Lean Angle as a Function of Speed

wle said:

this seems like a simple question.

assuming everything else equal,
[same radius, same rider, cyclist does not change
position relative to the bike] how does
lean angle vary as a function of speed?

The bike leans until the sum of the gravitational force and the
cetnripital force point in like with the frame, that is, if you had a
plumb bob with you while riding around the corner it would remain
palallel to the seat tube, rather than pointing straight down to the
ground.

The angle the bike leans is given by atan(v^2/g*r) where v is the
speed, g is a constant, and r is the radius of the turn. So, for a
constant turn the angle is proportional to atan(speed^2). This
function is almost linear from about 10 to 45 degrees. Which means
that once you are leaned at least 10 degrees every aditional x mph
will lean you over another y degrees until the bike falls over at 45
degrees.

wle asked:

Quoted message said:

this seems like a simple question.

assuming everything else equal,
[same radius, same rider, cyclist does not change
position relative to the bike] how does
lean angle vary as a function of speed?

Asymptotically.

Sheldon "Concise" Brown
+-----------------------------------------------------------------------+
|Dennis: "Listen, strange women, lying in ponds, distributing swords, |
| is no basis for a system of government. Supreme executive power |
| derives from a mandate from the masses, not from some farcical |
| aquatic ceremony" |
|King Arthur: "Shut Up!" |
| |
|Dennis: "I mean you can't expect to wield supreme executive power |
| just because some watery [censored] threw a sword at you" |
|King Arthur: "Shut Up!" |
|Dennis: "I mean if I went around saying I was an emperor, just because |
| some moistened bint lobbed a scimitar at me, they'd put me away" |
|King Arthur: "Shut Up! Will you Shut Up!" |
|Dennis: Oh, now we see the violence inherent in the syststem. COME |
| SEE THE VIOLENCE INHERENT IN THE SYSTEM. HELP! HELP! I'M BEING |
| REPRESSED" |
|King Arthur: "Bloody Peasant!" |
|Dennis: "Oh what a give away. What a give away huh? Did you see that? |
| You saw 'im repressin' me didn't you ? That's what I'm on about!" |
--Monty Python & the Holy Grail |
+-----------------------------------------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

Sheldon Brown said:

wle asked:

Quoted message said:

this seems like a simple question.

assuming everything else equal,
[same radius, same rider, cyclist does not change
position relative to the bike] how does
lean angle vary as a function of speed?

Asymptotically.

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

Lean angle is the tangent of weight of rider and bicycle divided by
the radial acceleration which is (r*omega^2) where r is the radius of
curvature and omega is the angular velocity in radians.

Boiled down, that becomes the lean angle for a given curve path is
proportional to the square of velocity.

This is treated in:

http://www.sheldonbrown.com/brandt/descending.html

# Braking at maximum lean

# For braking in a curve, take the example of a rider cornering with
# good traction, leaning at 45 degrees, the equivalent of 1G
# centrifugal acceleration. Braking with 1/10g increases the traction
# demand by one half percent. The sum of cornering and braking vectors
# is the square root of the sum of their squares,

# SQRT(1^2+0.1^2)=1.005 or an increase of 0.005.

# In other words, there is room to brake substantially during maximum
# cornering. Because the lean angle changes as the square of the
# speed, braking can rapidly reduce the angle and allow even more
# braking. For this reason skilled racers nearly always apply both
# brakes into the apex of turns.

As in picture:

http://www.sheldonbrown.com/brandt/

Jobst Brandt
[email hidden]

Quoted message said:

Lean angle is the tangent of weight of rider and bicycle divided by
the radial acceleration which is (r*omega^2) where r is the radius of
curvature and omega is the angular velocity in radians.

Boiled down, that becomes the lean angle for a given curve path is
proportional to the square of velocity.

This is an incorrect statement, and the sentence in the FAQ is
incorrect too. The cetripital force is proportional to the square of
the speed, of course, but the angle is proportional to the inverse
tangent of the square of the speed. Perhaps you are thinking of
flying, where the turn radius at a given bank angle is proportional to
the square of the speed?

In any case, you should update the faq.

Jim Smith said:

Quoted message said:

Lean angle is the [arc]tangent of weight of rider and bicycle
divided by the radial acceleration which is (r*omega^2) where r is
the radius of curvature and omega is the angular velocity in
radians.

Quoted message said:

Quoted message said:

Boiled down, that becomes the lean angle for a given curve path is
proportional to the square of velocity.

