Climbing steep hills on a bicycle is one of the joys and can
be one of the most uncomfortable aspects of the sport. Above, two of the great
climbers of the Tour de France make it look easy. Federico Bahamontes, the Eagle
of Toledo and Charly Gaul, the Angel of the Mountains during the 1959 Tour.
Gaul won the GC the previous year and 1959 would see Bahmontes’ as the overall
winner. Bahmontes won the King of the Mountains title an unprecedented six
times, possibly making him the greatest TdF climber of all time. Gaul was noted for
love for
inclement weather.
(TdF cognoscenti will quickly point out that after Bahamontes, Lucien Van Impe won six KOM titles plus one GC. Richard Virenque won seven KOM titles. Virenque's intimate association with performance enhancing substances taints his abilities and excludes him from the top climber title. So in all fairness it is between Bahamontes and Van Impe. Gaul, along with a host of others won two.)
So when climbing steep hills, other than working against
gravity, is there anything different about how a cyclist generates power, as compared
to riding on the flats? The answer is yes. During hill climbing the rider’s
efficiency in generating power is reduced. Why this is the case is one of the
topics covered in this post.
POWER
When commercial recumbent bicycles (Avatar, Hypercycle, Easy
Racer etc.) made their reappearance in the 1980’s, one of the reasons touted
for their superiority over upright bicycles was that the bracing of the rider’s
back allowed for much greater pedaling forces to be developed. The error here
is that greater pedaling forces may lead to greater accelerations, but they alone
do not lead to higher top-end speeds. In fact, the higher pedaling forces that
could be achieved blew out more than a few recumbent cyclists’ knees. Forces that
can be sustained for 10 reps using a leg-press machine quickly wear joints out
over the duration of century rides.
So, it is appropriate to discuss force, work and power. The
concept of force is simple, something that results in a pressure or a tension
on objects to which it is applied. Units are often Newtons or pounds. A force
moving through a distance is work or energy. Units are often Newton-meters,
Joules or foot-pounds. For something to change, the force must move through a
distance and do work.
The rate of doing work is power, work done per unit of time.
Power is the product of force times velocity. Units are often Joules/sec, Watts,
foot-pounds/sec or horsepower. For future conversations, one horsepower is
about 748 Watts. When a cyclist increases elevation going up a hill, work is done.
Riding up hill at a given velocity results in power being dissipated.
Similarly, a cyclist pedaling against air pressure at a particular velocity is
dissipating power. Power makes the bicycle move.
Peak power levels that are developed by elite athletes fall
off with time. For short durations, on the order of less than a minute the
power comes from the chemical stores within the muscles themselves. Since the
duration is too short for oxygen to be adequately delivered to the muscles,
this is anaerobic power. For durations of about six seconds, peak power levels
2hp. have been generated. Elite cyclists (Eddy Merckx for example) could
generate .5hp. for about an hour. These aerobic levels are limited by the rate
of oxygen consumption. For all-day activities, this level drops to about .25hp.
due to toxic byproducts building up in the muscles. Detailed graphs of power generation vs.
duration can be found in books like the Third Edition (2004) of Bicycling
Science by D.G. Wilson. Curiously, the unit of horsepower was supposedly the
power level that could be sustained by a work horse all day. The peak power
from a horse is about 12hp.
EFFICIENCY
One factor that limits mechanical power produced aerobically
is the rate of oxygen consumption, VO2. During the 1960s, when studies were
being done on the feasibility of man-powered flight, a relation between VO2 and
power production was empirically determined. 1 liter of oxygen consumed per
minute would result in .1hp or 75W of mechanical power production. What was overlooked was the fact that there
are various efficiencies assumed in this relation. Some of the efficiencies are
related to the bioenergetic processes going on in the muscle and some are
related to the mechanical conditions of the activity used to generate the
power. While it is difficult to modify the characteristics of the activity to
improve the efficiency, it is rather easy to modify them to reduce the efficiency.
