# Resources - [VD crash course](https://optimumg-s-school.thinkific.com/courses/take/vehicle-dynamics-lecture/lessons/13464891-load-transfer-vs-load-transfer-distribution) - [OptimumG Kinematics Parameters](https://optimumg.atlassian.net/wiki/external/483131815/YmMyNDY0ZDQ0MmViNDM3Mzk5YzE5ZjVlZWM5OGM2NDE) - [Multimatic Decoupled Manual](https://multimatic.wpengine.com/wp-content/uploads/2021/06/2021_06_17-FSAE-Damper-promotion-High-Res-Spreads.pdf) - [Ohlins Damper Manual](https://www.ohlinsusa.com/files/files/Inside_TTX_A4-Europe1.pdf) - [More than just bounce](https://www.autospeed.com/cms/a_112686/article.html) - [Racecar Tires](https://racingcardynamics.com/racing-tires-lateral-force/) - [Understeer and Oversteer](http://racingcardynamics.com/understeer-and-oversteer/) - [Weight Transfer](https://racingcardynamics.com/weight-transfer/) - [Clash Royale Balance Seminar](https://www.youtube.com/watch?app=desktop&v=6fyxlvDrx3M) - [Bump steer video](https://www.youtube.com/watch?v=Csw6FOHvHhs) - [Roll steer video](https://www.youtube.com/watch?v=hgx4xlGNIzo) - [Steady state response video](https://www.youtube.com/watch?v=hJY3eZU4US4) - [Steering, caster, and kpi article](http://www.driftforum.pl/viewtopic.php?f=125&t=22850) - [Toe and ackerman article](http://www.driftforum.pl/viewtopic.php?f=125&t=22852) - damping youtube series - [part 1](https://www.youtube.com/watch?v=wcUmzJP2jS8&list=PL4aBj62lK6_2egG_OLQ59C6XFrfjQ-A46&index=6) - [part 2](https://www.youtube.com/watch?v=CrkVFt-hc2s&list=PL4aBj62lK6_2egG_OLQ59C6XFrfjQ-A46&index=6) - [eigenvalues](https://www.youtube.com/watch?v=UovvU_noxWg&list=PL4aBj62lK6_2egG_OLQ59C6XFrfjQ-A46&index=7) - [bumpstops as a tuning tool](https://www.youtube.com/watch?v=uikpAQB6IYA) # Variables $a_{y} = \text{lateral acceleration } =\frac{V^{2}}{R}+\dot{v}= V(r+\dot{\beta})$ $r = \text{yawing velocity} = \frac{V}{R}$ $\beta = \text{vehicle slip angle} = \frac{v}{V}$ $v, v_{R}, v_{F} = \text{lateral velocity}$ $u = \text{longitudinal velocity}$ $V = \text{vehicle velocity} = \sqrt{ u^{2}+v^{2} } \approx u$ $R = \text{path radius}$ $b = \text{CG to rear axle distance} = l\cdot RWB$ $a = \text{CG to front axle distance} = l(1-RWB)$ $l = \text{wheelbase}$ $\delta = \text{steering angle}$ $\alpha_{F} = \text{front slip angle} = \beta + \frac{ar}{V} - \delta$ $\alpha_{R} = \text{rear slip angle} = \beta - \frac{br}{V}$ $Y = \text{lateral force} = C_{F}\alpha+C_{R}\beta$ $N = \text{yawing moment} = aC_{F}\alpha_{F} - bC_{R}\alpha_{R}$ # Tires ### Tire Model ### Load Sensitivity $\mathrm{F_{y}^{\prime}=C_{_{\alpha}}^{\prime}\alpha=}\left(a\mathrm{F_{z}}-b\mathrm{F_{z}^{2}}\right)\alpha$ ![[Pasted image 20241120230715.png|375]] ### Camber Thrust $\mathrm{F}_{\mathrm{y}}=C_{\alpha}\alpha+C_{\gamma}\gamma$ ![[Pasted image 20241120230802.png|375]] ![[Pasted image 20241120230827.png|375]] ### Cornering Stiffness ### Slip Ratio $ SR = \frac{\Omega R}{V\cos(\alpha)}-1 $ $\Omega =\text{wheel angular velocity [rad/s]}$ $R =\text{loaded radius [m]}$ $V =\text{road velocity [m/s]}$ $\alpha=\text{slip angle [rad]}$ - Free rolling at 0 slip ratio (tire surface speed same as ground speed) - Spinning at +/-1 slip ratio (arbitrary definition) - Peak grip at around 0.1 to 0.15 slip ratio ![[Pasted image 20241120230637.png|375]] ![[Pasted image 20240907160102.png|325]] ![[Pasted image 20240907160313.png|325]] # Car Parameters #### Rear Weight Bias Percent weight on the rear wheels - Typically around 50% - too low leads to understeer, too high leads to #### CG Height Height of the center of gravity - Typically 10” to 12” - want to be as low as possible to minimize load transfer #### Track Width Distance between the wheels in front view - Typically 50” or so - tradeoff between maneuverability through slaloms and reducing lateral load transfer #### Wheelbase 60.5” = 1.537 m Minimum by rules is 60” #### Brake Bias Ratio of front brake pressure to rear brake pressure - Typically around 70% - adjustable via brake bias bar in pedalbox - front tires need more braking pressure due to longitudinal load transfer when deccelerating ### Sprung vs Unsprung Mass ![[Pasted image 20241221163124.png]] # Kinematics ### Modes ![[Chassis-Modes-Illustration-1-Zapletal.-2000-Balanced-Suspension.png|500]] $ \begin{bmatrix} x_{H} \\ x_{P} \\ x_{R} \\ x_{X} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{bmatrix} \begin{bmatrix} x_{LF} \\ x_{RF} \\ x_{LR} \\ x_{RR} \end{bmatrix} $ ## Quarter Car Model ![[Pasted image 20241221162732.png|325]] ![[Pasted image 20241221162919.png|325]] ### Equations of Motion $ m_{s}\ddot{x} + C\dot{x}+K_{ride}x = 0 $ $ K_{ride} = \frac{K_{s}K_{t}}{K_{s}+K_{t}} $ $ \ddot{x} + 2\zeta \omega_n\dot{x} + \omega_n^{2}x = 0 $ $ \omega_n = \sqrt{ \frac{K_{ride}}{m_{s}} } $ $ \zeta = \frac{C}{2\sqrt{ m_{s}K_{ride} }} $ ### Motion Ratio $ MR_{\text{heave}} = \frac{\text{heave spring travel}}{\text{wheel travel in heave}} $ $ MR_{\text{roll}} = \frac{\text{roll spring travel}}{\text{wheel travel in roll}} $ With pushrod pivot arm length $a$, damper pivot arm $b$, and pushrod angle $\theta$, we have the following approximation: $ MR = \frac{a}{b}\sin(\theta) $ #### Spring Rate Backing out spring rate from ride frequency $ K_{\text{s}} = 4\pi^{2}f_{r}^{2}m_{s}(MR)^{2} $ ##### Shock Preload $ K_{\text{heave spring}} \cdot \text{preload}\cdot MR^{2} = W_{F} $ $ \text{preload} = \frac{W_{F}}{MR^{2} \cdot K_{hs}} $ #### Wheel Rate $ K_{\text{heave, w}}=K_{\text{heave}}\cdot(MR_{\text{heave}})^{2} $ $ K_{\text{roll, w}}=K_{\text{roll}}\cdot(MR_{\text{roll}})^{2} $ #### Roll Stiffness Torsional stiffness of the front axle $ K_{\phi, \text{ front}} = \frac{1}{2}K_{\text{roll, w}} \cdot T^{2} $ ### Damping #### Critical Damping Coefficient Minimum damping coefficient that results in zero overshoot and oscillations, in $Ns/m$ ##### Sprung mass $ C_{cr} = 2 \sqrt{ K_{\text{ride}}\cdot m_{s} } $ $ \zeta_{us} = \frac{2\sqrt{ k_{s}m_{s} }}{2\sqrt{k_{s}+ k_{t}m_{us} }} = \sqrt{ \frac{m_{s}}{1+3m_{us}} }=\sqrt{ \frac{305}{1+3\cdot 50} }= 1.42 $ ##### Unsprung mass $ C_{cr} = 2\sqrt{ K_{s}+K_{T}\cdot m_{us} } $ #### Damping Ratio Percent of critical damping $ \omega_n=\sqrt{ \frac{K}{m} } $ $ C = 2\omega_nm_{s}\zeta $ $ \zeta = \frac{1}{2}\sqrt{ K_{s} } $ ## Roll ### Instant Centers ### Roll Centers Higher roll center results in more load transfer for that axle. Generally want rears higher than fronts for a given TLLTD because this adds oversteer on corner entry when the roll angles and elastic weight transfer are small, increasing the car’s steering response. ### Roll Gradient Degrees roll per G of lateral acceleration $ K_\phi = \frac{- W_{s}h_{1}}{\left( K_{F}+K_{R}-\frac{W_{s}}{57.3}(h_{1} - \frac{t^{2}\text{rcm}_{F}}{4z_{\text{rcF}}}) \right)} $ The $W_{s}$ term in the denominator accounts for the rolling moment as a result of roll angle and roll center migration. ### Lateral Load Transfer Distribution Rear TLLTD is the ratio of rear load transfer to total load transfer: $ \text{TLLTD} = \frac{\text{LLT}_{R}}{\text{LLT}_{F}+\text{LLT}_{R}} $ ### Roll Center Migration ### Jacking $ \text{Jacking Force}=\frac{2W_sA_Y}{t(1+n_F)} (Z_{RCF}(n_F-1)+(RCM_{F}-1)\cdot(-n_F\cdot z_{w,FL}-z_{w,FR}) $ ## Pitch ### Pitch Center Point at which the longitudinal forces from the tires are reacted by the sprung mass $ \text{Pitch Center Height} = l\left( \frac{\tan\theta_{F}\tan\theta_{R}}{\tan\theta_{F}+\tan\theta_{R}} \right) $ $ \text{Pitch Center X} = \frac{h_{p}}{\tan(\theta_{R})} $ ### Pitch Moment $ M_{p} = W_{s}A_{X}(h-h_{p}) $ $ M_{\text{aero}} = -F_{DF} (x_{CoP}-x_{p})+F_{D}(z_{CoP}-h_{p}) $ $ M_{jack} = F_{jF}(l-x_{p})-F_{jR}(x_{p}) $ $ M_{\theta} = M_{p}+M_{\text{aero}}+M_{\text{jack}} $ ### Pitch Jacking $ \text{Jacking Force}= $ ### Sprung Mass Vertical Load $ F_{zs} = -F_{DF}+F_{jF}+F_{jR} $ ### Axle Heave $ F_{zF}=K_{hF}z_{F} = \frac{F_{zs}x_{p}-M_{\theta}}{l} $ $ F_{zR} =K_{hR}z_{R}= F_{zs} - K_{hF}z_{F} $ ### Pitch Angle $ \theta = \tan ^{-1}\left( \frac{z_{F}-z_{R}}{l} \right) $ ### Antis #### Anti-dive Percentage of geometric load transfer for the front tires when braking $ \%\text{Anti-dive} = \text{ Brake Bias} \cdot \tan\theta_{F} \left( \frac{l}{h} \right) $ #### Anti-rise Percentage of geometric load transfer