# The dynamics of a cornering wheel

Can a rolling wheel create a side force without first rotating on a vertical axis?

There is something wrong with the way we describe how a cornering vehicle wheel creates a cornering force. I think a rolling wheel must first rotate on a vertical axis to then create the cornering force. This is fundamentally different than what we think about cornering wheels. Can anyone else see this?

In Figs 1a -1b a rolling cylinder is shown to only roll along a straight path. To make it change direction it must rotate on a vertical axis Fig 1c. When you take a wheel off of a car and roll it along the ground (without leaning) it only rolls in one direction along a straight path Fig 2a. To make it change direction you must rotate it on a vertical axis Fig 2b just as with the cylinder. When you do this while it rolls, a force at its contact point makes it lean and possibly fall over Fig 2c. The cornering force changed the wheel velocity and it rolls in a new direction Fig 2d.

When we put the wheel on the car Fig 3, we believe the wheel travels along the circular path with a tangential velocity centered at the turn center. The wheel's motion is thought to be angular about the turn center as it traces a circular path. This is known as true rolling where the wheels will not experience scrub as long as the wheel axes intersect at the turn center.

The wheel’s travel is thought to be angular even though a wheel can only roll in a straight path as shown in Figs 1 and 2. This angular travel thought ignores the dynamics and geometry of the rolling single wheel. The cornering force is thought to be a centripetal force as defined with rotating bodies, but is actually a result of the change in direction of the straight path travel of a wheel.

Balanced and Unbalanced Forces at the Cornering Wheel Contact Patch

The tread at the tire contact patch of a rolling wheel that is not sliding is always stationary relative to the road. The cornering wheel has a static frictional contact patch and the constraint of an axle to allow travel in only one direction. The cornering force is perpendicular to that direction. How is the force created?

A car that travels straight while a wind blows at the side of the car will travel along a straight path because the forces from the wind and contact patch friction are balanced. The net lateral force at each contact patch is zero. This is proven because the wheels are not sliding laterally on the road. While the car corners, the horizontal forces at the contact patches are also balanced when the wheels are not sliding. But we know that unbalanced forces exist because the cornering car experiences lateral acceleration.

To accelerate a car in the forward direction, the engine creates the force that results in a torque at the wheel(s) and horizontal force applied at the contact patch of the wheel(s). Applying the brake creates the same horizontal force, but in the opposite direction. The unbalanced forces that result in the acceleration of the car are clearly identified in both examples above while a simultaneous balance of forces exists at the connection of the tires on the road. On the other hand, the accelerating lateral force at a cornering wheel occurs in a direction where only balanced forces can exist.

Because the road doesn’t move, the cornering wheel must create unbalanced forces while simultaneously creating balanced forces at the contact patch through static frictional contact with the road. The force at a cornering wheel accelerates the car laterally, changing the car’s travel direction while the wheels are in contact with the road and are not sliding. Because of the constraints of the wheel and its axle, the wheel rolls in one direction perpendicular to the wheel axle - wheel axis of rolling rotation. In the speed changing examples above, acceleration and deceleration forces are parallel to that wheel rolling direction. Yet while cornering, the wheel creates a force in a direction it doesn’t roll.

At any instant that the cornering wheel is in static frictional contact with the road and rolling, the wheel rolling direction and mass velocity are in alignment. Why? As explained above, when a force like a wind is applied to the side of the car the lateral forces are balanced when the tires are not sliding laterally. With the cornering wheel this seems contradictory since we know unbalanced lateral forces exist, yet the lateral forces are also balanced. Now we are left trying to explain the creation of this force by a wheel that cannot create a lateral cornering force by rolling and that’s simply because it doesn’t roll in that direction.

For a lateral force to occur at a wheel, either the motion of the car changes direction or the wheel changes direction. Since the wheel rolling direction determines the car’s direction of motion it must be the wheel that changes direction.

The only way a wheel can change direction is when it rotates on a vertical axis as demonstrated in Figs 1-2. After a vertical axis rotation the lateral force occurs because the wheel is then rolling in a different direction than the mass velocity direction Figs 2c-d. Note that the location of this vertical axis rotation is through the wheel contact patch. This turning/pivoting rotation of the wheel is not any different than what happens to the front wheels when you are turning the steering wheel. Because the lateral cornering force occurs after the wheel rotates on a vertical axis, the rule that the wheel must be in static frictional contact with the road to create a horizontal force is satisfied.

Centripetal Acceleration and Newton’s third law at the cornering wheel

Centripetal acceleration is the result of a force on a body in motion that is always perpendicular to its velocity creating circular motion. Because we think the turning car is in rotational motion about the turn center, it experiences an applied centripetal force from the road to the wheels. Newton’s third law would then say that the wheel applies a reactive force equal in magnitude on the road. The problem here is that the non-moving road cannot apply a force on a wheel to change its velocity.

