C[omp]ute

From: andrew cooke <andrew@...>

Date: Sat, 10 Jan 2015 11:10:37 -0300

[This is an attempt to clarify and extend some ideas that started with the
post http://www.reddit.com/r/MTB/comments/2rqn6r/physicists_leaning_bikes/
and which I deleted after it descended into name-calling.]


In Mastering Mountain Bike Skills (MMBS - a respected book on how to ride MTB)
riders are encouraged to "over-lean" the bike (ie lean it more than the body):

  This position gives you extra traction and stability in flat and off-camber
  corners, or any time your lean exceeds the angle of the ground. Weight your
  outside pedal. That levers your tires into the ground, so slipping doesn't
  happen. In almost every turn, this is the best, safest, and fastest way to
  go. Practice cornering this way. It'll never steer you wrong. This is the
  best cornering position for any turn whose traction might let you down.


Now one reason I think MMBS is a good book is because I can often (with some
work) understand the text in physical terms (ie using fairly simple,
first-year university dynamics).  And that has helped my riding, because it
sheds more light on exactly what was meant.  For example, doing a manual (a
"wheelie without pedalling") is intimately connected with angular momentum
about the axis perpendicular to the bike.

Interestingly, I've found that advice in the book that is initially
counter-intuitive makes more sense once you consider "dynamics".  By that, I
mean that it's not the physics of the equilibrium solution that is important,
but the addition of accelerations, typically from body movement, over short
time periods.  Indeed, much of "good MTB riding" is about moving your body
relative to the bike, and timing that movement to get a particular effect when
most important.  These accelerations are limited - you cannot continue to
"push down" on the bike once your arms and legs are fully extended, for
example.


Given the above, I started to wonder about the physics involved in cornering:
why is it better to lean the bike more?

The equilibrium solution has the centre of mass of the bike plus rider offset
relative to the contact position between bike and trail so that the vertical
force matches gravity and the horizontal force accelerates the rider towards
the centre of the curve.  In other words, the angle of the lean for the centre
of mass of the entire system is fixed by the mass of rider+bike, the speed,
and the radius of curvature of the turn.

In other words, from that viewpoint, leaning the bike relative to the body
doesn't seem to do much except change the angle with which the tyre meets the
ground (see later).


I'm not alone in noticing this.  Here is a nice image and related quote
http://advrider.com/forums/showpost.php?p=8190069&postcount=20 which explains
things nicely - the roll angle for the combined centre of mass does not
change.  I'll copy the text here:

  Cocco in "Motorcycle Design and Technology" basically says that staying
  inline with the bike vs. hanging off vs. pushing the bike down make no
  difference in the roll angle of the combined bike/rider CG with respect to
  the ground ([phi] in the illustration below) and assuming you aren't
  dragging parts and assuming the tire offers the same traction, no technique
  has an explicit speed advantage. So it's really just a rider preference of
  whether they want to motorcycle itself to be less leaned or more leaned
  (Δ[phi]) for any given speed which affects what parts drag and what profile
  of the tire is presented to the pavement. He views it as a rider style
  preference thing, not a a physics thing.


At this point I should acknowledge that the angle between tyre and ground is
important in its own right.  In fact, MMBS describes this on page 31 where it
says:

  A square tire requires more commitment in the corners: you really have to
  lean it onto the side knobs; otherwise the tire feels sketchy.

Which is consistent with the above, but makes the argument depend on the tyre
profile, while the quote I started with seems to be more general.

Perhaps I am misreading things.  Perhaps the instructions should be more like
"adjust the lean so that your tyre bites well; this may require more or less
lean than an 'inline' riding position"?

Anyway, for the sake of curiosity I will assume that there is some further
reason, which makes the "over-lean" a more general rule, and try to find out
what it might be.


Let's take a moment to summarise where we are and where we are going.  The
equilibrium solution for a cornering bicycle suggests that relative lean
(between rider and bike) is unimportant (except for tyre profile).  So if
there is an explanation we need to look at "dynamics" for an answer.  What
(relatively sudden) changes can happen, and how are they related to lean?

Below I discuss a couple of ideas.


