Opening up the roll moment
Lesson 3.1 made one big simplification: the whole car rolled about an axis on the ground, so every newton of lateral load transfer went through the springs and anti-roll bars. That gave a clean rule: total transfer \(m a_y h / t\), split front-to-rear by roll stiffness. It is a good first model, but real cars move load through three separate paths, and only one of them involves the springs at all.
Why care? Because the other two paths are tuning tools too. Suspension geometry (the roll-center height) changes how much load each axle transfers and how much the body rolls. Body roll matters because it changes camber (Lesson 2.4) and, on cars with wings, the aerodynamic platform.
Sprung and unsprung mass
- Unsprung mass \(m_{us}\): everything that moves with the wheel. That means the tire, wheel, upright, brakes and part of the links and springs. It sits low, at roughly wheel-center height \(h_{us}\).
- Sprung mass \(m_s\): everything carried on the springs. That means the chassis, engine, driver and fuel. Its CG is at height \(h_s\).
On a race car the unsprung mass is typically somewhere around a tenth of the total. Both masses are pushed sideways in a corner, but their lateral forces reach the tires by different routes.
Roll centers and the roll axis
Look at one axle from the front. The suspension links connect the wheels to the body, and their geometry defines a point called the roll center (RC). It has two meanings that turn out to be the same:
- It is the point about which the body rolls relative to the ground, at least for small motions.
- It is the point through which lateral forces pass from the tires into the body without making it roll.
Each axle has its own roll center. The line joining the front and rear roll centers is the roll axis. The figure below shows how to find it from the suspension links. Module 8 explores how it moves as the suspension moves. In this lesson, treat each roll-center height \(h_{rc}\) as a number the designer chooses, typically a few centimeters above the ground.
Why does this work? At any instant, each wheel moves relative to the body as if it were pinned at its instant center: the wishbones only allow motion about that point. A sideways force at the contact patch therefore reaches the body along the line from the contact patch through the IC. Where the left and right lines meet, the tire forces can act on the body without creating a roll moment about that point. That is the roll center.
Three paths for lateral load transfer
For each axle, with track \(t\), the per-wheel load change in steady cornering is the sum of three parts:
| Component | Per-wheel load change | What sets it |
|---|---|---|
| Unsprung | \(\Delta F_{z,us} = \dfrac{m_{us}\,a_y\,h_{us}}{t}\) | Wheel and brake mass, wheel size. Not adjustable at the track. |
| Geometric | \(\Delta F_{z,geo} = \dfrac{m_{s,\text{axle}}\,a_y\,h_{rc}}{t}\) | Roll-center height: transferred through the links, immediately, with no roll needed |
| Elastic | \(\Delta F_{z,el} = \dfrac{K_{\phi,\text{axle}}}{K_\phi}\cdot\dfrac{M_\phi}{t}\) | Springs and anti-roll bars, through body roll |
Here \(m_{s,\text{axle}}\) is the share of sprung mass carried by that axle, and \(M_\phi\) is the roll moment: the sprung mass's lateral force acting about the roll axis,
where \(h_{ra}\) is the height of the roll axis directly beneath the sprung CG. The distance \(h_s - h_{ra}\) is the roll moment arm. The springs and bars resist the roll moment, so the body rolls through
Engineers usually quote this per g of lateral acceleration, as the roll gradient in deg/g.
The three components always add up to the same total as Lesson 3.1. Moment balance about the ground still gives \(\Sigma \Delta F_z\,t = m a_y h\). Roll centers and unsprung mass don't change how much load is transferred; they change which axle transfers it and how much the body rolls while doing so.
Explore the split
Both models move the same total load: with equal tracks, the bars add up to the Lesson 3.1 total \(m a_y h / t\) (1966 N for the club racer at 1 g). What differs is how that total is split between the axles.
Lesson 3.1 put the roll axis on the ground and treated the whole car as one mass on springs. So all of the transfer went through the springs and bars, and it was shared in the roll-stiffness ratio: 58% front, 42% rear. In this explorer only the elastic part follows that ratio. The other two parts take their own routes:
- Unsprung transfer stays with the axle whose wheels and brakes created it. With 60 kg at each end it splits roughly 50/50 (118 N each), however stiff the springs are.
- Geometric transfer goes through each axle's links. It depends on that axle's sprung mass and its own roll-center height, not on the springs. The club racer's rear roll center (10 cm) is higher than its front (5 cm), so the rear takes more: 198 N against 107 N.
- Elastic transfer is what is left: the roll moment about the roll axis. That is smaller than in Lesson 3.1 because the moment arm is \(h_s - h_{ra}\) rather than the full CG height. Only this part (1426 N in total) is split 58/42.
About 540 N of the 1966 N bypasses the springs and is shared far more evenly than 58/42. So the front ends up carrying 53.5% instead of 58%: less than the simple model predicts at the front, more at the rear. To see the two models converge, set both roll-center heights to zero. The geometric part vanishes and the front share rises to 57%. The last 1% is the unsprung mass.
- With the club racer at 1 g, compare the front share of load transfer with the simple model's 58%. Which components cause the difference?
- Raise the front roll center from 5 cm to 15 cm. What happens to the front share, the body roll and the limit balance? Do the same for the rear.
- Set both roll centers to 0 cm. How close is the result to the simple model now? What is left over?
- Raise both roll centers until the roll moment arm is nearly zero. What happens to body roll? Read the warning that appears.
- Halve the total roll stiffness. Which components change, and which stay exactly the same?
The club racer has \(m_s = 630\) kg (52% on the front axle), \(m_{us} = 60\) kg per axle, \(h_s = 0.42\) m, \(h_{us} = 0.30\) m, track 1.50 m, roll centers 0.05 m (front) and 0.10 m (rear), and 58% front roll stiffness. Find the front axle's load transfer at 1 g.