This post will deal with three aspects that influence the efficiency of
pedal-power generation, impedance matching, cyclically-varying-speed-crank
systems and system kinetic energy.
IMPEDANCE MATCHING
The figure to the right is a plot of the force vs. velocity for the
muscle group that flexes the elbow under maximum stimulation. This was empirically
generated by D.R. Wilkie in 1950. The equation for the curve is to the right.
Notice that if the load is above 48lb., the load cannot be moved and no useful
work is done. At the other extreme, at zero load, a velocity of 21ft/sec can be
developed but again, no useful work is done.
Now suppose we have the device pictured right, where the
length “N” can be adjusted. The objective is to select an N that results in the
highest velocity of the 48lb. weight, based on the graph of elbow-flexion performance
shown above.
N, inches
|
P, pounds
|
V, ft./sec.
|
Power, ft.lb./sec
|
V of weight
|
1
|
48
|
0
|
0
|
0
|
2
|
24
|
4.2
|
100
|
2.1
|
3
|
16
|
7
|
112
|
2.33
|
4
|
12
|
9
|
108
|
2.25
|
5
|
9.6
|
10.5
|
101
|
2.1
|
6
|
8
|
11.7
|
93
|
1.95
|
7
|
6.9
|
12.6
|
86
|
1.8
|
8
|
6
|
13.4
|
80
|
1.68
|
The chart above shows the results for eight different values
of N. Notice that for N of 3, the maximum power is generated and the weight
moves the fastest. For this muscle group, the peak power is generated at 1/3
the maximum force and 1/3 the maximum velocity. Adjusting the length of N is
analogous to a cyclist shifting gears on a bicycle to find the “sweet spot”. The
“sweet spot” allows the cyclist to maximize the bicycle’s velocity for a given
level of effort. The cyclist can pedal in a less-than-optimal gear, but the
efficiency will be reduced and the oxygen consumption for a given power level
will be elevated over the minimal optimum.
CYCLICALLY-VARYING-SPEED
CRANK SYSTEMS
Cyclically-varying speed crank systems, CVSCS, for short,
like fashions, seem to come and go. Attached to rotary cranks, they increase
and decrease the crank speed (and inversely the gear ratio) in an attempt to
improve aerobic efficiency. Recently some of the Tour de France racers have
been using elliptical chainrings for the time-trial stages. The elliptical
chainring is probably the oldest of the CVSCS, having been used on early safety
bicycles over 100 years ago. Archibald
Sharp in his 1896 technical masterpiece, Bicycles & Tricycles, discussed
the technical aspects of elliptical chainrings. More recent CVSCS are BioPace
chainrings and the Powercam crank mechanism.
When I began my graduate research on optimization of human
power production under Prof. Ali A. Seireg at the University of Wisconsin, the initial
direction of the work was to focus on the effects of using
non-circular-pedaling motions generated by an adjustable four-bar linkage.
Since the curves generated by this linkage often had associated velocity patterns that
were not optimal or even desirable, it was necessary to insert another mechanism
between the linkage system and the load to produce adjustable velocity
fluctuations that could modify those generated by
the linkage. After some initial testing, it became clear that the velocity
patterns were more important for power optimization than were the shapes of the
pedal paths.
The photo, the table and the graphs that follow are from an
article in the April 1986 issue of Soma: Engineering for the Human Body,
published under the umbrella of the ASME. The graphs and photos were taken in
turn from my doctoral thesis in mechanical engineering.
The research apparatus for my thesis is shown above. I was
involved in designing all the components except the force-transducer pedal,
which was developed for a previous research project. The use of a supine rider
position was a result of directing the results along the lines of a commuter
vehicle design. The novel feature of the apparatus was you could measure oxygen
consumption, average and instantaneous force, velocity and power levels. The
majority of tests were run at an average power of .15hp., with selective tests
being run at .225hp.
