# Lean Angle == Turn Radius?

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Hi Carl,

Thank you very much for your input. As you can see, I've been chasing my tail a bit.

First, I didn't know that about cars. That is, that a car would follow a wider arc at a higher speed for the same turn angle of the front wheels... unless it was sliding a bit. For example, I know that a car will "understeer" and "oversteer", but, I thought of understeer as "pushing" the front to some degree, sort of a low level slide as it were where the front doesn't quite track. And oversteer meaning the rear is lighter and tends to come around. I don't know about the traveling further forward for the same amount of turning. I'll have to think about that.

In addition to that often a bike will steer a small bit into the turn once countersteering in is finished, more so at slower speeds, less at higher speeds, but if the bike is tracking and not sliding there is some amount of front wheel turn in is there not? While the rear may take the bulk of the turning on, I wouldn't be surprised to find the front contributing something. Sure you can wheelie mid turn, but that doesn't mean the front won't contribute a percentage if it's on the ground. It may be following along, but to contribute nothing it would have to be weightless. So perhaps the turned in front wheel also generates a small inward (turning) force as well, which also would be constant in rate, meaning speed would effect what radius arc it helped generate. If the front contributed nothing to cornering you couldn't lose the front in a turn.

Make sense?

Yes, this makes a lot of sense to me. And that is where I began in my own mind before reading the "Steer for the Rear" chapter in Twist of the Wrist II and trying to come at this from that angle so to speak. It just makes logical sense to my mind. I'm going to do a little more research and think about it some more.

racer

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Hi Carl,

Thank you very much for your input. As you can see, I've been chasing my tail a bit.

First, I didn't know that about cars. That is, that a car would follow a wider arc at a higher speed for the same turn angle of the front wheels... unless it was sliding a bit. For example, I know that a car will "understeer" and "oversteer", but, I thought of understeer as "pushing" the front to some degree, sort of a low level slide as it were where the front doesn't quite track. And oversteer meaning the rear is lighter and tends to come around. I don't know about the traveling further forward for the same amount of turning. I'll have to think about that.

Yes, this makes a lot of sense to me. And that is where I began in my own mind before reading the "Steer for the Rear" chapter in Twist of the Wrist II and trying to come at this from that angle so to speak. It just makes logical sense to my mind. I'm going to do a little more research and think about it some more.

racer

Well, I woke up this morning thinkng "no Carl, you're exactly, TOTALLY wrong, completely mistaken about a car." Very wrong and that is the key to the bike's behavior. A car will turn in the same amount at a higher speed because the rate of inward force will also increase with the speed so long as no sliding takes place. So I'm out in the garage pushing an upright bicycle at various speeds with the wheel turned in (you've got me crazy now too). I verify an increase in forward rate increases turn in rate and describes the same arc. Now the question may be will the rate of turn in increase be exactly proportional (linear) or not and if not why? but I think the car turn in IS LINEAR or nearly so and my original premise was indeed wrong with a car. Now to a bike.

However I'm thinking about riding two wheels, and going, wait a minute..... You lean 25-30 degrees with a motorcycle at walking speeds to go around a cone in a parking lot and you better have that front wheel turned in a whole dang bunch to help balance the forces!!!! If you do not turn it you don't get to turn sharp enough, period. So you balance forward momentum against lean/conical wheel turn in force/front wheel turned into the turn - turn in force/ against gravity. The TURNED IN FRONT WHEEL is doing a major portion of car like steering "making a tighter arc for a given lean angle."

The front wheel turning can also STILL even at 3mph adjust lean angle (countersteering) by the amount it turns in or doesn't turn in. At the same time forward speed can also adjust lean angle as well. To Illustrate that, if you are against the steering stop turned in on a low speed U-turn, and the bike is falling in, you can't stop that anymore by steering the front under yourself more. You can only arrest the fall in, and even stand it up more by gassing it, increasing forward speed. The forward force increases the rate of turning inward force at the front wheel (as it does with a car) which is turned into the turn like a car. The front wheel moves inward faster due to it's contact with the road, but the top of the bike and yourself (no contact with the road) try and stay at the previous rate of inward movement. The bottom of the bike moves in quicker, the top stays at the slower old inward movement, and the bike stands up more to a lesser lean angle. In effect steering the front back under yourself and stopping the bike from falling in without having to turn the bars tighter to do it.

So to go tightly around a cone at a given lean angle at slow speed the front wheel has to be turned into the turn a lot (even if you countersteered at first to get it to lean). The front will be heavily involved in influencing the rate of the front end inward movement due to the wheel pointing into the turn. However if you are doing that same lean angle at 80mph you WILL NOT be turning the front wheel into the turn the same amount to balance the forces (unless you had a sticky enough tire to lean at 98 degrees or something). It will be turned into the turn, but very little, contributing much, much, much less, to the inward movement of the front end relative to the rear than it did at slow speed. So I'm thinking countersteering and car steering are always operational to some degree on a bike regardless of speed. Countersteering to initiate leaning (unless you are going so slow bodyweight moved inside initiates the leaning, but countersteering can still do it).

