We’ve all been there. You’re leaving a party at a friend’s house located on a narrow dead end road. You get in your F-150 and your friend gets in her VW Golf. She turns the wheel and in one move turns around and heads home. You, on the other hand, do your best Austin Powers impression trying to get your F-150 facing the other way so you can get home yourself. What makes these two vehicles so different, and why was your friend able to get out of there so much easier? Of course, the answer is in the difference in turning radii between these vehicles. Let’s talk about what goes into that.
[Welcome to Huibert Mees’s column, where the former Ford GT/Tesla Model S suspension engineer gets to write whatever he wants on The Autopian. -DT]
Turning radius is the radius of the smallest circle a vehicle can make with the steering at full lock. It is a function of the wheelbase, the front tire width, and the angle the front wheels can be steered, but there are a number of factors that play into how automakers choose these dimensions. Before we get into that, here’s a visual:
There are actually two turn radii that are normally calculated: curb-to-curb and wall-to-wall. The wall-to-wall value will always be larger than the curb-to-curb value and depends on the shape and overhang of the front bumper; it basically represents the smallest circle you can turn without potentially hitting something with your bumper.
The most useful of the two radii is curb-to-curb since that is what most people will encounter in normal driving (as most front bumpers can just fly over curbs — this isn’t the case for something like a Lamborghini, of course), so that’s what we’ll focus on here. Our friend, Mr. Powers above, was clearly more interested in the wall-to-wall turn radius of his vehicle, though.
To calculate the turn radius we need to go back to our High School trigonometry class. Here is the formula:
Rturn = (Wheelbase / SIN(turn angle)) + ½ Tire Width
[Editor’s Note: Dammit Huibert, are you really not going to do the derivation on this? FINE, I’ll do it. (Things are about to get a tiny bit nerdy, so feel free to skip this note if you’re not feeling it).
So, here’s why the equation above makes sense. I’ll break it down in equation form in the image below (see red), but I’ll explain it all below the image.
- When you turn your tire, the tire is going to want to move in an arc. We can calculate the radius of this arc by creating an imaginary triangle. We’re doing this because the turning radius of a car is this radius plus half the tread width (have a look at the image above and you’ll see that the darker blue arc labeled “curb to curb turn radius” points to the arc created at the outside of the tire). So we’re going to calculate the arc created at the center of the tire during the turn, and just add half that tread width.
- We know that the sin of an angle is defined as the opposite divided by the hypotenuse (remember SOHCAHTOA?). So the sin of angle x shown in the image is equal to the wheelbase, b, over c, which is the turning radius minus that tire width (see equation 1).
- We want to know what c is in order to calculate the turning radius, so we rearrange equation 2 and we learn that c is equal to b (the wheelbase) divided by the sin of x.
- Again, b is the wheelbase.
- So what the heck is angle x? Let’s figure it out. First, we know that the sum of all angles in a triangle is 180 degrees, and we know that angle Y is 90 degrees. So to find angle X, we just need angle Z. What is angle Z? Well, we know that the radius of an arc is perpendicular to the tangent of that arc, so that little red angle next to angle Z, plus angle Z, has to equal 90 degrees. But what’s that little red angle? Well, it’s the turn angle, because we know that opposite angles are congruent. So angle Z is 90 minus the turn angle.
- Angle X, which we need to solver for to get c (which is what we add to half the tread width to get the turning radius), is 180 minus 90 (that’s angle Y) minus (90 minus the turn angle, which is angle Z).
- Solving 6, angle X is then equal to the turn angle
- So to find out what the turning radius is, you solve for c using the equation in 3 and add half the tire width. That ends up being the equation Huibert wrote:
Rturn = (Wheelbase / SIN(turn angle)) + ½ Tire Width
I asked Huibert how scrub radius factors in, since those front wheels don’t technically pivot exactly on an axis at the centerline of the tire. “Scrub radius would have a very small effect because it would change the wheelbase by the SIN of the angle. I think we can ignore it here,” he told me. “For a scrub radius of 15 mm, which is not out of the ordinary, the wheelbase would change by 0.04% with a 40 deg turn angle. I think it is reasonable to ignore this effect.”
Huibert added a simplified version of my derivation above, leaving out the geometry bits proving that the turn angle is the same as the bottom left angle in that triangle, so if you want a simplified explanation of how he got his equation, here it is:
Our turning radius calculation also ignores deformation/slip in the tires during the turn, but it should be fine as an estimation. I’m adding these notes because you dynamics engineers in the comments are relentless.
From this equation we can see that if the wheelbase gets bigger then the turn radius will get bigger. If the turn angle gets bigger then the turn radius will get smaller and if the tire width gets bigger then the turn radius also gets bigger. We can look at this graphically as well. Here is what happens if we just increase the wheelbase:
Now compare that top diagram to this one to see the effect of a turn angle increase:
Let’s put some numbers in so we can really see what’s going on here. Let’s assume that our turn angle is 30 degrees, our wheelbase is 110 inches, and we have 235mm wide tires. Putting those numbers into our formula (remembering to convert our tire width from mm to inches) we see that the turn radius becomes:
Rturn = (110 / SIN(30)) + ½ x (235/25.4)
Rturn = 224.6” = 18.7 ft.
