# Lean Angle == Turn Radius?

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and and and... here it is... holy ######.... it's the precessional forces that turn the rear outward due to the torque applied to the moment arm defined by the wheel itself!!!

Aaaahh!! Now I get it..! ....not

You totally lost me, dude. When you find the time, please explain this concept in English (layman's words)..

OK, I will. But I don't think tonight, I have to get to bed and I want to do a good job of it.

In a nutshell, when a torque is applied to the axle of a spinning wheel or gyroscope, the wheel tries to turn 90 degrees in the next plane of rotation. It's part of what makes a motorcycle lean over in the first place.

When you turn the front wheel right, the wheel tries to lean to the left which is a different plane x, y, z. Vertical, horizontal, and depth. Yaw, pitch and roll. Three dimensions or degrees of freedom.

In the meantime, this is where I learned most of my basic physics:

http://www.glenbrook.k12.il.us/gbssci/Phys/Class/BBoard.html

I also have some really good gyroscopic websites with some really cool experiments around here somwhere, I'll get them together for you too. In the meantime.... get the front wheel off your bicycle and hold the axle in your fingers and spin the wheel. Now try to move the wheel left and right and see what happens.

This effect is the basis of precession!

Google gyroscopes and precession and do some digging.

Be back soon.

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Keith Code also talks about it in TOTW II.

I don't have the page number right now but I will find it tomorrow.

Good night my friend!

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Keith Code also talks about it in TOTW II.

I don't have the page number right now but I will find it tomorrow.

Good night my friend!

Good night my @rse, it's 7:25 in the morning don't you know!!

Anyway, I remember doing that experiment with the bicyle wheel once. I sat in an office chair (one of those chairs that turn around), spun the wheel, and tried to "lean" the wheel over (that would be the z axis?). What happened was that the office chair would turn around (on the y axis, right?). I assume that you're moment of EUREKA has something to do with this phenomenon. Looking forward to hearing the full story from you. Sleep tight

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I fell asleep on the couch watching a DVD called The Doctor, The Tornado and The Kentucky Kid. (Story of 2005 Laguna Seca GP) I just woke up at 2:45 am here. Turned off the TV and came to turn off the computer to go to the bedroom and now I'm not so sure about my theory anymore... lol.

I was afraid of that. I knew I shouldn't get so excited.

The key I recall to those gyro moves is that the wheel deflects or precesses a given amount for torque applied... even if the torque doesn't cause motion in its own plane. At least that is what a NASA engineer told me once. But now I'm not so sure. What if the rear wheel has to actually stand up to be deflected or turn outward... ???

Hmm... back to the laboratory and my bicycle wheel tomorrow night...

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Anyway, I remember doing that experiment with the bicyle wheel once. I sat in an office chair (one of those chairs that turn around), spun the wheel, and tried to "lean" the wheel over (that would be the z axis?). What happened was that the office chair would turn around (on the y axis, right?). I assume that you're moment of EUREKA has something to do with this phenomenon. Looking forward to hearing the full story from you. Sleep tight

Yeah, that's the one! Try it just standing still and holding the wheel in your hands. Turn the wheel right and left and feel the wheel wants to lean over. Lean the wheel over and it wants to turn left and right. Rotate the whole experiment so that the wheel is nleaned over and try to un-lean it with the wheel turning. The wheel turns outward wrt the corner! So...

If centrifugal (cornering) force tries to make the rear wheel stand up (or lean less), gyro forces will make the wheel turn outward.

Possibly even if the wheel doesn't actually change lean angle but merely has force applied in that direction.

It still feels like the right answer in my gut.

PS - Also try tying a string to the axle of the wheel and hanging it from the ceiling or some form of truss and play with setting it at different lean angles. It will seem to defy gravity as it maintains degrees of vertical orientation while its only support is the string in the center.

Pretty freaky, dude!

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I removed the front wheel from my bicycle this morning to conduct this test against for myself and I am more convinced than ever now. Hodling the wheel at some degree of lean angle and attempting to reduce the angle of lean forces the wheel to turn outward with significant force.

Also, re-reading pages 58-59 of TOTW II, the points made in the "Steer for the Rear" chapter are much clearer and make more sense to me now.

I've run out of time to find the page where Keith discusses the gyro force and countersteering, but, I will find it later.

High ho, off to work I go!

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Soooo much great info in this thread.

Thanks to all for contributing so noobs like me can learn more!

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I removed the front wheel from my bicycle this morning to conduct this test against for myself and I am more convinced than ever now. Hodling the wheel at some degree of lean angle and attempting to reduce the angle of lean forces the wheel to turn outward with significant force.

Also, re-reading pages 58-59 of TOTW II, the points made in the "Steer for the Rear" chapter are much clearer and make more sense to me now.

I've run out of time to find the page where Keith discusses the gyro force and countersteering, but, I will find it later.

High ho, off to work I go!

