ENGINEERING NOTE · VEHICLE DYNAMICS · 23 SEPTEMBER 2026
Vehicle Suspension and Tire Physics
How does a wheel discover the ground, support a car, turn torque into acceleration, and lose grip? A mathematical guide to the shared physics and three different game-vehicle implementations.
This is an implementation study, not a controlled handling benchmark. Equations marked teaching model are simplified explanatory models; implementation claims are grounded in the inspected source revisions identified at the end. No commercial source code is reproduced here.
What must a vehicle simulation actually compute?
A rigid chassis alone can fall, bounce, and collide, but it cannot explain driving. At each wheel we need a contact state, a normal load from the suspension, longitudinal and lateral forces from the tire, and the opposing torque that changes wheel spin. Those forces act away from the center of mass, so they also produce pitch, roll, and yaw.
The useful abstraction is one rigid body plus four contact-and-wheel models. The tire is not simply a source of friction proportional to throttle: it can transmit only the forces allowed by its load, surface, slip, and chosen tire law. A suspension matters partly because it sets that normal load. “Better physics” therefore cannot be judged from spring motion alone.
Position p, orientation q, linear/angular velocity v, ω, and each wheel’s spin and suspension travel.
The road can push on a wheel, not pull it into the road: normal load is nonnegative.
Steering changes the local tire frame; engine and brakes change wheel torque, then slip and tire force.
This article concerns playable vehicle dynamics, not detailed deformation of each rubber tread block.
From a road query to one loaded wheel
Let aᵢ be wheel i’s chassis attachment, dᵢ its suspension direction, R its radius, and hᵢ the hit point reported by a road query. A ray or swept shape along dᵢ estimates the wheel center and the available suspension length ℓᵢ. Compression is xᵢ = clamp(ℓ₀ − ℓᵢ, 0, travel), where ℓ₀ is the uncompressed length. This is a geometric measurement, not a force yet.
Ray contacts are cheap and stable for smooth roads but a thin ray can miss a kerb beside the tire. A sphere/shape sweep samples a larger footprint but is costlier and still does not reproduce a deformable tire. A fully physical wheel is another choice: its collision body and constraints resolve contact instead of a single synthetic patch. Never confuse the visible wheel mesh with the contact representation.
Why the point of application matters
The first equation uses an inertial (world) frame; the second uses axes fixed to the chassis. The two lines are separate equations, not a single product. Here m is mass; a = dv/dt is linear acceleration; ω̇ = dω/dt is angular acceleration in the chassis frame; I is the inertia tensor. Accelerations, forces, lever arms, and angular quantities shown in these equations are vectors.
1. Why is the translational equation m a = ΣF + mg?
Start with the momentum of a body with constant mass: P = m v, where v is the velocity of its center of mass. Newton’s law says the time derivative of momentum equals the net external force. Thus dP/dt = m dv/dt = m a. Separating gravity from the other forces gives m a = Σᵢ Fᵢ + m g. The vector g points downward; gravity is not also included in the sum. In a world frame with upward z, the vertical component is m az = Σᵢ Fi,z − mg, with the last g interpreted as the positive magnitude 9.81 m/s².
2. Why does an offset wheel force rotate the body?
Choose the center of mass as the moment origin. A force Fᵢ applied at displacement rᵢ creates torque rᵢ × Fᵢ; its magnitude is |rᵢ||Fᵢ| sin θ, so only the perpendicular lever arm matters. In a rear view with x pointing right and z up, a right-side force with r = (0.8, 0, 0) m and F = (0, 0, 1,000) N gives r × F = (0, −800, 0) N·m: the minus sign follows the chosen axes, while the roll-moment magnitude is 800 N·m. Pure applied couples, if any, are added as τⱼ. An equal and opposite left–right pair can give zero net vertical force while creating a roll moment. The point rᵢ must be where the chosen rigid body actually receives the force: a chassis attachment for a separately simulated wheel, or an effective contact point in a simplified chassis model.