If the wheel was attached by a rope to the turn center, the rope would apply the lateral force to the wheel that changes the wheel’s velocity direction. The car sitting on a large spinning turntable would experience the applied centripetal force from the turntable at each wheel. Both examples have an applied horizontal force that then results in the reactive force on the rope and turntable. Since the road does not move while the car corners, it must be the wheel that applies the force on the road. The road then exerts the reactive force preventing the wheel from sliding. Now keep in mind that centripetal force is not a reactive force. It is the force that creates the circular motion and not the reaction to rotational motion of a body.

Centripetal force is the applied force on a body in motion creating the acceleration that results in circular motion of the body. The reactive force occurs as a result of the body’s inertia resisting that acceleration. If the force that creates the circular travel is a reaction to an applied force then it’s the applied force that changes direction and always points radially away from the center of the circular path. With the cornering car, each wheel applies this horizontal force to the road.

Since the wheel applies the force to the road, it is the wheel that determines the direction of the force. If you then say that the direction of the force exerted by the wheel on the road changes because the car is rotating around the turn center then you have forgotten that any change in the car’s velocity direction is the result of the acceleration created by the reactive force the direction of which is first determined by the wheel. In other words, the wheel changes travel direction not influenced by the car’s momentum and that’s because the wheel determines the direction of motion of the car and not the other way around, which is what we currently believe. We have the cart before the horse.

When and how does the wheel exert the perpendicular-to-its-rolling-direction force on the road? When the mass velocity is in a different direction than the wheel rolling direction as stated above. The only way that can happen is when the wheel rotates on a vertical axis located in its contact patch, not unlike turning your steering wheel. Why? Because the road does not move and the wheel only rolls in one direction. Also, the car is not rotating around the turn center as explained above.

If you disagree with this just keep in mind that nowhere in today’s vehicle dynamics do they say that the cornering wheel applies the force and the cornering force is then the reactive force. It’s easily proven, but would not work with how we currently describe cornering vehicles.

The cornering mechanism of cornering vehicles

Using the bicycle model: The axle constrains the wheel to roll in one direction and the contact patch friction prevents the wheel from traveling in a perpendicular direction. The contact patch friction also resists vertical axis rotation of the wheel.

The front wheel points in a different direction than the rear wheel and both wheels are biased to travel different directions Fig 4a. Because of the flexibility of the bicycle between both wheel contact points there will be some straight path travel of each wheel Fig 4b. ANY distance forward that a wheel rolls and there will be a vertical axis torque created. This is because of the contact patch friction and flexibility in every part of the bicycle through the frame to the rubber in the pneumatic tires.

As the bicycle rolls forward, at some point the torque overcomes the friction at the contact patch and the wheel changes travel direction Fig 4c. At that instant the cornering force occurs because the wheel travel direction is then different than the mass velocity direction. This is explained above.

This cornering mechanism can also be found in cornering boats or airplanes. The hull of a boat is biased to travel a different direction than the rudder. The boat and rudder resist vertical axis rotation and this results in a direction changing torque created at both the rudder and boat. This is a universal rule for any self-guiding vehicle with two or more contact points in or on any medium.

• The cornering mechanism described above is in agreement with the first point that the wheel must rotate on a vertical axis within its contact patch to change direction and create the cornering force. The cornering mechanism also explains how balanced lateral forces can exist at the wheel contact with the road at the same time that unbalanced forces are created to accelerate the car laterally. It is also in line with how the wheel first applies the horizontal force to the road and the cornering force is then the reaction force.

• About slip angle: There are many problems with applying slip angle concept to cornering wheels which is why it is not mentioned above. Slip angle describes how the cornering tire tread in contact with the road deflects the tire like a spring. The force from this spring is what they claim as the cornering force. One problem is that the slip angle concept does not address the cause for the tire deflection. Another problem is that no lateral acceleration of the wheel occurs from this spring force unless the deflected tire returns toward the relaxed/undeflected state (i.e. unless the spring is sprung). Otherwise, the lateral forces are balanced as described above whether the tire is deflected or not deflected. Another problem with applying the slip angle concept to cornering wheels is that it only defines straight path travel of a wheel while cornering wheels travel circular/curved paths. Additionally, slip angle does not play a fundamental role in the dynamics of cornering vehicles. Replace the car wheels with metal skates and slip angle does not exist yet everything stated in this paper would still be applicable to the cornering car with skates in place of wheels.

• Because of our perception of a cornering car’s circular motion we use our understanding of rotating bodies to describe its motion. We know rotating bodies have angular momentum and continue to rotate at a constant rate until a torque is applied. We assume that a wheel changes travel direction because the cornering car has angular momentum. This is a mistake. The reason the wheel travel is circular/curved is described above Figs 4-5. The changing wheel travel direction and subsequent circular/curved path is not influenced by the angular momentum of the cornering vehicle.

