Unsprung mass and wheel hop: the second resonance

Your car has two natural frequencies, not one. The body bounces around 1.3 Hz; the wheel bounces around 13 Hz on the tire's own stiffness — and that second one decides whether the tire is touching the road.

Everything so far has treated the corner as one mass on one spring. It isn’t. There are two masses and two springs, and the second pair behaves completely differently from the first.

Sprung mass is the body, riding on the suspension spring. Around 1.3 Hz, as chapter 12 established.

Unsprung mass is everything below the spring — wheel, tire, hub, brake, and roughly half the control arms. It rides on the tire, which is also a spring, and a very stiff one. That combination has its own resonance, an order of magnitude higher, and it’s called wheel hop.

The frequency

The unsprung mass sits between two springs — the tire below it and the suspension above — so both contribute:

Worked example, front corner. Tire stiffness 200 N/mm, wheel rate 23.2 N/mm, unsprung mass 35 kg:

Against a ride frequency of 1.30 Hz — the wheel resonance is 9.8 times higher than the body’s. The rear works out at 12.5 Hz, about 8.3× its 1.50 Hz body frequency.

Two things to notice in that equation, because they’re where the intuition comes from.

The tire dominates. 200 000 N/m of tire against 23 210 N/m of suspension — the suspension spring contributes about 10% of the total. Change your spring rate by 50% and wheel hop barely moves. Change your tire pressure and it moves properly.

Unsprung mass is under a square root. Halving it raises the frequency by only √2. Which is worth remembering next time someone claims lightweight wheels transformed their car — the effect is real but it is not linear, and its main benefit isn’t the frequency shift anyway.

Why you care: the tire has to stay on the road

Grip comes from vertical load on the contact patch. No load, no grip — and load varies as the wheel bounces.

Near the hop frequency the unsprung mass resonates, and the load fluctuation gets large. Push it far enough and the dynamic variation approaches the static load, at which point the tire is periodically unloaded to zero and grip goes with it. That’s what wheel hop looks like from the driver’s seat: a rough surface where the car suddenly loses steering or traction for no obvious reason.

This is also the honest reason unsprung mass matters. Not because a lighter wheel makes the frequency nicer, but because a lighter unsprung mass has less momentum to fight, so the tire follows the road more closely and load variation is smaller everywhere — not just at resonance.

The metric that captures this is tire load variation, and it predicts grip on rough surfaces far better than body acceleration does. A setup can be beautifully comfortable and still be losing the contact patch repeatedly.

What damping can and can’t do here

Wheel hop is damped by the same shock that damps the body, but the shock is badly positioned to do it. At 13 Hz the shaft velocities involved are in the fast region of the damper curve — the deliberately soft part (chapter 15).

So there’s a genuine three-way conflict:

• Soft fast damping → good ride, poor wheel-hop control.
• Firm fast damping → better wheel control, harsh ride.
• The tire’s own damping helps, and it’s small.

That conflict has no clean solution, which is why wheel hop is usually managed by avoiding it: keeping unsprung mass down, keeping tire pressure sensible, and not putting the hop frequency somewhere the road excites strongly.

Go look at it

Open the strut layout on the Springs/Shocks tab.

1. Run ▶ Play over Bumpy Back Road and watch the tire load trace rather than the body movement. Note how much it fluctuates around its static value.
2. In Car Design, raise the front unsprung mass from 35 kg to 55 kg — a heavy wheel and a big brake. Run the identical road. The tire-load trace gets visibly worse: bigger swings, longer to settle.
3. Put it back to 35 and change tire stiffness instead, from 200 to 260 N/mm — roughly what more pressure does. Run again. A stiffer tire raises the hop frequency and generally sharpens the load fluctuations, which is why over-inflation costs grip on rough surfaces.
4. Now try to damp it out. Raise Fast Comp and Fast Reb substantially and re-run. Tire load steadies — and RMS Accel (g) goes up. That’s the conflict, measured.
5. Finally, find the point where the trace touches zero. Any time the tire load reaches zero, the tire is off the ground and the grip at that instant is nothing at all.

What it costs you

Reducing unsprung mass costs money and, often, strength. Lightweight wheels, aluminum uprights, inboard brakes and carbon components all reduce it, and all trade against cost, durability or complexity. On a road car the durability question is real: a wheel that survives a pothole is worth more than one that shaves 2 kg.

The interesting design responses are the ones that move mass rather than remove it. Inboard brakes take the heaviest single unsprung item and make it sprung, at the cost of driveshaft complexity and cooling — the reason they appear on race cars and almost never on road cars.

And the trade nobody mentions: bigger wheels with lower-profile tires raise unsprung mass and raise tire stiffness simultaneously, pushing hop frequency up and load variation up together. Which is the engineering answer to why large-diameter wheels ride badly, and it isn’t only about sidewall compliance.

Rules of thumb

• Wheel-hop frequency: 10–15 Hz for most cars. The layout above is 12.7 Hz front.
• Ratio to ride frequency: roughly 8–12×. If the two get closer, the modes start interacting and the car becomes hard to tune.
• Unsprung mass: 30–45 kg per corner for a road car; 20–30 for a race car; 12–18 for a formula car.
• Unsprung fraction: aim under about 15% of corner mass. The front corner above is 35 of 384 kg, roughly 9%, which is good.
• Tire stiffness is your biggest lever on hop frequency, and pressure is how you change it in the paddock.
• Rally-raid and desert: unsprung mass is enormous — huge wheels, portal axles, massive brakes, sometimes 80–120 kg per corner. That drags the hop frequency right down, toward 6–8 Hz, uncomfortably close to the frequencies the terrain excites hardest. It’s a large part of why these cars need such sophisticated damping: they cannot solve the problem by being light, so they solve it by controlling the mass they’re stuck with.

Try this

1. Compute the wheel-hop frequency of your own car. Tire rate is the hard number — a rough guide is 150–250 N/mm for a road tire at normal pressure.
2. Find the unsprung mass at which the tire load trace first touches zero on Bumpy Back Road. How far is that from where you started?
3. Does lowering unsprung mass or lowering tire stiffness do more for the load trace? Does the equation predict which one wins?

That’s Module 2 complete. You can size a spring from a target frequency, separate ride height from rate with preload, split front and rear for flat ride, compute a damping ratio and recognize an over-damped car, read a digressive curve and place its knee, use a bump stop as a spring rather than a crash pad, and find the wheel-hop resonance that decides whether the tire is touching the road at all.

One closing note on the test surfaces. The roads in the app — Urban Road, Bumpy Back Road, Rally Gravel — are synthesized random profiles following the ISO 8608 roughness classes, each about four times rougher than the one before. They are broadband on purpose: a single bump excites one narrow band of frequencies and tells you almost nothing about a suspension, which is a filter you need to judge across a spectrum. Two consequences worth carrying forward. Compare setups only on the same class. And because each run uses random phases, two runs of the same setup differ slightly — so treat small improvements with suspicion and re-run before believing them.

Module 3 puts the whole car in a corner: weight transfer, roll stiffness, tires that get worse the harder you lean on them, and the one number that describes balance.

Preset values are illustrative starting points, not a measured setup for any particular car.