Reflections on the potential of human power for transportation

Sunday, March 30, 2014

Why Hill Climbing is Hard or Efficiency and Pedal Power Production




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

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