So, I'm now outside on my mountain bike 'cuase its easy to grab and check quick. Yes I can go one mph and IF I'm very careful NOT to shift any bodyweight whatsoever, I can countersteer the bicycle at 1mph. So, I'm thinking I am correct, a combination of countersteering followed by car type steering is present to differing degrees depending upon speed, because different speeds require a different combination to balance forces and carve arcs. There is no slow speed at which countersteering cannot work. just slow enough speeds where it is not needed with very light vehicles like bicycles where a shift in bodyweight will lean the light machine.

SO! I'm thinking at lower speeds the front's car type steering does influence the turn in rate and "might, maybe, probably" explain a tighter arc at the same lean angle as speeds slow.

Lean angle creates an inward movement due to a conical wheel radius at the inside and outside of the contact patch. Meanwhile lean also balances Gravity and speed/momentum I gather due to an upward force through the angle of lean from pushing at the contact patch. If you turn the front wheel into the turn (different from a skater or skier) it creates both more upward force as well as greater inward movement over time, as the front end tries to follow the pointing of the front tire alla a Car. You can steer the front end out from under you by steering to the outside of the turn creating more lean, or steer the front wheel back under you more by steering inside the turn creating less lean, or you can keep the status quo by steering the amount needed to maintain the lean (one of the things trail is supposed to do when set sweetly).

So at low speed if you tried to U-turn very tight on two wheels by lean angle alone (with no steering into the turn) you would not manage a very tight turn at all. IF you could do that without falling down (didn't have to steer the front wheel at all) I'm guessing now that you might travel the same arc at 10mph or 100mph. At this point I'm guessing the front steering has more work to do than we may think, and varying it's work as speeds increase. In the end I think the front wheel does more than it's given credit for to produce radius, especially so as speeds come down and less so as they go up. I agree it can be following along for the ride in the sense that it is set correctly for the forces in play, but I think it still maintains a significant influence even under throttle.

Under throttle with the back wheel exerting it's influence you are taking advantage of that stabilizing swingarm joint which has no vertical hinge to help settle the motorcycle. However, I still think the vertically hinged front with it's turned in (to whatever degree) front wheel and considerable force still going on at the front contact patch, I think that front still has a lot of say about the final arc taken (to whatever degree it is still loaded).

If you turned the front wheel into the turn at 100mph the same amount as a 5mph U-turn, AND IF the tire could hold... The bike would try and do the same tight arc at 100mph as it does at 5mph and the bike would highside massively, as the top kept going and the front wheel moved into a tight arc.

Maybe I'll wake up in the morning and go "Carl, you are all wet" again, but maybe not this time, but then again maybe so, I dunno.

Unfortunately I lent my copy of Twist2 to a friend in NH and can't look up the chapter in question. My guess is it refers to a "relative" amount of increased steering influence by the rear wheel ( I vaugly remember parts of the chapter). As an ancient old MotoXer, using weight bias to the rear, so that stiff swingarm joint stabilizes the bike is like breathing air, but it seems to me unless the front is off the ground, there is always some sort of influence from the front. It may be minimized again depending upon the amount of weight shift, but force at the front contact patch is force at the front contact patch I would guess.

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So I'm out in the garage pushing an upright bicycle at various speeds with the wheel turned in (you've got me crazy now too).

LOL!

Welcome to my asylum! Ha ha ha ...

Under throttle with the back wheel exerting it's influence you are taking advantage of that stabilizing swingarm joint which has no vertical hinge to help settle the motorcycle. However, I still think the vertically hinged front with it's turned in (to whatever degree) front wheel and considerable force still going on at the front contact patch, I think that front still has a lot of say about the final arc taken (to whatever degree it is still loaded).

This is where my logic takes me as well.

Unfortunately I lent my copy of Twist2 to a friend in NH and can't look up the chapter in question. My guess is it refers to a "relative" amount of increased steering influence by the rear wheel ( I vaugly remember parts of the chapter). As an ancient old MotoXer, using weight bias to the rear, so that stiff swingarm joint stabilizes the bike is like breathing air, but it seems to me unless the front is off the ground, there is always some sort of influence from the front. It may be minimized again depending upon the amount of weight shift, but force at the front contact patch is force at the front contact patch I would guess.

The chapter you refer to does not jump to mind right now. I will search through "my" copy of Twist II tonight for what you are on about here.

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I believe the front end geometry has a lot to do with what goes on at the front and that even at lean angle, the front is "trailing" due to the geometry. And, as you said, if the geometry is set-up or designed well at a "sweet" angle, it will trail in a 'neutral' state that will require no input at the handlebar to maintain lean angle.