The means of producing the velocity fluctuation was by using
two universal joints. A single universal joint produces a velocity fluctuation
when the input and output shafts are angled to each other. They were hooked together
in phase so the velocity fluctuation was squared instead of the conventional
orientation that cancels out that fluctuation. The cyclic angular fluctuation
was
Output speed/Input Speed=
((1-sin^2(C)*sin^2(B))/cos(B))^2
where C is the crank angle and B
is the offset angle. The maximum offset angle that was recommended for use was 45deg.
and this resulted on a velocity fluctuation of +/-62%. This corresponds to an
elliptical sprocket having a major/minor diameter ratio of 4.25/1. The velocity
fluctuation of +/-25% corresponded to a major/minor diameter ratio of 1.66/1.
Even the 25% setting would be considered extreme by current elliptical sprocket
fashions.
The multiple compliances in the
system deformed the fluctuation pattern in the manner shown below, shifting the
velocity peak earlier in the cycle.
The phase relation between the pedal path and the
fluctuation pattern could be changed over seven positions before repeating. The
zero-phase position was when the pedals were mid-stroke moving forward and the
fluctuation was at its lowest velocity point.
Since the readers may be more familiar with photos of
current elliptical sprockets, for convenience, the various phase relations are
represented as a crank arm that can be oriented in various positions on an
elliptical sprocket. The chains that would connect to the rear derailleur
extend horizontally to the left at the top and bottom of the chainring.
The powerful feature about the test apparatus is that by
playing with the phase of the fluctuation, one could cause and measure changes
in oxygen consumption while maintaining a constant average mechanical power
level. The efficiency of the activity was being changed. And, while the 1liter
VO2/.1hp. ratio was only improved on once, (.87 at 25% at a phase of 6 and
.15hp.) many higher (less efficient) ratios were achieved. In the subsequent
graphs, the velocities and forces plotted are in a normal direction moving away
from the subject while the power values are the total values for the normal and
tangential directions combined. A note about the selection of 50rpm as the
standard pedaling speed. Since the high fluctuation of 62% caused very high
peak pedal velocities, 50rpm was the highest average pedal velocity that could
be sustained over all conditions.
Above is a graph of
VO2 vs. phase for .15hp., 50rpm and a 62% velocity fluctuation
Here is the same test
sequence along with other measurements. Addition cases of 25% fluctuation for power
levels of .15hp. and .225hp. are listed.
Above is the graphical data for 50rpm, a phase
of 7 at 62% and .15hp.
Above is the graphical data for 50rpm, a phase of 3 at 62%
and .15hp.
Lastly is data for 50rpm, a .225hp. power level with a phase
of 6 at a 25% fluctuation. This is followed by a test at the same power level
with no fluctuation. The VO2 for the 25%-fluctuation case was 17% lower than
for the no-fluctuation case.
For use in subsequent discussions, it may be convenient to
refer to a pedal cycle for an upright cyclist viewed from the crank side. For
simplicity, divide it into four quadrants.
From 1:30 o’clock to 4:30 o’clock, the power stroke zone, this will be
referred to as the downstroke. From 4:30 to 7:30, this will be the backstroke.
From 7:30 to 10:30, this will be the upstroke and from 10:30 to 1:30, this will
be the forwardstroke.
Looking at this data, one could observe that we reinvented
the elliptical sprocket. Fluctuation phases that slowed the velocity (and
increased the gear ratio) through the power stroke (1.5 to 4.5 o’clock for
upright pedaling) produced significantly lower VO2s than those that increased
the velocity through the power stroke. Looking
at the fluctuation phase crank sprocket diagram above, having the crank arm in
positions 6 & 7 for lowest VO2 looks very similar to the orientation seen
in actual bicycles.
It is interesting that even though the most efficient phase
cases (6 & 7) tend to make the velocity profile more constant across the
stroke, the actual power generated was over a narrower zone than a constant
velocity system. We can consider this narrower power production zone as a
pulsatile power pulse, P3. The power is more pulsatile than in the no
fluctuation case. Since both the
Powercam and BioPace systems rely on producing P3, one may assume this is
more efficient than the more uniform power distribution case.