OK ok ok, so I'm aaalmost grasping the essence of what you're saying, but I'm still missing that final puzzle piece.. I'm with you on the whole gyro effect thing, I see how the rear will turn outwards if you apply a force to decrease lean angle (and vice versa). But how does this apply to real life riding? How would you apply that force to change the lean angle of the rear tyre? By countersteering the front wheel, or by changing BP or by throttle control...? Because you ARE talking about the REAR wheel all along..? Lets take the front wheel out of the quotation for a second: HOW would you "steer for the rear" if your front end was in the air as you exit a corner (GP style) ?

...

BTW, I was just thinking about how this phenomenon also affects the FRONT wheel as you turn on the bars. Interestingly, it seems that the "gyro effect" will contribute to leaning the bike to the left when countersteering to the right. So the front wheel actually "helps" the top end of the bike to lean over to the left when the lower end is turning right. Kind of like a team work between inertia and gyro forces.

PS. That gyro phenomenon only applies when the bike is "unstable" due to the rider giving steering inputs. It doesn't affect anything when the bike is trailing happily at perfect balance (straight forward or at a constant lean angle)..

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I removed the front wheel from my bicycle this morning to conduct this test against for myself and I am more convinced than ever now. Hodling the wheel at some degree of lean angle and attempting to reduce the angle of lean forces the wheel to turn outward with significant force.

Also, re-reading pages 58-59 of TOTW II, the points made in the "Steer for the Rear" chapter are much clearer and make more sense to me now.

I've run out of time to find the page where Keith discusses the gyro force and countersteering, but, I will find it later.

High ho, off to work I go!

OK ok ok, so I'm aaalmost grasping the essence of what you're saying, but I'm still missing that final puzzle piece.. I'm with you on the whole gyro effect thing, I see how the rear will turn outwards if you apply a force to decrease lean angle (and vice versa). But how does this apply to real life riding? How would you apply that force to change the lean angle of the rear tyre? By countersteering the front wheel, or by changing BP or by throttle control...? Because you ARE talking about the REAR wheel all along..? Lets take the front wheel out of the quotation for a second: HOW would you "steer for the rear" if your front end was in the air as you exit a corner (GP style) ?

...

BTW, I was just thinking about how this phenomenon also affects the FRONT wheel as you turn on the bars. Interestingly, it seems that the "gyro effect" will contribute to leaning the bike to the left when countersteering to the right. So the front wheel actually "helps" the top end of the bike to lean over to the left when the lower end is turning right. Kind of like a team work between inertia and gyro forces.

PS. That gyro phenomenon only applies when the bike is "unstable" due to the rider giving steering inputs. It doesn't affect anything when the bike is trailing happily at perfect balance (straight forward or at a constant lean angle)..

Lots to think about and not much time to do it yet. However I will mention from riding motocross. With the front wheel in the air on a left turn if I wanted to lean left more, I seem to remember turning the front wheel to the right, which made it easier to lean left more.

Also during a jump if I wanted to lean the bike left in the air to land pointing left some instead of landing straight (for a left turn placed right at the landing spot), I would steer the front right as I applied body english. The bike would lean left and I'd land sliding into the left turn ( which allowed more speed over the jump).

Gotta think about racer's questions and assertions this weekend.

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I'm with you on the whole gyro effect thing, I see how the rear will turn outwards if you apply a force to decrease lean angle (and vice versa). But how does this apply to real life riding? How would you apply that force to change the lean angle of the rear tyre?

I had the idea late last night that increased centrifugal force, regardless of a stable lean angle (ie. even if the lean angle didn't change), would apply increased upward torque to the rear wheel (whether enough to overcome gravity or not) and cause a wider radius or arc due to gyroscopic forces. I wasn't so sure when I woke up. I played with the bicycle wheel over coffee and decided to run with it anyway. I am afraid I am going to need to "cheat" a bit and brush up on gyroscopes and angular momentum, etc. to confirm or falsify this theory.

In the light of a new day, it seems just as reasonable that increased centrifugal "force" simply forces the rear wheel to "track wider" due to tangential force applied to the conically steering contact patch, hence, the contact patch rolling off center or off angle ... so to speak... if you know what I mean.

In any case, before I began doubting my latest theory, I was going to say...

You don't apply "that force" to change lean angle. You apply it to increase velocity (and rpm of the gyro). Higher velocity at the same lean angle applies increased centrifugal "force" as the bike travels through the corner. Regardless of whether the lean angle remains balanced or stable, that force still translates to vertical torque applied to the gyro of the rear wheel.

Or at least that is where I was thinking before. Oh well.. back to the drawing board.

BTW, I was just thinking about how this phenomenon also affects the FRONT wheel as you turn on the bars. Interestingly, it seems that the "gyro effect" will contribute to leaning the bike to the left when countersteering to the right. So the front wheel actually "helps" the top end of the bike to lean over to the left when the lower end is turning right. Kind of like a team work between inertia and gyro forces.

