Rolling Resistance

Rolling resistance and steady motion

Ideal rolling without slipping can occur with static friction doing no work at a rigid contact. Real wheels still require a forward driving force or a sustained downhill component to maintain speed. The dominant low-speed loss is often rolling resistance: deformation of the wheel, surface, or both shifts the contact-pressure distribution and dissipates mechanical energy internally. Its source differs from kinetic sliding friction. A wheel can roll with negligible visible slip while losing energy through hysteresis in rubber, soil compaction, bearing loss, and micro-scale deformation.

A simple resistance model writes $F_{rr}=C_{rr}N$, where $N$ is normal force and $C_{rr}$ is an empirical coefficient tied to the stated wheel, surface, and test conditions. It changes with tire pressure, load, temperature, speed, surface texture, wheel radius, and deformation history. Over a defined operating range, the model turns a measured resisting force into a power loss $P_{rr}=F_{rr}v$. Outside that range, a speed-dependent or load-dependent model may be required.

The contact resultant can act slightly ahead of the wheel centre for a freely rolling wheel that is being pulled forward. Its offset creates a resisting torque opposite the rotation. An equivalent force model places $F_{rr}$ opposite the translation at the axle. Both descriptions represent the same external power loss when applied consistently. Combining them would double-count loss unless the model explicitly separates the underlying mechanisms.

Slip, traction, and braking

The no-slip condition fails when the friction force required by the translational and rotational equations exceeds the available static-friction limit. At that point, material points on the wheel and surface have relative motion in the contact region, and kinetic friction, heat generation, wear, and altered traction become relevant. The transition depends on normal load, surface state, tire or wheel compliance, and the demanded braking or driving torque.

Slip is often summarized by a slip ratio. For a wheel moving with centre speed $v$ and rim speed $R\omega$, a common signed measure in straight-line motion is $s=(R\omega -v)/|v|$ when $v\ne 0$. Positive and negative conventions vary between driving and braking applications, so any reported value must state its definition. The physically direct quantity is the contact-speed mismatch $R\omega -v$. Zero mismatch corresponds to ideal rolling; nonzero mismatch indicates longitudinal creep or slip.

Traction rises only over a limited slip range. A compliant tire can develop increasing longitudinal force over a small slip range, reach a peak, then lose force as sustained sliding grows. Antilock braking and traction control use this behaviour. The objective is to remain near the traction peak instead of locking the wheel or spinning it freely. The curve is surface-specific and must be measured or modelled for the operating condition.

Energy loss and measurement limits.

A rolling-resistance measurement can use a level coast-down test, a controlled tow, or a driven wheel with measured torque and speed. A coast-down result must separate rolling loss from aerodynamic drag, bearing loss, slope, and any braking contact. A tow test should record normal load, speed, tire pressure, temperature, surface condition, and the distance used to average force fluctuations. The reported coefficient is meaningful only with those conditions.

Wheel angular speed and translational speed should be measured independently when slip is under study. A wheel encoder records angular speed; contact-state classification also needs video tracking, a ground-speed sensor, or a timing gate. Synchronization matters because a short acceleration or braking transient can create an apparent mismatch if the two speed signals are offset in time.

The energy balance gives a final check. On a level surface at steady speed, the drive power must equal rolling-resistance power plus other stated losses. During a coast-down, the decrease in translational and rotational kinetic energy should match the work done by the resisting forces within measurement uncertainty. A disagreement can indicate unmodelled drag, changing slip, sensor bias, or an incorrect assumption that the wheel remained in the no-slip regime.

Contact mechanics, traction, and energy balance.

Normal load sets the scale of the real contact. A nominally round wheel touches a surface over a finite contact patch because both bodies deform. The pressure within that patch is nonuniform and shifts when the wheel is driven, braked, or rolling over a compliant surface. A larger normal load generally increases patch size and can change the rolling-resistance coefficient, with tire construction or rail profile setting the relation.

Static friction and rolling-resistance torque have different roles. Static friction supplies the tangential contact force needed to enforce a no-slip acceleration or braking constraint; an ideal stationary contact dissipates no energy. Rolling resistance represents irreversible deformation and can oppose steady rolling even when the required static-friction force is zero. The contact resultant shift can be written as a resisting torque $Na$, where $a$ is a small offset, or as an equivalent resisting force $F_{rr}=Na/R$ at the axle. The two descriptions are alternatives for the same idealized loss.

