Visualize tire lateral force vs slip angle and the friction ellipse in real time using the Pacejka Magic Formula. Adjust B, C, D, E parameters to explore cornering stiffness and grip limits.
The operating point (red dot) animates through a straight→corner→brake scenario. Closer to the ellipse boundary = closer to grip limit.
4 Preset Comparison — Lateral Force Fy vs Slip Angle α
About the Tire Dynamics Simulator
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I'm really curious about the name 'Pacejka Magic Formula'. Why do you specifically call it 'magic'?
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When Professor Hans Pacejka (Netherlands) came up with an empirical formula that 'can reproduce almost all tire characteristics with this single equation,' a colleague remarked, 'It's like magic,' and that's the origin. It's a semi-empirical model derived not from theory but by fitting to measured data, featuring a unique structure combining sine and arctangent functions.
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When I moved the slider for 'B: Stiffness Factor', the slope near the origin of the graph changed. Is this what's called 'cornering stiffness'?
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Correct! Cornering stiffness is the slope of the curve at α=0, defined as $C_\alpha = dF_y/d\alpha|_{\alpha=0}$. Increasing B makes this steeper. In practical terms, it feels like 'the car responds strongly with just a slight turn of the steering wheel.' F1 slick tires have this 3 to 4 times higher than passenger car tires. If you select the 'Sport' preset in the simulator, B becomes 14, and you can see the slope at the origin get steeper.
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I was looking at the 'Friction Ellipse Animation' tab, and the red dot shifts from the top to the side when braking. Why is that?
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That's precisely the essence of the friction ellipse: 'the total force a tire can produce is constant.' The vertical axis is cornering force (lateral force), and the horizontal axis is braking force (longitudinal force). You can never go outside the ellipse. When you brake while cornering, you start using longitudinal force, reducing the margin available for lateral force, so the dot shifts sideways along the ellipse boundary. This is why in real cars, 'braking while cornering makes it easier to slip.'
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The E parameter often takes negative values, right? How does it differ when it's positive?
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When E is negative, the force drop after the peak is more gradual. This is typical of sport tires, which don't lose grip abruptly even beyond the limit, helping the driver maintain control. Conversely, if you set E > 0 (pull the slider to the right), the force drops sharply after the peak, resulting in a so-called 'abrupt slip' characteristic. On winter roads, the contact condition between the tire and road changes easily, so the sign of E significantly affects stability.
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When I overlay four types in the 'Preset Comparison' tab, the wet tire has the lowest curve. Is that because the D parameter is small?
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That's right. D is the peak coefficient multiplied by Fz×1000, corresponding to μ (friction coefficient). On wet roads, due to a water film, the effective friction coefficient drops to about 0.6–0.7. So D=0.7 for wet tires reflects the actual phenomenon well. However, note that wet tires also have a lower B, preventing abrupt force changes after slipping begins and ensuring a safety margin. It's a trade-off: 'Low B means lower cornering stiffness but a more predictable feel.'
Physical Model and Equations
Pacejka Lateral Force Magic Formula
$$F_y = D \sin\!\Bigl[ C \arctan\!\bigl\{ B\alpha - E(B\alpha - \arctan(B\alpha)) \bigr\} \Bigr]$$
$F_y$: lateral force generated by the tire [N] | $\alpha$: slip angle [rad]
$B$: stiffness factor - directly affects the slope near the origin (cornering stiffness)
$C$: shape factor - determines the saturated-region curve shape
$D$: peak factor - $D \approx \mu \cdot F_z/1000$ (determines maximum lateral force)
$E$: curvature factor - controls the post-peak drop-off (gentler when $E \lt 0$)
Cornering stiffness (tangent slope at $\alpha=0$)
$$C_\alpha = \left.\frac{dF_y}{d\alpha}\right|_{\alpha=0} = B \cdot C \cdot D \cdot F_z \cdot 1000 \quad [\text{N/rad}]$$
$F_x$: longitudinal force (drive/brake) [N] | $F_y$: lateral force (cornering) [N]
$\mu_x, \mu_y$: longitudinal and lateral friction coefficients | $F_z$: normal load [N]
If the combined longitudinal and lateral force lies outside the ellipse, the tire exceeds its slip limit.
