Hello fellow Autopians and welcome to Ask an Engineer. I’m Huibert Mees, the guy who helped design the Tesla Model S and Ford GT suspension, and I’m going to tell you about an incredibly important attribute of any vehicle’s suspension. It’s called the “roll center,” and while that may not sound sexy, this one parameter plays a massive role in determining how your car behaves.

A little while ago I received a message from someone who had heard about suspension “roll centers” but wasn’t quite sure what they were or why they were important. He’d found lots of videos online showing how to find the suspension roll center, but none of them really answered the question of why anyone would care. Today we’re going to answer these questions and explain how this one aspect of suspension design affects the way your car rides, the way it handles, and the amount of roll the body has in a turn.

What Is the Suspension Roll Center

As a vehicle enters a corner, the body will “roll” towards the outside of the corner, as shown in the Motor Week video above. We’ve all experienced this, and many of you have probably noticed that different types of vehicles roll more or less than others. I think we would all expect a sports car to roll less than an SUV or a large cushy sedan. What is it then that controls how much a vehicle rolls in a turn? As you might expect, the answer to that question is complicated and involves many different parts of the suspension and body, but one of the primary things that influence body roll is something called the roll axis.

Here’s how this works. Imagine a vehicle in a corner with the body rolling to the side. The rotation of the body happens about an axis that acts like a hinge between the body and the suspension, just like a door rotates about its hinge axis when it opens and closes. This is what is called the roll axis, and it is defined by a line through the front and rear suspension “roll centers,” which can be determined by figuring out the imaginary point about which a wheel pivots/hinges as it moves up and down.

Knowing where these roll centers are located is critical to understanding how the vehicle rolls in a turn. It is also critical to understanding some other vehicle behaviors, such as “jacking” and ride.

How do we find the roll center of a suspension? The method most commonly discussed, and which shows up in numerous places online, is the geometric method, which works well with some suspension designs but not all.

The Geometric Method

Let’s talk about the geometric method. It is the one most often described in textbooks and online, and it is particularly well suited to MacPherson strut and double wishbone designs that are so prevalent among modern cars. Basically, you draw lines that extend the upper and lower wishbones until the two lines intersect. The point where the lines intersect is called the “instant center.” For a double wishbone design, it looks like this:

Where the lines intersect is the instant center of the suspension at that particular position. You can think of the entire suspension pivoting about this point as it moves up and down. Be aware, though: The instant center will move as the suspension moves. It is only the instant center of the suspension at that particular instantaneous position.

Once we know where the instant center is, we then draw a line from there to the tire contact patch. We’ll call this line the roll center line. Like this:

Where this line crosses the center of the vehicle is the roll center, and the distance from there to the ground is called the roll center height. Do the same thing for the other side suspension and you will see that the two roll center lines cross at the centerline of the vehicle.

You can do a very similar geometric construction for a MacPherson strut but instead of having an upper control arm, you draw a line perpendicular to the strut axis. Like this (the top dotted line below isn’t horizontal, since the strut is leaning in a bit):

We need to keep in mind though, that this construction is only correct at that particular suspension position. As soon as the suspension moves up and down, these lines will move, and the instant center and the roll center will move. Take our double wishbone design, for example. Here it is with the wheels moved 40 mm up (for example, maybe you hit a speed bump, or loaded some cattle into your car):

Notice how the instant center has moved much closer to the ground compared to the previous position. This movement is very important to understand since it will affect us later when we talk about the effects of body roll.

WARNING: Things Are Going To Get Nerdy Here. Skip Ahead If You’d Like The Abridged Version

Problems With the Geometric Method

Using the geometric method is fine when we have a simple design like a double wishbone or a MacPherson strut, but what happens when there is no obvious upper or lower control arm to draw your lines from? Something like the multi-link that VW and Audi use for their front suspension or any of the many multi-link rear suspension designs out there?

You can see that it is not at all obvious which link to use for our extension lines, and the ones we choose will have a dramatic impact on the answer. But what if there were a method that could be used with any suspension regardless of the geometry? Fortunately, there is. I call it the Instantaneous Radius Method, and it works by understanding what it is that we are really looking for when we do the geometric constructions.

The Instantaneous Radius Method

The result of the geometric method is a line from the suspension’s instant center to the tire contact patch, but there is an infinite number of suspension geometries that would give an instant center along this same line. For instance, here we have a double wishbone geometry that is clearly very different from our previous example, but it has the exact same roll center because the instant center falls on the same roll center line as we had before.