Yep!

PS. That gyro phenomenon only applies when the bike is "unstable" due to the rider giving steering inputs. It doesn't affect anything when the bike is trailing happily at perfect balance (straight forward or at a constant lean angle)..

This is where I am hung up. I'm not sure this is true, but, I'm also not sure it matters. The more I think this theory through, the less realistic it seems and the more I am reverting to my original two possiblities of expansion of the contact patch imitating a contact patch higher up the crown of the tire or simply a degree of sliding or rolling off-angle. The last one really seems most plausible to me at the moment.

Hence, that would mean there is some validity to carl's original idea of the wheel traveling further forward for each measure of distance traveled inward on the circle.

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In any case, framing the angle change of the front wheel at increased speed as due to the centripetal force of the pavement applied tangentially or orthogonally to the circular path of the trailing contact patch really helped me to "feel" the "why" of the front turning out more at increased speed better.

I thought I had a good grip with my experiments with the trail bikes and framing the angle as dependent on the centrifugal or cornering force applied down through the forks which describes the same action from the opposite "node" back on page two of this thread, but, this really helps complete the picture.

Thanks for that guys.

That said, I'm still not ready to back off the idea of the rear being responsible for the radius and the front turning out as an effect, rather than a cause.

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"Tires

Because real tires have a finite contact patch with the road surface that can generate a scrub torque, and when in a turn, can experience some side slipping as they roll, they can generate torques about an axis normal to the plane of the contact patch.

One such torque is generated by asymmetries in the side-slip along the length of the contact patch. The resultant force of this side-slip occurs behind the geometric center of the contact patch, a distance described as the pneumatic trail, and so creates a torque on the tire. Since the direction of the side-slip is towards the outside of the turn, the force on the tire is towards the center of the turn. Therefore, this torque tends to turn the front wheel in the direction of the side-slip, away from the direction of the turn, and therefore tends to increase the radius of the turn.

Another torque is produced by the finite width of the contact patch and the lean of the tire in a turn. The portion of the contact patch towards the outside of the turn is actually moving rearward, with respect to the wheel's hub, faster than the rest of the contact patch, because of its greater radius from the hub. By the same reasoning, the inner portion is moving rearward more slowly. So the outer and inner portions of the contact patch slip on the pavement in opposite directions, generating a torque that tends to turn the front wheel in the direction of the turn, and therefore tends to decrease the turn radius.

The combination of these two opposite torques creates a resulting yaw torque on the front wheel, and its direction is a function of the side-slip angle of the tire, the angle between the actual path of the tire and the direction it is pointing, and the camber angle of the tire (the angle that the tire leans from the vertical).[17] The result of this torque is often the suppression of the inversion speed predicted by rigid wheel models described above in the section on steady-state turning.[14]

Because the front and rear tires can have different slip angles due to weight distribution, tire properties, etc., bikes can experience understeer or oversteer. Of the two, understeer, in which the front wheel slides more than the rear wheel, is more dangerous since front wheel steering is critical for maintaining balance.[17]"

...there's even more going on at the contact patch than I thought.

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I'm sorry but my brain hasn't digested the info in your previous posts yet, so there is no room for new information. I thought we we're coming to a conclusion, and you just keep on adding new aspects (contact patch) to the scenery.. Argh, my head hurts!

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I'm sorry but my brain hasn't digested the info in your previous posts yet, so there is no room for new information. I thought we we're coming to a conclusion, and you just keep on adding new aspects (contact patch) to the scenery.. Argh, my head hurts!

Conclusion? Where's the fun in that? Besides, we've barely scratched the surface in any meaningful scientific way.

As for conclusions... the gyro vs gravity vs angular momentum debate has been going on since physics first looked at two-wheeled vehicles. From counter-rotating gyro-canceling wheels to equations of motion, each side claims experimentally and mathematically supported victory on a regular basis.

In fact, each side claims the counter-rotating wheeled bicycle proves their theory. Can you believe that? Two groups of people ride the same machine. One group claims it is virtually as easy to balance and ride as a normal bike, the other claims it is virtually impossible to balance or ride at all.

Um... right. Somebody is full of spit... lol.

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Racer, you are jumping too far ahead for me, as trail was squatting in my brain. So I'm trying to get trail settled before I catch up to the things you have added.

Also you robbed me of my Sat. morning beauty sleep. I wake up at 5AM going “I have it!” (in regards to trail

Sorry about the length, this is mainly about the question of TRAIL and how the front wheel knows to point into the lean more or point in less, due to how fast the bike is going. We will see the front wheel is NOT JUST trailing along, but locked in a balance of forces similar to what we were already talking about. These forces POINT the front wheel at a specific angle depending upon the relationship of power/amount between the forces . They are side forces working at the contact patch, one turning the front wheel into the lean, the other turning the front wheel back to straight. How they balance when fighting each other, determines how far the wheel gets turned into the lean (removing rider input). We will see the amount of turn in does change with speed change, and we’ll see how that achieves a state of balance we talked about in earlier posts.