Frequently Asked Questions
It can fit tire manufacturer measurement data with very high accuracy (correlation coefficients of 0.99 or higher are not uncommon). It is standardly adopted in vehicle simulations by automakers (e.g., IPG CarMaker, CarSim), and F1 teams also use it. However, it does not account for temperature, aging, or sudden surface condition changes, so advanced models consider dynamic parameter variations.
The higher the cornering stiffness, the larger the lateral force generated for a small slip angle. This means sharper handling response. However, if it's too high, the driver may find it difficult to perceive changes in behavior. Passenger cars have 10,000–30,000 N/rad, sports cars over 50,000 N/rad, and F1 cars can exceed 100,000 N/rad.
ABS constantly monitors the longitudinal slip ratio κ and controls each wheel's brake to keep it in the 'peak longitudinal force region' around $\kappa \approx \pm15\%$. If κ becomes too large, the tire locks ($\kappa = -100\%$), lateral force drops to zero, and steering becomes ineffective. ABS prevents this, maximizing the longitudinal direction of the friction ellipse while retaining lateral force margin. In the simulator, moving the 'Longitudinal Slip Ratio' slider to $-100$ shows the operating point on the lateral force curve dropping sharply.
In the Pacejka formula, $D \approx \mu$ can be approximated (more precisely, longitudinal/lateral friction coefficients and load dependency are expressed with correction factors). In the simulator, $F_{y,max} = D \times F_z[\text{kN}] \times 1000$ is used. Typical values: dry asphalt μ≈1.0–1.2, wet 0.6–0.7, snow 0.2–0.3. Try matching these with the 'D' slider.
If $E \gt 1$, the curve becomes non-monotonic after the peak, potentially showing physically inconsistent behavior (lateral force increasing again). Therefore, the simulator limits $E \leq 1$. In actual tire fitting, $E \leq 1$ is usually imposed as a constraint.
The basic structure of the Pacejka formula can be applied to motorcycles, but since camber angle significantly affects motorcycles, an extended model including camber thrust is needed. This simulator assumes four-wheeled vehicles, but by adjusting the B, C, D, E parameters to motorcycle measurement data, it can be used for qualitative understanding of lateral force-slip angle characteristics.
What is Tire Dynamics Simulator?
Tire Dynamics Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Real-World Applications
Engineering Design: The concepts behind Tire Dynamics Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Set Pacejka B coefficient (0.8–2.0) to adjust curve shape and initial cornering stiffness response
Adjust C value (0.5–3.0) to control the shape factor; higher C makes the curve rise more sharply near peak grip
Configure D (peak friction, 0.8–1.5) which directly scales maximum lateral force in Newtons
Set E value (−1.0 to 1.0) to fine-tune the curvature at peak grip and post-peak behavior
Input slip angle range (typically 0–15 degrees) and observe the lateral force curve update in real-time
Worked Example
For a 1200 kg race tire with vertical load Fz=3000 N, set B=1.2, C=1.3, D=1.15, E=−0.5. The Magic Formula yields peak lateral force of 3450 N at approximately 8.5° slip angle. The friction ellipse shows remaining longitudinal grip of 2100 N when cornering at maximum lateral force. At 4° slip angle, lateral force reaches 2650 N with cornering stiffness of 331 N/degree, typical for high-performance street tires.
Practical Notes
Racing slicks use higher D (1.3–1.5) and lower B (0.9–1.1) for sharp peak grip; road tires use D=0.9–1.0 and B=1.5–1.8 for progressive, forgiving response
E parameter strongly affects post-peak rolloff; negative E (−0.8) creates abrupt understeer characteristics critical for stability simulation
Friction ellipse shrinks dramatically when combined slip exceeds 0.9; reduce D by 15–25% in multi-axis braking scenarios
Cornering stiffness (slope at small angles) approximates C×D×Fz/1000; verify against manufacturer datasheets for validation