If we are able to increase our turn angle to 40 degrees, then we get:
Rturn = (110 / SIN(40)) + ½ x (235/25.4) = 14.6 ft
Here is what happens if we instead reduce the wheelbase to 90”:
Rturn = (90 / SIN(30)) + ½ x (235/25.4) = 15.4 ft
If we can reduce the wheelbase as well as increase the turn angle then we get:
Rturn = (90 / SIN(40)) + ½ x (235/25.4) = 12 ft. That’s a big improvement over our initial 18.7-foot figure.
Looking at these numbers, we can now see why our friend’s Golf could make the turn while our F-150 couldn’t. It has a much smaller wheelbase and maybe can turn its wheels to a higher angle because its tires are smaller. I’m ignoring tire width here because while it is a factor, it is a minor player in this issue. The difference between the width of a Golf tire and a F-150 tire is too small to really worry about here.
[Editor’s Note: Since so many of our readers are engineers, I want to make sure we address Ackermann, which describes the concept of the two front tires not sharing the same steering angle. This is a part of steering design because, in order to avoid forcing the tires to slip sideways to negotiate a turn, the two tires need to share a common center of rotation.
I asked Huibert about this. Here’s his response:
Ackermann plays into it a little because the real steering angle will be some average of the inside and outside wheel angles, but then with weight transfer it would tend to favor the outside wheel but then there’s the tire slip angle, blah blah blah. It’s easier and really not that inaccurate just to use the outside wheel angle. At the end of the day, you have to measure it to really get an accurate number which includes all of these factors.
So, enginerds out there ready to send us a long email with numerous references to Thomas D. Gillespie, just understand, as I said before: We’re presenting a simplified version of what goes into turning radius. -DT]
Why Don’t All Cars Have Tight Turning Radii?
So why doesn’t every vehicle have a small turn radius? It would certainly make maneuverability much better. Unfortunately, as with everything related to car design, it’s a trade-off and it depends on many other priorities.
Reducing the wheelbase of a vehicle gets into one of the fundamental dimensions that determines what a vehicle is and can do. Could a pickup truck with the wheelbase of a Golf do the same things it can now? Could it carry a 4×8 sheet of plywood? Could it tow a 12,000 lb trailer? Could it have a 4-door cab AND a 6.5-foot bed? Probably not. It just wouldn’t be practical to have a full size pickup with the wheelbase of a Golf. [Editor’s Note: There are also emissions “Footprint” implications associated with wheelbase. I.e. vehicles with larger footprints have more relaxed standards; click that link to learn more. -DT].
But what about turn angle? Why can’t all vehicles have a high turn angle and a small turn radius? The answer has to do primarily with packaging. A large tire at full turn takes up a lot of space inside the vehicle. This is space that can’t be used for other things, like engines, batteries, headlights, exhaust pipes, etc. Those things would have to go somewhere else or get squeezed into smaller and smaller spaces if we had to turn our wheels more. The other part of the vehicle that has to fit into that space is the crash structure. With ever more stringent crash requirements, this structure becomes more and more important and in many cases larger. This really puts the squeeze on the space available for a tire at full turn.
Here you can see how the engine and the crash structure limits how far you can turn the wheels. If you wanted to get a smaller turn radius by getting more turn angle out of the front wheels, the only way to achieve that is to make the track width – and the rest of the car – wider or by making the tires significantly smaller. Neither of these choices may be possible or desirable for the type of vehicle you are designing so the whole thing becomes a trade-off between competing needs.
Another factor that limits turn angle is the outer CV joint in a front wheel drive or 4 wheel drive vehicle. Most modern axle-CV joints can be angled up to about 50 degrees. Anything more than that and the balls that are inside the joint are in danger of falling out or getting damaged. Now, while 50 degrees is a lot more than the 30 degrees we are using in our example, don’t forget that this angle also has to account for suspension travel. When the wheel is in full rebound, the angle of the CV joint is a combination of the steering angle as well as the suspension angle.
Here you can see how the CV joints are already at a sharp angle just from the suspension being in full rebound. Add steering angle to this and you can see how the outer CV joint could easily get to 50 degrees. The same thing can happen at full bump.
There are ways to mitigate this issue so that you can have a decent turn radius without causing problems at the limits of suspension travel. This is done by adding steering stops that limit the amount of steering angle you can get at full rebound or full bump.
This is an example from a Toyota that clearly shows the steering stop and the bracelet on the control arm it hits at full lock. Notice the curvature of the control arm bracket. Even though the stop block and steering stop move together, at different suspension travels the stop block will hit the control arm on a different part of this curvature, which will limit the turn angle differently (This happens because the angle of the control arm relative to the knuckle changes. That means the knuckle hits the control arm in a different spot, and by controlling the shape of that spot you can control at what turn angle the knuckle hits the control arm).