As an aside, HAH! Racer, I’m up at 5am not being able to sleep again. Then I’m out in the garage at 5:30AM taking the front wheel off of my wife’s bicycle just to confirm leaning it to the right will turn it out to the left, it does. Further turning it to the left leans it to the right. This is familiar yes? Yes, turning the front to the left also steers the front axle out from under the headstock, leaning the bike to the right. That means the gyroscopic forces help at the front with countersteering. They can help you lean the bike in the air too, which had been my experience.

THAT BEING SAID, I remember someone, somewhere, put reverse spinning discs on a two wheel vehicle to negate gyroscopic force and IT WAS STILL RIDABLE. Therefore, countersteering WILL WORK even without gyroscopic force, though it is aided by it. (PS I see you saw that somewhere too)

End of aside, and back to trail.

I’ve been trying to think how trail acts to steer the front wheel INTO the turn MORE going slow and less when the bike is going faster. I figured the front wheel could NOT just trail passively along behind conical steering force, in order for the bike to behave the way it does in the real world, so the answer had to lie in balancing forces at the contact patch. There had to be a force trying to turn the front wheel IN and a force trying to turn the front wheel OUT. These had to balance each other out in a way that helped describe the NEEDED path for balanced status quo of the bike in a turn.

In order for the front to turn in the CORRECT amount, creating centripetal inward force, to work with forward momentum, in balancing gravity, I figured the 3 forces we identified must be doing something at the contact patch with trail then. I figured we should look at our original 3 forces first. That line of thinking bore fruit in my sleep.

When you lean a motorcycle, the contact patch at the front wheel moves inside, off center. The pivot point for the front end is found thus; looking at the motorcycle from the side and drawing a line top to bottom through the headstock parallel with the forks, where that line contacts the ground is the pivot point. The actual contact patch of the tire is on the ground behind that, so It’s trailing behind the pivot point (trail).

So, you lean the bike, with the wheel pointing straight, the trailing contact patch moves inside the center (as viewed from the top, diagrams would be helpful) AND IT’S BEHIND the pivot. Gravity (one of our original three force friends) exerts its force on the patch making the patch want to ROTATE BACK IN LINE CENTERED behind the pivot (instead of staying inside center). Due to gravity pressing on the patch through a lean angle, the contact patch does rotate to the outside of the lean, and since the pivot is ahead of the patch this turns the front wheel INTO the LEAN (a picture would show this real easy). It does this quite a bit, as when I put the bike on the kickstand at not that much lean, the front wheel wants to turn in, all the way till it hits the steering lock when friction at the patch is overcome. That happens at not all that much lean even, and would do so more strongly at a steeper lean.

Important point, SO, GRAVITY is turning the front wheel strongly into the lean, due to trail and lean of wheel. You lean the bike, gravity wants to turn the front wheel sharply inward, the more you lean, the more strongly it wants to do that.

In our old formula centripetal (inward) force combined or added with forward momentum (speed)force, to balance(equal out) gravity. Remember this as we go on here. Speed+radius= Gravity that is during a constant lean, constant arced turn.

Once countersteering successfully has the bike leaning the amount the rider wants and he/she lets off pressure to the handlebar, the offset contact patch turns the front wheel into the turn from the force of gravity. Left to itself it would turn in TOO FAR to balance forces as the other two forces need to balance gravity for a status quo.

Conical steering along with the front axle of the bike following the pointing direction of the front wheel, creates centripetal (inward) force which is the 2d of our forces. The TRAIL TRIES TO MAKE the wheel point in the direction the front axle is TRAVELING (trailing along), working AGAINST theEXTRA inward pointing of the wheel from GRAVITY (gravity would make it turn too far in on its own). The trail is using centripetal force to balance what gravity is doing at the contact patch. That’s so far just like stabilizing lean angle, only it’s acting on stabilizing where the tire is pointing due to trail. There is however a third force, forward momentum (speed). So does speed come in to add it’s force and help balance things at the patch to steer, just like it helps balance lean angle? Yes.

The conical steering and turned front wheel are providing inward force.

At the contact patch speed trying to go straight ahead AGAINST a TURNED IN TIRE pushes backwards against the contact patch as the CONTACT PATCH tries to GO straight FOWARD in a direction the tire is not pointing in, the pressure at the patch is back.

The PATCH IS TRAILING the pivot point, SO the BACK pressure TRIES TO STRAIGHTEN the front wheel out more, FIGHTING the turn in from gravity ALONG WITH the centripetal inward force. THAT also FIGHTS GRAVITY which was turning the front tire in.

