Compute axle cornering stiffness Cα, understeer gradient K, characteristic and critical speeds, steady-state yaw rate gain and turning radius from a linear 2-DOF (bicycle) model. The tool follows Gillespie's formulation and classifies the vehicle into Understeer, Neutral or Oversteer based on the sign of K.
Parameters
Vehicle mass m
kg
Wheelbase L
m
CG to front axle a
m
Smaller a means more front weight bias
Front cornering stiffness C_f
N/rad
Rear cornering stiffness C_r
N/rad
C_r > C_f biases toward understeer
Speed V
km/h
Steering angle δ
°
Steering-wheel input (16:1 steering ratio)
Results
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Weight split W_f/W_r (%)
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Understeer gradient K (deg/g)
—
Characteristic / critical speed (km/h)
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Handling type
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Yaw rate (deg/s)
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Turning radius (m)
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Bicycle model plan view — trajectory & slip angles
Shows front/rear slip angles, lateral force vectors, body yaw and yaw centre. Blue arrow = front lateral force, orange arrow = rear.
Characteristic speed (US cars) and critical speed (OS cars). Yaw gain diverges and the vehicle spins above V_crit.
$$R = \frac{L\,(1 + K V^2)}{\delta_{\text{front}}}$$
Steady turning radius R [m]. At low speed R ≈ L/δ (Ackermann); for K > 0 the radius grows with speed (push-out).
Vehicle Cornering Stiffness — Designing the Understeer Gradient
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I've never really understood what "cornering stiffness" actually means. A tyre bends because rubber is soft, right? Where does the "stiffness" come in?
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Good question. Imagine rolling a tyre straight while tilting it slightly relative to the direction of travel — that angle is the "slip angle α". The contact patch then drifts sideways and a lateral force F_y builds up. In the linear regime you can write F_y = −Cα·α, and that proportionality constant Cα is what we call cornering stiffness. Units are N/rad. A typical passenger-car front tyre sits around 60,000 N/rad — meaning roughly 60 kN of lateral force per radian of slip.
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OK, and then how do C_f and C_r turn into the "understeer gradient" K? The tool already shows K = 3.65 deg/g with the defaults…
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That's the Gillespie formula: K = (m/L)·(b/C_f − a/C_r). With the defaults (1500 kg, L = 2.6 m, a = 1.1 m, C_f = 60k, C_r = 80k) we get b/C_f = 1.5/60000 (front term) and a/C_r = 1.1/80000 (rear term). The front term is larger, so the difference is positive. Physically, that means the front axle reaches its slip limit first, which is exactly what "understeer" feels like — the nose pushes out before the rear lets go.
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So if K > 0 is "safe", why would anyone want oversteer? I've heard race cars are intentionally oversteery…
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For production cars, yes — engineers target K = +1 to +4 deg/g so that the average driver doesn't snap the car around when they panic-steer. Race teams play a different game: they want corner-entry rotation (slight OS) and corner-exit traction (back to US). Cars like the rear-engined Porsche 911 are structurally close to neutral because of their weight distribution; the production cars compensate with PSM (Porsche Stability Management) and toe geometry to keep effective K > 0 on the road.
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The "critical speed" cell shows ∞ in the default config. What is that, exactly?
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Look at the yaw rate formula r = (V/L)·δ / (1 + KV²). For K < 0 (oversteer) the denominator can hit zero at some speed — that's V_crit = √(L/|K|). Above it, the gain diverges, and the car effectively spins without any steering input. So V_crit is a hard upper limit for OS cars. For US cars (K > 0), the denominator stays positive forever and V_char is just a "feel" speed where you need double the low-speed steering angle. The tool flips between the two displays based on the sign of K.
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Final question — real tyres lose grip in the rain, change with load, change with temperature. Is this simple model really enough?
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Sharp catch. The linear 2-DOF model is only valid up to about ±5° slip and 0.4 g lateral acceleration. Beyond that, the tyre saturates and you need Pacejka's Magic Formula, which is what CarSim, IPG CarMaker and Dymola use under the hood. In rain, friction μ drops to 0.3-0.5 and effective Cα roughly halves. Think of this tool as "early-stage sensitivity analysis" or "a reference model for an ESC controller". Real-world tuning still needs a full non-linear model plus on-track testing.
Frequently Asked Questions
Cornering stiffness Cα [N/rad] is the slope of lateral tyre force Fy versus slip angle α in the linear region: Fy = −Cα·α. Typical passenger-car values are 40,000-80,000 N/rad for the front axle and 50,000-100,000 N/rad for the rear. Cα varies strongly with vertical load, inflation pressure, camber and surface friction; the linear region only holds for slip angles up to about ±5°. Beyond that, non-linear models such as the Pacejka Magic Formula are required.
The sign of K = (m/L)·(b/Cf − a/Cr) defines vehicle handling. K > 0 is understeer (more steering needed at higher speeds, safe), K = 0 is neutral steer, and K < 0 is oversteer (too sensitive at high speed and diverging beyond the critical speed). Production cars are tuned to K = +1 to +4 deg/g, and ESC brakes individual wheels to virtually restore K > 0 whenever the vehicle drifts toward oversteer.
For an understeer car (K > 0), the characteristic speed V_char = √(L/K) is where twice the low-speed steering angle is needed to keep the same turn radius — a useful measure of handling 'feel'. For an oversteer car (K < 0), the critical speed V_crit = √(L/|K|) is the point at which the yaw rate gain diverges and the car spins even without steering input. This is a hard upper limit that must not be exceeded. The tool switches the displayed quantity automatically based on the sign of K.