There is a potential problem with this system though. If you are at full lock pulling into a driveway, for instance, and the suspension gets pushed into bump or drops into a pothole, it can yank the steering wheel out of your hands. Still, this trade-off has been deemed by many OEM’s to be acceptable since it improves turn radius under the normal conditions most customers will encounter and Toyota is certainly not alone in using this method.
[Editor’s Note: As Huibert says, tires take up a lot of space when you factor in the full range of motion both in the up-down direction (suspension travel) and rotated about that up-down axis (steering). In fact, when I was an intern in the packaging team at Fiat Chrysler, a friend of mine, Dominic, spent a lot of his time making tire envelopes using Computer Aided Design software. These big blobs basically represented a boundary that other packaging engineers had to be careful not to cross. Here’s a look at a tire envelope, courtesy of engineering firm PTC:
With this concept explained, let’s talk about a vehicle notorious for its humongous turning radius: The Ford Focus RS and ST, whose turning circles are a whopping 39 feet. These vehicles are a bit of a nightmare when it comes to turning radius, even though it may not seem that way since they’re just small hatchbacks.
First, these are transverse-engine front-wheel drive car; this alone isn’t a huge issue. But if you combine that with the fact that the Focus RS and ST are built on the same platform as a standard economy car (the base Focus) and that their sporting intentions require great stopping power and cornering grip, you’ll see the problem. Lots of stopping power means big brakes which means large-diameter wheels; a desire for grip means wide wheels and tires.
Throwing a huge wheel/tire package into a vehicle designed to have small tires (things like the strut tower location play a role in limiting where the wheel can end up relative to the body) can make things pretty tricky. But what can also make things a bit tricky is a steering geometry geared towards directness. Here, I’ll have Huibert talk about it:
I think one of the commenters [on the Focus-focused internet forum] hit the nail on the head which is that the wider 235 tires keep the suspension from being able to turn as far without the tire hitting the body rail.
It may also be that Ford ran into an issue called “toggle angle” when they shortened the steering arm. Toggle is what happens when you turn the wheel too far and the suspension then doesn’t know if it should turn back the way it came or keep rotating even further… Basically the tie rod and the steering arm of the knuckle get closer and loser to forming a straight line. Once that happens, and you try to steer back the other way, the tie rod and steering arm don’t know which way to go and could go either way. There is also the problem that as the tie rod and the steering arm approach a straight line, it takes more and more force from the steering system to rotate the knuckle back to straight ahead. This increasing force also leads to increasing friction in the ball joints and at some point the friction will be so high that the whole system will just lock in place and the steering just won’t move anymore.
Per Edmunds’ Ford Focus ST suspension walkaround, it seems like Ford wanted to improve steering directness by making the steering arm on the knuckle shorter; this way, a given steering wheel input will yield a larger angular displacement of the tires. (i.e. you won’t have to turn the wheel as much to steering the same amount; you can imagine this by thinking about a door. If you push the very outer edge of a door three inches, the door has barely opened. If you push the door closer to the door’s hinge, to make that part of the door move three inches requires opening the door quite a bit — so moving where the tie rod mates with the knuckle inboard means more steering per displacement of the steering rack). The issue is a smaller steering arm can cause a “toggle” condition to occur sooner in the steering travel.
You can think of the “toggle” concept as a point where you’ve “overturned,” and where trying to turn back the other direction binds up the steering system, since the tie rod no longer has a long lever arm to turn the knuckle about its axis. It’s almost like trying to open a door by pushing its edge (you know, that surface that the deadbolt juts out of) inboard towards the hinges; nothing would happen — you’d basically be trying to just compress the door.
Anyway, whether the toggle angle was the major factor behind the Focus ST/RS’s poor turning radius, I’m really not sure. But I do just want to keep reiterating Huibert’s point about packaging constraints.
The Scion TC in my driveway shows how tight things can get in front wheel-drive wheel housings:
I chatted with another dynamics engineer who worked at Stellantis. He said that, generally, 40-feet is the target above which drivers of regular-sized cars really become displeased by their steering radius. He talked about how this was a challenge on the front-wheel drive Chrysler PT Cruiser, as there was just not much packaging space. (Nowadays, he told me, rear-wheel drive cars are just as challenging thanks to the enormous tires carmakers are putting on everything).
Like Huibert, my friend mentioned newer crash standards as significant factors in turning radii. Specifically he talked about the Insurance Institute for Highway Safety’s small overlap crash test, which literally requires cars to have major structural components no farther inboard than 25 percent of the car’s width. This just adds additional packaging complexity to that wheel housing, and if I had to guess, it has had at least some effect on steering angles on certain vehicles in the industry.
So now that we understand why some vehicles have a small turn radius while others are much larger, we can decide if the trade-offs that lead to the large turn radius in our F150 example were worth it. A Golf may be more than sufficient to take us to the party, but if you need to carry large loads or tow a big trailer on other days, then a pickup or some other large vehicle may be the right one for you and you just have to live with the large turn radius. For those lucky enough to be able to own both a small car and a big car, this question is moot but for those who cannot, we get to pretend to be Austin Powers every time we make a U-turn.