3. Where does ω × (Iω) come from?
Angular momentum about the center of mass is H = Iω, and the angular-momentum law in the world frame is dH/dt|world = Σ torque. To keep the inertia tensor I constant, express H and ω in axes rotating with the chassis. The transport rule is dH/dt|world = dH/dt|body + ω × H. With a rigid body and constant I in those axes, dH/dt|body = I ω̇; substituting H = Iω gives the second equation. Express all forces, lever arms, and torques on its right-hand side in the same chassis frame. Uniform gravity acts at the center of mass, so it contributes no moment about that point. In principal axes where I = diag(Iₓ,Iᵧ,Iz), the first component is Iₓ ω̇ₓ + (Iz − Iᵧ)ωᵧωz = Mₓ. The second term shows how simultaneous angular velocities couple even without a new torque about x. For rotation about a fixed principal axis the cross term vanishes, leaving the familiar scalar I ω̇ = τ. For combined roll, pitch, and yaw it generally cannot be dropped.
In a game step, these equations yield a = (ΣF + mg)/m in world coordinates and ω̇ = I⁻¹[Σ(rᵢ × Fᵢ) + Στⱼ − ω × (Iω)] in chassis coordinates. Transform forces into the appropriate frame before each calculation, then integrate linear and angular velocity over the physics time step. Moving the same force vector to another point leaves a unchanged but can change roll angular acceleration.
Ride height, oscillation, and load transfer
A linear spring creates force proportional to compression. A damper resists speed of compression or rebound; it is not another spring and does not determine the final static ride height. With x positive in compression, an idealized support-force magnitude is Fs = max(0, kx + c ẋ), where k is in N/m, c in N·s/m, and ẋ in m/s. The tire normal load Fz is related to this support force but need not equal its magnitude on banked ground, with force offsets, load blending, or a separate constraint. Some implementations apply support along the suspension axis and others along the road normal.
At rest on level ground, a symmetric four-wheel vehicle has approximately Fz ≈ mg/4 per wheel; therefore x₀ ≈ mg/(4k) if the springs are identical and there is no preload. If m = 1200 kg and k = 30,000 N/m per wheel, x₀ ≈ 0.098 m. The spring travel must leave room both to compress further over a bump and to extend into a dip.
mₛ is the supported corner mass, y its displacement from static equilibrium, and r road displacement. Natural frequency fₙ = √(k/mₛ)/(2π); critical damping c₍crit₎ = 2√(kmₛ); damping ratio ζ = c/c₍crit₎. A one-mass model does not represent unsprung wheel mass, tire compliance, bump stops, or an anti-roll bar.
What happens when stiffness or damping changes?
Apply a small road step to one corner. Compare body height, road height, and the instantaneous support force.
Teaching model: a mass–spring–damper in continuous contact. The step’s instantaneous road velocity is ignored; impacts and tire deformation require a richer model.
A more complete quarter-car has sprung mass mₛ, wheel/axle mass mᵤ, suspension kₛ,cₛ, and tire stiffness kₜ. In one sign convention: mₛ z̈ₛ = −kₛ(zₛ−zᵤ)−cₛ(żₛ−żᵤ) and mᵤ z̈ᵤ = kₛ(zₛ−zᵤ)+cₛ(żₛ−żᵤ)−kₜ(zᵤ−r), after gravity is removed by measuring displacement from equilibrium. It predicts both body bounce and wheel hop; a single ray wheel approximates or omits the latter.
Load transfer is a torque balance, not extra weight
In a turn, lateral acceleration aᵧ at center-of-mass height h creates roll moment approximately m aᵧ h. With track width t, transferring load Δ from the inside side to the outside side creates opposing moment Δt, so Δ ≈ m aᵧ h/t for the entire car under a simple rigid, flat-road assumption. Total vertical load is still roughly mg: outside tires gain what inside tires lose. Front and rear load-transfer shares depend on roll stiffness, geometry, and unsprung effects; this formula alone cannot allocate the transfer to each axle.
