Investigating The Energy Needed to Stay in Balance
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Updated 4-28-05

Speculations on the unicycle by George Winters


One of the very fun things about unicycling, for me, is remembering how impossible it seemed when I first tried riding my homemade unicycle.  How is it that one can learn to move on such a completely slippery gizmo?

My next big surprise, after I actually started succeeding at riding the unicycle, was how tiring it was.

My initial investigation of the energy required to ride a unicycle suggests that there is Missing Energy.  This article attempts to resolve some of that disparity.  While looking at the mechanical process of riding the unicycle, I also see a hint that the ability to ride a unicycle builds on the often underestimated complexity of walking.

In the process of looking at the energy needed to unicycle I also propose that the challenge of riding a unicycle can help to balance the body and stretch the mind.  As I try to analyze this  complex mode of moving, it leads me appreciate a much wider array of

Click on image to find out more about unisk8tr and photos by John Childs*

systems in balance.

A friend asked me why I would want to ride a unicycle.  My flippant reply was, "Because it is as much fun as skiing, but I don't have to drive for 3 hours and spend $50 on a lift ticket."  After I said that, it slowly occurred to me: That's a fairly accurate assessment.

What is the energy demand to ride a unicycle?  
The skiing analogy seems apt to me (a novice at unicycling) because I have noticed that as I extended my riding to cover distances of a block or more, my thighs felt a similar stress to that which I have experienced when telemark skiing.

In his entertaining and informative book about his long distance unicycle adventure, One Wheel-Many Spokes, Lars Clausen notes that  a bicycle is one of the most energy efficient means of transportation.  A unicycle is not.  

Why not?

I did not find literature about testing the energy expenditure of a person riding a unicycle.

(I did read of a test where a pronghorn antelope was hooked up to a machine that measured the animal's oxygen consumption while it was running on a treadmill at 35 miles per hour.  Wow!  Imagine the challenge of doing that experiment!)

I am beginning to wonder if anyone has tested the biometrics of the energy requirement of unicycling.  Would it be possible to ride a unicycle on a treadmill while hooked up to a device so that a person's oxygen consumption and energy output could be measured?  Are there more general implications that could be learned from such an investigation?  

Maybe the answer has wider relevance than just having fun.

I can find lots of information on the energy demand to ride a bike, walk, run, or cross country ski.  What about the unicycle?

Unicycle vs Bike:  

• A unicycle does not coast.  Most bikes have brakes, but a unicyclist must supply all the energy to go, and then all the energy to stop.  
• A unicyclist provides energy to ascend a hill, and then must must absorb the energy of descending the hill.  If a long distance ride is made over hilly ground then every hill will demand twice as much energy from a unicycle rider without brakes as that hill demands from the standard bike rider.
• A bike moving at relatively slow speed gains stability from the angular momentum of the spinning wheels.  The bicyclist must provide energy to accelerate the system into motion, but once moving the bicycle rider can take advantage of a fundamental property of our universe and rest on the fact that all that motion not only tends to stay in motion, but the spinning wheels also provide inherent stability.  The same force that allows a spinning top to magically spin on a tiny tip also helps the bicycle rider stay upright.  The unicyclist does not get as much benefit from angular momentum, and must put more energy in to maintaining stability.   
• A unicycle is a simpler mechanism in its basic form with no gears or chain, only one wheel, and a very small frame structure.  
• The advantage of a simple frame may be diminished by the vibration instability of the structure.  Torque at the pedals cannot be re-directed by a stiff frame structure.  The frame and geometry of a unicycle cannot absorb or redirect disruptive forces the way the bicycle frame and geometry does.
• In bicycle racing it is well known that limiting rotating mass (wheels, pedals, gears, etc.) is more important for efficiency than limiting non rotating mass (frame, seat, body weight, etc.)  On a unicycle the entire mass of the rider and bike becomes a rotating mass that pivots about the point of the tire/road contact.
  

An Initial Measure of the Energy Difference

I have recently read three accounts of endurance rides on unicycles, people who attempted to test the limit of distance that can be covered in an hour or a day.  It occurred to me that long distance efforts such as these will generate parallel tests of human energy output that allow comparisons across a range of similar activities.  The energy needed to run or walk, ski or bike for extended time periods places demands on similar muscle groups, will produce similar tests of the human "engine," and should have the same limiting factors related to the amount of energy that can be stored and processed in the human "gas tank."

