How MRT Wheel–Rail Contact Works Using Mathematics: How Two Pieces of Steel Guide a Train Through a City

An MRT train can weigh hundreds of tonnes.

Yet almost all of that moving mass meets the railway through a surprisingly small set of wheel–rail contact patches.

Each steel wheel touches the steel rail over a tiny region rather than over the whole visible width of the wheel.

Inside that tiny contact region, an extraordinary number of jobs have to happen at once.

• The train’s weight must be supported.
• Traction force must accelerate it.
• Braking force must slow it.
• Lateral force must help guide it through curves.
• The wheelset must remain dynamically stable.
• Friction must be high enough for useful force but not so destructive that wear and noise become excessive.
• Wheel and rail profiles must continue to interact sensibly as both surfaces wear.

The passenger sees a wheel rolling on a rail.

The mathematician sees contact geometry, elastic deformation, pressure distribution, friction, creepage, curvature, oscillation, wear and feedback.

The entire railway vehicle is controlled through contact regions small enough that we normally never see them.

Singapore gives this physics a visible public consequence. LTA has explained that some curved sections of the Thomson-East Coast Line produce higher train noise because of track curvature. Mitigation includes reducing train speed in those areas and grinding the rail surface to smooth wheel–rail contact. SMRT similarly describes wheel profiling and rail grinding as preventive maintenance used to maintain smoother, quieter train operation.

This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then see How MRT Track Inspection Works Using Mathematics and How MRT Predictive Maintenance Works Using Mathematics.

Now we move to the physical interface underneath all of them.

The RFE — Why Does Wheel–Rail Contact Mathematics Exist?

The weakest possible answer is:

so the wheel can roll on the rail

The deeper answer is that the contact must transmit exactly the forces the train needs while controlling damage created by those same forces.

The Reason for Existence is:

support the train
+
transmit traction and braking
+
provide lateral guidance
+
allow efficient rolling
+
limit slip, wear and fatigue
+
remain stable through straight track and curves
+
continue working as wheel and rail profiles evolve

This immediately creates a trade-off.

Very low friction reduces available traction and braking force.

Very aggressive frictional interaction can increase wear, heat and noise.

A perfectly rigid wheel and rail would generate impossible local stresses.

A very soft interface would deform excessively and waste energy.

A wheel profile that strongly self-steers can improve curve negotiation but can also influence lateral stability.

So the contact problem is not to maximise one variable.

The RFE is to make the smallest physical interface on the train perform the largest collection of competing jobs.

Prompt 1 — How Can a Tiny Contact Patch Carry a Train?

Steel feels rigid to us.

Under railway loads, it is elastic.

The wheel and rail deform slightly where they touch.

That deformation spreads the load over a small contact patch.

Classical Hertz contact theory is a starting point for understanding this normal contact.

Under its assumptions, the contact region between two curved elastic bodies is approximately elliptical.

Let the ellipse have semi-axes a and b.

Area is:

Acontact = πab

If normal wheel load is N, average contact pressure is:

pmean = N/(πab)

For the Hertz elliptical pressure distribution:

p(x,y)
=
[3N/(2πab)]
√[1 − x²/a² − y²/b²]

Maximum pressure occurs at the centre:

p0 = 3N/(2πab)

So:

p0 = 1.5 pmean

The centre experiences greater pressure than the edge.

A deliberately simplified pressure example

Suppose one hypothetical wheel carries a vertical load of:

N = 60,000 N

Suppose an illustrative contact ellipse has:

a = 7 mm = 0.007 m
b = 5 mm = 0.005 m

Area:

A = π(0.007)(0.005)
  ≈ 1.10 × 10−4 m²

Average pressure:

pmean
≈ 60,000/(1.10×10−4)
≈ 5.46×108 Pa
≈ 546 MPa

Hertz peak pressure in this educational example is:

p0 ≈ 819 MPa

These are fictional dimensions and loads chosen only to show scale.

The important insight is that a large train load is concentrated into a very small elastic contact region, producing extremely high local stress.

Why the patch is not simply a dot

If steel were infinitely rigid, the ideal geometric contact might approach a point or line.