On a vehicle they are back-calculated from a steady-state circular test (ISO 4138) or a steering-ramp test. For tyres alone, flat-track machines sweep load, pressure and camber parametrically. In design, Pacejka 96/02 coefficients are fed to CarSim, IPG CarMaker or Dymola/VehicleDynamics to predict K and the characteristic speed. Tuning levers include roll-centre height, anti-roll bar stiffness, caster, toe, and bushing rates — particularly because roll-induced load transfer changes Cα non-linearly and shifts K.
Real-world applications
Passenger-car handling design: Production OEMs typically set a target K of +1 to +4 deg/g and reach it through roll-centre height, anti-roll bar rates, caster, and bushing stiffness. The linear 2-DOF model is used at concept stage for setting targets and for sensitivity studies; detailed evaluation switches to full-vehicle multi-body models (15-DOF or more) with Pacejka tyres in IPG CarMaker, dSPACE ASM or VI-grade VI-CarRealTime. A typical Toyota/Nissan/BMW workflow is: linear model targets → multi-body verification → proving-ground tests.
ESC/ESP (Electronic Stability Control): ESC compares the actual yaw rate from on-board sensors with the "driver-intended yaw rate" computed from a linear 2-DOF reference model. When the deviation exceeds a threshold, the controller brakes individual wheels to generate a corrective yaw moment and pull the car back toward understeer. The formula r_ss = (V/L)·δ/(1+KV²) used in this tool is exactly the core of every ESC reference model — Bosch ESP®, Continental EBC and others all embed an equivalent expression.
Motorsport setup: In F1, WEC and SUPER GT, engineers intentionally create slight OS on corner entry and revert to US on exit. The balance is shifted by axle Cα distribution and aerodynamic balance (front/rear downforce ratio). Pit changes to anti-roll bars, damper bump/rebound ratios and rear wing angle move K directly and translate to 0.1-0.5 s per lap. Teams pre-compute "setup sensitivity maps" with similar linear models to support real-time setup decisions.
Autonomous driving / ADAS path tracking: Lane-keeping assist (LKA) and self-driving path-following controllers (MPC, LQR) almost always use a linear 2-DOF internal vehicle model. The reason is computational efficiency and direct compatibility with linear-optimal-control theory. Tesla Autopilot, Mobileye, and Waymo stack a linear-model-based controller on top of a planner that draws the target trajectory, with the steering command then validated against a Pacejka-based plant model.
Common misconceptions and pitfalls
The single biggest trap is thinking the linear 2-DOF model is enough to finalise a setup. The Gillespie model is valid only up to about ±5° slip and 0.4 g lateral acceleration. Beyond 0.6 g the tyre saturates per the Pacejka Magic Formula, and Cα changes non-linearly with vertical-load transfer. During roll, the outer wheel sees more load and its effective Cα rises, while the inner wheel loses contact load and its Cα drops — this asymmetry is the main source of "unexpected oversteer near the limit". Conclusions drawn from a linear model never agree with real vehicle behaviour close to the grip limit.
The second pitfall is treating cornering stiffness as a single fixed tyre property. Real Cα depends on (1) vertical load Fz (roughly Cα ≈ k·Fz at low load, saturating as Fz grows), (2) inflation pressure (more pressure raises Cα but eventually reduces overall grip), (3) camber angle (negative camber boosts outer-wheel grip but hurts the inner wheel), (4) longitudinal slip ratio (drive/brake torque steals lateral capacity via the "friction circle"), and (5) road and tyre temperature, plus wear. The Cα number in a tyre data-sheet is for "fixed load, room temperature, new tyre" — on the road it can vary from half to twice that value. For critical setup decisions, always use measured tyre data.
The third misconception is the driver-side belief that "understeer is boring, oversteer is fun". For road cars with ESC off, a K < 0 vehicle often exceeds human reaction speed and creates a serious crash risk; OEMs treat this as unacceptable. Intentional OS is justified only on (a) closed circuits, (b) with professional drivers, or (c) when ESC can intervene at high frequency. Even a 911, with its rearward weight bias, ships with PSM permanently active to virtually maintain K > 0. Mixing up "passive OS" with "ESC-assisted virtual US" leads to confused setup discussions, so keep the two ideas separate.
How to Use
Enter vehicle mass (kg), wheelbase (m), and longitudinal CG position from front axle (m) to establish weight distribution
Input front and rear axle cornering stiffness values (N/rad) — typical ranges: 80,000–150,000 N/rad per axle for passenger vehicles
Set test lateral acceleration (g) or steering angle (deg); simulator computes understeer gradient K (deg/g), characteristic speed V_ch, and critical speed V_cr
Sedan with mass=1,500 kg, wheelbase=2.7 m, CG at 1.35 m from front, front cornering stiffness=120,000 N/rad, rear=100,000 N/rad. Weight split: 50.0% front/50.0% rear. At 0.4 g lateral acceleration with 15° steering input: understeer gradient K=0.8 deg/g, characteristic speed V_ch=78 km/h, critical speed V_cr=142 km/h, yaw rate=18.5 deg/s, turning radius=32.4 m. Classification: mild understeer (K > 0).
Practical Notes
Reducing rear cornering stiffness (worn rear tires or soft springs) increases K and shifts behavior toward understeer; racing setups target K ≈ 0 (neutral steer) for predictability
Critical speed V_cr defines the limit where yaw damping becomes negative; exceeding it causes sudden oversteer even in nominally understeer vehicles
For SUVs (higher CG), increase cornering stiffness estimates by 10–15% to account for roll stiffness coupling effects not captured in linear tire model
Validate cornering stiffness via tire test data or vehicle dynamics measurement; ISO 4138 lane-change maneuvers provide field validation at V_ch and V_cr