Tyres and rails impose different limits. A pneumatic tire has a relatively large compliant patch, so deformation hysteresis and surface texture contribute to both rolling loss and traction. A steel wheel on a steel rail has a much smaller patch and often low rolling resistance, but its available adhesion can limit acceleration and braking on contaminated track. In either case, a torque demand above the available traction produces creep or slip; lower deformation loss leaves usable longitudinal force dependent on the contact response.

The simple numerical model has measurement limits. Normal force can change with load transfer, road slope, or suspension motion. The coefficient may vary with temperature and speed, and a coast-down includes aerodynamic drag that rises much faster than rolling loss at high speed. A defensible result reports the operating speed, normal load, surface, temperature, slip condition, and method used to separate losses. These details determine whether a measured force is a rolling- resistance value, a traction-limited braking value, or a mixture of both.

Contact pressure is also a diagnostic quantity. Dividing normal load by a nominal patch area gives only an average; local pressure can be substantially larger near an edge, tread block, rail crown, or surface asperity. Those local peaks influence wear, heating, and the onset of material damage even when the average normal force is unchanged. A compliant wheel can spread load over a larger region, but it may also dissipate more energy as the material is repeatedly compressed and released.

A braking test should compare wheel torque, angular deceleration, vehicle deceleration, and contact force should be checked together. A locked wheel has angular speed near zero while the vehicle still translates, so its energy loss includes sliding work at the interface. A freely rolling wheel has a changing angular kinetic energy if its speed changes, even when the contact remains static. These distinctions prevent a measured temperature rise or force trace from being assigned to rolling resistance when the dominant process was actually braking slip.

Slip transients and braking.

The rolling constraint is kinematic. Contact forces arise from the translational and rotational equations. With forward centre speed $v$ and clockwise spin magnitude $\omega$, the contact-point speed relative to a stationary surface is $v-R\omega$. Ideal rolling requires this quantity to vanish. Differentiating while the contact remains stuck gives the matching acceleration condition $a=R\alpha$, where $\alpha$ is the clockwise angular acceleration magnitude. A mismatch between linear and angular acceleration creates relative contact motion unless static friction can supply the force and torque needed to restore the constraint.

During driven acceleration, applied axle torque can make $R\omega$ exceed $v$. The bottom of the tire then tends to move backward relative to the road, so static friction on the wheel acts forward. That forward friction accelerates the centre of mass while its torque opposes the excessive spin. During braking, brake torque can make $R\omega$ fall below $v$. The contact point tends to move forward, so static friction on the wheel acts backward. Its translational effect slows the vehicle, while its torque tends to maintain wheel rotation against the brake torque.

If demanded friction stays below ${\mu }_{s}N$, the wheel can remain in static rolling through the transient. If the demand exceeds that limit, the contact enters a slip regime. Under hard braking, angular speed may fall toward zero while the vehicle continues translating. A locked wheel has $R\omega \approx 0$ and a large negative contact-speed mismatch in this convention. The interface then experiences kinetic sliding friction, which often provides less controllable longitudinal force than the traction peak available at modest slip.

Lock-up removes the wheel's ability to roll and can remove steering control for a road vehicle because lateral tire force also depends on the contact state. Antilock systems reduce brake torque when wheel deceleration indicates approach to lock-up, then reapply it to keep slip near a traction-producing range. The control target is a surface- and tire-dependent compromise between longitudinal braking force, lateral control, heat, and sensor delay.

The rigid no-slip model has limits near the transition. A real tire contact patch can contain regions with different local slip, and compliant deformation delays the buildup of force after a torque change. Road texture, temperature, water, normal-load transfer, and wheel inertia all affect the transient. A calculation that treats friction as an instantaneous switch from static to one constant kinetic value can identify the direction of slip. Detailed braking force, heat, noise, and control response require a contact model and measured parameters.

Diagnostics should compare wheel speed, independent ground speed, brake torque, and normal-load estimate on one synchronized time base. A wheel-speed trace alone can show rapid angular deceleration. Ground speed separates a vehicle stopping from a wheel locking while the vehicle continues moving. The relative contact speed and its rate of change provide the direct evidence for the transition. Repeating the same brake command on different surfaces or loads also separates a control-system limit from a change in available tire-road traction.

The energy record is equally informative. A locked or strongly slipping wheel can produce rapid interface heating and visible wear, whereas a controlled near-peak slip event distributes more of the braking work through the intended tire response and brake hardware. Temperature, stopping distance, and force data should be interpreted together because each alone can conceal the underlying contact regime.