for the rear tires when braking $ \%\text{Anti-rise} = (1-\text{ Brake Bias}) \cdot \tan\theta_{R} \left( \frac{l}{h} \right) $ #### Anti-lift Percentage of geometric load transfer for the front tires when acclerating $ \%\text{Anti-lift} = \text{Drive Ratio} \cdot \tan\theta_{F} \left( \frac{l}{h} \right) $ #### Anti-squat Percentage of geometric load transfer for the rear tires when acclerating $ \%\text{Anti-squat} = (1-\text{Drive Ratio}) \cdot \tan\theta_{F} \left( \frac{l}{h} \right) $ ![[Pasted image 20241230071831.png|500]] ![[Pasted image 20241115123240.png|500]] ## Camber ### Camber Gain in Bump $ \Delta \gamma_{b} = -\tan ^{-1}\left( \frac{z_{b}}{\text{fvsa}} \right) \approx -\frac{z_{b}}{\text{fvsa}} $ ![[Pasted image 20241112160628.png|325]] ### Camber Gain in Roll $ \begin{align} \Delta \gamma_{r} &= \tan ^{-1}\left( \frac{\phi t}{2\text{ fvsa}} \right) - \phi \\ & \approx \phi\left( 1-\frac{t}{2 \text{ fvsa}} \right) \end{align} $ ### Camber Gain in Steer $ \Delta C_{\delta} = \frac{KPI\times\delta^{2}}{2} + \text{caster}\cdot\delta $ ![[Pasted image 20241110205649.png|325]] ![[Pasted image 20241120230537.png|325]] ## Steering ![[Untitled.png|300]] ![[Pasted image 20241120232120.png|375]] ### Scrub Radius ### Mechanical Trail ### Kingpin Angle ### Caster Angle ### Bump Steer Change in toe of a wheel during suspension travel Usually want to make this close to zero because it’s difficult to control effectively 1. Tie Rod should point at the instant axis of the wheel 2. Place tie rod points on the lines connecting upper and lower AArms in front view ### Roll Steer ### Toe ### Ackerman Geometry (100% Ackerman) All four tires follow trajectories about a shared instant center at low speed $ \begin{array}{l}\\ \delta_{o} = \tan ^{-1}\left( \frac{l}{R+\frac{T}{2}} \right) & & & \delta_{i} = \tan ^{-1}\left( \frac{L}{R-\frac{T}{2}} \right) \end{array} $ $ \delta=\text{Ackerman Angle}=\frac{L}{R} $ ![[Pasted image 20241026211158.png|475]] ### Percent Ackerman - negative ackerman is known as anti-ackerman or reverse ackerman - 0% ackerman is parallel steer - 100% ackerman is known as “Ackerman Geometry” or “Perfect Ackerman” $ \%\text{Ackerman} = \frac{\delta_{i} - \delta_{o}}{\delta_{i}} \cdot 100\% $ ![[Pasted image 20250117024729.png|550]] ### Slip Angle ![[Pasted image 20241004221524.png|]] ## Compliance # Dynamics ## Bicycle Model ![[Pasted image 20240929140253.png|350]] ![[Pasted image 20240929140429.png|250]] ### Axle Forces $ \begin{array}{l}\\ F_{F} =C_{F}\alpha_{F}= M \frac{b}{L} \cdot \frac{V^{2}}{R} \\ F_{R} =C_{R}\alpha_{R}= M \frac{a}{L} \cdot \frac{V^{2}}{R} \end{array} $ ### Steering angle Required to maintain constant radius: $ \begin{align} \delta &= 57.3\frac{l}{R} + \alpha_{F}-\alpha_{R} \\ &= 57.3\frac{l}{R} + KA_{Y} \end{align} $ ### Understeer Gradient $ K=\frac{\Delta\delta}{\Delta A_{Y}}=\frac{\alpha_{F} - \alpha_{R}}{\Delta A_{Y}}= D_{f} - D_{r} $ Understeer if ${} K> 0 {}$ Neutral steer if $K=0$ Oversteer if $K<0$ ### Axle Cornering Compliance $ D_{f} = \frac{d\beta_{f}}{da_{y}} =57.3\cdot \frac{mgb}{a+b}\cdot \frac{1}{C_{F}} $ $ D_{r} = \frac{d\beta_{r}}{da_{y}}= 57.3\cdot \frac{mga}{a+b}\cdot \frac{1}{C_{R}} $ ### Equations of Motion $\begin{array}{c}{{\mathrm{I}_{z}\dot{\mathrm{r}}=\mathrm{N}_{\beta}\beta+\mathrm{N}_{\mathrm{r}}\mathrm{r}+\mathrm{N}_{\delta}\delta}}\\ {{\mathrm{m}\mathrm{V}\Bigl(\mathrm{r}+\dot{\beta}\Bigr)=\mathrm{Y}_{\beta}\beta+\mathrm{Y}_{\mathrm{r}}\mathrm{r}+\mathrm{Y}_{\delta}\delta}}\end{array}$ $Y = \text{lateral force} = C_{F}\alpha+C_{R}\beta$ $ Y = (C_{F}+C_{R})\beta + \frac{1}{V}(aC_{F}-bC_{R})r-C_{F}\delta $ $\mathrm{Y}_\beta=\text{Damping-in-Sideslip Derivative} = C_{\mathrm{F}}+C_{\mathrm{R}}$ $Y_{r} = \text{Yaw Coupling Derivative} = \frac{1}{V}(aC_{F}-bC_{R})$ $Y_{\delta}=\text{Control Force Derivative}=-C_{F}$ $N = \text{yawing moment} = aC_{F}\alpha_{F} - bC_{R}\alpha_{R}$ $ N = (aC_{F}-bC_{R})\beta+\frac{1}{V}(a^{2}C_{F}+b^{2}C_{R})r-aC_{F}\delta $ $N_{\beta} =\text{Static Directional Stability Derivative} = aC_{F}-bC_{R}$ $N_{r}=\text{Yaw Damping Derivative} =\frac{1}{V}(a^{2}C_{F}+b^{2}C_{R})$ $N_{\delta} =\text{Control Moment Derivative} = -aC_{F}$ #### Laplacian Form $ \begin{bmatrix} mVs + C_{F} + C_{R} & mV+\frac{1}{V}(aC_{F}-bC_{R}) \\ aC_{f}-bC_{R} & I_{z}s+\frac{1}{V}(a^{2}C_{F}+b^{2}C_{R}) \end{bmatrix} \begin{bmatrix} \bar{\beta_{v}} \\ \bar{r} \end{bmatrix} = \begin{bmatrix} C_{f} \\ aC_{f} \end{bmatrix} \delta $ ### Characteristic Speed In the understeer case, the vehicle requires twice the ackerman angle to take a corner at its characteristic speed $ V_{\text{char}} = \sqrt{ 57.3 \frac{Lg}{K_{us}} } $ This is the speed at which the vehicle is most responsive in yaw ### Critical Speed In the oversteer case, vehicle will be unstable for speeds greater than $V_{\text{crit}}$ $ V_{\text{crit}} = \sqrt{ -57.3 \frac{Lg}{K} } $ ## Transients ### Corner Entry ### Corner Exit ### Damping ## Corner Speed ### Low Speed Corners ### Mid Speed Corners ### High Speed Corners ## Stability Peak grip but no stability - if you increase steering angle and front slip angles, you lose grip in the front and the car understeers! ![[Pasted image 20241106040906.png|450]] ### Stability Index ## Ride Height ### Static Ride Height ### Aero Maps Plot downforce and aero balance as a function of front and rear ride height - used to optimize dynamic ride height in cornering ![[Pasted image 20241004224903.png|325]] ![[Pasted image 20241004224944.png|325]]