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A Theory of Holonomic Scleronomous Gyrators,
or
The Virtues of Riding Fixed
by
James Bernard Lee
Portland, Oregon
Prepared for
1st International Fixed Gear Symposium
Traverse City, Michigan
August 2005

If it’s broken, fix it.

My first local fixed gear street ride on FGG 2219 was a
familiar route, in a 66 inch gear. Climb like that in 66
inches? Must be some mistake! So I re-rode the route in my
nice 27 speed Campy equipped road machine...yes, 44 inch gear
for the same climbing ability as the fixer!
Now Dick DiGennaro, Mr. Solo Velo, who had built the frame
and persuaded me to try riding fixed on street and road, is a
very nice guy. Besides being a master frame builder he works
every day with crazy physicists at Lawrence Berkeley
Laboratory, which is why we get along well. To return some
part of the great favor he had shown me I decided to work out
the peculiar and perplexing physics of fixing. It took two
years of meditating on the subtleties of classical mechanics
to ascertain a set of simple truths:
All cyclists are biomechanical gyrators. Fixed gear bicycles
operate under a scleronomous holonomic constraint. Free gear
bicycles operate under a rheonomous holonomic constraint.
Gibberish, yes, but here are two real examples:
Watch Lance climb out of the saddle: he jackhammers the
bottom of the stroke just as every tyro does; this is because
human anatomy of itself cannot rotate the cranks at constant
angular velocity. The presence of a free wheel in the drive
train allows this by decoupling the forward rotation of the
rear wheel from the forward rotation of the cranks. The
effect is not so apparent when the rider is seated, but
riding a freebie, seated or standing, a roadie’s--or anyone
else’s--legs cannot catch up to the rear wheel until the
leading crank is about at two o’clock.
Then watch world class trackies, say the Uruguayan Olympic
team at Alpenrose Velodrome in 2004: their smooth loping
cadence is perfectly even, seated or standing, because their
cranks always are synchronized with the rear wheel; there is
no dead spot at the top and bottom of their stroke.
This is a very big deal!
It means that the fix rider can apply force to the crank
wherever she or he chooses, and the human system of optimal
adaptive control soon will figure out how best to do so. The
fix rider can get into the power stroke at 12 o’clock, or a
little before, and so utilize the entire downward stroke for
propulsion. Way strong!
The free rider can apply downward force beginning only at two
o’clock, and so has wasted fully one-third of the downward
stroke. This is not good!
Looked at the other way, the fixer has a 50% longer power
stroke than the freebie, which equates to 50% more torque
and therefore 50% more power at a given cadence.
I’m all steamed up.
There is another effect working here.
A cyclist’s legs operate like the two opposed pistons of a
steam locomotive: both generate greater forces at lower
speeds, and maximum force at “stall,” or zero revs.
As a fixed gear rider begins to climb he or she necessarily
slows, and so can apply more force to the cranks. But at
some speed the knee joints protest, and then the rider rises
out of the saddle; this is less efficient than being seated,
but more forceful. Given the 50% torque advantage of riding
fixed, one can climb grades with surprising ease and
quickness this way. Of course a lower gear allows one to
climb a steeper grade either seated or standing; this is a
matter of terrain and ability for every rider.
But a free gear rider has a necessary decision upon beginning
to climb that we fixers do not: way down in torque, she or he
wants to shift to a lower gear, usually several times; the
tendency is to chase lower and lower gears, because the lower
gearing is more than offset by the decreasing torques at the
higher cadence after each shift. Therefore she or he applies
progressively less force to the road and continues to slow.
This process stops only when the rider’s cadence has slowed
enough to regain useful torque in some lower gear; depending
upon the grade, that gear can be far removed from the initial
gear.
(Trained road-racers know about this trap: when beginning an
ascent they shift up a cog or two and stand, staying in the
big ring; when reaching terminal speed they sit, shift to the
small ring and climb efficiently--but not so smoothly and
easily as a fixer in a considerably higher gear.)
Loss of a third of one’s power stroke is a big price to pay
for fitting a free wheel! So if one fits a free wheel one
needs lots of gears, and needs to know how to use them
effectively. I estimate that a single fixed gear is the
equivalent of three or four widely spaced gears on a generic
freebie, and about six or seven gears on a close ratio road
setup.