SO, DUE TO TRAIL, at the contact patch you have CENTRIPETAL FORCE, AND SPEED (forward momentum) BALANCING out GRAVITY at the patch and telling the front tire where to point. SO, speed+centripetal force=gravity at the patch to steer the wheel due to the fact trail is built into the system AND the fact the bike leans and makes the contact patch move inside center and gravity is working at the patch through a leaning angle.

DUE TO TRAIL, due to trail, due to trail:

To understand the following REMEMBER, forward momentum (Speed) COMBINE with centripetal (inward force from following an arc or radius), needs to match or equal the force of gravity (lean angle). That MATCHING of the combined two forces to the third force, achieves a balanced status quo (constant arc/radius/lean angle).

Said another way for a constant corner, you want speed AND turn in angle, to match the singular falling down force of lean angle. For steering at the contact patch you want speed AND turn in angle, to match the turning in pressure of gravity.

If you increase the lean (gravity, which is trying to turn the tire in more) keeping speed the same, the tire WILL turn in more, increasing centripetal (inward) force to balance the extra gravity gained from more lean.

If you decrease the lean (gravity, which is turning the tire in), keeping the speed the same, the tire will turn out more (gravity trying to turn it in, speed trying to turn it out). This lessens centripetal (inward force) to match the decrease in lean (gravity)

If you increase the speed (speed trying to turn the tire out), keeping the lean (gravity) the same, the tire will straighten out more. That lessens centripetal force which combines with the extra speed to still retain balance with gravity (lean angle) in use.

If you decrease the speed, keeping the lean the same , the tire will turn in more. Since speed is less you need the tighter turn to balance the same lean angle.

The competing forces of gravity turning IN and speed turning OUT at the TRAILING contact patch, turn the tire the required amount to dial in the needed centripetal force for balanced status quo.

I’m guessing there is a sweet spot for the amount of rake and trail needed to get this to work in a way that dials in the CORRECT amount of turn in, WITHOUT the rider having to do much, or without the rider having to do anything. So, It’s probably no accident rake and trail numbers are pretty similar on good handling bikes.

Of course the bike goes through an unbalanced state to get to the next balanced state (whether rider induced or otherwise). With the right amount of trail it should seek balance on its own due to what I just described above.

SO, THAT is my idea of how the trail knows how much to turn the front wheel into the turn given a lean angle and a speed.

THE REASON YOU NEED A DIFFERENT RADIUS AT DIFFERENT SPEEDS for the SAME LEAN ANGLE, is that is necessary or the bike cannot keep either a constant arc or a constant lean angle. That is because speed and turn sharpness forces must combine to equal the force of the lean angle used. The ONLY way the bike can vary the radius is by following the direction the front tire is pointed in, and there is only ONE CONSTANT radius possible for any given speed and lean angle combination.

It seems the front with its vertical hinge is VERY needed to change and achieve the status quo in turning at a wide variety of speeds.

On the other hand it seems the rear with its horizontal hinge is VERY needed, when stability is important.

I’m thinking as far as stability is concerned you can weight shift for one hinge or the other. However while the front is on the ground rolling free of large slip angles, the axle is going where the front wheel is rolling along and pointed, trying to haul the rest of the bike after it and so heavily involved in steering. That’s even if the rider is not touching the bars and the speed and gravity through trail is doing the steering instead.

Got weekend stuff to do, then I'll come back and look at your other stuff.

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THE REASON YOU NEED A DIFFERENT RADIUS AT DIFFERENT SPEEDS for the SAME LEAN ANGLE, is that is necessary or the bike cannot keep either a constant arc or a constant lean angle. That is because speed and turn sharpness forces must combine to equal the force of the lean angle used. The ONLY way the bike can vary the radius is by following the direction the front tire is pointed in, and there is only ONE CONSTANT radius possible for any given speed and lean angle combination.

I believe we all agree on that part. And you've brought a lot of interesting in-depth info that supports this "theory". I must admit that I haven't read carefully through your entire post yet, but I will as soon as I have the time. I reckon I need at least half an hour to grasp all of that info..

Have a nice weekend y'all

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THE REASON YOU NEED A DIFFERENT RADIUS AT DIFFERENT SPEEDS for the SAME LEAN ANGLE, is that is necessary or the bike cannot keep either a constant arc or a constant lean angle. That is because speed and turn sharpness forces must combine to equal the force of the lean angle used. The ONLY way the bike can vary the radius is by following the direction the front tire is pointed in, and there is only ONE CONSTANT radius possible for any given speed and lean angle combination.

Well, call me pedantic, but, the "bike" doesn't do anything, only the rider does.

Hence, the bike can not "vary the radius", only the rider can. That said...

You have done a good job of outlining forces that control the direction of the front wheel while a bike is leaned over, but, I do not believe you have demonstrated that the front wheel is responsible for the lean angle or direction of the bike (aside from rider input at the bars).

The way I see it, under acceleration, weight is biased to the rear, hence, without rider input at the bars, the rear dictates the lean angle of the bike and the front wheel is "forced" along for the ride.