A friend has suggested that my proposition would be best presented if I could chose performance efforts from comparable population samples.  If I had a large range of performance samples in each activity and I chose, say, the 90th percentile effort in each activity, it would be realistic to say that each of these human "engine" efforts produced the same amount of energy for some long period, an hour or a day.  But, I don't have access to that extent of data, especially in the activity of riding a unicycle where the population sample is relatively very small.

I am choosing a very subjective limited comparison while trying to keep in mind a larger relationship and primarily using limited data to show that a realistic comparison can be made without trying to claim a high level of precision.  I am trying to be accurate, while not necessarily precise.

The Basic Plan:

• The energy needed to move a unicycle at a particular speed can be calculated initially by using a Bike Energy Calculation.   This calculation allows putting in factors for the shape, wind resistance, mechanical efficiency, and weight of the system.
• A comparison with other activities can be made using this energy calculation.  If similar "engines" of equal power move a regular bike at say 20 mph while only moving a unicycle at 15 mph, this indicates the size of the loss of energy and/or an inefficiency of using the power. 
• I have tried to model some of the basic actions of moving on a unicycle and the use some properties from my college physics text to see where the energy loss occurs.
  
• A list of basic physics equations I have used.
  

The Comparison Subjects:

• Lars Clausen wrote of riding across the US both ways in one long summer.  Along the way he decided to attempt to set a new Guinness World record by riding his touring unicycle on the highway for 24 hours.  He covered slightly over 200 miles.  His book about this six month long ride is a great read: One Wheel-Many Spokes
• Ken Looi improved on this record riding in New Zealand recently, covering 235 miles in 24 hours on an outdoor track.  In that same ride, Mr. Looi also tried to set a record for 100 miles, which his account records as just over 6 and one half hours.
• Unisk8r has a posting on Unicyclist.com listing his effort at a one hour record ride, covering 14.278 miles.

Each of these people either have, or are applying for Guinness recognition.   The performances they have established represent evolving and limited data.  Certainly they are very good athletes with training and experience, but they probably do not represent the extreme upper limit of the human engine capability.

• It is relevant to note in this discussion that the standard food consumption energy unit is called a calorie but this represents a Kilo calorie, or 1000 of the standard calories that are used in chemistry and physics measurement.
• The food energy requirement for humans doing measurable mechanical work are adjusted using a factor that assumes the human body is 25% efficient in converting food energy into actual work.  So, if a person needs to move an object that will require 100 calories of energy output, this person will need 400 calories of food energy to perform that work.
  

The green line in the graph on the right represents the energy demand for the unicyclist evaluated with a Bike Energy Calculation as if it were a bike of appropriate weight and shape going the same average speed of the three endurance trails referenced.  The "George's Max." line represents the energy output I have managed (as a younger and slimmer self) riding a lightweight bike for each of those time periods.  I am not a stellar athlete.  The comparison with my bicycle energy output shows that as a bare minimum each of the individuals above must have been putting out at least 100 Kilo calories more per hour than the green line estimate to accomplish their feats of similar duration.  

The calculation of effort to propel a unicycle in a long distance effort will require at least the addition of 100 food calories per hour above what is accounted for by the simple dynamics of moving the machine and system over level ground through the resistant air.

I have made an effort to find a comparable performance level in each of three other activities, running, walking, and biking to show energy outputs that would represent people who would be similar to the three individuals noted above.  My choice of comparisons is very sketchy, and is primarily listed just to show that there is an obvious disparity between the simple bicycle comparison and the real world energy demand.

The 24 hour energy requirements tend to converge because it is reasonable that a 24 hour event will tax the total energy storage limit of the individual.  The graph comparison shows that using the bike energy calculation for the unicyclist greatly under estimates the energy requirement.  So, if one uses the speed of a unicycle to calculate energy demand, the value attained by calculations of wind resistance, friction, and efficiency may miss  by half the real energy demand.  

Keeping an unstable dynamic system in balance can double the energy demand!