So... THAT is one thing. And it needs to be sorted for neutral and/or non-neutral (?) geometry.

The OTHER thing in my mind is what happens a} under acceleration... b} at an equivalent delta v or rate of acceleration at one speed range or another, ie. taking Turn 3 at 50 mph or taking Turn 3 at 65 mph.

And, finally, c} ... in the same way that you experimented at very low speeds and compared them to high speeds, consider speed ranges in between those extremes such as 2bigalow alluded to in his post... and THEN... apply this to different lean angle ranges, ie. minor leaning at low speed driving around town vs major leaning at low speed or bending off on the interstate at 100 mph vs dragging your footpeg through a 100mph sweeper.

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Yes, this makes a lot of sense to me. And that is where I began in my own mind before reading the "Steer for the Rear" chapter in Twist of the Wrist II

That is the chapter I was trying to remember, and can't remember exactly what was said. The swingarm not having a vertical hinge for a joint like the front, being the reason for increased stability under acceleration (when weight shifts to the rear) is my own idea, not found in the book. Weight biased towards the front has the chassis weight funneling through the vertical swiveling headstock joint and so is less stable in a turn than weight shifted to the rear wheel which cannot turn left and right as it is attached by a swingarm with a horizontal joint. I think that is a major difference in stability under throttle on/ throttle off conditions. The rear can't deflect left and right like the front so when the rear is a bigger influence in a turn, it is a stabilizing influence.

And, finally, c} ... in the same way that you experimented at very low speeds and compared them to high speeds, consider speed ranges in between those extremes such as 2bigalow alluded to in his post... and THEN... apply this to different lean angle ranges, ie. minor leaning at low speed driving around town vs major leaning at low speed or bending off on the interstate at 100 mph vs dragging your footpeg through a 100mph sweeper.

I do think the difference here is the amount the front wheel is turned into the turn and the turning influence that has. I haven’t heard turn in mentioned much but certainly it is there. More turn in at low speed going to little to no turn in at very high speed.

Once you tip in with countersteering, turn in then plays in to combine with lean angle, to balance out the force of momentum and gravity so the amount of turn in will differ at the same speed with a change in lean angle. The more you lean, the more the front tire can turn in at any speed, but it still will have to do so less and less as the speed increases. Major lean at very low speed can see a full lock turn in. Major lean at high speed can see a turn in you may barely notice happening or not notice happening Get the speed high enough I’d guess it is not turning in any more at all as the rear pivots around the headstock turning axis under throttle?

I kept thinking the front contact patch may play a part in the pivot point, but if that was the case, a bikes tendency to self correct during a rear wheel slide would not happen. The rear is pivoting around the steering head.

Wouldn’t it pivot around a line drawn from the steering head at the rake angle through the forks, to the road (ending ahead of the contact patch)?

The OTHER thing in my mind is what happens a} under acceleration... b} at an equivalent delta v or rate of acceleration at one speed range or another, ie. taking Turn 3 at 50 mph or taking Turn 3 at 65 mph.

Doesn't the rear tend to pivot a bit around the headstock/rake line where it intersects the road ahead of the contact patch, but does it through the contact patch (since that's the contact point)? In addition the leaned rear wheel has to exert some amount of turning inward movement due to the conical forces of smaller tire radius on the inside of the contact patch, and larger tire radius on the outside of the contact patch, no? If the bike is accelerating it would also accelerate the rate the front is turning in, effectively steering the front under the top of the bike some and so decreasing lean angle (if the rider keeps the bars turned a constant amount). If the rider let the bars do their own thing, perhaps the trail would steer the front out of the turn a bit as the rear tried to pivot.... maybe, keeping lean more constant. so many variables between possible rider input and what the bike might do on it's own.

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Yes, this makes a lot of sense to me. And that is where I began in my own mind before reading the "Steer for the Rear" chapter in Twist of the Wrist II

That is the chapter I was trying to remember, and can't remember exactly what was said.

OK, I just read "Steer for the Rear" again and caught an important detail that I was misinterpreting. What Keith actually said is that after the bike is fully leaned into the corner, the rear wheel determines the lean angle that the bike will hold. He did NOT say that it determines the radius that the bike will hold. Only the lean angle.

*light bulb goes off inside racer's head*

But... whatever the lean angle, won't the radius of the wheel(s) ultimately determine the radius the bike follows?

*light bulb goes off again*

Grrr...

2big said that shortening the wheelbase causes the bike to turn sharper in the hook turn, and intuitively, I agree, but, if the wheels are closer to each other, they are closer together on the arc of the turn and that would seem to imply that the front wheel must be turned out more, not in more.

Which brings me back to the front suspension and what happens when increasing speed at a given lean angle. We know that the bike raises up (both front and rear) under acceleration. But what about higher relative speed for a given turn?