Here is Keith Code's take on this from pages 58 and 59 of Twist of the Wrist II:

According to the laws of physics and engineering principles, the following is true: As long as you apply force to the bars, the bike continues to lean further over. However, once the bike is fully leaned into a corner, the rear end "steers" the machine. The front end "turns" the bike or changes the lean angle but the moment the motorcycle is leaned over and stable, the main mass of the bike--from the steering head back--determines the lean angle it will hold.

[snip]

If your throttle control is standard, the only things that will change the lean angle of the bike to any great degree are a slide/catch action or steering input. The most convincing illustration of this is doing a wheelie while coming off a corner. The lean angle of the bike remains the same even though the front wheel has left the ground!

Front-End Duties

Once leaned over in a turn, the front end is no longer steering the bike: It helps stabilize it but does not steer it. But the front end's function is still important. The 30 or 40 percent of the cornering load it is carrying accounts for about that same percentage of cornering speed. In other words, if you added 30 or 40 percent more load on the rear wheel at speed, you would certainly slide it.

So, when the rider turns the throttle more at a stable lean angle, increased cornering force causes the radius to increase. But I believe that the angle of the front wheel changes because the radius increases, not vice versa. IMO, by your own description of the forces involved, it is the forces applied to the front wheel caused by the changing radius that turns the front wheel wider after the fact.

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So, when the rider turns the throttle more at a stable lean angle, increased cornering force causes the radius to increase. But I believe that the angle of the front wheel changes because the radius increases, not vice versa. IMO, by your own description of the forces involved, it is the forces applied to the front wheel caused by the changing radius that turns the front wheel wider after the fact.

I must admit that the more I think about it, the more I'm convinced that you are right. Once the bike is stabilized at a constant lean angle and corner radius (no rider steering input), the rear wheel will "command" the front wheel . Which means that the radius will increase when more throttle is applied. But why does the radius increase if you roll the throttle off as well? Is it possible to TIGHTEN a radius whilst steering with the rear?

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So, when the rider turns the throttle more at a stable lean angle, increased cornering force causes the radius to increase. But I believe that the angle of the front wheel changes because the radius increases, not vice versa. IMO, by your own description of the forces involved, it is the forces applied to the front wheel caused by the changing radius that turns the front wheel wider after the fact.

I must admit that the more I think about it, the more I'm convinced that you are right. Once the bike is stabilized at a constant lean angle and corner radius (no rider steering input), the rear wheel will "command" the front wheel. Which means that the radius will increase when more throttle is applied.

Forgive me if I am misunderstanding you, but, that still sounds like you think the radius increases because the front wheel turns more outward.

I am saying that the radius increases because the rear wheel steers or turns the bike more outward and the front wheel is forced to turn to "follow" or trail the rest of the bike.

And I believe that conclusion is what Keith's words above must lead to.

However, you and Carl may have a point here. Let's look at it.

Carl said he believes that the radius will always be the same for a single wheel at a given lean angle no matter the velocity. Period. Hence, if the front wheel is off the ground (wheelie) while coming out of a corner, no matter how fast you are traveling, the radius will always be determined by the lean angle. (And it is only the front wheel turning outward that alters the rear wheel radius to follow the front with both wheels on the ground.)

I disagree. I believe that the nature of the rubber/pavement contact patch at the rear allows increased cornering force to alter the path or radius of that single rear wheel.

Here is some logic to support that assertion.

If a bike is cornering with standard throttle (accelerating 60/40) at a given lean angle, radius and velocity with both wheels on the ground, why doesn't the radius 'tighten up' if you lift the front wheel? Why doesn't the rear revert to the "single wheel" radius dependent only on lean angle without the front wheel to steer it wider?

See what I mean?

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Well, call me pedantic, but, the "bike" doesn't do anything, only the rider does. smile.gif

We mostly agree here for sure, I only meant the rider does it through the bike. Also a riderless bike on it's own will sometimes correct itself and run on a long ways by itself until it slows enough to fall over so it does do some things "itself" due to forces acting on it and since the bike is riderless and no longer under throttle, it does them with the front wheel as well as the rear.

So far, "I think" our difference is that you think the front wheel only trails along in the direction the rear points it due to the conical steering and forward momentum along a path of the rear. I think the front continues to play a significant part in the direction the bike takes, according to it's load, while receiving it's due influence from the rear.

I may be mistaken, but it is my memory that at slower speeds when I lifted the front on a dirt bike in a turn the arc the rear dictated was not the same as the arc being traveled when the front was on the ground. Some adjustment in CG and or lean had to be made. The thing is I never did this as an experiment, so I only did it occasionally on exit that I remember. I'm thinking that the arc only remains the same during a wheelie at higher speeds where conical steering is doing a larger percentage of the steering.