Vehicle response and material mechanisms

Wheel inertia changes the force required for acceleration. In ideal rolling, every wheel gains translational kinetic energy with the vehicle and rotational kinetic energy about its axle. For a vehicle mass $m$ with wheels of inertias $I_i$ and effective radii $R_i$, the no-slip acceleration model can be written

The added terms behave like an effective mass. A heavy rim increases the required drive force more than the same mass concentrated near the hub because inertia weights mass by radius squared. The relation applies while the wheels roll without appreciable slip and their radii are well defined. A deforming tire has a rolling radius that can differ from its unloaded geometric radius.

Effective mass helps separate vehicle response from contact loss. On level ground, a drive-force measurement that exceeds the measured translational mass times acceleration can include the force needed to accelerate the wheels. During braking, wheel rotational energy is returned to the brake system or dissipated at the contact depending on the braking method and slip state. Regenerative braking can recover some rotational energy, but tire-road loss and drivetrain conversion loss remain separate terms in the energy balance.

Coast-down tests estimate resistance from deceleration after drive torque is removed. With no active braking, a measured speed trace gives acceleration. The effective-mass model then gives total resisting force. Repeating the test over several speed ranges helps separate a nearly speed-independent rolling term from aerodynamic drag, which often rises strongly with speed. The road must be level or its measured slope must be included, since a small grade can imitate a substantial rolling-resistance force.

Torque and force measurements provide complementary checks. A hub torque sensor combined with wheel speed estimates mechanical power delivered at the axle. A ground-speed measurement and a longitudinal force sensor estimate vehicle power at the contact or tow point. Their difference includes drivetrain loss, wheel inertia effects during transients, and slip. The sensors must be synchronized; an accelerating wheel can make a phase offset look like a real energy discrepancy.

Radius and slip uncertainty can dominate the inferred rotational contribution. Since the effective term contains $I/R^2$, a relative radius error appears twice in that term. Using a wheel encoder with an assumed radius can also bias ground speed when tire pressure or load changes the rolling radius. During slip, wheel speed requires an independent translational measurement. A coast-down analysis uses a distance or ground-speed reference whenever the contact state is uncertain.

Report the total vehicle mass, wheel inertias or their estimation method, loaded radius, speed range, slope correction, air conditions, and synchronization method. Those details identify whether a fitted effective mass represents wheel rotation, unmodelled drag, or an artifact of the measurement system. A result that changes with acceleration direction or test order may indicate thermal drift, slope bias, or changing contact conditions, each of which can bias an inertial estimate.

Loaded radius can be calibrated by counting wheel revolutions over a surveyed distance at the same load and speed used in the dynamic test. This directly tests the encoder scale and avoids assigning an unloaded workshop radius to a deformed rolling contact.

Tyres, rails, and material loss mechanisms.

Rolling resistance in a pneumatic tire is largely a material-loss problem. Rubber entering the contact patch is compressed and sheared; it returns only part of the stored elastic energy when it leaves the patch. The stress-strain path forms a hysteresis loop whose enclosed area is energy dissipated per deformation cycle. The wheel continuously carries material through this cycle, so the loss appears as a resisting force or torque even when the contact point has no macroscopic sliding speed relative to the road.

Temperature, load, inflation pressure, and speed alter this loss. Temperature changes rubber viscoelastic response. Higher load changes patch shape and strain amplitude. Lower pressure generally increases deformation, while high speed can change both material response and aerodynamic loss. A coefficient measured on a cool smooth drum at one load is tied to that test condition. Report the coefficient over a stated operating range, with its geometry and conditions.

Rail-wheel contact uses different geometry and terminology. Steel wheels and rails have a small elastic contact region and usually much lower rolling resistance than pneumatic tyres, but they still develop small relative motions inside the patch. Rail engineering often calls this creep: the wheel may roll almost exactly at the kinematic speed while elastic deformation produces a small longitudinal or lateral mismatch across the contact. Creep generates the tangential force needed for traction, braking, and guidance before gross sliding begins.

The creep-force relation initially rises with creepage, reaches a contact-dependent limit, and then approaches a sliding regime. Rail creep and tire slip ratios use different normalizations of the mismatch between wheel surface motion and vehicle motion. Rail creep is often expressed as a small relative velocity or strain-like quantity; tire slip ratios commonly normalize the mismatch by vehicle or wheel speed. Their numerical values require their definitions and normalization.