During straight-line acceleration or braking, a first approximation for load transferred between axles is Δfront↔rear ≈ m aₓ h/L, where L is wheelbase. An anti-roll bar couples left and right suspension travel: a schematic axle model applies opposing increments Fbar,L = kbar(xL − xR) and Fbar,R = −Fbar,L, with signs chosen for the local axes. It changes the distribution of load, not total grip for free. Many tires exhibit load sensitivity: doubling normal load produces less than double peak lateral force, so an uneven load split can reduce axle grip.
This balance assumes no aerodynamic downforce change, road bank, curb impact, or vertical chassis acceleration. A suspension force may point along the suspension axis or the road normal; the tire law should use its actual normal component.
Stiffness primarily sets static compression and body frequency. Damping sets transient oscillation. Travel, bump stops, anti-roll coupling, center of mass, and tire contact determine whether the wheel stays loaded.
Why a tire needs slip to make force
At the contact patch, let Vx be forward ground-relative speed, Vy sideways speed, Ω wheel angular speed, and R effective radius. A common pedagogical longitudinal slip ratio is κ = (RΩ − Vx)/max(|Vx|, vₑ); slip angle is α = atan2(Vy, max(|Vx|, vₑ)). Here vₑ is a low-speed regularizer. Different implementations use different denominators, clamping, and sign conventions, so these are shared concepts, not a claimed literal formula from every codebase.
Positive drive torque tends to raise Ω, producing κ > 0; braking lowers it. Steering changes the wheel’s local frame, creating α even while the chassis moves forward. The tire force is a function of slip, normal load, surface, camber, and sometimes pressure and speed: (Fx,Fy,Mz) = T(κ,α,Fz,γ,p,V). Mz is self-aligning torque. A game tire model may omit some arguments; no universal single “correct” tire curve exists.
A friction ellipse is a useful budget model: using more force to accelerate leaves less available to turn. Actual tire data may have asymmetric, load-dependent peaks and non-elliptical combined-slip behavior. On no contact, Fz = 0 and this tire-force budget vanishes.
Brake and turn on the same contact patch
Request forward and lateral forces; the dot is the feasible force after clipping to a simplified friction circle.
Teaching model: isotropic circle, fixed Fz. Not the exact combined-slip law of any one implementation.
Close to zero slip, the tire’s initial lateral slope is cornering stiffness Cα = |∂Fy/∂α|₍α≈0₎, measured in N/rad. Near the peak, the response saturates; after the peak, some fitted tires lose force as slip grows. Load sensitivity, camber, pressure, road material, temperature, and wetness can shift the curve. Transient tire response can be modeled with a relaxation length λ: approximately τ ≈ λ/max(|Vx|,vₑ) and τ Ḟy + Fy = Fy,steady(α). This prevents an instantaneous jump to the steady-state curve; whether a particular game implementation actually models it must be checked separately.
For a driven wheel, an idealized angular equation is Iʷ Ω̇ = Tdrive − Tbrake − Fx R − Trolling. Newton’s third law couples the ground force to wheel spin. Integrating wheel rotation solely from chassis speed would hide wheelspin and wheel lock. At almost zero speed, avoid dividing by Vx = 0; low-speed blending or a static-friction treatment is necessary.
What mathematics does each system choose?
The core laws are shared; what differs is how contact is represented, how spring load is coupled to the chassis, and especially how slip becomes tire force. These observations describe the inspected source, not the full set of possible configurations or a ranking of measured realism.
A / CHAOS VEHICLES
Spring load plus a grip-limited wheel
- Contact. Each wheel queries a road along its suspension axis; the inspected path supports raycasts and sphere sweeps. The chassis carries suspension targets into a constraint path, with a direct-force fallback.