• Our perceived turn center rotational motion of the cornering vehicle and its wheels has a history that can be documented with the invention of Ackermann steering. The Ackermann mechanism creates vehicle wheel rolling axes orientations that intersect at the perceived turn center/center of rotation. It was invented to avoid wheel scrub (sliding) on the road. This paper reveals that all cornering wheels must go through periods of sliding when the wheel changes travel direction. Ackermann steering is proven useless in this effort.

• Now reposted verbatim here : engineering.stackexchange.com/q/25601/10902
– user207455
Jan 21 '19 at 9:03
• Your post is quite long and has several parts. If you could emphasize what your question is, it might generate some additional answers. Right now it scans more like an argument than a question. Feb 6 '19 at 4:17
• @BowlOfRed Just one question at the top. Can…axis? I see that it has to first rotate on a vertical axis to then create the lateral cornering force. It is currently seen another way. Two parts to the post: 1. Deals with the geometry of a wheel changing rolling direction. 2. Deals with the horizontal forces at a contact patch of a wheel that only rolls one direction at any instant. Both parts reveal a wheel needs to rotate on a vertical axis to change its rolling direction. Feb 6 '19 at 11:22

We tend to ignore the (relative) rotation of the wheel to the car for several reasons. The rotation itself doesn't cause the forces that are under analysis, so examining them isn't always useful.

I don't exactly know what your question is, but I'll give you two things to think about that might help you be more comfortable with the scenario.

The first is that you might want to think about an (ideal) wheel as a device that has zero coefficient of friction in one direction and infinite coefficient of friction in the perpendicular direction. So when a force is applied that has components in both directions, you get acceleration (motion) and you get a reaction force. Finally, if a torque is applied to turn the wheel, it can turn as well.

Can a rolling wheel create a side force without first rotating on a vertical axis?

Yes. If you apply a side force to a wheel, it will supply a counter side force up to the friction limit. This is independent of the rotation or direction of the wheel.

The other thing is to imagine rather than a real car where (some) wheels can rotate, a toy car where the wheels are fixed. We'll mount the wheels similar to a car turning. Now there will never be any (relative) rotation of the wheels to the car.

When the car sits at rest, there are no (horizontal) forces or torques. Then without changing the axis of the wheels, we start to push the car forward. The angle that the wheels have causes forces that push the front of the car left, and it causes torques that spin the entire vehicle (and with it, the wheels) counter-clockwise.

a wheel only rolls in one direction along a straight path.

That's only true if no (vertical) torques are applied to the wheel. As soon as a torque is applied, the wheel will turn, which could modify the path that it may travel.

I am describing the ideal wheel and how a force ONLY occurs when the rolling wheel rotates/pivots on a vertical axis.

For an ideal wheel, there are no forces required to pivot the wheel. A real rubber tire has a contact patch and friction which detracts from the ideal, but other solutions are possible.

Turning forces come about when a force is applied to the wheel that is not in line with the movement direction of the wheel.

We currently believe that when we attach the wheel to a vehicle with other wheel(s) pointing a different direction it travels a circular path by way of a different principle.

The reason I gave the example of a car where the pivot angle is fixed is to show that you can go from no turning happening (car is stationary) to turning just by applying a force forward on the car. You don't have to change the pivot angle.

You have asked twice about seeing a problem, and I must say that I don't think I understand your specific concern. I'm probably missing the critical point.

• Thanks BowlOfRed. I think in your statement “The angle that the wheels have causes forces…” you are describing the steady state cornering scenario in my question. I am describing the ideal wheel and how a force ONLY occurs when the rolling wheel rotates/pivots on a vertical axis. We currently believe that when we attach the wheel to a vehicle with other wheel(s) pointing a different direction it travels a circular path by way of a different principle. It no longer needs to first rotate on a vertical axis to create the side force. Do you see the problem? Jan 17 '19 at 22:25
• Using a simple bike model: both wheels point different directions and we find the turn center and predicted path. From a stop we move the bike forward. ANY motion of each the front or rear wheel will be linear motion because of the constraints of the axles. While the wheels travel their straight paths, which may be a very very short distance, no lateral cornering forces will occur. If it’s hard to grasp, just remember that there is flexibility from contact point to contact point of all vehicles. Jan 18 '19 at 12:28
• Considering the definition and our understanding of rotational motion, ANY motion of a rotating object is angular motion. This is how we understand and calculate the cornering wheel’s circular travel as if physically connected to the turn center. This is our mistake. Wheels only roll straight paths until forced to rotate on a vertical axis. This is fundamentally different than what we have always thought about cornering vehicles. My question is, can you see this error in our current understanding? Jan 18 '19 at 12:30