static df drag rl i} &= \Delta_{Fz drag rl i} + F_{z static df rl i} \\
F_{z static df drag rr i} &= \Delta_{Fz drag rr i} + F_{z static df rr i}
\end{aligned}
$$


## Static Normal Force Axle Split

$$
\begin{aligned}
F_{z static f i} &= \frac{W_{i} \left(l_{wb i} - x_{CG i}\right)}{l_{wb i}} \\
F_{z static r i} &= \frac{W_{i} x_{CG i}}{l_{wb i}}
\end{aligned}
$$


## Aerodynamic Downforce (Total)

$$
\begin{aligned}
F_{z df i} &= \frac{A_{ref i} C_{L i} \rho_{i} v_{i}^{2}}{2}
\end{aligned}
$$


## Aerodynamic Downforce Per Wheel

$$
\begin{aligned}
F_{z df fl i} &= \frac{F_{z df f i}}{2} \\
F_{z df fr i} &= \frac{F_{z df f i}}{2} \\
F_{z df rl i} &= \frac{F_{z df r i}}{2} \\
F_{z df rr i} &= \frac{F_{z df r i}}{2}
\end{aligned}
$$


## Drag-Induced Normal Load Shift (Axle)

$$
\begin{aligned}
\Delta_{Fz drag f i} &= \frac{M_{pitch drag i}}{l_{wb i}} \\
\Delta_{Fz drag r i} &= - \frac{M_{pitch drag i}}{l_{wb i}}
\end{aligned}
$$


## Drag-Induced Normal Load Shift (Per Wheel)

$$
\begin{aligned}
\Delta_{Fz drag fl i} &= \frac{\Delta_{Fz drag f i}}{2} \\
\Delta_{Fz drag fr i} &= \frac{\Delta_{Fz drag f i}}{2} \\
\Delta_{Fz drag rl i} &= \frac{\Delta_{Fz drag r i}}{2} \\
\Delta_{Fz drag rr i} &= \frac{\Delta_{Fz drag r i}}{2}
\end{aligned}
$$


## Maximum Friction Force

$$
\begin{aligned}
F_{f max i} &= F_{z i} \mu_{i}
\end{aligned}
$$


## Load-Dependent Friction Coefficient (Generic Model)

$$
\begin{aligned}
\mu_{load dep i} &= f_{load i} \mu_{0 i}
\end{aligned}
$$


## Maximum Longitudinal Acceleration (Friction-Limited)

$$
\begin{aligned}
a_{long max i} &= \frac{F_{z i} \mu_{i}}{m_{i}}
\end{aligned}
$$


## Maximum Lateral Acceleration (Friction-Limited)

$$
\begin{aligned}
a_{lat max i} &= \frac{F_{z i} \mu_{i}}{m_{i}}
\end{aligned}
$$


## Maximum Cornering Speed (Friction-Limited)

$$
\begin{aligned}
v_{max corner i} &= \sqrt{\frac{F_{z i} R_{turn i} \mu_{i}}{m_{i}}}
\end{aligned}
$$


## Resultant Tangential Force

$$
\begin{aligned}
F_{tan i} &= \sqrt{F_{x tire i}^{2} + F_{y inertia i}^{2}}
\end{aligned}
$$


## Friction Circle Utilisation

$$
\begin{aligned}
friction_{util i} &= \frac{F_{tan i}}{F_{z i} \mu_{i}}
\end{aligned}
$$


## Friction Ellipse Model

$$
\begin{aligned}
\frac{F_{x tire i}^{2}}{F_{x tire max i}^{2}} + \frac{F_{y inertia i}^{2}}{F_{y max i}^{2}} &= 1
\end{aligned}
$$


## Tractive Power Definition

$$
\begin{aligned}
P_{trac i} &= F_{x tire i} v_{i}
\end{aligned}
$$


## Braking Deceleration with Aerodynamic Downforce

$$
\begin{aligned}
a_{brake aero i} &= \frac{\mu_{i} \left(F_{z df i} + F_{z static i}\right)}{m_{i}}
\end{aligned}
$$


## Centripetal Force Requirement

$$
\begin{aligned}
F_{centripetal i} &= \frac{m_{i} v_{i}^{2}}{R_{turn i}}
\end{aligned}
$$


## Road Slope Longitudinal Force

$$
\begin{aligned}
F_{slope i} &= g_{i} m_{i} \sin{\left(\theta_{slope i} \right)}
\end{aligned}
$$


## Maximum Longitudinal Acceleration (Traction-Limited)

$$
\begin{aligned}
a_{x max i} &= \frac{F_{x tire max i}}{m_{i}}
\end{aligned}
$$