The reason there are many different suspension geometries that give the same roll center height is because the roll center really has nothing to do with what is happening in the suspension. Instead, it has to do with what is happening at the tire contact patch, and that is what we are really after here. We want to know how the tire contact patch is moving as the suspension moves because this is where the cornering forces are acting on the suspension. As we will see later, knowing where the roll center is, or more specifically, where the roll center line is, tells us how the suspension will respond to those cornering forces.

So, let’s look at the Instantaneous Radius Method. This is where a computer model becomes essential, but fortunately, it can be done with a very simple model using software that is readily available and not too expensive. I use Fusion 360 since it lets me build models that I can move around onscreen.

Once the model has been built, it is just a matter of moving the tire up and down a small amount and tracking the motion of the contact patch. In this example, I’ve moved the suspension up 10 mm and drawn a point where the tire contact patch was. I then moved the suspension down 10 mm and drew a point where the contact patch was. Next, I drew a line through those two points:

The last step is to draw another line perpendicular to the line through our points. This is the roll center line, and as in the geometric method, it defines the line from the suspension instant center to the contact patch. The difference is that, in this new method, we just don’t know where along that line the instant center actually is. But, as we noted earlier, that is not really that important:

As a final check, let’s compare our new method with the geometric method to see how good they are:

As you can see, the suspension instant center falls exactly on the line we created with the instantaneous radius method. Both methods in this case gave us the same results. That shows how good the geometric method is for a simple double wishbone design, but it also shows that we now have a very versatile method for finding the roll center that we can use with any suspension design that anyone could ever come up with.

So, now that we know how to find the suspension roll center, we need to talk about why we even care. There are basically three reasons to care about roll centers: Jacking, Ride, and Body roll.

IF YOU SKIPPED AHEAD: You can continue reading here:

Jacking

In the old days when solid axle rear suspensions ruled the land and independent suspensions were still a novelty, some enterprising car companies tried a type of suspension design called a Swing Axle. The idea was fairly simple: Take a solid axle, split it on either side of the differential, add CV joints where the axle stubs meet the differential, a tension link to keep things under control, and call it good. The problem was that it wasn’t very good as many customers soon found out. The design was used by VW in the Beetle and by Triumph in a number of their cars including the Spitfire:

Notice what’s happened here. The rear wheels have tucked themselves way under the body and jacked the body well up into the air. Looks decidedly unstable. In fact, in many cases, even after the driver had straightened the wheel out, the suspension stayed jacked up and the car would be stuck on the side of the road until a tow truck came to the rescue. Let’s look at why this was happening.

[Editor’s Note: Ralph Nader criticized this suspension design in his famous auto-safety book “Unsafe at any Speed”:

The image above is either from that book or from Chevrolet. -DT]

If we look at a model of a swing axle suspension, we can see where the roll center lines would be and where the resulting roll center would be:

A swing axle design uses an axle shaft rigidly attached to the knuckle, pivoting around the inner CV joint. The inner CV joint therefore is the instant center of the suspension and is the point from which we draw our line to the tire contact patch. Notice how high that puts the roll center, i.e., the point where the roll center line crosses the center of the car.

When we know the instant center of a suspension, we can think of the entire suspension being reduced to a single link attached to the body and pivoting around the instant center at one end and rigidly attached to the knuckle at the other. The knuckle pivots around the instant center instantaneously and all the links behave as if they are one component. This is easy to picture in the case of a swing axle, since it really is just a rigid link from the instant center to the knuckle, but it takes a bit more imagination in the case of a double wishbone or MacPherson strut.

In either case though, when we think of the suspension instantaneously as a single link, it doesn’t really matter where we attach this imaginary link to the knuckle as long as it pivots at the instant center. We could attach it where the upper control arm attaches, or we could attach it where the lower control arm attaches. We could even attach it to the knuckle at the tire contact patch since it is purely imaginary anyway. Of course that would be physically impossible, but we’re talking imaginary here. Thinking of this link as being attached at the tire contact patch helps in understanding what Jacking is all about.

If we think about a car in a corner, there is an inboard force trying to push the tire contact patch towards the middle of the car. This force is resisted by our suspension, and in particular by our imaginary link where it attaches at the suspension instant center. If our imaginary link is sitting at an angle, the force from cornering and the resistance force from the suspension will be at different heights and this difference causes a moment that wants to stand the link upright. In other words, this moment is trying to push the link down at the contact patch and up at the instant center, end but since the contact patch is constrained by the ground, the only end of the link that could conceivably move is the instant center end. And that is exactly what happens. [Editor’s Note: Imagine the eraser side of a pencil touching a grippy floor, and the pencil leaning at an angle like the Tower of Pisa, but a little more of an angle. Now imagine that floor moves like a conveyor belt pushing the eraser (the base of the Tower of Pisa) laterally in the direction of the tip of the pencil. That tip is going to be pushed upwards. The tower of pizza will become taller as its base moves closer to a position inline with its roof. This is jacking. -DT].