Again, perhaps I'm wrong, but I also think at times under good throttle when some slip angles develop at the back, the rear tries to move outside, pivoting as it pushes forward around the front steering pivot, which steers the bike on a tighter" path not a wider path. There may be some of this effect at all times under throttle, but in general I'm thinking if you purposely do not change lean angle, and you add throttle, that will increase speed, which will require a correspondingly larger radius to be run, or lean angle will have to alter. This can be a very small effect on radius and speed at small acceleration amounts like 10%, especially since the friction of cornering must be overcome with some acceleration. So, at 10% there may be no discernible change in speed and therefore radius.

If a bike is cornering with standard throttle (accelerating 60/40) at a given lean angle, radius and velocity with both wheels on the ground, why doesn't the radius 'tighten up' if you lift the front wheel? Why doesn't the rear revert to the "single wheel" radius dependent only on lean angle without the front wheel to steer it wider?
It DOES revert to the single radius dependent only on lean angle with the front off the ground. It couldn't tighten up because the front was doing it's part to run a bit tighter radius than conical steering of the rear alone would do when both wheels were on the ground. The higher the speed, the more likely only conical steering of both tires finally comes into play. The lower the speed the more likely you would see a change in path because the front was steering more sharply.

I don't know that Keith really wanted his words to be taken quite the way you're interpreting them Racer (though maybe he did), I'm guessing he may have been referring more to the stability the rear wheel gives to the equation. Again I dunno though. Everyone is evolving in understanding of this stuff, a two wheel vehicle's ability to do what it does is NOT simple (at least for me) because SO MANY darn things are going on at once and contributing to the situation. What Keith may have thought a few years ago could be somewhat changed through new discovery today. Heck what I thought a few hours ago can change in a flash with new information and thinking. Perhaps again, I'm way off base, but I'm still standing fast until something comes to light that fits "better" for me. One thing that could change my mind is IF something happening at the contact patch could change it's amount of conical steering so radically it could explain the differences of radius at slow and high speed. I am doubting right now. That means that a wooden or steel wheeled bicycle would behave radically differently with respect to radius and speed. Again I'm doubting.

I disagree. I believe that the nature of the rubber/pavement contact patch at the rear allows increased cornering force to alter the path or radius of that single rear wheel.
This may be correct though the amount it alters it may not be radical enough as I just wrote. However perhaps this needs to be explored more (though I still, so far, say the front has more effect than you are giving it credit for). HOW do you think this works?
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So far, "I think" our difference is that you think the front wheel only trails along in the direction the rear points it due to the conical steering and forward momentum along a path of the rear. I think the front continues to play a significant part in the direction the bike takes, according to it's load, while receiving it's due influence from the rear.

The front wheel plays a significant role in keeping the bike up on two wheels according to its load. I don't think it plays a significant part in the direction the bike takes while leaned over in a stable state with standard throttle.

I may be mistaken, but it is my memory that at slower speeds when I lifted the front on a dirt bike in a turn the arc the rear dictated was not the same as the arc being traveled when the front was on the ground. Some adjustment in CG and or lean had to be made. The thing is I never did this as an experiment, so I only did it occasionally on exit that I remember. I'm thinking that the arc only remains the same during a wheelie at higher speeds where conical steering is doing a larger percentage of the steering.

Right. Higher speeds on a road bike is what I am talking about. But, I would humbly suggest that you try that experiment rather than rely on your occasional "memory". I think you may find that the arc remains the same.

Again, perhaps I'm wrong, but I also think at times under good throttle when some slip angles develop at the back, the rear tries to move outside, pivoting as it pushes forward around the front steering pivot, which steers the bike on a tighter" path not a wider path. There may be some of this effect at all times under throttle, but in general I'm thinking if you purposely do not change lean angle, and you add throttle, that will increase speed, which will require a correspondingly larger radius to be run, or lean angle will have to alter. This can be a very small effect on radius and speed at small acceleration amounts like 10%, especially since the friction of cornering must be overcome with some acceleration. So, at 10% there may be no discernible change in speed and therefore radius.

Well, now you are talking about sliding the rear. And, actually, I think the front wheel will continue to track while effectively turning into the slide. Hence, the bike may continue at the same angle (unless it leans further), and/or go straight... it will not tighten up. However, once the slide stops, the rear will then be pointing in a new direction... which will effectively be tighter. So, to answer Leftlaner's question: yes, there is a way to tighten the radius with the rear... by sliding it... and then catching it.

If a bike is cornering with standard throttle (accelerating 60/40) at a given lean angle, radius and velocity with both wheels on the ground, why doesn't the radius 'tighten up' if you lift the front wheel? Why doesn't the rear revert to the "single wheel" radius dependent only on lean angle without the front wheel to steer it wider?
It DOES revert to the single radius dependent only on lean angle with the front off the ground. It couldn't tighten up because the front was doing it's part to run a bit tighter radius than conical steering of the rear alone would do when both wheels were on the ground. The higher the speed, the more likely only conical steering of both tires finally comes into play. The lower the speed the more likely you would see a change in path because the front was steering more sharply.

