The key formulas from every finished module, grouped by lesson. Each lesson title links back to the full explanation. Symbols follow the course's SAE convention (x forward, y right, z down) unless a note says otherwise. For definitions of the terms, see the glossary.
| Symbol | Meaning | Units |
|---|---|---|
| The car | ||
| \(m,\ W = mg\) | Total mass and weight | kg, N |
| \(m_s,\ m_{us}\) | Sprung mass; unsprung mass (per axle) | kg |
| \(a,\ b,\ L = a + b\) | CG to front axle, CG to rear axle, wheelbase | m |
| \(t\) | Track width | m |
| \(h,\ h_s,\ h_{us}\) | CG height; sprung-mass CG height; unsprung-mass CG height | m |
| \(h_{rc},\ h_{ra}\) | Roll-center height (front or rear); roll-axis height under the sprung CG | m |
| \(K_\phi\) | Roll stiffness (\(K_{\phi f}\), \(K_{\phi r}\) per axle) | N·m/deg |
| \(I_z\) | Yaw moment of inertia | kg·m² |
| Motion | ||
| \(V\) (or \(v\)) | Speed | m/s |
| \(R\) | Turn radius | m |
| \(a_x,\ a_y\) | Longitudinal and lateral acceleration | m/s² or g |
| \(a_{CG}\) | Deceleration of the CG under braking (Lesson 3.3) | g |
| \(r\) | Yaw rate | rad/s |
| \(\psi,\ \theta,\ \phi\) | Yaw (heading), pitch and roll angles | rad or deg |
| \(\delta\) | Front-wheel steer angle | rad or deg |
| \(\beta\) | Sideslip angle at the CG. In Lesson 3.3 only, \(\beta\) is the front brake bias. | rad or deg |
| \(K\) | Understeer gradient (negative = oversteer) | rad/(m/s²) |
| Tires | ||
| \(F_x,\ F_y,\ F_z\) | Longitudinal, lateral and vertical (normal) tire force. \(N\) is also used for normal force on FBDs. | N |
| \(\mu\) | Peak friction coefficient (\(\mu_x\), \(\mu_y\) for each direction) | — |
| \(\mu_0,\ F_{z0},\ s\) | Load-sensitivity model: friction at the reference load, the reference load, the sensitivity | —, N, — |
| \(\alpha,\ \kappa,\ \gamma\) | Slip angle, slip ratio, camber angle | deg, —, deg |
| \(C_\alpha,\ C_\kappa,\ C_\gamma\) | Cornering stiffness, longitudinal slip stiffness, camber stiffness | N/deg, N, N/deg |
| \(\Omega,\ r_e,\ I_w\) | Wheel spin rate, effective rolling radius, wheel inertia | rad/s, m, kg·m² |
| \(T_b,\ p\) | Brake torque; tire pressure | N·m; kPa |
A plain \(a\) is always the CG-to-front-axle distance; accelerations always carry a subscript (\(a_x\), \(a_y\), \(a_{CG}\)).
| Quantity | Conversion |
|---|---|
| Speed | km/h \(\div 3.6\) → m/s; mph \(\times 0.447\) → m/s |
| Acceleration | 1 g = 9.81 m/s² |
| Angle | 1 rad = 57.3° |
| Pressure | absolute = gauge + 101.3 kPa; 1 psi = 6.895 kPa |
| Temperature | K = °C + 273.15 |
| Power | 1 hp ≈ 746 W |
| Quantity | Formula | Notes |
|---|---|---|
| Newton's second law | \(\Sigma \mathbf{F} = m\,\mathbf{a}_{CG}\) | Real forces only on an FBD: no “centrifugal force”. |
| Yaw rotation | \(\Sigma M_z = I_z\,\ddot\psi\) | Steady turn: \(\Sigma M_z = 0\). |
| Circular motion | \(a_y = \dfrac{V^2}{R} = V r\) \(r = \dfrac{V}{R}\) | Know any two of \(V, R, r, a_y\) and you know the rest. |
| Steady turn, forces | \(N_{\text{in}} + N_{\text{out}} = mg\) \(F_{y,\text{in}} + F_{y,\text{out}} = m\dfrac{V^2}{R}\) | Vertical and lateral balance. |
| Steady turn, roll moment | \((N_{\text{out}} - N_{\text{in}})\,\dfrac{t}{2} = (F_{y,\text{in}} + F_{y,\text{out}})\,h\) | Moments about the CG: why the outside tires carry more load. |