By trying to stand our imaginary link upright, the moment caused by the cornering force lifts the instant center up which in turn pushes the body up with it. This is the jacking force, and the size of this force is determined by the following formula (going back to our high school trigonometry classes):

Jacking Force = Cornering Force x SIN(Roll Center Line Angle)

As we can see from this formula, the larger the roll center line angle, the larger the jacking force will be. If the roll center line angle were zero, i.e. the roll center was on the ground, there would be no jacking force. This makes sense since our imaginary link would be lying flat on the ground and there would be no tendency for it to want to stand up due to a cornering force.

Getting back to our swing axle design, we can see how the roll center line angle is very large and therefore the jacking force is going to be very large. This was the fundamental problem with swing axle designs. There is no way to get the roll center low in the car so that the jacking forces could be reduced. The only solution was to change the suspension design to another concept, which is exactly what both VW and Triumph, and every other company that used swing axles, did a few years later.

The Total Jacking Picture

We’ve learned now that there is a jacking force coming from the outside tire in cornering, but the suspension has two tires. What’s happening at the other tire and how does that affect jacking? If we expand our force diagram, we see that there is indeed a cornering force that comes from the inside tire.

There’s a lot going on in this diagram, but what it boils down to is that while the outside cornering force (right cornering force above) pushes on its imaginary link and tries to make it stand up, the cornering force on the inside tire (the left cornering force above), pulls on its corresponding imaginary link, causing a moment that wants to make that link lay down instead of standing up. The result is an anti-jacking force at the left instant center which counteracts the jacking force coming from the right-side cornering force.

Unfortunately, the inside cornering force is always going to be smaller than the outside cornering force due to the weight transfer that happens during cornering. The same force that causes the body to roll in a corner causes the weight of the vehicle to transfer to the outside of the corner, effectively increasing the weight on the outside tires and decreasing the weight on the inside tires. This means the outside tires have to do more of the cornering work than the inside tires which in turn means the outside cornering force and corresponding jacking force will always be greater than the inside cornering force and its corresponding anti-jacking force.

The Effect Of Body Roll On Jacking

There is yet another factor that we need to consider when talking about jacking and that is the effect of body roll. As the vehicle body rolls, the suspension on the outside of the turn and the suspension on the inside of the turn will be in different positions. As we saw earlier, the instant center and the roll center will move as the suspension moves, and while we have so far defined the roll center as the point where the roll center line crosses the centerline of the vehicle, this method breaks down when we look at body roll. Let’s look at a vehicle that is turning to the left so that the body has rolled to the right:

Notice how the instant centers of the two sides of the suspensions have moved differently. The right side instant center has moved considerably closer to the vehicle centerline while the left side instant center has moved way off the page to the right. The result is that the two roll center lines cross the vehicle centerline in different places. If we use the traditional definition of where the roll center is, which of these two lines would we then use? The answer is neither. The actual roll center is where the two roll center lines cross as indicated in the diagram above. Notice how the roll center has moved towards the inside of the turn. This is called the roll center migration and it is a very important characteristic of roll centers. Here we can see why this matters:

We can see here how the right roll center line angle is now much smaller than the left roll center line angle. Remember that in a left turn, the jacking force was a function of the right roll center line angle, and the anti-jacking force was a function of the left roll center line angle. Since the left roll center line angle is now larger, it means the anti-jacking force is now also larger than it would be without the body roll. Similarly, the jacking force has been slightly reduced since the right roll center line angle has been reduced. 

The reality is that while body roll certainly has an effect on jacking forces, the outside cornering force is still so much larger than the inside cornering force, there is still a net jacking force on the body that must be controlled as much as possible by keeping the roll centers reasonable low.

Solid Axles Are A Different Animal

So far, everything we’ve discussed has been for independent suspension vehicles. For a vehicle with a live/solid axle, the roll center picture is quite different. Since the left and right side suspensions are rigidly connected to each other by the axle tube, there is no way for the outside tire to tuck up under the body like it does in that picture of the Triumph Spitfire. The result is there is no jacking force generated by a live axle and you can have the roll center as high as you want. In reality, it is very difficult to get a low roll center in a live axle, even if you wanted to, and they tend to be about 3-4 times higher than in an independent suspension. This can have a distinct advantage which we will discuss later in another article.