Coefficient definitions must be kept separate. A tire rolling-resistance coefficient often means $F_{rr}/N$, dimensionless. A rolling-resistance moment parameter can instead be written as $M_{rr}/N$ and has dimensions of length; it becomes a force coefficient only after division by an effective radius. Rail resistance data may combine bearing, aerodynamic, grade, curvature, and contact terms into a total specific resistance. A traction or creep coefficient describes available tangential force relative to normal load; it may represent adhesion or dissipative rolling loss.

Measurements should identify the coefficient definition, contact state, material temperature, normal load, speed, surface, and whether the result came from a steady rolling test, a transient traction test, or a brake test. This prevents a low rail rolling-resistance number from being compared incorrectly with a tire traction coefficient or a total vehicle coast-down resistance.

Test preparation can change the result before the first measurement. Tire warm-up raises compound temperature and may alter hysteresis over repeated runs. A rail surface can acquire moisture, oxide, or contaminants that change adhesion without greatly changing bulk rolling loss. Record conditioning time, prior loading, and surface preparation. A controlled temperature change tests the coefficient over more than one service condition. These controls also distinguish a material-loss trend from drift in force sensors or speed calibration.

Slip ratio and control

Slip ratio compares wheel peripheral speed with vehicle ground speed, but several definitions are used. Let $v$ be forward ground speed and let $R\omega$ be the forward peripheral speed associated with the driven wheel. For braking at positive vehicle speed, a common convention is

It gives zero for ideal rolling and approaches one for a locked wheel with $R\omega =0$. For driving, a common convention instead normalizes by wheel speed,

It gives zero for ideal rolling and approaches one for a freely spinning drive wheel. Other conventions use one signed expression with a different denominator. The numerical sign and scale require the stated definition and driving or braking mode.

The unambiguous physical quantity is the longitudinal contact-speed mismatch $R\omega -v$. It has units of speed and remains defined when the wheel changes direction. Ratios become ill-conditioned near zero ground speed or zero wheel speed because their denominators become small. A controller can avoid this singularity by switching to a low-speed rule, using a regularized denominator, or controlling wheel deceleration and measured contact force instead of a raw slip ratio. A reported low-speed slip value should identify the convention and any threshold used by the algorithm.

Longitudinal tire force usually rises from the rolling condition as modest slip develops, reaches a surface-dependent peak, and then declines or saturates toward sliding. This peak sets the braking-control band instead of exact zero slip or lock-up. In braking, a controller reduces brake torque when wheel deceleration or estimated slip moves beyond the selected band, then reapplies torque as the wheel recovers. In driving, it reduces drive torque when the wheel begins to spin faster than the vehicle can accept traction.

The band shifts with wet pavement, ice, gravel, rail contamination, tire temperature, normal load, and tread condition. A value calibrated on one surface can be unsafe or ineffective on another. Vehicle-control systems combine wheel- speed signals with estimates of ground speed, acceleration, yaw, and sometimes surface state; wheel-speed comparison omits parts of the traction condition.

Measurement limits begin with ground-speed reference. Wheel encoders measure rotation. Using them alone assumes the ground speed under investigation. Radar, optical flow, inertial integration with correction, or a non-driven reference wheel can supply an independent estimate, each with its own noise and failure modes. Encoder quantization dominates at low speed, where one pulse represents a large apparent speed increment and differentiation magnifies the noise into a spurious acceleration or slip signal.

Signals also need a common time base. Brake pressure, motor torque, wheel speed, and ground speed can change on short time scales. A delay between channels shifts the computed mismatch and can make a stable rolling wheel appear to slip. Filtering reduces noise but delays the signal, so the filter and its latency are part of the control and measurement model. A complete test record states the slip convention, wheel radius model, ground-speed method, sampling rate, synchronization, filter, surface condition, normal load, and the criterion used to classify lock-up or spin.

Controller calibration should be checked against deliberately varied braking and drive demands. A band that suppresses wheel oscillation on dry pavement may react too slowly on ice or too aggressively on loose gravel. Logging commanded torque, estimated slip, measured deceleration, and wheel-speed recovery exposes whether a limit arose from contact physics, sensor latency, or a control-law choice.