Limits to propelling a Unicycle at speed:

• It is interesting to note that the investigation of the energy requirements needed to ride a unicycle suggests that a huge range of performance improvement may be possible.  The skill of the individual in managing balance becomes a dominant factor in the overall performance.  If a rider can gain the skill to increasingly minimize the energy needed to balance, or the wasted energy used fighting balance, then the potential energy available for increasing speed is dramatic.  I would guess that if large numbers of people practiced long hours, similar to the way people train for running, biking, or skiing, then the 1 hour speed record could probably approach 20 mph or more.
• The unicycle may have inherent mechanical limitations that prevent the effective transferal of power into forward motion.  For example: On a bicycle it is reasonably possible to momentarily exert more force on the pedal than the rider's actual weight by taking advantage of the handle bars and toe clips.  The amount of torque that this effort produces and the resulting force of forward push produced by the wheel on the road may not be possible on a unicycle.
  

What is using up the extra energy?  
Modeling the System As An Inverted Pendulum:

In the simplest analogy, the unicycle is like a long uniform rod balanced on one end.  The center of mass will pivot about the balance point moving in an arc, similar to an inverted pendulum.  As the center of mass tips away from the balance point, the arc of motion lowers the mass relative to the ground.  Each time the system tips, the energy needed to move back to the balance point will be equal to the energy needed to lift the mass of the system back up to the highest point.  This proposition applies the principle of  the conservation of energy.  The kinetic energy due to the center of mass changing height will equal the kinetic energy of the tipping rod which can be analyzed as a rotating mass.  
 Mgh = 1/2 I w²
M is the mass, g is the acceleration of gravity, h is the changed height, I is the inertial constant of the rotating body, and w is the angular speed of rotation.

Imagine the rider is in a hallway and regularly pushes off the adjacent wall to return to balance.  The energy represented here is the energy needed to continually push back off the walls.  In actual riding this energy is developed by steering, subtle changes of speed, and moving the body and arms.  Ultimately, the energy has to come from using the legs to push against the ground through the leverage action of the pedals to the wheel.  The energy required is the same as the push off the wall.  The direction of motion out of balance will be changing in  360 degrees of the horizon, but each individual return to balance can be analyzed as a rotation in two dimensions.

The question then becomes two fold:

1. How far has the center of mass pivoted on the arc?
2. How frequent is the need to apply energy to return to the balance point?

These two questions interact in the dynamic motion.  

• If the arc of motion is kept small, then the frequency of adding the correcting energy increases.  
• If the frequency of adding correcting energy is reduced, then the arc of motion is longer and the needed energy to return to balance increases.

The relationship between frequency and arc angle is not linear because the effect of gravity, g, is an accelerating contribution, and the speed of motion in the arc angle, w, is a squared factor.  So if the changing balance angle can be minimized by frequent corrective action, the energy output will be minimized.  This fact tends to show that the skill of the rider in correcting the balance change is very critical to total energy efficiency.  Energy not needed for balancing can then be available for propulsion in the desired direction.

Balancing Energy Requirement 1st Case
angle of tilt (radians)	time	angular velocity (radians/sec)	h meters	watts of energy needed to return to balance	Kilo calories per hour
0	0	0	0	0	0
0.02	0.32	.063	0.00021	0.1834	2.0
0.04	0.48	0.12586	0.00083	0.7337	5.3
0.06	0.58	0.18878	0.00187	1.6506	9.7
0.08	0.66	0.25168	0.00333	2.9337	15.2
.1	0.73	0.31455	0.00520	4.5826	21.7

As the representational long rod tips the time it takes the tipping mass to reach each wider angle is shortened as the angular speed increases.  The force of gravity has a larger pull on the mass relative to the vertical support of the unicycle.  At the balance point all of the weight is vertical over the pivot point and there is no inherent motion to tip.  As the mass leans away from the balance point, it moves faster and faster.
 

The table at right shows kilo calorie energy requirements if various tipping motions were repeated regularly in the natural time it takes for gravity to move the mass to each of the listed tilt angles.  The angles are shown in radians, and 0.1 radian is approximately 6 degrees.  This list indicates that small angular displacement for balance does not require a lot of energy.  As I noted above, I am guessing that a minimum of 100 kilo calories per hour is used in the balancing activity.  This graph is only showing an energy requirement of at most 22 kilo calories per hour.  The angle size and frequency of balancing action could be measured with stop action photography.

Balancing Energy Requirement 2nd Case with overbalance velocity
angle of tilt (radians)	time	angular velocity (radians/sec	h meters	watts of energy needed to return to balance	Kilo calories per hour
0	0	0.08	0	*	*
0.02	0.2	0.102	0.00021	3.95	13.6
0.04	0.33	0.15	0.00083	4.01	13.8
0.06	0.43	0.205	0.00187	5.24	18.0
0.08	0.5	0.264	0.00333	7.00	24.1
0.1	0.57	0.325	0.00520	9.15	31.5

• The return to balance will probably not be an exact application of the required force.
  