Unsprung: \(60 \times 9.81 \times 0.30 / 1.50 = 118\) N.
Geometric: front sprung mass \(0.52 \times 630 = 327.6\) kg, so \(327.6 \times 9.81 \times 0.05 / 1.50 = 107\) N.
Roll axis under the CG: the CG is 48% of the wheelbase behind the front axle, so \(h_{ra} = 0.05 + (0.10 - 0.05)\times 0.48 = 0.074\) m. Roll moment \(M_\phi = 630 \times 9.81 \times (0.42 - 0.074) = 2138\) N·m.
Elastic: \(0.58 \times 2138 / 1.50 = 827\) N.
Front total \(= 118 + 107 + 827 = 1052\) N. Doing the same for the rear gives 914 N, so the front carries 53.5% of the transfer. That is noticeably less than the 58% the simple model assumed, because the rear roll center is higher.
High or low roll centers?
If roll-center height is a tuning lever, why not raise it until the body doesn't roll at all? Because the geometric path has side effects. When the roll center is above the ground, the lateral tire forces acting through the links also push the body up, an effect called jacking. A jacked-up car has a higher CG just when it needs a low one, and the jacking varies as grip varies, which drivers feel as unpredictability. Roll centers also move as the suspension moves (Module 8).
So most race cars use fairly low roll centers and control roll with springs and anti-roll bars, then use small roll-center changes to fine-tune the balance between the axles.
“The body is pinned at the roll center and swings about it.” The roll center is not a physical joint. It is a construction from the link geometry that is only exact for small motions, and it moves as the car rolls and heaves. It is a very useful idea, but treat it as an approximation.
Check your understanding
A Formula SAE car has 120 kg of sprung mass on its front axle, a front roll center 4 cm above the ground and a front track of 1.20 m. What is the geometric load transfer (per wheel) on the front axle at 1.5 g?
\(\Delta F_{z,geo} = m_{s,\text{axle}}\,a_y\,h_{rc}/t\), with \(a_y\) in m/s².
\(120 \times (1.5 \times 9.81) \times 0.04 / 1.20 = 58.9\) N, small because the roll center is low.
The club racer (\(m_s = 630\) kg, \(h_s = 0.42\) m, \(h_{ra} = 0.074\) m) corners at 1.2 g. Its total roll stiffness is 1500 N·m/deg. How far does the body roll?
Find the roll moment \(M_\phi = m_s a_y (h_s - h_{ra})\), then \(\phi = M_\phi / K_\phi\).
\(M_\phi = 630 \times 11.77 \times 0.346 = 2566\) N·m, so \(\phi = 2566/1500 = 1.71^\circ\). That is a roll gradient of about 1.4°/g.
For the same car and corner, the front axle has 58% of the roll stiffness and a 1.50 m track. What is the elastic load transfer on the front axle?
\(\Delta F_{z,el,f} = (K_{\phi f}/K_\phi)\,M_\phi / t_f\). Reuse your roll moment from the previous question.
\(0.58 \times 2566 / 1.50 = 992\) N.
At the track, an engineer can change springs and anti-roll bars quickly. Which components of lateral load transfer can those parts change?
- All three.Springs and bars act only on the roll moment, so they cannot change the unsprung or geometric paths.
- Only the elastic component, specifically how it is split between the axles.The unsprung part depends on unsprung mass and height, and the geometric part on roll-center height. Springs and bars only share out the roll moment.
- Only the geometric component.The geometric path goes through the suspension links, which springs and bars don't change.
With everything else unchanged, the front roll center is raised. What happens?
- The front axle takes a larger share of the load transfer and the body rolls less, so the car tends toward more understeer.More geometric transfer at the front, and a higher roll axis, so a smaller roll moment. With load-sensitive tires, more front transfer means less front grip.
- The total lateral load transfer falls.The total is still \(m a_y h\) divided among the tracks. Only the split and the roll change.
- The rear axle takes more of the transfer, giving more oversteer.Raising the front roll center adds geometric transfer at the front.
A designer puts the roll axis at the height of the sprung CG. What is the main consequence?
- The car has no lateral load transfer.Load transfer still happens, entirely through the links instead of the springs.
- There is no roll moment, so the body doesn't roll. But large jacking forces push the body up and make the car harder to control.With zero moment arm, nothing makes the body roll. The lateral forces acting through high roll centers lift the car instead.
- The springs carry all of the load transfer.It is the opposite: the geometric path carries it all, and the springs see no roll moment.
Going further with Milliken
Read: RCVD Chapter 18, Wheel Loads, for the full breakdown of lateral load transfer. Then read Chapter 16, Ride and Roll Rates, for how roll stiffness is built up, and skim Chapter 17, Suspension Geometry, for how roll centers are found.
As you read, look for answers to these questions:
- How does the book's treatment handle different front and rear track widths and unequal unsprung masses?
- What does it say about roll centers below ground level, and about how roll centers move in roll?
- How does tire (vertical) stiffness enter the roll-rate calculation?
Summary
| Idea | Key relation |
|---|---|
| Unsprung transfer | \(m_{us} a_y h_{us} / t\) |
| Geometric transfer | \(m_{s,\text{axle}} a_y h_{rc} / t\) |
| Roll moment | \(M_\phi = m_s a_y (h_s - h_{ra})\) |
| Elastic transfer | \((K_{\phi,\text{axle}}/K_\phi)\,M_\phi / t\) |
| Body roll | \(\phi = M_\phi / K_\phi\); roll gradient in deg/g |
| Total | Unchanged: the paths only redistribute it |
Next: load transfer in the other direction, and how it decides which wheels lock first under braking.