---
title: Rolling Resistance
module: Rotation
moduleNumber: 5
lessonNumber: 5
order: 505
summary: >
  Ideal rolling should coast forever, yet every real wheel slows down. The reason
  is that a deformable tire and road do not press through a single point: the
  contact patch spreads, the normal-force resultant shifts ahead of the axle, and
  that offset is a resisting moment even with no gross sliding. We package it as an
  equivalent force $F_{rr}=C_{rr}N$, tie the coefficient to load, surface, speed,
  and temperature, and use coast-down, towing, and traction tests to separate this
  contact loss from aerodynamic drag, bearing friction, and the adhesion limit
  where rolling gives way to skidding.
topics: [Rotation]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 9 — Rotation; §9-6"
---

## 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.

$$
% caption: Deformation shifts the contact-pressure resultant $N$ ahead of the axle, so its moment opposes rotation; the equivalent model replaces it by an axle force $F_{rr}$ opposing translation, removing energy without visible sliding.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.4,0)--(5.9,0);
\draw[thick,fill=black!5] (2.9,1.15) circle (1.1);
\draw[fill=white,draw=black] (2.9,1.15) circle (1.8pt);
\draw[black,thick] (2.1,0.05) .. controls (2.5,-0.28) and (3.3,-0.28) .. (3.7,0.05);
\draw[->,acc,very thick] (3.25,0.05)--(3.25,0.95);
\node[acc,right] at (3.32,0.6) {$N$};
\draw[->,black,very thick] (2.9,1.15)--(1.85,1.15);
\node[black,above] at (2.35,1.15) {$F_{rr}$};
\draw[->] ([shift={(2.9,1.15)}]105:1.1) arc (105:-25:1.1);
\node[black] at (4.3,2.15) {rotation};
\node[black,below] at (2.9,-0.45) {deformed contact};
\end{tikzpicture}
$$

## 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 $v\ne0$. 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.

$$
% caption: Longitudinal traction rises from the no-slip condition to a surface-dependent peak, then falls as sustained sliding grows; the slip-ratio convention must accompany any measurement.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.5,0.5)--(5.8,0.5) node[right] {slip ratio};
\draw[->,black] (0.5,0.5)--(0.5,3.2) node[above] {traction};
\draw[acc,thick] plot[domain=0.62:5.35,samples=160] (\x,{0.5+2.32*((\x-0.6)/1.85)^2*exp(2*(1-(\x-0.6)/1.85))});
\draw[black,dashed] (2.45,0.5)--(2.45,2.82);
\draw[fill=acc!20,draw=acc] (2.45,2.82) circle (1.8pt);
\node[acc,above] at (2.45,2.82) {peak};
\node[black,below] at (0.75,0.5) {roll};
\node[black,below] at (5.2,0.5) {slide};
\end{tikzpicture}
$$

**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 $N a$, 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.

$$
% caption: A finite deformed contact patch carries the load through a resultant $N$ offset a small distance $a$ ahead of the axle; static friction $f_s$ supplies traction while the moment $Na$ is the rolling-resistance torque.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.4,0.6)--(5.8,0.6);
\draw[thick,fill=black!5] (2.9,1.7) circle (1.02);
\draw[fill=white,draw=black] (2.9,1.7) circle (1.7pt);
\draw[black,thick] (2.05,0.67) .. controls (2.48,0.35) and (3.35,0.35) .. (3.72,0.67);
\draw[black,dashed] (2.9,0.6)--(2.9,1.7);
\draw[->,acc,very thick] (3.2,0.68)--(3.2,1.5);
\node[acc,right] at (3.27,1.12) {$N$};
\draw[->,black,very thick] (2.25,0.68)--(3.0,0.68);
\node[black,above] at (2.58,0.68) {$f_s$};
\draw[<->,black] (2.9,0.28)--(3.2,0.28);
\node[black,below] at (3.05,0.3) {$a$};
\node[black,below] at (2.9,-0.08) {contact patch};
\end{tikzpicture}
$$

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.

> **Worked example.** One wheel supports $N=600\ \mathrm N$ with $C_{rr}=0.012$
> at $v=10\ \mathrm{m\,s^{-1}}$. The model gives $F_{rr}=C_{rr}N=7.2\ \mathrm N$
> and a rolling-loss power $F_{rr}v=72\ \mathrm W$; over $100\ \mathrm m$ of level
> travel the loss is $F_{rr}d=720\ \mathrm J$. At steady speed the drive must
> supply that work on top of bearing, aerodynamic, and drivetrain losses; during
> acceleration it also supplies the translational and rotational kinetic-energy
> increase.

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 $\mu_sN$, 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 $R\omega\approx0$ 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.

$$
% caption: Under braking the rim speed $R\omega$ falls below the vehicle speed; lock-up occurs when the spin reaches zero while the vehicle still translates, converting rolling contact into sustained sliding.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.5,0.5)--(5.8,0.5) node[right] {time};
\draw[->,black] (0.5,0.5)--(0.5,3.2) node[above] {speed};
\draw[acc,thick] (0.72,2.82)--(5.28,1.32);
\node[acc,right] at (4.35,2.1) {vehicle};
\draw[black,thick] (0.72,2.82)..controls (1.85,2.2) and (2.72,0.92)..(3.45,0.62)--(5.28,0.62);
\node[black,right] at (3.65,0.88) {rim};
\draw[black,dashed] (3.45,0.5)--(3.45,0.62);
\node[black,below] at (3.45,0.5) {lock};
\end{tikzpicture}
$$