| Quantity | Formula | Notes |
|---|---|---|
| Vehicle axes | SAE: \(x\) forward, \(y\) right, \(z\) down | ISO 8855: \(x\) forward, \(y\) left, \(z\) up. |
| Rotations | roll \(\phi\) about \(x\), pitch \(\theta\) about \(y\), yaw \(\psi\) about \(z\) | Right-hand rule. |
| Course angle | \(\psi + \beta\) | Direction of travel = heading + sideslip. |
| Slip angles | \(\alpha_f = \beta + \dfrac{a\,r}{V} - \delta\) \(\alpha_r = \beta - \dfrac{b\,r}{V}\) | SAE signs, small angles. |
| Tire lateral force | \(F_y = -C_\alpha\,\alpha\) | SAE: a negative slip angle gives a positive (rightward) force. |
| Quantity | Formula | Notes |
|---|---|---|
| Force to turn | \(F_y = m\,\dfrac{V^2}{R}\) | Supplied entirely by the tires. |
| Maximum corner speed | \(V_{\max} = \sqrt{\mu\, g\, R}\) | No aerodynamic downforce. |
| Friction circle (one tire) | \(\sqrt{F_x^2 + F_y^2} \le \mu F_z\) | Grip is shared between braking/driving and cornering. |
| g-g circle (whole car) | \(\sqrt{a_x^2 + a_y^2} \le \mu\) | Accelerations in g. |
| Friction ellipse | \(\left(\dfrac{a_y}{\mu_y}\right)^2 + \left(\dfrac{a_x}{\mu_x}\right)^2 \le 1\) | Forward acceleration is also capped by the drive (power) limit. |
| Quantity | Formula | Notes |
|---|---|---|
| Critical speed | \(U_{\text{crit}} = \sqrt{\dfrac{L}{|K|}}\) | Oversteering car (\(K < 0\)) is unstable above it. An understeering car has none. |
| Quantity | Formula | Notes |
|---|---|---|
| Cornering stiffness | \(C_\alpha = \left.\dfrac{dF_y}{d\alpha}\right|_{\alpha = 0}\) | Slope of the tire curve at zero slip. |
| Linear tire | \(F_y \approx C_\alpha\,\alpha\) | Small slip angles only. |
| Peak lateral force | \(F_{y,\max} = \mu(F_z)\,F_z\) | Reached near the peak slip angle. |
| Load sensitivity | \(\mu(F_z) = \mu_0\left[1 - s\,\dfrac{F_z - F_{z0}}{F_{z0}}\right]\) | \(\mu\) falls as load rises. |
| Pair of tires, load transfer \(\Delta F\) | \(F_{y,\text{pair}} = 2\mu_0 F_{z0} - \dfrac{2\mu_0\, s\, \Delta F^2}{F_{z0}}\) | Load transfer always costs an axle grip. |
| Magic Formula | \(F_y = D \sin\!\big(C \arctan \Phi\big)\) \(\Phi = B\alpha - E\,(B\alpha - \arctan B\alpha)\) | \(D\) = peak force, \(C\) = shape, \(E\) = curvature; slope at the origin \(BCD = C_\alpha\). |
| Quantity | Formula | Notes |
|---|---|---|
| Slip ratio | \(\kappa = \dfrac{\Omega r_e - V}{V}\) | \(\kappa > 0\) driving, \(\kappa < 0\) braking, locked wheel \(\kappa = -1\). |
| Linear tire | \(F_x \approx C_\kappa\,\kappa\) | Small slip ratios only. |
| Peak longitudinal force | \(F_{x,\max} = \mu_x F_z\) | Typically at 5–15% slip, depending on the tire. |
| Wheel spin dynamics | \(I_w\,\dot\Omega = F_x\,r_e - T_b\) | Past the peak, lock-up runs away. |
| Stopping distance | \(d = \dfrac{V^2}{2\mu g}\) | Use the peak \(\mu\) for ABS, the sliding \(\mu\) for a locked wheel. |
| Quantity | Formula | Notes |
|---|---|---|
| Friction ellipse (tire) | \(\left(\dfrac{F_x}{F_{x,\max}}\right)^2 + \left(\dfrac{F_y}{F_{y,\max}}\right)^2 \le 1\) | Combined braking/driving and cornering. |
| Lateral force left while braking | \(F_y = F_{y,\max}\sqrt{1 - \left(\dfrac{F_x}{F_{x,\max}}\right)^2}\) | On the ellipse boundary. |