Real elastic deformation creates finite area.

This is essential because:

pressure = force / area

Without deformation spreading the force, theoretical pressure would become unbounded.

Contact mechanics therefore begins with a paradox:

The wheel is able to support a huge train precisely because the steel is not perfectly rigid.

Prompt 2 — If a Steel Wheel Rolls, Why Is Friction Needed?

An ideal wheel rolling at constant speed on level track could seem to need almost no friction.

An MRT needs friction whenever it changes its motion or direction.

• Acceleration requires longitudinal contact force.
• Braking requires longitudinal contact force in the opposite direction.
• Curving requires lateral force.
• Disturbances require corrective forces.

A simple adhesion limit is represented by Coulomb friction:

|Ft| ≤ μN

where:

• F_t is tangential contact force,
• μ is effective adhesion coefficient,
• N is normal force.

If the demanded force exceeds available adhesion, gross slip or slide can occur.

Traction requirement

Suppose total train mass is m and desired acceleration is a.

Ignoring resistance for a moment:

Ftraction ≈ ma

That force must ultimately be transmitted through the driven wheel–rail contacts.

If total available adhesive normal load on driven wheels is N_d:

ma ≤ μNd

So a simplified adhesion-limited acceleration is:

amax ≤ μNd/m

The exact vehicle distribution is more complicated, but the structure shows why adhesion matters.

Friction is not constant

The effective wheel–rail friction condition can change with:

• water,
• contamination,
• surface roughness,
• temperature,
• wear debris,
• applied friction modifiers,
• and contact conditions.

A clean dry laboratory value is therefore not a universal operating constant.

Research on wheel–rail contact emphasises that measured creep-force curves can differ significantly from idealised contact theories because real surfaces contain third-body layers and changing friction conditions.

The friction paradox

Too little adhesion:

poor traction
poor braking force transmission
wheel slip or slide

Too much frictional stress in difficult contact conditions:

greater wear
higher tangential stresses
possible noise and fatigue growth

The objective is not maximum friction.

It is sufficient, controlled adhesion.

The wheel needs enough friction to control the train, but the same friction is one of the mechanisms that slowly consumes the wheel and rail.

Prompt 3 — What Is Creepage, and Why Can a Rolling Wheel Still Slip?

This is one of the strangest ideas in railway mechanics.

A wheel can be rolling normally while tiny relative slip exists inside the contact patch.

This small mismatch is called creepage.

Suppose wheel rolling speed based on angular velocity is:

vwheel = ωr

Vehicle forward speed is V.

A simple longitudinal creepage is:

ξx = (ωr − V)/V

If ξ_x = 0, ideal rolling speed matches vehicle speed.

If ξ_x is small but non-zero, microscopic stick and slip regions develop inside the contact patch and create longitudinal force.

Linear creep-force region

For sufficiently small creepage, Kalker-type linear contact theory gives relationships of the form:

Fx ≈ −f11 ξx

and lateral contact force can be represented conceptually as:

Fy ≈ −f22 ξy − f23 ξspin

The f coefficients depend on contact geometry, material properties and contact-patch dimensions.

The equations say that small relative motion produces restoring tangential force.

Force saturation

The linear relationship cannot increase forever.

Eventually contact approaches the friction limit:

√(Fx² + Fy²) ≲ μN

Beyond the linear region, more creepage produces proportionally less additional force and more slip.

This gives the characteristic creep-force curve:

small creepage
→ force grows nearly linearly

larger creepage
→ contact partly slips

high creepage
→ force approaches adhesion limit

Why creepage is useful

Without a small elastic mismatch in the contact patch, the wheel could not generate controlled traction or lateral guidance in the way real railway dynamics require.

But more creepage increases dissipated energy.

A simple contact-power loss is:

Ploss ≈ Fx vx,slip + Fy vy,slip

Normalised by vehicle speed, wear-energy measures often relate contact force to creepage.

The contact therefore needs small controlled slip to create useful force, but that same slip becomes a source of heat and wear.

A railway wheel is not perfectly rolling or simply sliding. Much of railway control lives in the narrow mathematical region between those two ideas.