Forgive me for not being more specific. I was thinking along the lines of the ongoing discussion... wider arc due to faster speed. I should have said, "why doesn't the arc change... tighter or wider?" And, to reiterate, I do not think it will do either.

I don't know that Keith really wanted his words to be taken quite the way you're interpreting them Racer (though maybe he did), I'm guessing he may have been referring more to the stability the rear wheel gives to the equation. Again I dunno though. Everyone is evolving in understanding of this stuff, a two wheel vehicle's ability to do what it does is NOT simple (at least for me) because SO MANY darn things are going on at once and contributing to the situation. What Keith may have thought a few years ago could be somewhat changed through new discovery today. Heck what I thought a few hours ago can change in a flash with new information and thinking. Perhaps again, I'm way off base, but I'm still standing fast until something comes to light that fits "better" for me. One thing that could change my mind is IF something happening at the contact patch could change it's amount of conical steering so radically it could explain the differences of radius at slow and high speed. I am doubting right now. That means that a wooden or steel wheeled bicycle would behave radically differently with respect to radius and speed. Again I'm doubting.

I do not think I interpreted Keith's words. I quoted them directly from his book. I think they are quite clear and speak for themselves. Hence, I am taking them at face value. However, I believe it would be interpreting them to "guess" what he "really meant".

...once the bike is fully leaned into a corner, the rear end "steers" the machine.

Once leaned over in a turn, the front end is no longer steering the bike: It helps stabilize it but does not steer it.

Those declarative statements seem fairly air-tight to me. Not much wiggle room from where I'm sitting. In any case, I wouldn't deign to "guess" what The Guru "really meant". And as far as whether or not Keith may have changed his mind since he wrote those words, I believe he can speak for himself and guess that he would do so.

I disagree. I believe that the nature of the rubber/pavement contact patch at the rear allows increased cornering force to alter the path or radius of that single rear wheel.
This may be correct though the amount it alters it may not be radical enough as I just wrote. However perhaps this needs to be explored more (though I still, so far, say the front has more effect than you are giving it credit for). HOW do you think this works?

I am still not entirely clear about the mechanics of why a motorcycle runs a wider arc at a higher speed.

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Those declarative statements seem fairly air-tight to me. Not much wiggle room from where I'm sitting. In any case, I wouldn't deign to "guess" what The Guru "really meant". And as far as whether or not Keith may have changed his mind since he wrote those words, I believe he can speak for himself and guess that he would do so.

Hei Keith, that was your cue!

..Enter the guru himself..

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Those declarative statements seem fairly air-tight to me. Not much wiggle room from where I'm sitting. In any case, I wouldn't deign to "guess" what The Guru "really meant". And as far as whether or not Keith may have changed his mind since he wrote those words, I believe he can speak for himself and guess that he would do so.

Hei Keith, that was your cue!

..Enter the guru himself..

Yeah, I've reached the end of my suppositions, surmises, and guestimations for now at least. Never know what tomorrow brings, but I would need additional information to go forward from here. I'll do some searching, but would be interested to know Keith's thoughts on the subject one way or another.

Racer, It's an interesting discussion for sure, thanks for getting me thinking about it.

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I'll do some searching, but would be interested to know Keith's thoughts on the subject one way or another.

Have you read Keith Code's Twist of the Wrist books? I got both my copies of Twist I and Twist II from my public library, but, you can order the books, DVD's or videos from this website here. If nothing else, it will give you a good idea of what Keith thinks and why (assuming he hasn't changed his mind ). At best, the \$20 might save your life. The books and schools have saved my life many times over.

In general, the Twist books have been my riding bible(s) for fifteen and twenty-five years respectively. I can't recommend them highly enough and do, in fact, recommend them to every rider I encounter over my parts counter at the dealership everyday. Just like affording a good helmet, I ask them, "how much is your life worth?"

I wish you good hunting in your search, sir!

racer

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You know, Carl, you said something that has stuck in my mind. And it has led me to think about what is happening at the front while a bike is accelerating. To paraphrase, I believe you said something about needing a certain degree of acceleration just to maintain a balanced state between all the forces (lean, radius, velocity). And, to extrapolate, further acceleration (or deceleration) might be forcing the front wheel to be attempting to countersteer the bike even though the rear gyro doesn't let the bike come up, but that would seemt to be turning the front wheel inward. Perhaps that force (centripetal behind the steering angle due to trailing geometry) is turning the wheel outward and creating a steering force at conditions (of accel/decel) near a stable velocity? I believe a higher rate of acceleration has more effect on the rake: lifting the front, extending the forks and wheelbase. Though that seems different than a stable velocity that is merely higher. But maybe it isn't.

Does it take an increased rate of (limited) acceleration to maintain a balanced (60/40) state when traveling at a higher velocity?

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