Racing fixed?
Fixers rule on the track because they are faster when geared
appropriately for restricted circumstances. Specifically,
their 50% torque advantage means 50% greater acceleration in
the appropriate gear, and acceleration is nearly everything
in track racing.
Fixers would be great for criterium racing too, if it were
not for those nasty pedal strikes in tight corners. The
field likely would not complete a single lap, much to the
consternation of riders and promoters!
Not good at all for road racing, where slow ascents must be
followed by blazing descents. But if the speed range on any
day of a stage race were not great the fixer’s high and wide
torque range could conquer a multitude of conditions. Might
be good for time trials, especially team time trials, for
pace lines would be easier to maintain and probably faster
too.
Early every summer in Portland we have six weeks of racing on
a closed course at Mount Tabor, a (very old) local volcano:
137 feet of climb and descent for each 1.3 mile lap. The
descent is continuous for 3/4 mile, followed by a short flat
straight, a 4.3% climb, a very short flat straight, and a
7.5% climb. As the only tight turn is at the top of the 7.5%
grade and so is very slow, pedal strikes are not a problem
and the venue is suitable for racing fixed, which we do for
four laps at the beginning of each weekly session.
There are several really strong fixers (I not one) whose laps
are in the 3:05-3:30 range, which is about the same as the
Cat 1-2 riders later in the evening. The freebies clearly
are faster on the descent, which means that the fixers are
faster on the climbs. The strong guys ride about 75 inch
gears, climbing the lesser grade seated, standing on the
steeper grade, and spinning 150 revs or more downhill.
Fixing the rest:
The same high torque and wide range of the fixed gear that
make it a viable racer for certain events, even on the road,
make it a more than viable utility bike as a daily driver on
the street. As a daily driver it simply is the best!
After all, few of us ascend long steep grades, speed along in
a peleton, and make harrowingly fast descents on our daily
grinds. Even if ordinary riders knew how to operate 20 or 30
speed machines, only 5 or 6 would be useful to them, for
which a planetary hub shifter is more than adequate and much
easier to work.
Most street riding is at 10-15 mi/hr, which on a fixer with a
63 inch gear means cadences of 60-90 rev/min. This just the
“sweet spot” for smooth spinning, easy climbing, and great
acceleration. Once one gets the hang of riding fixed on the
street it actually is much easier and more convenient all
around. And there is the bonus of exquisite control at low
speed: one can control one’s position as well as one’s speed.
What do those horrid words in the title mean?
“Gyrator:”
This is akin to an electrical “transformer,” which converts
one cyclic voltage into another cyclic voltage, and one
cyclic current into another cyclic current; proportionality
is determined by the numbers of windings on the primary and
secondary; higher voltage always means lower current, and
vice-versa.
But a gyrator converts a cyclic voltage into a proportional
cyclic current, and vice-versa; for example, a garden variety
electrodynamic loudspeaker converts the cyclic voltage from
an amplifier into an acoustic current, or “volume velocity”.
(Gyrators are very tricky to analyze, which is why it is hard
to design good loudspeakers.)
A bicycle converts the cyclic forces from our leg muscles
into progressive speed, so it is only a “quasi-gyrator,” and
let’s leave it at that.
Except! Ever thought about how big, how really really big,
the crank torques on a bicycle can be? I weigh 170 pounds,
so when I stand on a 6 inch crank arm in the horizontal
position I produce 85 pound-feet of torque. This is about
the same as the engine of a Harley-Davidson, or even a small
car! We produce our torques at low speed, and so do not
generate much power; an internal combustion engine produces
its torques at very high speed and so generates much more
power.
An internal combustion piston engine must be geared down to
generate useful torque at the road, and even then needs some
kind of clutch to get its load moving, for it has zero torque
at zero speed.
We gear our bikes up to get useful speeds from our very
strong but slowly moving legs. And because, like steam
locomotives, our greatest torque is at zero speed, we need no
clutch at all.
Unless we need to shift gears, and who would want to do that?
(Archibald Sharp, who designed a two speed fixer, with a
neutral for coasting, in 1896, that’s who.)