I do think the difference here is the amount the front wheel is turned into the turn and the turning influence that has. I haven’t heard turn in mentioned much but certainly it is there. More turn in at low speed going to little to no turn in at very high speed.

I thought I had mentioned it earlier regarding my experiments with compressing the front fork, but, perhaps I wasn't clear. In any case, the front definitely turns into the corner and to what degree depends on the radius

Once you tip in with countersteering, turn in then plays in to combine with lean angle, to balance out the force of momentum and gravity so the amount of turn in will differ at the same speed with a change in lean angle. The more you lean, the more the front tire can turn in at any speed, but it still will have to do so less and less as the speed increases. Major lean at very low speed can see a full lock turn in. Major lean at high speed can see a turn in you may barely notice happening or not notice happening Get the speed high enough I’d guess it is not turning in any more at all as the rear pivots around the headstock turning axis under throttle?

I'd like to simplify this a bit and concentrate on figuring out what happens to the front with higher relative cornering loads and "standard throttle" (60/40). Does it, can it compress more? If so, and the longer wheel base forces the wheel to turn in or out more to track the same arc, what about the geometry itself, ie. the rake and trail? And what about the compression of the tire or the contact patch being rolled up more toward the center line of the tread?

I kept thinking the front contact patch may play a part in the pivot point, but if that was the case, a bikes tendency to self correct during a rear wheel slide would not happen. The rear is pivoting around the steering head.

I'm not certain exactly what you mean here re: the front contact patch playing a part. Can you be more specific?

Thanks,

racer

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OK, I just read "Steer for the Rear" again and caught an important detail that I was misinterpreting. What Keith actually said is that after the bike is fully leaned into the corner, the rear wheel determines the lean angle that the bike will hold. He did NOT say that it determines the radius that the bike will hold. Only the lean angle.

*light bulb goes off inside racer's head*

But... whatever the lean angle, won't the radius of the wheel(s) ultimately determine the radius the bike follows?

*light bulb goes off again*

Grrr...

2big said that shortening the wheelbase causes the bike to turn sharper in the hook turn, and intuitively, I agree, but, if the wheels are closer to each other, they are closer together on the arc of the turn and that would seem to imply that the front wheel must be turned out more, not in more.

Or is the degree the front wheel is turned in dependent on the geometry (rake) and bringing the front wheel closer to the rear forces the bike to turn tighter because the amount the front wheel is turned in remains consistent ... ?

Hmm...

Which brings me back to the front suspension and what happens when increasing speed at a given lean angle. We know that the bike raises up (both front and rear) under acceleration. But what about higher relative speed for a given turn?

I do think the difference here is the amount the front wheel is turned into the turn and the turning influence that has. I haven’t heard turn in mentioned much but certainly it is there. More turn in at low speed going to little to no turn in at very high speed.

I thought I had mentioned it earlier regarding my experiments with compressing the front fork, but, perhaps I wasn't clear. In any case, the front definitely turns into the corner and to what degree depends on the radius.

... that the bike is turning. ??? Will the radius the bike is turning determined by the rear and speed dictate the amount the front is turned in?

Or the radius depends on what degree the wheel is turned in which depends ultimately on the geometry... which changes with rake angle... which changes with compression... ?

Once you tip in with countersteering, turn in then plays in to combine with lean angle, to balance out the force of momentum and gravity so the amount of turn in will differ at the same speed with a change in lean angle. The more you lean, the more the front tire can turn in at any speed, but it still will have to do so less and less as the speed increases. Major lean at very low speed can see a full lock turn in. Major lean at high speed can see a turn in you may barely notice happening or not notice happening Get the speed high enough I’d guess it is not turning in any more at all as the rear pivots around the headstock turning axis under throttle?

I'd like to simplify this a bit and concentrate on figuring out what happens to the front with higher relative cornering loads and "standard throttle" (60/40). Does it, can it compress more? If so, and the longer wheel base forces the wheel to turn in or out more to track the same arc, what about the geometry itself, ie. the rake and trail? And what about the compression of the tire or the contact patch being rolled up more toward the center line of the tread?

I kept thinking the front contact patch may play a part in the pivot point, but if that was the case, a bikes tendency to self correct during a rear wheel slide would not happen. The rear is pivoting around the steering head.

I'm not certain exactly what you mean here re: the front contact patch playing a part. Can you be more specific?

Thanks,

racer

So maybe I've been going at it backwards, putting the cart before the horse, so to speak. But still... under an equal rate of acceleration, higher speed will demonstrate a wider arc. Or at the end of the day, have I been confusing increased speed with increased acceleration? That would explain a lot. Anyway...

... is the front wheel turn-in dependent on the rake/trail regardless of speed? And is the turn radius dependent on the amount the wheel is turned in or something else?