Prompt 4 — How Can Two Wheels Fixed to One Axle Steer Around a Curve?

A railway wheelset usually has two wheels rigidly connected to the same axle.

They rotate at the same angular velocity.

On a curve, however, the outer rail has a longer path than the inner rail.

How can equal wheel rotations cover unequal distances?

The answer lies partly in wheel-profile geometry.

Rolling radius difference

Railway wheels are not simple cylinders.

The tread profile allows effective rolling radius to change as the wheelset moves laterally.

For a simplified conical wheel model:

rR = r0 + λy
rL = r0 − λy

where:

• r₀ is nominal rolling radius,
• y is lateral wheelset displacement,
• λ is an idealised conicity parameter.

The rolling-radius difference is:

Δr = rR − rL = 2λy

The wheel on the outer side can therefore roll on a slightly larger effective radius than the inner wheel.

Pure-rolling curve condition

Let track half-spacing between rolling paths be a and curve radius of the wheelset centre be R.

Outer and inner path radii are approximately:

Router = R + a
Rinner = R − a

For ideal pure rolling with equal angular velocity:

rR/rL
≈
(R+a)/(R−a)

Using the small-ratio approximation gives:

λy/r0 ≈ a/R

so:

y ≈ ar0/(λR)

This simplified equation says tighter curves require greater rolling-radius difference and therefore greater lateral shift or stronger profile action.

Real railway wheel profiles are more complicated than perfect cones, and bogie suspension, creepage, flange contact and contact geometry all participate.

But the mathematical idea is powerful:

A wheelset can steer partly because moving sideways changes how large each wheel effectively becomes.

Lateral acceleration

A train moving at speed v through curve radius R requires centripetal acceleration:

ac = v²/R

Required net lateral force for mass m is:

Fc = mv²/R

Again velocity is squared.

Double speed through the same radius and required lateral force becomes four times as large.

This helps explain why tight curves are sensitive to speed.

LTA’s public explanation of higher TEL noise on curved sections and the use of reduced speed in those areas is therefore intuitively consistent with the wheel–rail mechanics: curvature increases the lateral-contact challenge, and lower speed reduces v²/R.

Prompt 5 — Why Can Wheel Conicity Both Guide the Train and Make It Oscillate?

The same rolling-radius difference that helps a wheelset steer also creates a self-correcting lateral mechanism on straight track.

Suppose the wheelset moves slightly to one side.

One wheel encounters a larger effective rolling radius.

The other encounters a smaller one.

Because both wheels rotate together, the wheelset tends to steer back across the track.

But inertia can carry it past the centre.

Then the radius difference reverses.

An oscillation can develop.

left displacement
→ rolling-radius difference
→ steering correction
→ cross centre
→ opposite rolling-radius difference
→ opposite correction

This geometric oscillation is historically associated with the kinematic behaviour of coned wheelsets.

A simple oscillator

At a teaching level, lateral wheelset motion can be compared with a damped oscillator:

mÿ + cẏ + ky = F(t)

where:

• m represents effective lateral mass,
• c represents damping,
• k represents effective restoring behaviour,
• F(t) represents track and contact excitation.

A real railway vehicle has multiple wheelsets, bogie frames, suspension elements, yaw motion and nonlinear wheel–rail contact, so its dynamics are far richer than this equation.

But the central stability question remains:

after a disturbance,
does lateral motion decay,
remain,
or grow?

If amplitude decays:

stable

If amplitude grows:

unstable

Railway-dynamics research studies how equivalent conicity, suspension properties, wear and speed influence this stability.

Equivalent conicity

Real wheel and rail profiles are curved, worn and non-conical.

Engineers therefore use equivalent conicity to describe the effective rolling-radius difference produced by a given lateral displacement.

A conceptual definition is:

λeq ≈ Δr/(2y)

for the relevant displacement range.

Wheel and rail wear change λ_eq.

That means maintenance changes vehicle dynamics even if suspension hardware remains untouched.

The wheel profile is part of the steering system even though it has no steering motor.