The second table shows an alternative likely case in which with each balancing motion, a small excess energy causes the next cycle of moving out of balance to begin with an extra initial velocity.  Even a small initial velocity will significantly change the energy demand because the time available to reach each wider angle is shortened, and the increased angular velocity creates more energy that must be countered to prevent tipping.  This second table shows that at even small angles of moving balance, a very small over balance effort can add a large percentage increase in energy demand.

• The balancing action should also be modeled in a second way, as the rider's hips, arms, shoulders, and the unicycle twist horizontally around a vertical axes running lengthwise through the center of the body.  My initial simplified model of this action accounts for energy consumption of less than 5 Kcal per hour.
• These two rotations, modeled as a tipping rod and as twisting motion, are fundamental extra actions that are not happening on a bicycle.

Timing is Everything

• On a unicycle the timing of corrective balance effort is directly correlated to each pedal stroke and the time it takes at a given speed for the next pedal stroke to be in a position where the subsequent corrective force could be applied.  Here I am only considering direct drive unicycles where the pedal is always at the exact same spot with each turn of the wheel.
• On a firmly packed dirt path I have been able to trace the track of my riding motion.  The track on the ground has the shape of a repetitive wave, a sine curve, with crests that are exactly spaced in correspondence to each stroke of a pedal.  By measuring my speed as the track is made, I know the length of time between each pedal stroke.  
• In the situation where riding on the soft packed ground challenges my balance, the crest of each wave pattern on the ground will show the amplitude of my motion correcting for  balance.  
• The time of each pedal stroke can be matched with the same value in the time column in the first table above and this corresponds to a particular angle of tilt.  The amplitude of my motion on the ground matches the displacement of my center of gravity corresponding to that same angle.  
• More measurements need to be done, but my initial suggestion is that a correlation exists between the length of time between each pedal stroke and the angle of of tilt that can be potentially corrected.  At very slow speed the longer time of each pedal stroke allows for correcting a larger tilt away from balance.  At higher speed there is either a limit to the potential angle of tilt that could be corrected for, or the wheel has to be dramatically accelerated as a compensation, or multiple pedal strokes would be needed to correct for a larger angle of tilt.
  

Learning to Ride the Unicycle Depends on Having Learned to Walk.

• Part of the subtle magic of being able to learn to unicycle is due to the fact that we have put so much time and effort and neural development into leaning to walk.  All of the basic demands on the body's sense of position and moving adjustments are just refinements of the basic motions of walking.
• I have made a similar simplified model of the energy needed to walk.  The walking motion moves the center of mass up and down.  The arms and legs swing.  The body will move up and down (m*g*h in the equation above) and the arms and legs swing as pendulums (1/2 I *w*w above) as repetitions determined by the pace.   The same model used above when applied to these actions gives the estimate of energy demand for walking as 70 Kilo calories per hour.  Most reference standards I found state an energy demand of 90 Kilo calories per hour for walking.  My simplified model works as a reasonable first order estimate.
  

Bumps in the Road

A bicycle is an incredibly efficient transport device over relatively smooth and level roads.  The size of the wheels, long wheelbase, and stiff frame structure allow the bike to carry its moving momentum quite successfully over small obstacles.  When a small bump is struck, the geometry of the bike transfers most of that energy into a vertical lift that is not counter productive to the forward momentum.  The light weight bicycle will effectively rock over the obstruction without much change in the rider's larger independent momentum.


The unicycle cannot absorb and re-direct this energy so effectively.  In fact, the obstruction in the road is almost certain to transfer a large amount of the force into the angular velocity as listed by the factor w in the tables above.  For a unicyclist traveling at 16 mph, a change of speed of 1 mph imparted to the angular velocity w would introduce an instantaneous demand for balancing effort that is equal to 100% of the energy that is being applied to forward motion.  This energy demand is not the same as the constant rhythmic motion proposed in the tables because the repetition over times depends on how often the bumps are encountered.  In addition to energy loss to bumps, similar losses will occur due to disturbances such as a change of wind, change of turning angle, and other dissipating forces.  These disturbances will potentially add large increases to the angular velocity that must be countered by applying energy to remain in balance.

Successfully applying energy to forward motion while on a unicycle is a constant challenge of sustaining momentum.