I'm losing it...lol. Going in circles. I need sleep.

G'night.

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We know that the bike raises up (both front and rear) under acceleration.

Here's a quote from Cobie Fair in the thread 'Feeling for rear traction': .."many think adding throttle will bring the bike up, but it doesn't."

Check it out for yourself: http://forums.superbikeschool.com/index.php?showtopic=899

By the way, I think that the RADIUS is a direct result of a correlation between LEAN ANGLE and SPEED. This only applies when the rider is in line with the bike. If he's hanging off, you need to make a correction based on what the lean angle would have been if he wasn't hanging off..

I also think that you can find the SPEED if you know the LEAN ANGLE and RADIUS, and that you can find the LEAN ANGLE (hanging off not considered) based on a given SPEED and RADIUS. In other words, I think these three elements are linked together and you can't calculate one of these elements without knowing the two other elements..

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We know that the bike raises up (both front and rear) under acceleration.

Here's a quote from Cobie Fair in the thread 'Feeling for rear traction': .."many think adding throttle will bring the bike up, but it doesn't."

Check it out for yourself: http://forums.superbikeschool.com/index.php?showtopic=899

You misunderstood me. The suspension rises or comes up under acceleration. I thought that was clear from the context of the discussion. Sorry.

By the way, I think that the RADIUS is a direct result of a correlation between LEAN ANGLE and SPEED.

Yup.

This only applies when the rider is in line with the bike. If he's hanging off, you need to make a correction based on what the lean angle would have been if he wasn't hanging off..

Nope.

It always applies. Hanging off merely alters the lean angle in relation to the other two elements.

And... the question I am asking is: why does speed alter radius when lean angle remains the same.

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You misunderstood me. The suspension rises or comes up under acceleration. I thought that was clear from the context of the discussion. Sorry.

No, I'M sorry!

Since English isn't my first language, I sometimes screw up when it comes to technical jargon and such.. o_O

EDIT: Note to myself: There is a difference between "rising up" and "standing up".. Got it!

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You misunderstood me. The suspension rises or comes up under acceleration. I thought that was clear from the context of the discussion. Sorry.

No, I'M sorry!

Since English isn't my first language, I sometimes screw up when it comes to technical jargon and such.. o_O

EDIT: Note to myself: There is a difference between "rising up" and "standing up".. Got it!

PS - Yup = yes, nope = no.

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PS - Yup = yes, nope = no.

That would be Norwegian, sir!

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Your English 'sounds' perfect to me. It has none of the unusual syntax or grammatical goofs normally noticed with non-native speakers. I would never have known if you didn't tell me.

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Your English 'sounds' perfect to me. It has none of the unusual syntax or grammatical goofs normally noticed with non-native speakers. I would never have known if you didn't tell me.

Thanks for the compliment!

OK, so now that I (hopefully) grabbed the essence of your question, here's my theory:

At a given lean angle, the radius will change as a function of speed BECAUSE the front wheel is turned more inwards the slower you go (and less inwards the faster you go).

Once the desired lean angle is acquired (by countersteering), the front wheel will turn INTO the corner. At slower speeds this is easy to see. At higher speed (say at 120mph or so) it's more difficult to see because the front wheel is turned in by no more than a fraction of an inch. The bike is going where the front wheel is pointing! Makes sense?

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OK, so now that I (hopefully) grabbed the essence of your question, here's my theory:

At a given lean angle, the radius will change as a function of speed BECAUSE the front wheel is turned more inwards the slower you go (and less inwards the faster you go).

Once the desired lean angle is acquired (by countersteering), the front wheel will turn INTO the corner. At slower speeds this is easy to see. At higher speed (say at 120mph or so) it's more difficult to see because the front wheel is turned in by no more than a fraction of an inch. The bike is going where the front wheel is pointing! Makes sense?

In essence, your "theory" sounds good.

But... can you say "why" this happens? That is 1) why the front wheel is turned in more or less at different speeds and 2) considering that each wheel will independently describe a circle when rolling at a specific lean angle, even if the front wheel is turned in more or less, and if the front is dictating the direction the bike goes, that means the rear wheel must roll a wider arc to follow it. Or perhaps that both wheels are rolling a wider arc independent of which wheel might be dominant, or, I could say that the front wheel is always in "trailing" mode (due to the geometry of the front end) and that the rear wheel rolling a wider arc is causing the front wheel to turn or "trail" wider to follow the wider line being traveled by the rear wheel.

See my dilemma?

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OK, so now that I (hopefully) grabbed the essence of your question, here's my theory:

At a given lean angle, the radius will change as a function of speed BECAUSE the front wheel is turned more inwards the slower you go (and less inwards the faster you go).

Once the desired lean angle is acquired (by countersteering), the front wheel will turn INTO the corner. At slower speeds this is easy to see. At higher speed (say at 120mph or so) it's more difficult to see because the front wheel is turned in by no more than a fraction of an inch. The bike is going where the front wheel is pointing! Makes sense?