Measurement and parameter identification

A coast-down experiment estimates resistance by releasing the drive torque and recording speed as the vehicle or wheel assembly slows. The test begins only after the wheel, bearings, and tire have reached a controlled thermal state. Runs should cover a stated speed range on a level surface or over repeated opposite directions so that grade and wind effects can be estimated. The vehicle must remain in the same contact regime: braking contact, steering corrections, or a change from rolling to slip invalidates a simple rolling-resistance fit.

In a no-slip coast-down, the effective mass converts measured deceleration into total longitudinal resistance. A common low-speed model combines a nearly constant rolling term with an aerodynamic term that grows approximately with speed squared:

The fitted constant term can include tire hysteresis, bearing drag, seal friction, brake rub, and a small uncorrected grade. The interpretation depends on independent checks, such as a free-spinning wheel test, a bearing-temperature record, or a repeated run with altered aerodynamic area. A rolling-resistance coefficient requires these other contributions to be bounded or explicitly included.

Torque-versus-speed fitting provides a complementary procedure. Measure axle torque and wheel speed during steady towing or controlled driving, convert torque to an equivalent longitudinal force using the loaded rolling radius, and fit the speed dependence over the measured range. The fit can separate a low-speed offset from a higher-speed rise, but correlated radius, torque-sensor, and speed errors must be retained. Drivetrain asymmetry, sensor zero drift, or a slope bias produce directional residuals.

Temperature conditioning is essential for viscoelastic tyres and lubricated bearings. Record ambient temperature, tire or wheel temperature when practical, inflation pressure, normal load, and time since the previous high-load run. A sequence of tests can warm the assembly and make later runs appear to have a different coefficient even on the same surface. Randomizing run order or repeating an initial condition after the sequence tests whether the system has drifted.

Uncertainty enters strongly through differentiated speed data. Small timing noise can become large acceleration noise, so smoothing or fitting is often required. The smoothing rule must be stated because it can suppress genuine changes in drag or contact state. Grade uncertainty, wind, wheel radius, torque calibration, ground-speed scale, and synchronization each contribute to the fitted parameters. Repeated runs estimate scatter while leaving a common slope or calibration bias. Report confidence intervals for both the resistance offset and the speed-dependent term, including their covariance.

Validation uses withheld data. Predict a coast-down trace at a new initial speed, a new load, or a reversed direction and compare the residuals. If the low-speed residual changes after temperature conditioning, a constant- coefficient model is inadequate for that use. If high-speed residuals grow with wind direction, the aerodynamic term requires a better environmental correction. The resulting parameter set is an empirical description of the stated apparatus and operating range.

Parameter tables should retain units and reference conditions. Report fitted force or torque offsets separately from dimensionless coefficients, state the loaded radius used in every conversion, and preserve raw time-series data with the fitting script or method. The retained data permit recalculation after a temperature, radius, or drag-model correction changes.

The rolling-resistance coefficient represents the stated test configuration only. The table keeps contact loss, aerodynamic loading, and fixture losses separate before they are combined in a vehicle-level energy estimate.

Regime maps and design verification

Rolling performance varies across regimes. At low speed on a hard smooth surface, deformation and bearing losses can dominate. As speed increases, aerodynamic drag and material-rate effects grow. At higher normal load, contact-patch size, stress amplitude, and temperature rise can change the rolling-loss contribution. A coefficient measured in one corner of this space supports interpolation near that condition and becomes unreliable when load, speed, or temperature changes substantially.

Temperature strongly affects viscoelastic tyres. A compound selected for low hysteresis in a narrow operating range may become stiff and lose traction when cold, or become soft and dissipative when hot. Tire pressure changes the tradeoff again: increased pressure can reduce deformation loss on some surfaces while reducing contact conformity and changing traction on rough or wet ground. Efficiency, braking force, ride, wear life, and damage resistance compete.

Material selection begins with the duty cycle. A long-distance road tire may prioritize low loss and thermal stability at sustained speed. A racing tire may accept greater hysteresis and wear to obtain high traction in a short thermal window. A rail wheel and rail system may prioritize low rolling loss, contact fatigue resistance, and stable creep behaviour under large loads. The material, geometry, inflation or preload, and surface treatment are selected as a coupled system.

Energy and traction are coupled but distinct. Reducing rolling hysteresis lowers steady energy loss, yet a compound or contact geometry that minimizes hysteresis may fail to produce the largest controllable longitudinal or lateral force. Braking and cornering require a favourable shear response in the contact patch; very low deformation can reduce the ability to conform to a rough surface. The correct comparison uses the required energy, traction, temperature, and wear targets for the actual mission instead of one laboratory resistance number.