Quoted message said:

This is an incorrect statement, and the sentence in the FAQ is
incorrect too. The cetripital force is proportional to the square of
the speed, of course, but the angle is proportional to the inverse
tangent of the square of the speed.

Of course the tnagent function returns a ratio from an angle while the
arctangent returns an angle from a ratio. I'll get on that as soon as
I get that properly reworded.

Quoted message said:

Perhaps you are thinking of flying, where the turn radius at a given
bank angle is proportional to the square of the speed?

I think in flying the dynamics are identical so this applies to that
as well.

Quoted message said:

In any case, you should update the faq.

.... as well as the current posting. However the relationships are
correct, only the tan or 1/tan are reversed. Thanks,

Jobst Brandt
[email hidden]

wle asked:

Quoted message said:

Quoted message said:

Quoted message said:

this seems like a simple question.

assuming everything else equal,
[same radius, same rider, cyclist does not change
position relative to the bike] how does
lean angle vary as a function of speed?

I replied:

Quoted message said:

Quoted message said:

Asymptotically.

Zog wrote undeniably:

Quoted message said:

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

Sheldon "Whoooooooosh!" Brown
+------------------------------------------+
| There was a young lady named Bright |
| Whose speed was far faster than light; |
| She set out one day |
| In a relative way |
| And returned on the previous night. |
| -- A.H.R. Buller |
+------------------------------------------+
Harris Cyclery, West Newton, Massachusetts
Phone 617-244-9772 FAX 617-244-1041
http://harriscyclery.com
Hard-to-find parts shipped Worldwide
http://captainbike.com http://sheldonbrown.com

Sheldon Brown said:

wle asked:

Quoted message said:

Quoted message said:

> this seems like a simple question.
>
> assuming everything else equal,
> [same radius, same rider, cyclist does not change
> position relative to the bike] how does
> lean angle vary as a function of speed?

I replied:

Quoted message said:

Quoted message said:

Asymptotically.

Zog wrote undeniably:

Quoted message said:

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

But I have to wonder - what color would your bike appear to be at the
speed of light?

Mark Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame

Mark Hickey said:

Sheldon Brown said:

wle asked:

Quoted message said:

>>this seems like a simple question.
>>
>>assuming everything else equal,
>>[same radius, same rider, cyclist does not change
>> position relative to the bike] how does
>>lean angle vary as a function of speed?
>

I replied:

Quoted message said:

>Asymptotically.

Zog wrote undeniably:

Quoted message said:

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

But I have to wonder - what color would your bike appear to be at the
speed of light?

Mark Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame

Blue if it's coming towards you and red if it's moving away.

Marty

"Mark Hickey" <[email hidden]> wrote in message
news:[email hidden]...

Quoted message said:

Sheldon Brown said:

wle asked:

Quoted message said:

>
>> this seems like a simple question.
>>
>> assuming everything else equal,
>> [same radius, same rider, cyclist does not change
>> position relative to the bike] how does
>> lean angle vary as a function of speed?
>

I replied:

Quoted message said:

>
> Asymptotically.

Zog wrote undeniably:

Quoted message said:

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

But I have to wonder - what color would your bike appear to be at the
speed of light?

Mark Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame

Same as it always does to him - he's on it.

Cal

CEarly said:

"Mark Hickey" <[email hidden]> wrote

Quoted message said:

Sheldon Brown <[email hidden]> wrote:

Quoted message said:

Quoted message said:

Quoted message said:

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

Quoted message said:

Quoted message said:

But I have to wonder - what color would your bike appear to be at the
speed of light?

Quoted message said:

Same as it always does to him - he's on it.

Don't you ever read Fabrizio? What's important is the impression you
make on others. The red/blue shift visible to those you're passing or
being passed by isn't really significant, but you have to keep up your
image with the huddled masses along the side of the road, y'know.

Mark Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame

Mark Hickey said:

CEarly said:

"Mark Hickey" <[email hidden]> wrote

Quoted message said:

Quoted message said:

Quoted message said:

But I have to wonder - what color would your bike appear to be
at the speed of light?

Quoted message said:

Quoted message said:

Same as it always does to him - he's on it.

Quoted message said:

Don't you ever read Fabrizio? What's important is the impression
you make on others. The red/blue shift visible to those you're
passing or being passed by isn't really significant, but you have
to keep up your image with the huddled masses along the side of
the road, y'know.

If they're along the side of the road they'll see your bike as being
its usual color, at least at the moment when they're looking
perpendicular to your path.

--
Ray Heindl
(remove the Xs to reply)

In article <[email hidden]>,
[email hidden] says...

