# Where is vertical axis of rotation when we observe car yaw?

Where is located vertical axis of rotation when we observe only car yaw, is this axis at different places for different angles of steering wheels?

• if you think about it, the logical place is between the rear wheels ... unless the car has four wheel steering Commented Oct 26, 2023 at 15:39

I have been an Accident Reconstructionist for over 25 years and have been professionally accredited for 12. I also have a degree in mathematics with a concentration in physics. Yes, The vertical Axis of rotation can and does translate during a real-world yaw event. Not only that, but there is no assurance that the axis will be orthogonal to the ground plane. It's location depends upon several factors, including weight distribution, the friction between the car and the surface at the contact points (typically at the tires), and the amount of lean a vehicle experiences during the yaw.

But your question is incomplete: Are you asking about the center of rotation of the vehicle with respect to itself or with respect to the world? A vehicle experiencing a yaw is, in fact, rotating about 2 vertical axes - the center of its turn and the center of its yaw. An old-school critical speed analysis assumes that the only consideration is the center of the vehicle's rotation with respect to its intended path, not its yaw velocity. But empirical testing has shown this type of analysis to be very imprecise.

In the real world, steering will influence the angle of the front wheels, which then affects the friction (and thus the cornering force) at the front tires. When the tires are more parallel to the path of travel, they offer less cornering force than if they are turned sharply. So the axis of rotation is definitely affected by the steering angle. But can it be predicted by the steering angle alone? Usually not.

Think of it like a pitcher throwing a curveball. The ball has a lot of spin on it so that it will negotiate a curved path. The ball is spinning about its central axis, but it is also traveling along a curved path toward home plate. We can estimate the center of the ball's rotation about itself, but where its center of rotation with respect to the world is constantly changing.

In most cases during post-impact travel, a vehicle will tend to yaw about one of the ground contact points - especially if a particular wheel is experiencing greater friction than the others or there is a lot of weight shifted to one wheel. In such a case, steering has no appreciable influence upon either center of rotation.

But in a steer-induced yaw, one of two things must happen. On a high-friction surface, the steering circle can become smaller as the vehicle slows down until it slows enough to recover lateral stability and the yaw ends; or on something like ice, the steering circle becomes larger as the vehicle "wants" to travel in a straight line. In both of these cases, the steering angle can and does have an affect upon the center of rotation, but how much influence can be difficult to say.

Unfortunately, there are so many variables involved in a yawing vehicle, a precise estimate of the center of rotation almost always must be determined empirically or at least iteratively using very small time increments and a robust tire/vehicle mathematical model (this is how reconstruction programs do it). Theoretically, we can pick a point to get past many problems, but we probably should recognize that our center of rotation is a potential source of error.

The best way to establish the center of rotation is good documentation of tire mark evidence and a scale model of the vehicle. Using CAD or even paper and pencil (like we old guys used to do it), the vehicle can be plotted along the tire marks at incremental positions. Given sufficient positions, the axes of rotation become more clear. Steering angle is a poor predictor of the axis of rotation unless the vehicle is in a controlled turn.

• "the center of its turn and the center of its yaw" But where is center of its yaw? Commented Oct 26, 2023 at 19:33
• @user628075 I agree that your question is incomplete. When the car is turning it can be considered to be turning around the center of the turning arc or about the car's center of gravity. Which you use depends on what you are trying to do. It's really just a matter of referencing. Commented Oct 26, 2023 at 22:16
• @user628075 The phrase "The vertical Axis of rotation can and does translate during a real-world yaw event." means that the "center of the car's yaw" (really, the yaw component of the axis of rotation) can be just about any place. For a car that hasn't been skewered by an I-beam or something the center of rotation will probably be within the bounds of the car, but I'm sure that Mr. Eisenbeisz has seen all sorts of exceptions to this. Commented Oct 27, 2023 at 23:45

If you have a "typical" car, with fixed rear wheels pointing straight ahead, "perfect" Ackermann steering, traveling on a perfectly flat smooth surface and traveling slowly enough that there's no tire scrub, then the axis of rotation in a turn from the point of view of the car will be normal to the support surface and extending up through the middle of a line between the rear wheels.

If the car isn't typical, or if it's moving in such a way that the tires are slipping, then the from the point of view of the car the axis of rotation will be, well, somewhere else.

From the point of view of an external viewer, with all of the above "typical slow car" conditions, the axis of rotation will be the center of the turn, and it will be on a line projected between the centers of the rear wheels.