Rake trail and handling, please critique my analysis.

This is the way I see things:

If we are dealing with the simple case of a perfectly straight
fork/steering axis that intersects the wheel axle, then the trail angle
(measured between the tires contact point with the ground, the wheel
axle and the point on the tire that intersects the steering axis) is
equal to the deviation of the head tube from vertical.

If we have a slack head tube, we have more trail, and leaning the bike
to one side will produce a torque about the steering axis that turns the
wheel and steers the bike in the direction of the lean, tending to right
the bicicle and contribute to stability. This is the dominant righting
mechanism for slow-speed handling. The magnitude of this torque is
proportional to the linear distance of the tires contact point to the
steering axis, and is therefore proportional to (1-Cos[trail angle]) as
well as the square of the radius of the wheel (from the cosine law). A
more vertical head tube means less trail angle means less torque means
less stability, more trail = more stable at slow speeds. (at higher
speeds the bike is felt to be more stable, is this the contribution of
velocity dependant gyroscope forces?)

So far so good, but how come cornering gets wider as the trail is
increased? The way I see it, if the trail angle is zero and you rotate
the stem through S degrees, there is a 1:1 correspondence at the tire
contact path and the front wheel travels at an angle of S degrees to the
axis of the frame. If the trail angle is 90 degrees, then turning the
fork will make no difference to the direction of travel. Based on this,
I conclude (guess) that the angle that the front wheel travels at is
equal to S*Cos[trail angle], or a similar trig expression where the
steering angle is proportional to S and decreases with increasing trail.
The turning radius could then be derived based on the angle of front
wheel travel and the wheelbase length.

My problem is how to prove this last point more rigorously (correctly),
and how to extend my conclusions to the case where there is a non-zero
offset of the wheel axle from the steering axis, such as with any fork
with rake. The geometry is stumping me, and none of the websites I have
browsed have explained this. I would like to be able to quantitatively
compare all aspects of bicycle handling between differing frame
geometries. Thanks for any tips or pointers!

Adam

Jacobe Hazzard wrote:
This is the way I see things:


Etc., etc...

Adam/Jacobe: perhaps you should consult Prof. Bill Patterson:
http://www.calpoly.edu/~wpatters/
He used to teach a class about bicycle steering geometry. He's put
numbers to the things you've thinking about.

He's retired now, so the site hasn't been updated in a while. It's
still a good source.

Jeff

I've got some numbers now. These seem to work..

According to my calculations, if the steering axis intersects the wheel
axle and makes an angle of T with vertical, then a rotation of the
steerer of R degrees about its axis will result in the tires contact
path rotating by

ArcTan[ ( Cos[T]*Sin[R] )/( 1-(Cos^2[T])*(1-Cos[R]) ) ] degrees

For a vertical head tube, T=0 and the expression reduces to R, for a
horizontal head tube, T=90 degrees and the expression reduces to 0. For
values of T in between 0 and 90 degrees, the graph of Tire Angle vs.
Stem Angle is an almost straight line, with a slope of between 1 and 0
for the extreme cases. I plan on redoing the analysis for offset forks,
I expect the answer to be similar in form.

If anyone is interested I could scan my diagrams and calculations, but I
would need to clean them up first.

wrote:
Jacobe Hazzard wrote:

This is the way I see things:



Etc., etc...

Adam/Jacobe: perhaps you should consult Prof. Bill Patterson:
http://www.calpoly.edu/~wpatters/
He used to teach a class about bicycle steering geometry. He's put
numbers to the things you've thinking about.

He's retired now, so the site hasn't been updated in a while. It's
still a good source.

Jeff

Very interesting stuff, thanks.

I will definitely go over it in more detail, once i've completed my own
analysis, which I am making progress with now, and compare.

Jacobe Hazzard wrote:
I've got some numbers now. These seem to work..

According to my calculations, if the steering axis intersects the wheel
axle and makes an angle of T with vertical, then a rotation of the
steerer of R degrees about its axis will result in the tires contact
path rotating by

ArcTan[ ( Cos[T]*Sin[R] )/( 1-(Cos^2[T])*(1-Cos[R]) ) ] degrees
snip

Would you mind contacting me off-list?

I have an Excel file I would like to send you that you may find of
interest. I also have some thoughts along these lines; perhaps we could
compare some notes and theories.

Regards,

Tom