Wheels, camber thrust and slopes.

[mingmongmang]

My understanding of 'camber thrust' is that when a wheel rolls with it's axle not parallel to the ground, it will act as a rolling cone and try to follow an arc around the point at which it's rotation axis intersects the ground.
I've seen this effect on flat ground for instance being able to roll a child's hoop in a circle, but how about on sloping ground? If I roll a wheel 'across' a slope (on a path perpendicular to the downhill direction) should camber thrust make the wheel climb the slope?

 

[rcgldr]

In the case of a hoop, it's light and relatively rigid, so the contact patch area is so narrow that I doubt "camber thrust" is significantly involved. If "camber thrust" isn't significant, the hoop will roll straight across an inclined plane.

When rolling in a circle, the hoop turns inwards because it applies an outward force to the pavement, which reacts with an inwards centrepital force on the hoop. The hoop yaws because the contact path is long enough, and there's enough friction, to create enough torque force to keep the hoop rolling in the direction it's traveling.

 

[mingmongmang]

...When rolling in a circle, the hoop turns inwards because it applies an outward force to the pavement, which reacts with an inwards centrepital force on the hoop. The hoop yaws because the contact path is long enough, and there's enough friction, to create enough torque force to keep the hoop rolling in the direction it's traveling.

Ok...I don't understand the yaw part. What's causing the hoop to change direction?

 

[rcgldr]

Ok...I don't understand the yaw part. What's causing the hoop to change direction?

Gravity pulls down on the hoop at it's center of mass. The pavement pushes up on the hoop at the point of contact. This creates an "inwards" torque. The torque creates an outward force at the point of contact with pavement. The pavement responds with and equal and opposite inwards (centripetal force). This is what causes the hoop to change direction or follow a curved path.

By "yaw", I'm referring to the fact that as the hoop changes direction, it changes it's orientation so it points in the direction that it's travels. The yaw axis is a veritical axis, pependicular to the pavement. Looking down on the hoop, it yaws clockwise if its path is a clockwise spiral, and coutner-clockwise if its path is a counter-clockwise spiral.

To help clarify the terminology, imagine that the x-y plane is horizontal and the z axis is vertical, and that an aircraft is flying in the direction of the y axis. The x-axis is parallel to the wings, and the z-axis is still vertical. The x-axis is the "pitch" axis (nose up and down), the y-axis is the "roll" axis (wingtip up and down), and the z axis is the "yaw" axis (nose side to side).

 

[mingmongmang]

Gravity pulls down on the hoop at it's center of mass. The pavement pushes up on the hoop at the point of contact. This creates an "inwards" torque. The torque creates an outward force at the point of contact with pavement. The pavement responds with and equal and opposite inwards (centripetal force). This is what causes the hoop to change direction or follow a curved path...

But wouldn't exactly the same set of forces occur with a stationary hoop that's falling over? The component you're calling centripetal would just be a simple horizontal force in a stationary wheel so why does it cause a rolling wheel to change direction?
I'm still confused.

 

[rcgldr]

But wouldn't exactly the same set of forces occur with a stationary hoop that's falling over?

Yes, in this case the force from the pavement would move the center of mass in the direction of the fall.

The component you're calling centripetal would just be a simple horizontal force in a stationary wheel so why does it cause a rolling wheel to change direction?

If the wheel is rolling, the force is perpendicular to the direction of the hoop which causes the hoop to turn.

 

[JeffKoch]

If I roll a wheel 'across' a slope (on a path perpendicular to the downhill direction) should camber thrust make the wheel climb the slope?

It would be one force trying to make the wheel climb the slope, but that doesn't necessarily mean it will (I don't know, probably depends on the details) because there are other forces involved too. Taking a case I know something about, a cornering motorcycle, you find that camber thrust can't account for the bar angle in most situations - the required bar angle just follows from the turn radius, and you generally have to push or pull on the bars to keep that angle stable. In that case, the tire is actually slipping and sliding as it grips the pavement.

I think a child's hoop probably has negligible camber thrust - this is only an issue for tires with significant contact patches.

 

[mingmongmang]

If the wheel is rolling, the force is perpendicular to the direction of the hoop which causes the hoop to turn.

But how? All the forces you have described are acting along the same vertical plane that the axle lies on and that is perpendicular to the direction of travel. how can this 2-dimensional set of forces cause motion (yaw) in the third dimension?

 

[rcgldr]

But how? All the forces you have described are acting along the same vertical plane that the axle lies on and that is perpendicular to the direction of travel. how can this 2-dimensional set of forces cause motion (yaw) in the third dimension?

I'm not sure, but I suspect that the forwards rolling rate combined with an inwards torque results in a yaw reaction because of gyroscopic precession. The turn won't be coordinated unless the yaw reaction corresponds to the lean angle.

 

[wysard]

Not exactly. The problem is the frame of reference.

The wheel rolls in reference to the centre of axis. When you roll it across a slope imagine that the "slope" is actually the flat plane and you are a relative outside observer tilted in a different plane.

It's the same physical phenomena that makes a hippodrome work.

The wheel's axis is calculated parallel to the plane the wheel is on, not the observer. So, the wheel will "precess" about the centre relative to the slope it is on all other things (like, say gravity...lol) being equal. From the wheel on a slope's perspective it is on the flat ground, not on a slope. The only factors are the angle of intescetion of the axis of the wheel with the slope, the distance to to the centre of that circle, the wheel's angular velocity and it's mass.