1 - Rotation on slipping

This is my best attempt at understanding
http://mspaintbike.blogspot.com/2013/10/why-you-cannot-corner-mountain-bike.html
which I find rather vague.  The argument there seems to be that if the wheels
start to slide then the position of the rider alters how the bike rotates (it
moves upright rather than falling towards the inside of the curve).

Consider the following diagram (bike and rider viewed head-on) showing the
centre of mass (rider and bike) M, the contact between tyre and ground T, the
vertical force Y that counteracts gravity, and the horizontal force X that
accelerates the bike and rider towards the centre of the curve.

          Y ^  M
            | /
            |/
            T----->
                  X

(This is the same as http://www.whizmoandgizmo.com/Misc/MCLeaning.jpg but I'm
considering only the joint centre of mass and being more careful about
directions to avoid any discussion about whether "centrifugal" is "real" or
not.)

The vector sum of X and Y must pass from T to M (or the entire system would
rotate).  This is the argument I made above about the angle being fixed by
mass, speed, and radius of curvature.

Now, if X suddenly disappears (or decreases), what happens?  The resulting
force becomes more vertical (Y dominates X), implying a rotation about the
centre of mass that causes bike and rider to fall into the turn.  This is the
opposite of what is argued in the link above, and is independent of the
relative position of bike and rider.


2 - Space to react

I've already noted that good MTB riding involves accelerating the body
relative to the bike at critical moments.  This is like a spaceship firing
thrusters for control, but instead of thrusters we have to use our bodies.
For example, if you want to push the bike down for more grip, you push your
body upwards (and the "equal and opposite" force on the bike is downwards).

If we look at cornering from this point of view, we can ask "how should we
position the body relative to the bike so that it is ready to move in the
direction we need?".  In other words: "where should we point our thrusters?"

This is important because, as I already noted, our movement is restricted.  We
can only move our body a limited amount, so these temporary adjustments are of
a limited duration.  For example, this explains why the attack position is low
- so that we have space to move the body upwards - but not too low, because we
also need space to "suck up" bumps by moving the body down (relative to the
bike).

OK, so how would we want to move our body?  And does that explain how we
should lean the bike?

Let's assume that the tyre slips.  Momentarily we are faced with:

          Y ^  M
            | /
            |/
            T
             
which is going to be ugly.  In terms of thrusters, what we would like is
something like this:

            |
	    | Thruster
            V 

          Y ^  M
            | /
            |/
            T

In other words - a downwards vertical force, over the contact patch.  Which we
would achieve by firing the thruster vertically upwards.

That would push the tyre down, achieve grip, hopefully, and save our sorry
ass.

Now, we don't have thrusters, we have our body.  So we want our body to be
moving upwards, over the contact patch.  Which means that we want our body "on
top" of the cycle.  Which means that we need to "over-lean" the bike...


So far I have tried to avoid any imaginary dialogue with other riders, to
avoid being called "defensive".  But at this point I cannot ignore a chorus of
"but that's what we said" and "everyone said you should push down on the
outside pedal".

So what is new here?  The important difference is that this is a sudden,
momentary correction.  It's not the "steady state".  Putting your static
weight on the pedal is not enough (is, in fact, irrelevant - no more important
to the physics than the force between bones in your knee, for example).  What
is important here is being able to give an extra push, firing your body
upwards, when grip is lost.

Now in practice there are likely multiple slips in a corner, and the
distinction between static weight and extra push is likely blurred.  Of
course.  That's why this problem was hard to understand...

Anyway, my apologies to anyone who thinks I am making a big fuss over a tiny
distinction.  I hope that mental cost is balanced by someone, somewhere, with
a background like mine, understanding this a little more.


In summary, then, I suggest that the reason it is recommended to over-lean an
MTB bike on cornering is for at least two reasons:

First, many tyres have a profile with distinct outer knobs.  Leaning the bike
helps these "dig in" and so grip better.

Second, placing the body above the bike lets you launch yourself upwards,
pushing the bike downwards, when you lose grip.  This is not a static thing -
it's to give you room to actually, physically, and perhaps violently, move
your body upwards in response to slip.

Andrew (aka science boy).