Can you say "why" that happens?

I think it must be due to the fact that the sentripal force on the CG is increased at the same rate as the bike velocity at a given radius. (wow, that sounds fancy)

So if you're doing 60mph at a given radius, and you increase the speed to 70mph, you'll have to:

A: Lean the bike further in (to "counterweight" the increased sentripal force)

B: Maintain the lean angle but at a larger radius (and thus maintaining the sentripal force)

From this we can derive that the lean angle is directly correlated to sentripal force (and sentripal force is correlated to velocity and radius).

Here's another way to look at it. Let's say you're doing 60mph at a given radius, and you lock the steering at that particular "steering rate" (which means that you are forced to stay on that particular corner radius). If you increase the speed by 5mph, the bike will "fall out". If you decrease the speed by 5mph, the bike will "fall in".

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We are cross posting. I have significantly edited my original post while you were replying.

Sorry, I frequently re-edit my posts for clarity moments after making my original post.

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Can you say "why" that happens?

I think it must be due to the fact that the sentripal force on the CG is increased at the same rate as the bike velocity at a given radius. (wow, that sounds fancy)

So if you're doing 60mph at a given radius, and you increase the speed to 70mph, you'll have to:

A: Lean the bike further in (to "counterweight" the increased sentripal force)

B: Maintain the lean angle but at a larger radius (and thus maintaining the sentripal force)

From this we can derive that the lean angle is directly correlated to sentripal force (and sentripal force is correlated to velocity and radius).

Here's another way to look at it. Let's say you're doing 60mph at a given radius, and you lock the steering at that particular "steering rate" (which means that you are forced to stay on that particular corner radius). If you increase the speed by 5mph, the bike will "fall out". If you decrease the speed by 5mph, the bike will "fall in".

Yeah, I get that part.

I'll leave the mechanics of how the front wheel "knows" to turn in more for now and get right down to the heart of the matter.

How does a wheel rolling at a specific lean angle roll a wider arc at higher speed?

Does the contact patch change shape, ie. due to higher cornering force, does it "roll" or "squish" to a different shape that mimics less lean angle or being higher on the tire profile?

Or is the tire "pushing" and not rolling cleanly or precisely?

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We are cross posting. I have significantly edited my original post while you were replying.

Sorry, I frequently re-edit my posts for clarity moments after making my original post.

Well, I think that my "theory" points out the most important reason why you need to lean the bike further in as velocity is increased, at a given radius.

But I'm sure there are a lot of other factors that also contibute to the "front wheel turn-in angle theory" that I described further up.

The steering geometry (rake angle, wheelbase, front/rear height) might have an impact on corner radius..?

Tyre profiles might affect the trail of the bike..? Perhaps gyroscopic forces from the crankshaft plays a minor role too..?

But essentially, I think all these factors are of relatively marginal importance compared to where the front wheel is pointing..

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We are cross posting. I have significantly edited my original post while you were replying.

Sorry, I frequently re-edit my posts for clarity moments after making my original post.

Well, I think that my "theory" points out the most important reason why you need to lean the bike further in as velocity is increased, at a given radius.

But I'm sure there are a lot of other factors that also contibute to the "front wheel turn-in angle theory" that I described further up.

The steering geometry (rake angle, wheelbase, front/rear height) might have an impact on corner radius..?

Tyre profiles might affect the trail of the bike..? Perhaps gyroscopic forces from the crankshaft plays a minor role too..?

But essentially, I think all these factors are of relatively marginal importance compared to where the front wheel is pointing..

Saying this detail and that are not predominant or marginal is not answer. In effect, you are saying that you don't know why the front is turned in nor what it is doing there.

Yes, I know the front wheel points in the direction the bike is traveling, but, we have no evidence that it is steering the bike as opposed to following what the rear is doing. With all due respect, your position is an arbitrary choice to "believe" or have "faith" with or because of no evidence.

I'm aware of the forces involved and the observations of the matter at hand. What I'm after is the mechanics of how, as it were.

Forget the front wheel for a minute and think about the rear wheel. Or simply ONE wheel. Roll it through a corner at lean angle x and you get radius y. Now in crease velocity v and radius r increases. Yes, it is due to centrifugal force. But what exactly is happening to allow that to happen. A wheel rolls a circle due to the conical section of its contact patch, just like a styrofoam cup on its side. So, either the wheel is sliding sideways to accomplish this wider arc, or something else is going on.

Do you understand the concept of rake and trail?

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Saying this detail and that are not predominant or marginal is not answer. In effect, you are saying that you don't know why the front is turned in nor what it is doing there.