I see this as a close analogy to cross country skate skiing, where successful propulsion is a constant challenge of controlling balance and momentum.  Every action that takes away balance has a doubly counterproductive result by losing forward momentum and demanding that energy be used to regain balance rather than being applied to forward speed.

Notice in the drawing that the center of mass, the x, tends to be over the crank axle.  This is a logical expectation, since one's weight is what allows the application of force to the pedals.






How is energy applied to maintain balance?

• Steering on a bicycle.  Very subtle steering motions convert the angular momentum of the spinning wheels into torque that rights the leaning bicycle.  Hold a front bike wheel in your hands and have someone spin it rapidly.  Now make a small steering like turn.  The wheel will also tip with a strong force.  The direction of this tip relates directly to the nature of the angular momentum of the wheel mentioned above.  A surprising property of this angular momentum in relation to steering is the fact that when steering the bike in a situation that needs a slight lean, the front wheel is initially steered opposite the turn, the proper lean will result as the conservation of angular momentum leans the wheel in the desired tilt.  This action is counter-intuitive.  To test this fact, ride at 10 mph or so in open flat pavement and place your hands in such a way that you can only push but not grasp the handle bar.  You will discover that a slight turn of the wheel to the right precedes the action of making the bike lean to the left which is necessary for a left hand turn.  The same counter motion is true in a turn to the right.
• Steering on a unicycle.   Much of the balance on a unicycle comes from steering in the direction needed to counteract a lean.  At slow speeds this is accomplished by pivoting the body in such a way as to get the hips to pivot the bike and use the legs to accelerate in the desired direction.  At slow speeds and when doing lots of maneuvering such as trying to ride in a small figure 8 pattern, one quickly feels the energy demand with the motion of turning hips, arms, and shoulders.  Ultimately the resistive force that one acts against is the contact of the wheel with the ground.  There is significant side pressure applied to the wheel, more so than on a bike, and this requires a somewhat stronger wheel and spoke configuration than is usually needed on a bike.
• I have proposed that the motion of a person on a unicycle can be simplified into a case of an inverted pendulum motion.  If one thinks of the bike and person as a long narrow rod, a secondary action is also happening as one turns.  The person will turn about an axis parallel and running through the length of the rod, like a drive shaft turning.  So the steering motions are somewhat activated by turning motion about this long axis.  Energy is applied to make changes in twisting motion about this long axis, then as the turn reacts properly to the lean, the legs can apply the next necessary energy through the pedals to move the wheel under the lean and lift the center of mass back up to the balance point.
• These corrective balancing motions are not greatly different in general range of actions than the normal actions used when walking or running, but the spread out platform of the feet in stride makes a larger leverage base to counter the changing balance.  
• When walking, one senses a slight tilt out of balance and the foot step swings out slightly to control the lean.  The center of mass is lowered with the step and lifted as the opposite leg swings by.
  
• On a unicycle, one senses a slight tilt and the cycle is steered slightly to come under the lean.  The center of mass lowers and gets lifted.
  
• On the bicycle, the frame geometry does the most complete job of allowing the center of mass to remain at an almost unvarying height and a greater proportion of the natural weight of the body can be applied to forward propulsion.  That is one reason why moving the extra weight of a bike on a smooth level road can be more efficient than walking.  The bike minimizes wasted energy and maximizes forward propulsion.  As one climbs steeper hills, at some point the extra weight of lifting the bike is no longer offset by the efficiency of the propulsive effort and most people would rather leave the bike behind and walk.  In mountainous terrain the advantage of wheels is of little use until gently sloping roads are constructed.
• As a beginning unicycle rider, one of my first leaps of improvement in riding ability was when I could let my weight rest more completely on the seat of the bike and then expend less energy twisting and moving my body and fighting the motion with perpetually tense legs.  At the early learning stage, the energy demand of riding the unicycle seems very high because of the range of twisting and turning motion in every part of the body. 