Prompt 6 — Why Do Curves Create Noise, Wear and Flange Contact?

Ideal rolling-radius steering cannot perfectly satisfy every curve.

On tighter curves, bogie geometry, wheelbase, wheel profile and available rolling-radius difference may not match the required path geometry perfectly.

The wheelset then develops larger creepage and lateral contact forces.

The wheel flange may also contact the gauge side of the rail.

Angle of attack

The wheelset may not point exactly tangent to the rail curve.

Let small yaw mismatch be angle ψ.

For small angles, lateral creepage contains a term related to ψ.

ξy ≈ vy/V − ψ

Larger mismatch means more lateral sliding tendency in the contact patch.

Frictional power

Contact energy dissipation can be estimated from creep forces and relative slip velocity.

Pfriction
≈
Fx vx,slip
+ Fy vy,slip
+ Mspin ωspin

More dissipated energy generally means more opportunity for:

• wear,
• heat,
• surface damage,
• and noise excitation.

Curve squeal

Wheel–rail noise on tight curves can involve stick–slip, lateral creepage, flange interaction and structural vibration.

The detailed acoustics are complex, but the feedback structure can be represented conceptually:

lateral creepage
→ frictional force
→ wheel/rail vibration
→ contact condition changes
→ friction force changes again

If this feedback supplies energy at a structural vibration frequency, audible squeal can develop.

Singapore’s TEL example makes the principle concrete: LTA says higher noise occurs on certain curved sections because of track curvature, and mitigation includes reduced train speed and rail grinding to smooth wheel–rail contact.

Why lower speed helps

Centripetal acceleration is:

ac = v²/R

Reducing v reduces required lateral acceleration quadratically.

It also changes dynamic excitation and the time-scale of the contact interaction.

This does not mean slower is always better for the railway, because journey time and capacity matter.

It means curve speed is another optimisation variable.

Prompt 7 — How Do Wheel and Rail Wear Each Other into New Shapes?

Every train passage applies repeated contact stress.

Over enough cycles, material state changes.

Two major damage families are:

• wear, where material is removed,
• rolling contact fatigue, where cyclic stress initiates and grows cracks.

These mechanisms interact.

Wear can remove material containing shallow fatigue cracks.

Too little wear can allow some fatigue damage to accumulate.

Too much wear consumes wheel and rail profile rapidly.

Research reviews therefore treat wear and rolling contact fatigue as coupled rather than independent.

Archard wear law

A classical simplified wear relationship is:

Vwear
=
k Ws/H

where:

• V_wear is worn volume,
• k is a dimensionless wear coefficient,
• W is normal load,
• s is sliding distance,
• H is material hardness.

The exact wheel–rail wear process is much more complex because contact pressure, creepage, material state, lubrication and surface conditions vary.

But the law reveals the basic structure:

more load
+
more sliding
→ more wear

Wear number

Railway wear models often use frictional work or a wear number related to contact force and creepage.

A simplified tangential work per unit rolling distance is:

Twγ
≈
|Fx ξx| + |Fy ξy|

Higher Tγ generally indicates more frictional work available to drive wear, though the relationship depends on wear regime.

Profile feedback

Wear changes wheel profile.

Changed wheel profile changes contact geometry.

Changed contact geometry changes forces and creepage.

Those forces change future wear.

profile
→ contact patch
→ creepage and force
→ wear
→ new profile
→ new contact patch

This is a feedback loop.

A wheel and rail therefore co-evolve.

Wheel profiling

SMRT publicly states that wheel profiling is used to maintain even wheel roundness and diameter for smoother and quieter operation.

Mathematically, profiling attempts to bring measured wheel profile r(y) back towards a target profile r_target(y):

eprofile(y)
=
r(y) − rtarget(y)

Machining removes material so that profile error is reduced while preserving allowable wheel size and life.

Rail grinding

Rail grinding reshapes and smooths the rail head.

SMRT says uneven track surface is a major contributor to track noise and uses rail grinding to create smoother, quieter operation. LTA also cites rail grinding as a mitigation for wheel–rail noise on curved TEL sections.