- Vertical force. A displacement-based spring uses Fspring = kx and two damping coefficients selected by the direction of displacement change, of the form Fdamp = −c ẋ under the code’s displacement sign. Normal load can blend computed spring force with a resting-force setting. An anti-roll calculation couples the two wheels on an axle.
- Tire. Drive/brake torque divided by radius proposes longitudinal force. Lateral force comes from cornering stiffness or a lateral-slip graph. A circle-like clamp scales the joint force when √(Fx²+Fy²) > μ Fz, with additional low-speed and assist logic.
- Trade-off. Easy to tune through direct grip and steering quantities; the inspected default tire path is not a calibrated Pacejka law. Constraint-based suspension and force fallback should not be conflated.
B / ADVANCED VEHICLE SYSTEM (AVS 1.4.3)
Custom slip dynamics, with two wheel modes
- Contact. The inspected vehicle code line-traces from a compressed-wheel start to an extended-wheel end, subtracts wheel diameter from the hit distance, and clamps spring length.
- Vertical force. It explicitly converts configured spring rate from N/mm to N/m and compression from cm to m; the force is k·compression + c·compression speed. In this path it applies the force along the hit normal. Excess-compression handling adds an extra support term.
- Raycast tire mode. It computes contact-plane forward/right axes; a target longitudinal slip is driven by torque versus Fz R μx, lateral target slip blends a low-speed sign response with slip angle, and both slips are interpolated over time. A normalized combined-slip vector and a square-root lateral shaping turn Fz μx and Fz μy into force.
- Physical wheel mode. A wheel collision body receives the opposite suspension force. Contact friction and drive torque then rely on the rigid-body contact solution instead of the preceding synthetic tire-force branch. These modes should not be described as one tire law.
C / FGEAR VEHICLE PHYSICS
Spring preload plus interchangeable tire curves
- Contact. The inspected wheel code can use ray contacts or shape-based detection and computes wheel travel from the hit geometry. The contact normal participates in spring force.
- Vertical force. Configurable up/down travel, preload, spring rate, and separate compression/relaxation dampers produce load; the effective damping coefficient depends on the direction of travel. An optional suspension-limit constraint can prevent excessive chassis penetration.
- Slip. Longitudinal slip starts from RΩ − Vx; slip angle starts from atan(Vy/(|Vx|+ε)). Ratios and angles are bounded near difficult low-speed cases. The tire object receives load, slip, camber, pressure, radius, and speed.
- Tire choices. The inspected Simple tire uses a load-scaled sine/arctangent curve; Tire96 and MF61 include load-dependent Magic Formula terms and combined-slip adjustments. MF61 additionally includes pressure and moment terms. These are model families, not proof that a bundled parameter set matches any measured tire.
This schematic Magic Formula explains FGear’s Simple tire family: s is slip (ratio or angle), B controls slope, C shape, D peak, and E curvature. The detailed 1996/MF6.1 families add many load, camber, pressure, shift, and combined-slip coefficients. This is not the exact equation used for Chaos or AVS.
How does longitudinal force change with slip?
Switch an illustrative tire response and move the slip marker. The plot teaches curve shape, not measured force from a particular tire or an exact code replay.
Normalized illustrative curves. Grip-limited and slip-state curves are explanatory abstractions, not extracted tire curves. The Magic Formula coefficients here are illustrative, not fitted test data.