 Torque

Torque is force applied in a rotational reference.   A person standing is applying force on the ground due to the attraction of gravity acting between the person and the earth.  If the person stands on a lever such as a pedal and crank arm, the force is then described as torque.  As force or torque is applied over changing position, work is done.  As noted above, the bicycle frame geometry combined with efficient chain and gear connections is very good at using the pedal force to create the ability to do work at the wheel, propelling the mass of the bike and rider along the road.  The same torque at the pedals of the unicycle will result in a slight twist of the bike and wheel.  In the photo at the top of the page one can notice that Unisk8r has changed the geometry of the unicycle to add a handle structure.  This probably helps the rider control some of the torque at the pedal and more effectively direct the force into forward motion.  The long shape of a unicycle seat is a help in counteracting the torque of pedaling as well as being necessary to apply torque to steering.  My first attempt at learning to unicycle was on a home made unicycle.  I started with a normal touring bike seat.  With my fist effort on my homemade cycle, I realized that a long narrow seat was needed so that one's body can keep the unicycle from twisting (or make it twist as needed).

Unbalanced torque on a spinning wheel will cause precession, or an added rotation around a pivot point.  In an example above I suggested holding a spinning bike wheel to experience the effect of the conservation of angular momentum when the spinning wheel is turned.  In a similar way, if you hold the spinning bike wheel with one had supporting just one end of the axle, there is an added motion created where the spinning wheel wants to rotate horizontally (precession) about the support point at the end of the axle.  This precession is due to the torque caused by gravity that works at a right angle to the angular momentum of the spinning wheel.

On a unicycle there will be a counter productive torque applied to the spinning wheel by the leverage of the pedal and crank mechanism.  A bicycle frame with the stiff gear transmission structure and the added benefit of toe clips allows this torque to be applied very efficiently to forward motion.  On a unicycle some strategy of adding structure to the frame might diminish the otherwise counter productive effects of torque.  Toe clips might seem to make a logical structural improvement in the application of this torque, but as Lars Clausen describes in his book One Wheel-Many Spokes, Lars' attempt to use toe clips had other major drawbacks.  It is possible that in the specific case of riding a unicycle over a level track for long distance, having a heavier wheel would create more balancing torque from the spin of the wheel and allow the rider to put more pedaling effort into effective torque.  The increased angular momentum of a heavier wheel could act like a stiffer frame on a racing bike.  The extra effort needed to move the wheel up to speed would be offset by the increased ability to apply force to the pedal to maintain speed.

• Counter Balance
• Take a unicycle and hold it by the seat.  Momentarily push and lift so the wheel spins quickly while holding if off the ground by the seat while the wheel spins.  A very dynamic vibration will twist the wheel back and forth.  This is another example of the effect of unbalanced torque as the cranks and pedals move as offset weights.  If counter weights were added, similar to the way a piston engine has counter balance weight to smooth the action, the stability of the unicycle might be improved for faster riding.  These counter weights might be able to compensate even for the vibrating weight of the moving legs.
  

On the unicycle, torque at the pedal from the legs pushing is going to cause a rhythmic rotation of the wheel as a steering action.  This is a disrupting and energy consuming balance problem.  Using shorter crank arms and spinning faster might mitigate some of this torque.    The unicycle rider is probably constantly applying some extra force to both pedals simultaneously to counteract some of the unbalanced or uncontrolled torque of the driving leg.


Where The Rubber Meets the Road vs The Liquid Air

Torque (which is force) has another important relation to the force needed to move the unicycle.  The tire on the road is where all the effort has to work to finally spin the energy into motion.

Air resistance is a major force that resists the rider as speed increases.  The power needed to push through the air increases with the square of the speed.  The rider on the unicycle is not very aerodynamic compared with a fast bike rider.  

The graph shows the difference in force that must be developed by the unicycle rider and a bike rider to move at a range of likely speeds.  The rider must apply force at the pedal, but the pedal has a reduced leverage ratio with the wheel pushing on the road.   The actual force applied at the pedal will be higher depending on the size of the wheel, the length of the crank, and the gear ratio.  A lightweight bicycle in high gear often will have an effective gear of about 104 inches.  This means that in high gear the ratio of pedal motion is the same as if the rider were pedaling a 104 inch diameter wheel directly.  The unicycle riders I have noted here were all using a 36 inch wheel.  Ken Looi and Lars Clausen write of changing the length of the crank to change their effective gear, and in general they use a crank that is shorter than the standard bicycle crank.  