Grinding therefore does several possible jobs depending on the maintenance objective:

• remove surface irregularity,
• restore desired profile,
• manage corrugation,
• remove shallow damaged material,
• reduce adverse contact conditions.

But grinding itself removes rail material.

Again the optimum is not:

grind as much as possible

It is:

remove enough material
at the right time
to improve contact condition
without wasting rail life

Wheel and rail maintenance is controlled wear: remove a small amount deliberately so uncontrolled wear and fatigue do not decide the shape instead.

Prompt 8 — How Does the Whole Wheel–Rail System Become an Optimisation Problem?

By now, the contact has many objectives.

• low rolling resistance,
• adequate traction and braking adhesion,
• stable straight-running behaviour,
• good curve negotiation,
• low flange contact,
• low wear,
• low rolling-contact fatigue,
• low noise and vibration,
• long wheel and rail life,
• good passenger comfort.

These objectives conflict.

A conceptual design objective can be written:

minimise J
=
w1 wear
+ w2 fatigue risk
+ w3 noise
+ w4 rolling resistance
+ w5 lateral instability
+ w6 flange contact
+ w7 maintenance cost

subject to:

required traction force
required braking force
safe wheel unloading limits
allowable vehicle dynamics
track geometry
wheel and rail material limits
passenger comfort
operating speed

Decision variables can include:

• wheel tread profile,
• rail profile,
• suspension stiffness and damping,
• speed through curves,
• friction management,
• wheel reprofiling interval,
• rail grinding interval.

This becomes a multi-objective optimisation.

Pareto frontier

Suppose Design A has low wear but slightly higher rolling resistance.

Design B has lower resistance but greater curve wear.

Neither dominates the other.

Both can lie on a Pareto frontier.

An engineering choice then depends on:

route curvature
service intensity
maintenance access
noise constraints
asset cost
reliability objectives

This explains why there is no universal “best wheel profile” independent of the railway it must operate on.

Robust optimisation

A wheel profile is new only once.

A rail profile changes with traffic.

Friction changes with environment.

Passenger load changes wheel force.

A good design should therefore perform adequately over a set of scenarios rather than one nominal condition.

minimise worst-case or expected J
across:
new wheel
worn wheel
new rail
worn rail
high load
low load
straight track
curve
varying adhesion

This is robust wheel–rail design.

The maintenance feedback loop

Inspection measures the evolving profiles.

Vehicle monitoring measures vibration and ride behaviour.

Noise monitoring records acoustic consequences.

Maintenance changes the surfaces.

The next measurements show whether contact improved.

wheel/rail geometry
→ contact mechanics
→ force and vibration
→ wear/noise
→ inspection
→ grinding/profiling
→ new geometry

The wheel–rail problem is therefore never fully finished.

It is continuously managed.

A railway does not design the wheel–rail interface once. It keeps redesigning it slowly through inspection and maintenance as the surfaces wear.

A Complete Fictional Wheel–Rail Calculation

Let us build one simplified wheel-contact example.

All values are invented for teaching and do not represent Singapore MRT specifications or operational limits.

Step 1 — Normal contact

N = 55,000 N
a = 6.5 mm
b = 4.8 mm

Area:

A = πab
≈ π(0.0065)(0.0048)
≈ 9.80×10−5 m²

Average pressure:

pmean
≈ 55,000/(9.80×10−5)
≈ 561 MPa

Hertz peak:

p0 ≈ 1.5(561)
   ≈ 842 MPa

Step 2 — Traction creepage

Suppose small longitudinal creepage is:

ξx = 0.004

and illustrative linear creep coefficient is:

f11 = 2.0 MN

Linear force estimate:

Fx = f11 ξx
   = 2,000,000 × 0.004
   = 8,000 N

Suppose effective adhesion coefficient is 0.20.

Friction limit:

μN = 0.20 × 55,000
   = 11,000 N

The 8,000 N demand remains below the simplified adhesion limit.

Step 3 — Increase creepage

Suppose ξ_x doubles to 0.008.

Linear theory would predict:

Fx,linear = 16,000 N

But the simplified friction limit is only:

11,000 N

Therefore the linear model is no longer valid.