 

[rcgldr]

When you roll it across a slope imagine that the "slope" is actually the flat plane and you are a relative outside observer tilted in a different plane. So, the wheel will "precess" about the centre relative to the slope it is on

If I remember correctly, I tried this before and the hoop just rolls straight across the slope if there is no initial "lean" to the hoop. Start off with the hoop perpendicular to the slope, and it ends up rolling down hill. I'll test this again with a bicycle wheel / tire or one of the neighborhood kid's hoops.

 

[wysard]

rolling straight

Jeff. You could very well be right about it rolling straight.

The point I was making was about the expectation of the frame of reference. Ie: which way you can expect it to roll. What you have added is a personal observation of experimental data, which is perfect. What I mean by that is that the theory must answer observational data or it doesn't work right?

So. As you pointed out, the hoop moved straight. Which should never happen on an flat plane right? But it did. So, what does that tell us? That the hoop was trying to travel uphill, but remember the Earth has a gravity well. So if the hoop moved straight it means the factors for the hoop you used would have tried to move up the slope but the force acting on the hoop was counteracted by gravity dragging the hoop downhill. As a result it moves straight. Spin the hoop faster, or make the rim heavier and it would slowly climb. Make it lighter, or a little slower and it would actually go downhill, just slower than expected.

Make sense?

 

[rcgldr]

Jeff. You could very well be right about it rolling straight. As you pointed out, the hoop moved straight. Which should never happen on an flat plane right?

My point was that camber thrust wasn't a signficant factor in the case of a hoop. Whether the hoop turns or not depends on it's initial orientation when released. Eventually, since the hoop is unstable, it will start to turn, but not because of camber thrust.

If I were to roll a cone on it's side it will roll in a circle, angled plane or not because of "camber thrust".

 

[wysard]

Camber Thrusting Cone

If I were to roll a cone on it's side it will roll in a circle, angled plane or not because of "camber thrust".

True. A cone has the potential for a lot more traction and a huge difference in camber angle than most wheel applications. Wheels typically use a .5 degree camber +/- 30 min which would take a pretty impressive cone to actually see the true amount of camber thrust applied. Nevertheless it should still turn slowly uphill unless traction or some other factor is in play. Like gravity. Especially with a hoop or wheel with a comparatively heavy rim and a decent diameter where gyroscopic progression will make a much larger impact to turning characheristics.

I just did a quick google on camber thrust and found an appalling number of references likening camber thrust to turning a bike by leaning, which is clearly not camber thrust at all, but the aformentioned gyroscopic progression. Next thing you know counter steering a motorbike will be an act of voodoo. Sheesh.

 

[rcgldr]

Nevertheless it should still turn slowly uphill.

Ok, but the surface of a hoop is very stiff, the hoop is very light, so the contact patch area is very small, almost approximating a point, and coefficient of firction is small, so the amount of turn due to camber thrust is virtually unnoticable. I'd have no idea if I started a hoop roll with a non-vertical lean versus any camber thrust effects involved.

Regarding the cones and camber thrust, if two cones are used, one in front of the other, and the axis of the cones are parallel, then the twin cone setup goes in a straight line, with a lot of slippage. I'll have to find the website that did this experiment with a pair of styrofoam cups.

 

[wysard]

Regarding the cones and camber thrust, if two cones are used, one in front of the other, and the axis of the cones are parallel, then the twin cone setup goes in a straight line, with a lot of slippage.

Obviously. But the OP is not asking about that, just about a single wheel or hoop.

Ok, but the surface of a hoop is very stiff, the hoop is very light, so the contact patch area is very small, almost approximating a point, and coefficient of firction is small, so the amount of turn due to camber thrust is virtually unnoticable. I'd have no idea if I started a hoop roll with a non-vertical lean versus any camber thrust effects involved.

If you make the contact patch approximate a point camber thrust is not virtually unnoticable it is zero, and if friction is almost zero the hoop will slip and fall flat on the ground as soon as you tilt it over making the test equally moot. I do not argue that it is a small vector, the point I am trying to make is that camber thrust exists in any real world scenario where the contact patch is trapezoidal and that if it is, the hoop, wheel, rim, or whatever is constrained to turn proportionally irrespective of whether the plane is tilted or not with respect to the observer.

I think we are on the same track, just with a slightly different perspective and I'm trying to figure out exactly how yours fits into mine with respect to the question. That way I learn something too!

 

[rcgldr]

If you make the contact patch approximate a point.

I was just pointing (bad pun) out the fact that the contact patch area is small. Also the smaller still central area of the contact patch gets the higher normal force (pressure), while the outer area of the contact patch gets less, so the outer area can slip while the central area doesn't.

I now recall that on a somewhat slick surface (like a waxed gym floor), a hoop can be launched forward at a slight angle, while spun backwards, and there's still visual evidence of gyroscopic precession before the hoop falls over onto it's side. The hoop will be sliding across the floor while yawing at the same time. In this case, the yaw reaction has virtually no impact on the path of the center of mass of the hoop, it just continues to slide across the floor.

Maybe a better example would be a bicycle wheel and tire (more friction).