| Normalized combined slip | \(n = \sqrt{\left(\dfrac{\kappa}{\kappa_{\text{peak}}}\right)^2 + \left(\dfrac{\alpha}{\alpha_{\text{peak}}}\right)^2}\) | \(n = 1\) is the combined peak. A sliding tire's force points along its sliding direction. |
| Quantity | Formula | Notes |
|---|---|---|
| Camber thrust (linear) | \(F_y \approx C_\alpha\,\alpha + C_\gamma\,\gamma\) | \(C_\gamma \ll C_\alpha\). |
| Camber at the limit | \(\gamma_{\text{at limit}} = \gamma_{\text{static}} + \Delta\gamma_{\text{roll}}\) | Aim near the tire's optimum camber. |
| Contact patch area | \(A \approx \dfrac{F_z}{p}\) | Membrane model, \(p\) gauge. |
| Hot pressure | \(p_{2,\text{abs}} = p_{1,\text{abs}}\,\dfrac{T_2}{T_1}\) | Absolute pressure and absolute temperature (K). |
| Normalized tire curve | \(F_y / F_z\) | Curves that don't collapse onto one show load sensitivity. |
| Quantity | Formula | Notes |
|---|---|---|
| Static axle loads | \(W_f = W\,\dfrac{b}{L}\) \(W_r = W\,\dfrac{a}{L}\) | Each wheel carries half its axle's load. |
| Longitudinal load transfer | \(\Delta F_{z,\text{long}} = \dfrac{m\,a_x\,h}{L}\) | Change in each axle's load; to the front under braking. |
| Total lateral load transfer | \(\Delta F_{z,f} + \Delta F_{z,r} = \dfrac{m\,a_y\,h}{t}\) | \(\Delta F_{z}\) = load gained by each outside wheel (lost by each inside wheel). |
| Front share (simple model) | \(\dfrac{\Delta F_{z,f}}{\Delta F_{z,f} + \Delta F_{z,r}} = \dfrac{K_{\phi f}}{K_{\phi f} + K_{\phi r}}\) | The axle with more load transfer saturates first. |
| Quantity | Formula | Notes |
|---|---|---|
| Unsprung transfer | \(\Delta F_{z,us} = \dfrac{m_{us}\,a_y\,h_{us}}{t}\) | Per axle, using that axle's unsprung mass. |
| Geometric transfer | \(\Delta F_{z,geo} = \dfrac{m_{s,\text{axle}}\,a_y\,h_{rc}}{t}\) | Through the links, immediately, with no body roll needed. |
| Roll-axis height under the CG | \(h_{ra} = h_{rc,f} + (h_{rc,r} - h_{rc,f})\,\dfrac{a}{L}\) | Straight line between the two roll centers. |
| Roll moment | \(M_\phi = m_s\,a_y\,(h_s - h_{ra})\) | \(h_s - h_{ra}\) is the roll moment arm. |
| Elastic transfer | \(\Delta F_{z,el} = \dfrac{K_{\phi,\text{axle}}}{K_\phi}\,\dfrac{M_\phi}{t}\) | Through the springs and anti-roll bars. |
| Body roll | \(\phi = \dfrac{M_\phi}{K_\phi}\) | \(K_\phi\) in N·m/deg. Roll gradient \(\phi / a_y\) in deg/g. |
| Total | unsprung + geometric + elastic | Same total as Lesson 3.1: the paths only redistribute it. |
| Quantity | Formula | Notes |
|---|---|---|
| Axle loads under braking | \(\dfrac{N_f}{W} = \dfrac{b}{L} + a_{CG}\,\dfrac{h}{L}\) \(\dfrac{N_r}{W} = \dfrac{a}{L} - a_{CG}\,\dfrac{h}{L}\) | \(a_{CG}\) in g. |
| Brake force split | \(F_{xf} = \beta\,W a_{CG}\) \(F_{xr} = (1-\beta)\,W a_{CG}\) | \(\beta\) = front brake bias (front share of brake force). |
| Ideal brake bias | \(\beta_{\text{ideal}} = \dfrac{b}{L} + a_{CG}\,\dfrac{h}{L}\) | At the limit \(a_{CG} = \mu\); moves rearward when grip falls. |
| Front lock-up | \(a_{CG}^{\text{front lock}} = \dfrac{\mu\,b/L}{\beta - \mu h/L}\) | The front never locks if \(\beta \le \mu h/L\). |
| Rear lock-up | \(a_{CG}^{\text{rear lock}} = \dfrac{\mu\,a/L}{1 - \beta + \mu h/L}\) | Maximum deceleration is the smaller of the two lock values. |
| Set-up rule | \(\beta\) slightly forward of \(\beta_{\text{ideal}}\) | Fronts lock first and the car stays stable. |