Design margins must account for variation. Load transfer changes normal force from one wheel to another during acceleration, braking, and turning. Surface water, contamination, and roughness change available traction. Repeated flexing changes temperature and may age the material. A component that meets a target at a nominal condition can fail a braking or thermal requirement at a boundary condition. Regime maps identify combinations that require direct testing beyond an extrapolated coefficient.

Coefficient definitions also carry hidden choices. A force coefficient normalized by normal load, a torque parameter normalized by load, and a total coast-down resistance can all be reported as rolling resistance while representing different physical combinations. Keep aerodynamic drag separate from a contact coefficient unless the speed range and fit model combine them explicitly. Similarly, a traction coefficient measured at finite slip describes a different mechanism from a steady rolling-loss coefficient. Keeping the terms separate allows a design model to change one mechanism without silently changing another.

An adequate design report names the regime map axes, the tested surface, the thermal condition, wheel or rail geometry, normal-load range, speed range, and acceptance criteria. It then states which tradeoff was chosen: lower energy use, higher traction reserve, longer wear life, lower noise, or reduced thermal stress. This statement defines the selected operating condition and its design margin.

Vary one condition at a time around the selected design point, then test combined boundary conditions where load, temperature, and speed move together. This tests whether the chosen margin survives realistic service variation.

Design calculations and verification.

Heat accounting prevents the torque result from being misread as stored wheel energy. At steady speed, wheel rotational energy is constant, so the resistance torque removes mechanical work continuously. Most of that work becomes internal heating in the deforming tire and surface, with smaller contributions from bearings and air motion depending on the system. During a long descent or sustained drive, the average heat-generation rate is the rolling-loss power. Local contact temperature can be much higher than a bulk tire or rail temperature, so a thermal design check needs a heat-transfer model or measurement alongside energy input.

Model verification uses independent quantities. A coast-down test predicts a speed decrease from the same resistance range. A torque sensor predicts axle power at a measured wheel speed. A timed energy or fuel measurement over a controlled route provides a third comparison, after aerodynamic and grade terms have been estimated. Agreement among these methods supports the parameter range. One speed-and-load agreement leaves the range untested for other conditions.

The comparisons use the same load, surface, temperature, and speed interval as the fitted coefficient. A match at one condition does not validate transfer across a different contact regime.

State limits before design decisions are made. The constant-coefficient calculation becomes weak if speed-dependent hysteresis, temperature rise, large load transfer, significant slip, or rough-surface deformation changes the contact state. Adequate traction and thermal life require separate checks. The final design check compares the calculated energy range with traction reserve, braking requirement, component temperature, wear target, and the uncertainty of each measured input.

For reporting, preserve the coefficient definition, normal-load distribution, loaded radius, speed range, temperature, surface, and all losses excluded from the calculation. Rounding the resulting force or energy to more precision than the coefficient range supports would conceal the real design margin. A calculation identifies the expected loss and the conditions requiring a replacement model.

A verification run tests both ends of the calculation range. At the lower coefficient condition, check whether sensor resolution and bearing drag are large enough to obscure the predicted contact loss. At the upper coefficient condition, check whether tire or wheel temperature, drive-torque capacity, and braking reserve remain within the design limits. If the measured force leaves the predicted interval, inspect surface state, pressure, loaded radius, and wind or slope correction before redefining the coefficient. The sequence keeps the parameter range tied to identified mechanisms.

In an accelerating design case, use the same measured resistance range together with the effective mass and target acceleration to determine required axle force. Then compare the implied contact force with the available traction range. A design can have sufficient motor torque and still spin the wheel. It can also meet steady energy targets while exceeding a thermal or adhesion limit during transients.

Verification records should retain the residuals as well as the fitted parameters. Residuals ordered by speed, temperature, load, or travel direction often expose a condition-dependent mechanism before it appears in a summary coefficient. A flat residual pattern over the tested range supports the compact model for that range; a systematic pattern identifies the variable that requires explicit treatment in the next design iteration. This preserves the connection between the calculation, the experiment, and the operating condition used to justify the result.

The same discipline applies when results are transferred between vehicles or test rigs. Match normal-load distribution, wheel construction, surface, thermal state, speed range, and slip condition before comparing coefficients. If one condition changes, preserve the original parameter set and record the new result separately. The comparison then shows whether the change belongs to contact deformation, aerodynamic environment, instrumentation, or an interaction among them.