Contact enters a nonlinear saturation region with more slip rather than proportionally more useful force.

Step 4 — Curve force

Suppose an illustrative vehicle mass associated with a bogie group is 30,000 kg, speed is 15 m/s and curve radius is 300 m.

Lateral acceleration:

ac = 15²/300
   = 0.75 m/s²

Associated lateral force scale:

Fc = 30,000 × 0.75
   = 22,500 N

This force is not carried by one wheel contact in this simple way in a real bogie. Suspension, multiple wheels, cant and vehicle dynamics distribute it.

The calculation only demonstrates the v²/R dependence.

Step 5 — Reduce curve speed

Reduce speed from 15 m/s to 12 m/s.

ac,new = 12²/300
       = 0.48 m/s²

Reduction:

(0.75 − 0.48)/0.75
= 36%

A 20 per cent speed reduction produced a 36 per cent reduction in this lateral acceleration measure because speed is squared.

Step 6 — Estimate wear tendency

Suppose tangential contact force is 8,000 N and relative longitudinal slip speed is 0.06 m/s.

Frictional power:

Pfriction
= 8,000 × 0.06
= 480 W

If another contact condition doubles slip speed while force remains similar:

Pfriction = 960 W

More frictional work is being dissipated at the interface, increasing potential wear energy.

Step 7 — Wear changes profile

Suppose equivalent conicity changes from fictional value:

λeq = 0.08

to:

λeq = 0.13

The same lateral displacement now produces a larger rolling-radius difference.

Vehicle steering and lateral dynamic response change.

Step 8 — Maintenance closes the loop

Wheel profiling and rail grinding restore target geometry.

Inspection measures the profiles again.

Noise and vibration monitoring checks operational effect.

profile error falls
→ contact distribution changes
→ creepage changes
→ vibration/noise changes
→ wear rate changes
→ next maintenance interval is updated

The steel surfaces have become part of a feedback-controlled maintenance system.

The Wheel–Rail Deletion Tests

Remove elastic deformation

The theoretical point contact creates unrealistic infinite pressure.

Remove friction

The train cannot transmit the required traction, braking and lateral guidance forces.

Remove creepage

The model loses the small relative motion through which real tangential wheel–rail forces develop.

Remove wheel-profile geometry

The wheelset loses an important passive steering mechanism.

Remove suspension

Contact geometry alone must control dynamics that actually belong to the whole bogie.

Remove wear

The model assumes contact geometry never evolves.

Remove fatigue

Cyclic stress is treated as harmless as long as immediate wear is low.

Remove inspection and profiling

Wheel and rail shapes drift until contact behaviour is decided only by uncontrolled wear.

The contact patch works only because the entire railway around it—geometry, suspension, materials, maintenance and control—keeps returning it to a usable operating region.

The Wheel–Rail Paradoxes

Paradox 1 — Tiny contact creates huge pressure

The small contact patch makes efficient rolling possible but concentrates enormous local stress.

Paradox 2 — Slip helps a wheel roll properly

Microscopic creepage is needed to create controlled tangential forces.

Paradox 3 — Friction is both control and damage

The same contact force that accelerates the train contributes to wear and heat.

Paradox 4 — Wear can sometimes remove fatigue-damaged material

Very low wear is not automatically the optimum if shallow rolling-contact-fatigue cracks accumulate.

Paradox 5 — The profile that helps steering can contribute to oscillation

Rolling-radius difference gives passive guidance but also participates in lateral dynamics.

Paradox 6 — Grinding away rail can extend rail life

Controlled removal can restore profile and remove damaged surface before uncontrolled wear or fatigue grows.

Paradox 7 — A quieter curve can require a slower train

Reducing speed lowers lateral dynamic demand, but timetable and capacity consequences must also be managed.

How Wheel–Rail Mathematics Grows from School to Research

Primary Mathematics and Science

• force,
• area and pressure,
• speed and distance,
• circles and radius.

Secondary Mathematics and Physics

• friction,
• centripetal force,
• quadratic speed relationships,
• oscillation,
• graphs of force and displacement.