One last observation on the research. Since it was clear
over various fluctuations that VO2 was not proportional to average power, was there
any measure to which VO2 was proportional? The average mechanical power was the
average of the product of instantaneous force and the instantaneous velocity
summed for both the normal and tangential directions. There is an arbitrary power
calculation equal to the product of the average total force and the average
total velocity. We christened this product physiological power. It appeared
that VO2 was proportional to physiological power, high average forces times
high average velocities require high oxygen consumptions.
I also have had experiences using a Durham elliptical
sprocket, a Powercam and several variations of BioPace chainrings. No formal
testing was done but I will share a few subjective observations.
I rode a 25mi. time trial with the Durham Elliptical. It
made pushing big gears more comfortable than round chainrings. I purchased the
60T version. It worked very well standing up pedaling but its large size
prevented it from being used on long steep hills.
The Powercam (a.k.a. the Biocam and Selectocam) has produced some impressive competition
results. Scott Dickson finished second in the 1979 Paris Brest Paris cycling
marathon, the first American to finish that well. I used the Powercam briefly on an upright
bike. The most noticeable performance feature was it was very difficult to
stand up and pedal. When I bought an Avatar recumbent several years later, I
mounted the Powercam on it. It had been suggested that it might work well for
recumbent pedaling because you couldn’t stand up and pedal on a recumbent. It
was good advice, and the Powercam improved on the notoriously poor hill
climbing performance.
The cam allows a more rapid change in pedal velocity than
a non-circular chainring. The explanation for how the PC functions is that the
gear is very high through the forwardstroke. As the downstroke is entered the
gear drops rapidly and before the leg muscles can reduce the high force they
needed previously, a large power pulse is generated. (Large force*high
speed=large power pulse). When pedaling hard going up hill, you could feel the
bike surging, speeding up and slowing down as the extra-instantaneous power from
the pulse was put into the system. To absorb the excess power, the bicycle must
speed up slightly and slow down after the pulse. To a much lesser degree, this
speedup was also associated with the more extreme-shape versions of the BioPace
chainrings. When used on mountain bikes, this power pulse could break the tire
loose and the rider would loose traction. The tire breaking loose is often the
kiss of death on steep climbs because it could end forward progress and cause
the cyclist to put down a foot to prevent falling over. Hence the subsequent term
of derision for suspension systems that bob during pedaling, “biopacing”.
I used the BioPace chainrings on both an upright and a
recumbent. Correct phasing when used on a recumbent was questionable because
the chainrings needed to be rotated ¼ of a revolution forward and the holes
spacing only allowed for moves of 1/5 of a revolution. I thought that the
increasing ovalarity from the large to small sprockets made sense, since the
smaller ring would be used for hill climbing. Like the Powercam, but to a
lesser degree, standing and pedaling in the smaller chainring didn’t seem to
work as well as sitting. Using the small ring on a recumbent appeared to
improve hill climbing speed.
KINETIC ENERGY AND
CYCLIC ENERGY STORAGE
The Powercam, BioPace chainrings and to a lesser extent
elliptical and circular chainrings produce a P3 (pulsatile power pulse) interspersed
with rest periods for the muscles. Since the external power demand is
essentially constant, and the power production is intermittent, the excess
energy (pulse energy less power demand energy) must be stored and recovered to
accommodate the difference. The energy is stored in the kinetic energy of the
bicycle/rider system. The excess energy storage is a function of the system
mass and the square of the velocity change.
KE= ½ M*(V2^2-V1^2)
Where V1 is the system velocity before the power pulse and
V2 is the system velocity after the power pulse.
I have come up with a measure of the effectiveness of the
energy storage capacity of the bicycle/rider system in relation to the power
demand of the exercise. An activity with a lower power demand requires the
storage of less kinetic energy to modulate the P3. My measure is a time
constant, Tes (energy storage) equal to the kinetic energy of the system
divided by the power required for the exercise.