“Scleronomous:”
According to Cornelius Lanczos (who got it from Ludwig
Boltzmann) a “scleronomous kinematic condition,” or
“constraint,” is one that does not involve time. In our case
it simply means that our fixed gears require the cranks to
rotate at constant speed throughout one complete cycle.
Unless the rear wheel is accelerating or decelerating, which
causes the cranks to slowly follow suit.
(“Rhenomous” describes the opposite kinematic condition, in
which the cranks are determined to rotate at varying speed
during a complete cycle. Here the constraint is imposed not
by the mechanics of the bicycle but by the physiology of the
rider.)
“Holonomic:”
If our wheels roll on a surface without slipping they obey a
holonomic constraint, able to move only forward or backward.
(But if a ball rolls on a surface without slipping its motion
is non-holonomic, because it can move in any direction.) All
bicycles are holonomic unless something nasty has happened.
A few more words on stroking!
I noted above that fitting a freewheel means one needs more
gears, because the power stroke is much shorter. Fast
roadies learn to pull back at the bottom of the stroke to
extend it a little; this works, for one can feel the bike
accelerate slightly in response. But there is little more
one can do to improve what is fundamentally a short stroke,
so the stroke tends to change little and one modulates speed
by changing gears. This is basically mechanical modulation.
Also I said that the fixed rider learns to modulate the
stroke in response to changing conditions. I am not certain
how we do this: do we keep the longest possible stroke and
modulate the force; do we use a constant force and modulate
the length of the stroke? Probably a mixture of both. In
casual riding on the street I feel that I am modulating the
length of the stroke while keeping the force pretty much
constant, but I really would like to get the opinions of
other fixers on this matter.
A coach at Alpenrose Velodrome taught me that the best way to
finish off a power stroke is to drop the heel at the bottom.
This is just the opposite of finishing the stroke on a
freebie, but it has a similar effect, causing a noticeable
acceleration.
I think it is this direct physiological modulation of power
to the bicycle which is at the core of the feeling that
riding fixed is simply an wonderful extension of the rider:
the intimate cyclist!
A modest dissent:
As a cyclist and a physicist I am dismayed that I have been
required to do all this! After all, the bicycle has been
around for well more than a century and many smarter riders
than I have thought about its propulsion.
Near the beginning there was the redoubtable Archibald Sharp.
Now there is Mike Burrows. Both Brits, and both well worth
reading.
I cannot in good faith recommend other authors, for many seem
to violate my basic principle of physics: do not worry about
what is happening in the second decimal place until you
understand what is happening in the first! There are
infinitudes of disquisitions upon the niceties of the pedal
stroke, and further infinitudes of odd mechanical devices
portending to optimize it. But has anyone ever delimited the
gross difference between the stroke of a fixer and the stroke
of a freebie?
Particularly I am perturbed by what many consider the
cannonical work in the field: “Bicycling Science,” by Whitt
and Wilson, in its several editions. Books of this sort
address the extreme aspects of competitive cycling as if such
were the chief interest of all of us. Certainly less than 5%
of the world’s cyclists race. 95% care only about safe,
efficient, and comfortable transportation.
Read Whitt and Wilson if you wish to enter HPV competitions.
There are many other books if you wish to train for racing.
An essential disclaimer:
Racers and HPV people are interested in generation and
application of high power over artifically defined distances
and times. Whitt and Wilson and their ilk address this
problem admirably, but treat the subject of development of
torque tangentially, if at all. But it is torque, not power,
that determines how a bicycle is built, why it works as it
does, and ultimately, how it is ridden!
Dick DiGennaro and I agree that a graph of torque versus
cadence begins high at zero cadence and decreases to zero at
some cadence above 100 rev/min that depends upon the ability
of the rider. We also agree that it is not a straight line.
But we have no solid experimental evidence for its actual
shape. So crucial parts of this discussion are based on
“anecdata.”
I would very much like to see good experiments that measure
torque versus cadence for fixed gear and freewheel bicycles
under well controlled conditions.
A couple of final minor points: a rear brake reduces stresses
on the knees; a fixer’s chain wears more quickly because it’s
always moving and routinely transmits larger forces.
Enough said!
©
James Bernard Lee
2005