Quoted message said:

Sheldon Brown said:

wle asked:

Quoted message said:

>
>> this seems like a simple question.
>>
>> assuming everything else equal,
>> [same radius, same rider, cyclist does not change
>> position relative to the bike] how does
>> lean angle vary as a function of speed?
>

I replied:

Quoted message said:

>
> Asymptotically.

Zog wrote undeniably:

Quoted message said:

So if you're cornering at the speed of light, you're almost, but not
quite, horizontal?

That's correct. Try it yourself if you don't believe me! (You'll need
unusually good tires for this)

But I have to wonder - what color would your bike appear to be at the
speed of light?

I'm not sure, but the wheels would be out of round, and you'd be plagued
with a wicked toe overlap.

;-)

Rick

Java Man said:

[email hidden] says...

Quoted message said:

But I have to wonder - what color would your bike appear to be at the
speed of light?

I'm not sure, but the wheels would be out of round, and you'd be plagued
with a wicked toe overlap.

That's okay; the amount you'd have to turn the handlebars in order to
corner would be minuscule at that speed.

(Actually, to the outside observer, the bicycle would appear to be
proportionately shorter, so the relative overlap wouldn't change).

Furthermore, even if you had toe overlap it wouldn't matter, since your
cadence would approach zero.

🙂

--
Benjamin Lewis

I regret to say that we of the FBI are powerless to act in cases of
oral-genital intimacy, unless it has in some way obstructed interstate
commerce. -- J. Edgar Hoover

Benjamin Lewis said:

Java Man said:

[email hidden] says...

Quoted message said:

But I have to wonder - what color would your bike appear to be at the
speed of light?

I'm not sure, but the wheels would be out of round, and you'd be plagued
with a wicked toe overlap.

That's okay; the amount you'd have to turn the handlebars in order to
corner would be minuscule at that speed.

(Actually, to the outside observer, the bicycle would appear to be
proportionately shorter, so the relative overlap wouldn't change).

To clarify, I meant shorter wheelbase, not height.

--
Benjamin Lewis

I regret to say that we of the FBI are powerless to act in cases of
oral-genital intimacy, unless it has in some way obstructed interstate
commerce. -- J. Edgar Hoover

Benjamin Lewis said:

Benjamin Lewis said:

That's okay; the amount you'd have to turn the handlebars in order to
corner would be minuscule at that speed.

(Actually, to the outside observer, the bicycle would appear to be
proportionately shorter, so the relative overlap wouldn't change).

To clarify, I meant shorter wheelbase, not height.

Good thing you clarified that, I could have made a serious mistake on
tomorrow's light speed time trial attempt.

Mark "that's "light speed", not "Litespeed" Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame

Mark Hickey axed:

Quoted message said:

what color would your bike appear to be at the speed of light?

It would appear its usual colour to the rider, but *at* the speed of
light, bystanders would see it neither coming nor going, witnessing only
a bright blue flash of Cerenkov [or Cherenkov] radiation as it whizzed past.

--
"Science is a wonderful thing if one does not have to
earn one's living at it." -- Albert Einstein

LioNiNoiL_a t_Y a h 0 0_d 0 t_c 0 m said:

Mark Hickey axed:

Quoted message said:

what color would your bike appear to be at the speed of light?

It would appear its usual colour to the rider, but *at* the speed of
light, bystanders would see it neither coming nor going, witnessing
only a bright blue flash of Cerenkov [or Cherenkov] radiation as it
whizzed past.

it's a long time ago but...... Chereknov radiation is emitted when a
particle is travelling *faster* then the speed of light in the medium and is
a slowing (energy loss) mechanism.

So if the cycle is *at* the speed of light...

pk

pk said:

LioNiNoiL_a t_Y a h 0 0_d 0 t_c 0 m said:

Mark Hickey axed:

Quoted message said:

what color would your bike appear to be at the speed of light?

It would appear its usual colour to the rider, but *at* the speed of
light, bystanders would see it neither coming nor going, witnessing
only a bright blue flash of Cerenkov [or Cherenkov] radiation as it
whizzed past.

it's a long time ago but...... Chereknov radiation is emitted when a
particle is travelling *faster* then the speed of light in the medium and is
a slowing (energy loss) mechanism.

So if the cycle is *at* the speed of light...

.... then the spokes going over the top every revolution of the wheel
would be going nearly twice the speed of light. Whoo hoo...

Mark Hickey
Habanero Cycles
http://www.habcycles.com
Home of the $695 ti frame