And I never claimed to be an expert or a rocket scientist either! I thought this forum was all about sharing ideas and thoughts on stuff. I can only offer some ideas from a practical point of view. If you want the mathematical forumula that explains this phenomenon, you'll have to wait for someone else to chime in..

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I'm aware of the forces involved and the observations of the matter at hand. What I'm after is the mechanics of how, as it were.

Forget the front wheel for a minute and think about the rear wheel. Or simply ONE wheel. Roll it through a corner at lean angle x and you get radius y. Now in crease velocity v and radius r increases. Yes, it is due to centrifugal force. But what exactly is happening to allow that to happen. A wheel rolls a circle due to the conical section of its contact patch, just like a styrofoam cup on its side. So, either the wheel is sliding sideways to accomplish this wider arc, or something else is going on.

Do you understand the concept of rake and trail?

Yes, I understand the concept of rake and trail. And with all due respect, I think you'd stand a better chance at getting the answers you seek in a physics forum rather than a bike forum. If we take one of the wheels out of the equation, then whatever answer you get to your question won't be related to cornering technique.

I thought your question was why the radius increases as speed increases, at a fixed lean angle? But now you're asking why the front wheel "knows" how much to turn in? Well I "believe" and "have faith in" that it has to do with the trail (correlated to rake angle).

There's a lot on the subject here: http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_geometry

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I have a question...if more speed causes the bike to run a wider radius at a given lean-angle, why is that when you get on the gas after flicking the bike over, the bike doesn't start drifting wide?

If I remember right, CSS teaches that chopping the throttle and/or grabbing the brakes will make the bike run wide and that more gas will make it hold its line. If you say that more speed increases turning radius, then why would the bike run WIDE when slowing DOWN and keep a CONSTANT RADIUS when accelerating? There have to be other forces at work, right?

Maybe I'm just confused, but no one's ever really explained the physics of it to my satisfaction...rake, trail, precession, torque-steer, gravity vs. centrifugal force...it seems like everyone has their theories and the "solution" is somewhere in the middle, or a combination of all of these.

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At a given lean angle, the radius will change as a function of speed BECAUSE the front wheel is turned more inwards the slower you go (and less inwards the faster you go).
This is what I've been trying to say, even if poorly with too many words.

I think I've stated the last half of this post more clearly than the first, but here's what I've come to.

You have many things going on here at once. You have "conical steering" AND "car type steering" (where the rear wheel follows the front) , ALONG with "countersteering" (where the front wheel gets steered either 'out from under' or gets steered 'more underneath' the top of the bike and rider), and all these are happening at once. They are all used to establish a state of equilibrium between momentum, centripetal force, and gravity in a leaned state when cornering.

Lest say you are leaning 35 degrees while going 10mph.

At that slow speed, the amount of turning from conical tire turning forces is not adequate enough to balance gravity at a 35 degree lean angle. If the front does not turn into the turn and tighten the radius further, "car steer" in a sharp enough radius, the bike and rider will fall on their arse as gravity sucks the bike down further and further. Conical tire steering turns too large a radius over what is needed to overcome gravity at that lean, at that 10mph speed. Some additional radius tightening method is needed to strike a balance of forces.

SO, the front tire turns into the turn and tightens the radius car steering style, enough to balance gravity against velocity at 35 degrees of lean at 10mph. This is a form of countersteering going on (steering the front back under the bike) at the SAME TIME that car steering is operating (rear wheel following where the front wheel is pointed) AND conical steering is doing what it can to contribute to the situation.

I'm defining countersteering here as either A. steering the front wheel out from under the top of the bike and rider, thus initiating lean angle or leaning the bike in more. OR B. steering the front wheel BACK under the bike reducing lean angle, or standing the bike up more. THIS IS DONE with the front wheel (which does not say you can not wheelie and use rear wheel conical steering to corner IF you have the speed needed to balance gravity at a given lean.

Since the rear wheel can only steer at lean through conical steering alone, I'd bet you dollars to donuts that in a wheelie lean angle DOES determine radius, regardless of speed. However we have two wheels and one can steer the front under or out from under as well as steer the whole machine in a direction.

At any rate back to the 35 degree lean turn at 10mph.

Let us say we are doing this turn as described now. The front wheel is turned in very sharply doing MOST of the actual turning in this low speed situation. It is turned sharply because the radius must be tight to balance gravity at that weak speed force. In order to make it easier to think about, let's say it is turned in to full lock, and the bike is doing a 25 foot radius at 10mph (arbitrary full lock, arbitrary radius numbers) and momentum and gravity is in balance and the bike will do this all day at this setup, 35 degree lean, full lock, 10mph speed.

NOW, let's instantly accelerate the bike to 100mph keeping the lean angle at 35 degrees and the front wheel turned to full lock. In our make believe setup we'll say the tires can NOT slide. the bike WOULD still try and describe a 25 foot radius following the front wheel, conical steering effect would still be operating at a given effect regardless of speed.