As Tes decreases, it becomes more difficult to sustain P3. I
inadvertently happened to determine what may be a minimum value of Tes for
efficient power generation. I purchased a Monarch bicycle ergometer to use for
off-season training. When I tried to pedal at a power level of about 200W it was
very uncomfortable to sustain. I concluded that the system didn’t have enough
inertia to modulate my power pulses. The ergometer had a 20” diameter aluminum
flywheel and a gearing of 44/14. I decided to switch to 3/8” pitch industrial
drive components because it allowed me to fit a 72T crank sprocket within the
chain covers. I used a 10T cog on the flywheel, thus doubling the drive ratio.
This resulted in the pedaling becoming reasonably comfortable. The kinetic
energy of the system at a cadence of 70rpm was 538 Joules. Tes was therefore
538J/200W=2.7sec. So let us say a minimum value for Tes is 3sec.
Let’s compare this 3sec Tes to values for two on-the-road
cases, a top athlete riding on the flats and climbing a steep hill. Our cyclist has a system mass of 200lb. For
the on-the-road case, our cyclist is riding at 25mph (36.7ft/sec) and is
generating ½ hp. The Tes is 5671 Joules/375W or 15.2sec. For the 30% grade hill
climb our cyclist is moving at 2.5mph (3.75ft/ sec). The Tes is 59.2J/375W or .16sec.
So the Tes can vary by almost two orders of magnitude over the extremes of
riding conditions.
When the Tes drops below a value of about 3sec, the rider
must begin producing power during the forwardstroke, the backstroke and the
upstroke of the pedal cycle in addition to the downstroke, where the power is usually produced. The power generation under these conditions is
much less efficient than the power production during the downstroke. This is
the reason that climbing steep hills becomes so difficult. The aerobic
efficiency of power production has been significantly reduced. I have been
passed by runners of lesser athletic ability while pedaling up steep hills.
So if anything, what can be done to improve the efficiency
during climbing steep hills?
One approach that most riders uses is to stand up on the
pedals. If you allow your body to sink when the pedal is passing through the
downstroke, the center of gravity for the system with respect to the road sinks
as well and the speed of the system moving vertically fluctuates. Since the
power demand is now non-constant, if allows rest periods in power production
and improves aerobic efficiency.
Clearly, P3 systems do not work because sufficient kinetic
energy can not be stored in the moving mass of the system. A coworker has had
some success with placing a spring in series between the pedals and the drive
system to periodically store and release energy.
A few years ago, I added a
system that stored the energy in a long ½” diameter rubber cord to my EcoVia
commuter trike. I could adjust the spring rate by adding or removing wraps of
rubber. However, over a range from very soft to a very stiff rubber spring, my
hill climbing performance was not as efficient as when using a round chainring.
My only conclusion was that while my coworker used a metal spring, my rubber
spring may have dissipated too much energy.
The hill drive system in the uncharged state. In the picture above, the bungee cord at the very right represents the energy storage medium and the sprocket at the very top has a one-way ratchet.
The same system is the charged state. Notice the displacement of the crank from the discharged to charged state. This shows the windup for the cyclic energy storage. The bungee has moved to the left, shortening the length of the drive side of the chain.
Lastly we come to that old favorite, the constant-torque
treadle.
We know from the historical record that these systems can
not be pedaled very quickly and consequently cannot produce high power levels.
However, they do produce a constant torque throughout the stroke and this has
to be more efficient than applying a large force through the forwardstroke, backstroke
and upstroke of the pedal cycle. And there is some historical evidence that
such systems excelled at hill climbing.
And what of Bahamontes and Gaul? Both used unorthodox climbing styles. Gaul used small gears and would spin up the hills. Bahamontes would alternate between sitting for 16 revolutions and standing for 16 revolutions. Their performances showed that each benefited from their preferred techniques.
Wow, did you really just read all that? You must almost be
as big a bike-tech nerd as I am. On the upside, you will have plenty to think
about as you slog up the next long hill!
Hephaestus