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

$$
F_{\rm drive}-F_{\rm loss}=
\left(m+\sum_i\frac{I_i}{R_i^2}\right)a.
$$

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.

$$
% caption: In no-slip acceleration each wheel gains both translational and rotational kinetic energy, so its inertia adds an effective mass $I/R^2$; rim-weighted mass raises the drive force beyond the translational term.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.4,0.6)--(5.8,0.6);
\draw[thick,fill=black!5] (2.7,1.65) circle (0.98);
\draw[fill=white,draw=black] (2.7,1.65) circle (1.7pt);
\draw[->,acc,very thick] (2.7,1.65)--(3.85,1.65) node[right] {$a$};
\draw[->,acc] ([shift={(2.7,1.65)}]105:0.98) arc (105:-30:0.98);
\draw[->,black,very thick] (1.9,0.66)--(2.62,0.66);
\node[black,above] at (2.26,0.66) {$f_s$};
\node[black,below] at (2.7,0.1) {wheel inertia $I$};
\end{tikzpicture}
$$

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.

$$
% caption: Tire rolling loss is viscoelastic hysteresis: material entering and leaving the deformed patch follows different loading and unloading stress-strain paths, and the enclosed loop area is dissipated as heat.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.5,0.5)--(5.8,0.5) node[right] {strain};
\draw[->,black] (0.5,0.5)--(0.5,3.2) node[above] {stress};
\fill[acc!12] (0.72,0.62)..controls (1.75,1.58) and (2.85,2.72)..(4.92,2.92)..controls (3.58,2.18) and (2.15,0.98)..cycle;
\draw[thick] (0.72,0.62)..controls (1.75,1.58) and (2.85,2.72)..(4.92,2.92);
\draw[thick] (4.92,2.92)..controls (3.58,2.18) and (2.15,0.98)..(0.72,0.62);
\node[black] at (4.1,2.68) {load};
\node[black] at (3.85,1.6) {unload};
\node[black] at (2.5,1.95) {loss};
\end{tikzpicture}
$$

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

$$
s_b=\frac{v-R\omega}{v}.
$$

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,

$$
s_d=\frac{R\omega-v}{R\omega}.
$$

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:

$$
F_{\rm res}=F_{rr}+c_dv^2.
$$

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.

> **Worked example.** A $1200\ \mathrm{kg}$ car coasts on level ground. At low
> speed, where the aerodynamic term $c_dv^2$ is negligible, its measured
> deceleration is $0.10\ \mathrm{m\,s^{-2}}$. The longitudinal resistance is then
> $F_{rr}=ma=120\ \mathrm N$, giving a rolling-resistance coefficient
> $C_{rr}=F_{rr}/(mg)=120/(1200\times9.81)=0.0102$. Extrapolating this low-speed
> value to highway speed would understate the total resistance, because the
> aerodynamic term then dominates and must be fitted separately.

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.

| test | fitted or measured quantity | terms that require control | independent check |
| --- | --- | --- | --- |
| coast-down | $F_{\rm res}=F_{rr}+c_dv^2$ | grade, wind, contact regime, thermal state | reversed-direction run and predicted speed trace |
| steady towing or drive | axle torque and loaded radius | torque zero, wheel speed, drivetrain contribution | force-equivalent comparison with coast-down |
| free-spinning wheel | bearing and seal drag | preload, temperature, brake clearance | bound non-contact contribution to $F_{rr}$ |
| repeated conditioning run | change in fitted parameters | tire pressure, load, surface, elapsed time | return to the initial condition |

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.**

> **Worked example.** A vehicle carries a total normal load of $12\ \mathrm{kN}$
> on level ground. With a rolling-resistance coefficient between $0.008$ and
> $0.015$ over the intended conditions, the contact-loss force is
>
> $$
> F_{rr}=C_{rr}N=96\text{ to }180\ \mathrm N.
> $$
>
> At $20\ \mathrm{m\,s^{-1}}$ that is $1.92$ to $3.60\ \mathrm{kW}$ of rolling-loss
> power, and over $10\ \mathrm{km}$ of level travel it is $0.96$ to
> $1.80\ \mathrm{MJ}$ of work — excluding aerodynamic drag, grade, bearing, and
> drivetrain losses, which the full energy budget adds. Expressed as wheel torque
> across four equally loaded wheels of loaded radius $0.32\ \mathrm m$, the
> resistance torque per wheel is $(F_{rr}/4)R=7.7$ to $14.4\ \mathrm{N\,m}$, the
> steady torque needed at each wheel to balance rolling loss; acceleration adds
> wheel and vehicle inertia, drag, grade, and drivetrain terms.

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.

| comparison | prediction from the resistance model | measured quantity | mismatch that changes the model |
| --- | --- | --- | --- |
| coast-down | $m_{\rm eff}\,\d v/\d t=-F_{\rm res}(v)$ | speed trace over a controlled route | speed-dependent residual, grade or wind bias |
| axle measurement | $P_{\rm axle}=\tau_{\rm axle}\omega$ | torque and wheel speed | radius, drivetrain, or torque-zero error |
| route energy | $W_{\rm roll}=\int F_{rr}\,\d s$ | fuel, electrical energy, or dynamometer work | omitted aerodynamic, grade, or thermal term |
| thermal check | average loss power becomes contact heating | tire, rail, or bearing temperature trend | heat-transfer or contact-state change |

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.