It turns out that when I plot the actual force needed on the pedal for speeds of 10 to 15 mph, the choice of crank lengths used by Lars and Ken will create leverage ratios such that the force at the pedal closely matches the standard bicycle.  At 15 mph the torque needed at the pedal is about 22 pounds.  Note that this is a continuous force.  Neither the bike rider or the unicycle rider can apply continuous force.  Torque will be lost due to counter productive leverages and flex in the system.  It is also not possible to constantly apply the force perpendicular to the moving crank.  Toe clips can help, but in any event the rotating crank is putting some of the pedal force into the vertical push on the ground and some of the pedal force into tipping or turning the wheel.  These limits imply that the rider is trying to push with episodic forces in the range of 30 to 40 pounds in order to produce the 22 pound force needed to travel at 15 mph.  

The unicycle rider may approach a limit as to what force is mechanically possible, given the counterproductive losses noted above.  At 20 mph the continuous force at the pedal will need to be well over 30 pounds and momentary real torque at the pedal may be double that at perhaps 60 pounds.  

• I have experimented with riding a unicycle up steep paved roads to estimate the limit of effective pedal force I can produce.  On a hill at slow speed one can calculate the force needed at the pedal by using a basic principal of leverage: The applied force times the distance it moves will equal the resistant force times the distance it moves.  For example: a 10 pound force moving 10 inches can lift a 20 pound resistance 5 inches.   On a hill, if the percent gradient is 10 percent, then each foot of travel lifts the cycle one tenth foot.  If I know how far each turn of the crank moves the cycle, then the radius of the crank arm allows me to calculate the force I need to apply at the pedal to lift my weight that one tenth foot.
• To slowly climb a hill, the Force needed at the pedal = (my weight) x (percent grade) x (wheel radius) / (crank arm radius)
• On a 15 percent grade I need an effective force of 30 pounds at the pedal to cycle up hill.  At that effort level the unicycle is going very slowly and the wheel steers so dramatically from left to right with each pedal effort that I cannot long sustain the climb without loosing balance.  I can use the rate and magnitude of this steering motion to calculate the uncontrolled torque affecting the steering angle. 
• I have no way to measure the actual force I am applying at the pedal to get the effective force of 30 pounds, but the off center torque due to the width of the pedal will be similar to the condition of a unicycle rider trying to go 20 mph on level ground as noted above where the resistance is wind rather than hill slope.  The angular momentum of the spinning wheel at the faster speed will help control this out of balance torque,  but the potential uncontrolled torque at the pedal is still present as the same force.
• The length of the seat post is long relative to the off center length of the pedal, approximately 7 to 1.  This allows the rider to counteract most of the off center torque at the pedals with the legs and body.  The energy going into steering vibrations becomes very small relative to the kinetic energy of a faster moving wheel.  For a 36 inch wheel traveling at 20 mph this vibration would be applied alternately left and right 3 times per second and create a steering vibration of 1 degree or less.
  

How the tire can create the needed push,
Good News/Bad News
The ability of the tire to push forward depends on friction between the rubber tire and the road.  The weight on the rear wheel of a bicycle will be some percentage (more than half) of the total weight of the bike and rider.  The ability of the tire to push against the road with the torque needed for forward motion is limited by the weight on the wheel and the coefficient of friction between the tire and the road.  The coefficient of friction between rubber and concrete is about 1.  Therefore the tire can provide almost as much forward push as the weight on the wheel.  The good news is that the limits of push that the tire can provide for forward propulsion is not reached at any realistic speed.  This is one factor that gives the advantage to the unicycle.  Since all of the weight of the system is on one wheel, the possible forward push is higher than is the case for the bike.  The bad news is that because of the unfavorable aerodynamics and lack of gears, the unicycle rider will need to put out more torque at the pedal than the rider's body weight at somewhere around 50 mph.


Spinning Your Wheel(s)

  As noted above, rotating mass is a primary factor in the energy demand of a bicycle or unicycle.  That is why racing bikes have expensive light weight wheels and tires.  Every time some resistance dissipates the rotating speed of any part in the system, it will take energy to increase the rotation speed of those parts in addition to the energy needed to return the entire mass to the desired speed.  Reducing the weight of rotating objects is paramount.  The mass and the radius of the rotating wheel are factors in this equation of energy. (See  A list of basic physics equations I have used.)  In the wheels, this same moment of inertia that will absorb energy when the rotating velocity is increased is also the moment of inertia that helps provide stabilizing balance.  One can ask two useful questions:

1. Is there an optimal wheel diameter?
2. How does one wheel vs two wheels play in this energy and balance process?

In Question One, the radius, mass, and geometry of the wheel do affect the energy it takes to rotate the wheel.  Consider a simple example when a wheel is allowed to roll down a ramp.  The potential energy of the wheel at the top of the ramp (due to the mass of the wheel) will be converted into the rotational energy of the wheel and forward motion at the bottom of the ramp.  The potential energy is based on mass and the height of the ramp.  The rotational energy is proportional to mass and the radius squared.  To make a long story short, it turns out that if the proportional geometry of the wheel is kept constant, it will not matter how large the radius is.  All wheels will have the same velocity along the ground at the bottom of the ramp.   The velocity over the ground is the radius multiplied by the angular velocity.  If the radius is increased, the angular velocity will proportionally decrease and the velocity over the ground remains the same.   At the end of the ramp the speed will only depend on the change in height h.  This is the theoretical answer: Radius does not mater.  

With a real wheel spinning along the road, diameter will mater.  A larger wheel will roll over small bumps more easily (see above).  The structural features of making the wheel do come into play as the size of the wheel changes.  There is a practical limit of wheel sizes that will work with the human dimensions.

 
With Question Two, I have made rough calculations for the rotating moment of inertia for various

	racing bike wheel set	mountain bike wheel set	36" wheel	24" wheel
angular momentum at 4 mph in kg * m2/sec	2.8	5.2	1.9	0.9
KE at 4 mph in joules	6	11.5	7.35	5.1
angular momentum at 15 mph in kg * m2/sec	4.3	8	7	3.3
KE at 15 mph in joules	42	81	52	36

wheels.  I have estimated the weight of the Coker type wheel by extrapolating on the dimensions of a light mountain bike wheel.    The table shows some comparisons of stabilizing momentum and energy requirements to move the wheels to 4 mph and 15 mph.  This table indicates numerically what one would intuitively guess.  At slow speed, a small unicycle wheel does not give much stability from its rotational momentum.  The 36" unicycle wheel does potentially provide some stabilizing balance.  The kinetic energy needed to rotate the 36" wheel from rest to 15 mph falls between the values for a racing bike and a mountain bike.  (Since writing this, I have had a short note from Lars Clausen  suggesting that I have underestimated the weight of the 36" wheel, but his note also reinforces my suggestion above, that extra weight in the wheel is helpful, giving much more stability.)



How to make the Unicycle Less Fun

The stabilizing effect of two wheels on a bike frame will have an additional benefit due to the length of the wheel base.  This is like adding stability to a table with a wider spread of legs.  The frame of the bike can use the extended leverage of the two spinning wheels to add another resistance to some of the adverse effects of torque at the pedals.  If one wants to make a more stable unicycle that more effectively transmits the human engine into forward motion, I propose a Hypothetical Unicycle as shown at right.  Widely spaced spinning wheels could add balance and torsional stiffness.  Small wheels closely spaced and spun at high rotational velocity could also ad stability.   Counter balances could be built into the drive wheel to minimize vibration from the unbalanced weight of cranks and pedals and perhaps even the moving legs.  (This would not work if the wheel has a gear ratio drive system.)

There is probably some structure that could improve the balance and usable energy of the unicycle, but that brings me back to my larger speculation hinted at above.  Riding the unicycle is partly fun because it doesn't stay in balance.  

A large part of the fun, and value, of skiing or biking or running or unicycling is the way these activities  demand an absolute connection to the dynamic pull of the world around us, being open to the subtle powers and the fine adjustment of the universe, and having a wide open view of space and environment.  The process of interacting with that dynamic balance within the environment gives lots of opportunity to appreciate the quality of the road and the need for assistance along the way.  Every change of light, wetting rain, and adjustment in the slope of the world tips ones attention into the here and now.  

On a unicycle, a passing glance or an unexpected "Hello" can alter one's fine tuned leverage with the world.  The balance is so precise, even changing one's mind can cause a fall.  

It seems to me that having a keen appreciation for the demand of balance and the necessity of putting effort into maintaining balance is a fine metaphor for the world around us.  Machines such as bicycles and unicycles have a very aesthetic place within that metaphor.




For further investigation:

Koyaanisqatsi: A Hopi word meaning life out of balance.  There is a fascinating artistic movie with this name that plays with the rhythm and pattern of modern life, check it out.
One Wheel-Many Spokes, the book noted above by Lars Clausen.
Physics for Scientists and Engineers, by Raymond A. Serway.  Second Edition.  The college physics text I used to find equations and methods.
Bike Energy Calculation
Unicyclist.com
Unicycle.com

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