Where the models actually diverge
| Question | Chaos Vehicles | AVS 1.4.3 | FGear (inspected version) |
|---|---|---|---|
| Road contact | Ray or sphere sweep | Ray; alternate physical collision wheel | Ray or shape-based wheel queries |
| Vertical model | Spring + separate damping, constraint or force path; load blend and axle coupling | Explicit spring/damper + excess-compression support; physical-wheel alternative | Spring + preload + asymmetric damping; optional penetration constraint |
| Tire force | Torque/radius + lateral stiffness/graph, shared grip clamp | Interpolated target slip, shaped normalized longitudinal/lateral force; physical wheel uses contact friction | Interchangeable tire models: Simple, 1996, MF61, with combined slip |
| Useful strength | Direct, low-parameter baseline and force budget | Choice of synthetic or physical contact and explicit transient slip behavior | Broad tire-model controls, including camber/pressure effects in richer models |
| Model limitation | Grip law is stylized unless tuned from data | Ray and physical modes do not share a single behavior; custom heuristics need validation | Many coefficients create calibration burden; elaborate formulas cannot repair poor input data |
No published head-to-head accuracy or CPU benchmark is claimed here. To compare them fairly, hold chassis mass/inertia, wheelbase, radius, surface, control inputs, fixed step, and vehicle speed constant; record tire force, normal load, skidpad lateral acceleration, brake distance, bump response, and missed contacts. A system with a longer parameter list does not automatically look or feel better.
A practical simulation loop and tuning order
- 01 · Establish units and chassis state
Use consistent mass, length, force, torque, time, and inertia units. Put the center of mass and inertia tensor in a plausible location. Start with a fixed physics step.
- 02 · Measure contact
For every wheel: determine contact, hit normal/position, suspension length, road-relative velocity at the hit, and surface properties. A contact point on a moving platform must use relative velocity.
- 03 · Compute suspension
Set travel and spring rate from static sag; choose damping from a target fraction of 2√(km). Limit compression at bump stops and set load to zero when airborne. Add left–right anti-roll coupling only after individual wheels work.
- 04 · Compute wheel spin and tires
Transform contact velocity into each steered wheel frame; compute slip with a low-speed rule, evaluate tire forces from the chosen model, enforce combined-slip behavior, then update Ω using drive/brake/contact torques.
- 05 · Apply and integrate
Apply force at each contact or prescribed suspension point; integrate rigid-body linear and angular motion. Use substeps when wheel rotation, contact, or stiff springs change faster than the render frame. Interpolate graphics separately from physics.
- 06 · Instrument before adding assists
Plot per-wheel compression, Fz, κ, α, Fx, Fy, angular speed, and stability-control interventions. Tune the bare car before ABS, traction control, camera shake, or visual wheel motion.
These inspected implementations calculate their core forces in processor-side simulation code; a GPU draws the result and may compute unrelated visual effects. A GPU-based vehicle solver is possible, but is not implied by high visual quality. Fixed-step cost scales approximately with wheel count, contact-query complexity, constraint iterations, and substeps—not only the number of tire coefficients.
What the sources establish—and what they do not
| Test | What to measure | What would reveal a problem |
|---|---|---|
| Static load and drop | Corner load ≈ mg/4; sag, overshoot, settling | Incorrect units, bottoming, oscillation or negative ground load |
| Skidpad and steering ramp | Lateral acceleration, slip angle, load transfer | Unlimited turning force or abrupt sign changes at low speed |
| Acceleration and hard braking | κ, wheel speed, stopping distance, combined force | No wheelspin/lock, traction increasing with zero load |
| Single-wheel kerb | Contact hit rate, suspension travel, body roll | Ray misses, excessive impulse, chassis penetration |
| Variable frame rate | Same maneuver at different render rates | Physics changes with visual frame duration |
Source inspection confirms specific algorithms but does not provide calibrated real-world error, equal-hardware runtime, tire test data, or a controlled comparison among the three. Formula-rich FGear settings can be physically motivated yet still wrong without fitting; simplified AVS or Chaos settings can be more convincing for a target game after careful tuning. All three abstract away distributed rubber deformation, temperature, wear, and many contact-patch effects. Contact query choices can fail on sharp geometry; numerical substeps and solver constraints influence stability.
Choose the contact representation and tire law according to the phenomenon you need to reproduce, then validate the same maneuvers and telemetry under the same conditions. Evaluate physical behavior and control feel separately.