In this case the front wheel would still follow the 25 foot radius, but this would overwhelm gravity and bring COUNTERSTEERING (as I defined it) into play. The front would steer back under the top of the bike and rider standing it straight up and down (because the radius is the same, but the momentum was increased way past balancing gravity[lean angle]), it would further continue on the radius and after the bike got straight up and down it would lean the OTHER WAY... performing a massive high side. This is what would happen in the real world as well but sliding or losing traction would be involved (causing a front traction break lowside, if the front could hold the bike would highside).

THAT"S WHY a bike's radius increases with speed. To successfully turn a bike, you have to BALANCE the forces with lean and radius (or centripetal force).

We are using a changeable momentum (speed) and a changeable centripetal force (radius) in order to balance the constant force of gravity.

At low momentum (10mph), for a given gravity (35degrees of lean), we need a higher centripetal force (tighter radius) to balance that gravity since the speed is low.

At a high momentum (100mph), to balance that same force of gravity (35 degree lean angle) we can only use a very small centripital force (large radius) to add to momentum (speed) for balance of gravity (35 degrees of lean) to be struck, and a successful turn made.

If we fix the gravity (35 degrees lean angle) We can either vary the speed (momentum), or the radius (centripital force). If we increase one, we must reduce the other. We increase the speed the radius must be expanded or balance goes away. If we lower the speed the radius must be tightened to make up for it to balance the constant force of gravity.

How do we vary the radius on the bike for a given lean angle? The conical steering is constant. the only variable available to actually change radius is how much the front wheel is turned in or not (again car type steering principal)

It matters not if the front wheel is trailing along. If it is turned in, unless it is sliding, it will roll in the direction of it's turn AND it will also be doing some conical steering as well. If it turns that way the forks follow. the headstock will try and lag until a status quo is set (countersteering). IF the forces are balanced, and ONLY if the forces are balanced will the bike hold it's lean. Otherwise it will be falling in or standing up. One can countersteer to "fix" the unblanced forces with a change in centripital force (radius) OR a change in speed (momentum) Gravity is the same for a given lean angle.

The thing is ALL these things work in flux, in changing combinations. Gravity is the same until the lean angle changes, then it is different.

SO, gravity (lean angle) is a constant for a given lean angle but changes as soon as the lean angle alters (the same gravity force acting through an angle).

Conical steering force is constant and the same regardless of speed, for a given lean angle but will change radius with changing lean angles.

Momentum (speed) can change.

Countersteering is always in operation until a balance of forces is found that sets the lean angle to constant, then no countersteering is taking place as long as the lean and forces stay the same.

Car steering (front of bike goes in direction front wheel is pointed, rear follows along) is ALWAYS in play but is ALSO acting as countersteering UNLESS a balance of forces has been established that maintains lean angle.

In countersteering the front wheel steers car style (the axle moves in the direction the front wheel is pointed), BUT the headstock lags behind, trying to continue it's former direction. This changes lean angle (either lean, more lean, less lean, or total change in which side you lean to).

AS SOON as a balance of "momentum, centripetal force, gravity," IS established thus holding a constant unchanging lean angle, THEN BOTH conical steering AND car steering are in total play (so long as the front wheel contacts the ground). Countersteering isn't being used at all (unless the rider or other force makes a steering change upsetting things), and the axle AND the headstock BOTH DO follow the direction the front wheel is pointed in to the same degree. THIS is how a bike can describe a tight turn at slow speed with the front turned in sharply and a wide turn at high speed with the front barely turned in, or not turned in (totally relying on conical steering). The front can even turn out to balance a rear wheel slide car style IF the forces of momentum (speed), centripetal force (radius), and gravity (lean angle), remain in BALANCE.

Again at slow speeds (weak momentum force) you need a tighter radius (front wheel turned in a lot to increase centripetal force) to balance the gravity of a 35 degree lean angle.

At high speed (strong momentum force) the speed is doing a lot of the balancing of gravity (35 degree lean angle) so to keep forces balanced, you can only add a small amount of centripital force (open radius), making a large radius necessary for the same lean angle. Otherwise forces would be out of balance, countersteering would be active, and lean angles would be changing (as would all the things in play change until balance was established again or if not, then a crash occurred).

To summarize, The radius only changes with speed due to the front wheel changing how much it is turned into the turn. If conical steering was relied on solely (at two wheel bike with welded headstock) lean WOULD determine radius regardless of speed. Of course you couldn't balance a bike with welded headstock because you could not steer the front wheel of a falling bike back under yourself fast enough (conical lean turning alone only turns very slowly, not fast enough to maintain balance).

As to how trail comes into play with the set it and forget it thing I'll think on that some more.

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Wow!

I'm going to have to take some time to digest all of that.

PS - I must say I believe that I have finally met my match for verbosity!

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