Junior College

• calculus and curvature,
• differential equations,
• energy and power,
• simple harmonic and damped motion,
• optimisation.

University and Research

• elasticity and Hertz contact,
• Kalker rolling-contact theory,
• tribology,
• vehicle dynamics,
• finite-element analysis,
• nonlinear stability,
• rolling-contact fatigue,
• wear simulation,
• acoustics and friction-induced vibration.

The railway turns school mathematics into physical consequence.

Area becomes contact pressure.

Quadratics become curve force.

Differential equations become lateral stability.

Integration becomes frictional energy.

Optimisation becomes wheel profile and maintenance strategy.

The World Return — When the Real Wheel Answers the Contact Model

The engineer predicts contact pressure.

The wheel returns wear.

The model predicts stable lateral behaviour.

The train returns measured vibration.

The model predicts a wear pattern in a curve.

Inspection returns an actual profile.

The model predicts a quieter contact after grinding.

Noise monitoring returns the acoustic result.

Let predicted wheel profile after mileage M be r̂(y,M).

Measured profile is r(y,M).

Profile error is:

e(y,M)
=
r(y,M) − r̂(y,M)

If one region repeatedly wears faster than predicted, the model may be missing:

• actual friction conditions,
• curve forces,
• rail profile,
• suspension behaviour,
• traffic load,
• or material behaviour.

The maintenance loop becomes:

model wheel–rail contact
→ predict force and wear
→ operate
→ measure wheel/rail profile
→ measure noise and vibration
→ compare
→ reprofile or grind
→ measure again
→ update model

This is why track inspection and predictive maintenance belong next to wheel–rail mathematics.

The contact model describes why deterioration occurs.

Inspection tells us what actually occurred.

Maintenance changes the boundary conditions for what occurs next.

The wheel–rail theory is never complete until steel worn by the real railway answers it.

RFE Return — What Does Good Wheel–Rail Contact Owe the Passenger?

The passenger should not have to know any of this.

They should experience:

smooth acceleration
+
controlled braking
+
stable straight running
+
comfortable curves
+
manageable noise
+
reliable wheels and rails
+
efficient rolling

The wheel–rail interface owes the passenger something almost invisible:

the ability to transmit enormous forces without making those forces feel enormous.

The train may carry hundreds or thousands of people.

The contact patch remains tiny.

The wheel may travel tens of thousands of kilometres.

The profile is slowly changing the whole time.

So the RFE is not simply:

keep wheel on rail

It is:

keep force, guidance, wear, noise and stability inside a usable envelope while millions of wheel revolutions slowly change the surfaces doing the work.

Conclusion — The City Rides on a Patch of Mathematics

The MRT seems to move because motors turn wheels.

That explanation stops too early.

Motor torque must become contact force.

Contact force must remain inside available adhesion.

Wheel geometry must convert lateral displacement into rolling-radius difference.

The wheelset must negotiate curves while remaining stable on straight track.

The contact patch must carry high pressure without unacceptable damage.

Friction must create useful force without consuming the surfaces too quickly.

Wear changes the profiles.

Changed profiles change contact.

Inspection discovers the change.

Grinding and profiling reshape the system.

The cycle begins again.

load
→ elastic contact
→ pressure
→ creepage
→ traction and guidance
→ frictional work
→ vibration and wear
→ profile evolution
→ inspection
→ maintenance
→ renewed contact

Singapore’s public rail maintenance gives us this physics in everyday form.

Track curvature can increase wheel–rail noise.

Lower speed changes lateral demand.

Rail grinding changes surface contact.

Wheel profiling changes rolling geometry.

Automatic track inspection watches what happens afterwards.

The passenger sees none of the equations.

They feel the result through the floor.

The train accelerates.

It enters a curve.

It stays guided by two rails.

It stops accurately at the next platform.

And beneath all of it, a tiny region of elastic steel carries the city forward.

The MRT does not ride on rails in the abstract. It rides on a moving sequence of tiny contact patches where geometry, force and friction agree for a fraction of a second at a time.

