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

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

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)

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

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

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

Wheel Displacement from Heave and Pitch

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

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

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

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

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

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

\[\begin{split} \begin{aligned} \Delta_{Fz long f i} &= - \Delta_{Fz long i} \\ \Delta_{Fz long r i} &= \Delta_{Fz long i} \end{aligned} \end{split}\]

Longitudinal Load Transfer Per Wheel

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

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

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

Lateral Load Transfer Per Wheel (clockwise positive turn)

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

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

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

Drag-Induced Normal Load Shift Per Wheel

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

Normal Force: Static + Downforce (Axle)

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

Normal Force: Static + Downforce (Per Wheel)

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

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

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

Static Normal Force Axle Split

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

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

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

Drag-Induced Normal Load Shift (Axle)

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

Drag-Induced Normal Load Shift (Per Wheel)

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

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)

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