Ride Quality

The second reason to care about roll centers is for ride. It may be counterintuitive to think about ride in terms of roll centers because so far, we’ve only been talking about them relative to cornering, but they most definitely have an effect on ride. As a vehicle drives down the road, the body will move up and down over bumps and undulations in the road. As the tires move up and down relative to the body, the height of the roll center causes a side-to-side motion of the tire contact patch. Let’s use our swing axle model to show how this works since it exaggerates the effect:

Notice how the tire contact patch moves as the wheels move up and down. While it has moved up and down with the suspension, the tire contact patch has also moved side to side. This is called track change. The issue is that as the car moves down the road, the tires really just want to roll in the direction they are facing. They don’t want to move side to side because they are stuck to the road, but they are being forced to by the suspension. Since the tires don’t want to move side to side, they resist the up and down motion of the suspension. Unfortunately, good ride is all about letting the body move up and down freely while being controlled by the springs and dampers, not constrained by any forces coming from the tires. If instead we look at our double wishbone design, we see how a much lower roll center results in much less track change:

There is still some track change, but it is much less and the resulting impact on ride would be much less.

In summary then, for good ride, we want a roll center as low as possible.

Body Roll

We all know that when a vehicle is in a turn, the body wants to roll towards the outside of the turn. Why is this? What is the mechanism that causes this roll motion and how can we control it?

As we said earlier, if you draw a line through the front and rear roll centers, this line represents the hinge axis the body rolls around in a turn. The relationship of this line to the location of the vehicle center of gravity is critical to understanding how and why a body rolls in a turn. Naturally, the height of the front and rear roll centers will determine the height of this line and where this line sits relative to the vehicle center of gravity. Let’s look at this in our double wishbone suspension design. Keep in mind that we are looking at this in a 2-dimensional diagram although this is really a 3-dimensional problem. We’re simplifying things here, but the principle is the same.

Notice where the roll center is relative to the vehicle center of gravity. The difference in height between the center of gravity and the roll center is called the roll moment arm. Since the roll center represents the hinge the body rolls around in a turn, it also represents the point where the total cornering forces coming from the left and right suspensions act on the body. The paths that the cornering forces coming from the left and right tires take through the left and right suspensions are extremely complex, but you can think of them all coming together at the roll center. Of course, since the cornering forces are trying to push the vehicle into the turn, there is a reaction force coming from the body at the center of gravity. Something like this:

Again, we are looking at a vehicle making a left turn. There’s a lot going on here so let’s break it down. You can see how the total cornering force acts on the roll center while the body reaction force acts at the center of gravity. The distance between them, the roll moment arm, creates a torque or moment which wants to roll the body towards the outside of the turn (towards the right in this example).  Of course, the size of the moment that wants to roll the body is directly related to the size of the roll moment arm. If we can make this distance smaller, the roll moment will be smaller and the tendency of the body to roll in a turn will be smaller. There are two ways we could do this: lower the height of the center of gravity or raise the height of the roll center. The height of the center of gravity is dependent on a large number of factors that suspension engineers have no control over but raising the roll center is certainly possible. Let’s suppose we could design a suspension where the roll center was at the same height as the center of gravity. The roll moment arm would be zero and there would be no roll moment at all. In fact, if we could place the roll center above the center of gravity, the roll moment would go the other way and the car would roll into the turn like a motorboat. Sounds like fun doesn’t it! The reality is that we do want some amount of body roll so we can use things like anti-roll bars, springs, and dampers it to tune the vehicle dynamics of the vehicle. With little or zero body roll, these components lose their effectiveness and tuning the vehicle for understeer and oversteer becomes impossible.

In summary then, to help control body roll we want a high roll center, within reason.

Conclusions

We’ve learned how to find the roll center of a suspension using two different methods and we’ve learned why we need to care about roll centers. We learned there are two factors that would like us to keep the roll centers low: jacking and ride. And we’ve learned there is one factor, body roll control, that would like us to have a high roll center (within reason). In reality though, the jacking concern outweighs the others. We MUST control jacking, and this means we need to keep the roll centers relatively low. This has the added benefit of helping our ride, but it does mean we may have more body roll than we really wanted. Fortunately, there are ways to control body roll without raising the roll centers, such as anti-roll bars and dampers.

Top image: Honda/Huibert. Edited by The Bishop