Key Equations

Acontact = πab
Elliptical contact-patch area

pmean = N/(πab)
Average contact pressure

p(x,y) = [3N/(2πab)]√[1−x²/a²−y²/b²]
Hertz elliptical pressure distribution

p0 = 3N/(2πab)
Maximum Hertz pressure

|Ft| ≤ μN
Simplified adhesion limit

Ftraction ≈ ma
Basic traction-force requirement

ξx = (ωr−V)/V
Longitudinal creepage

Fx ≈ −f11ξx
Linear longitudinal creep force

Fy ≈ −f22ξy − f23ξspin
Conceptual lateral/spin creep force

√(Fx²+Fy²) ≲ μN
Tangential-force saturation envelope

rR = r0+λy
rL = r0−λy
Idealised conical rolling radii

Δr = 2λy
Rolling-radius difference

rR/rL ≈ (R+a)/(R−a)
Ideal curve rolling condition

y ≈ ar0/(λR)
Simplified lateral shift for pure rolling

ac = v²/R
Centripetal acceleration

Fc = mv²/R
Lateral force scale

mÿ+cẏ+ky=F(t)
Simplified lateral oscillator

λeq ≈ Δr/(2y)
Conceptual equivalent conicity

Pfriction ≈ Fxvx,slip + Fyvy,slip + Mspinωspin
Frictional power dissipation

Vwear = kWs/H
Archard wear relationship

Tγ ≈ |Fxξx| + |Fyξy|
Simplified wear-energy measure

eprofile(y)=r(y)−rtarget(y)
Wheel-profile error

J = weighted wear + fatigue + noise + resistance
    + stability + maintenance costs
Multi-objective wheel–rail optimisation

Reader-safety note: All numerical loads, contact-patch dimensions, friction values, profiles, speeds and thresholds in the worked examples are fictional educational values. They do not reproduce Singapore MRT wheel profiles, rail profiles, adhesion limits, maintenance tolerances, derailment criteria or security-sensitive operating parameters. Actual wheel–rail engineering uses validated train-specific and track-specific models, measurements and professional safety standards.

Continue the MRT Mathematics Series

• How MRT Timing Works Using Mathematics — the full system pillar.
• How MRT Braking Works Using Mathematics — stopping distance and regenerative braking.
• How MRT Headway Works Using Mathematics — train spacing and throughput.
• How MRT Station Dwell Time Works Using Mathematics — queues, doors and platform flow.
• How MRT Delays Propagate and Recover Using Mathematics — instability, regulation and recovery.
• How an MRT Timetable Is Built Using Mathematics — demand, train cycles and robustness.
• How MRT Energy Use Is Optimised Using Mathematics — traction, coasting and regenerative energy.
• How MRT Passenger Capacity Is Calculated Using Mathematics — vehicle, station and delivered capacity.
• How MRT Routes and Transfers Are Optimised Using Mathematics — graph theory, transfer cost and route choice.
• How MRT Network Resilience Is Measured Using Mathematics — redundancy, cascading overload and recovery.
• How MRT Predictive Maintenance Works Using Mathematics — anomaly detection, failure probability and maintenance timing.
• How MRT Track Inspection Works Using Mathematics — geometry, sensors and continuous health mapping.

The next natural leg is How MRT Noise and Vibration Work Using Mathematics: sound pressure, decibels, frequency spectra, resonance, wheel–rail roughness, tunnel acoustics, vibration transmission and why a small change in rail surface can become a sound heard throughout a carriage.

Sources and Further Reading

• Land Transport Authority — TEL track curvature, wheel–rail noise, speed reduction and rail grinding
• SMRT — wheel profiling, rail grinding, daily track inspection and Rail Vision
• Urban Rail Transit — Review of wheel–rail wear, contact and rolling contact fatigue
• Friction — Recent advances in wheel–rail rolling contact fatigue and wear testing
• Railway Engineering Science — Review of wheel and rail profile wear simulation
• Tribology International — Transient wheel–rail rolling contact theories
• Railway Engineering Science — Wheel–rail contact and Hertzian contact theory review

Return to the complete MRT Mathematics hub: How MRT Works | It’s Mathematics.