Bicycle performance

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In both biological and mechanical terms, the bicycle is extraordinarily efficient. In terms of the amount of energy a person must expend to travel a given distance, investigators have calculated it to be the most efficient self-powered means of transportation.¹ From a mechanical viewpoint, up to 99% of the energy delivered by the rider into the pedals is transmitted to the wheels, although the use of gearing mechanisms may reduce this by 10-15% ^{2 3}. In terms of the ratio of cargo weight a bicycle can carry to total weight, it is also a most efficient means of cargo transportation.

Contents 1 Energy efficiency 1.1 Typical speeds 2 Weight vs power 2.1 Kinetic energy 2.2 Convert to calories 2.3 Advantages 2.4 Explanations 3 Aerodynamics vs power 3.1 Power required 3.2 Energy cost of acceleration 3.3 Why experience and the math might disagree 4 Notes 5 References 6 See also

[edit] Energy efficiency

A human being traveling on a bicycle at low to medium speeds of around 10-15 mph (16-24 km/h), using only the energy required to walk, is the most energy-efficient means of transport generally available. Air drag, which increases with the square of speed, requires increasingly higher power outputs relative to speed. A bicycle in which the rider lies in a prone position is referred to as a recumbent bicycle or, if covered in an aerodynamic fairing to achieve very low air drag, as a streamliner.

On firm, flat, ground, a 70 kg man requires about 100 watts to walk at 5 km/h. That same man on a bicycle, on the same ground, with the same power output, can average 25 km/h, so energy expenditure in terms of kcal/kg/km is roughly one-fifth as much. Generally used figures are

• 1.62 kJ/(km∙kg) or 0.28 kcal/(mile∙lb) for cycling,
• 3.78 kJ/(km∙kg) or 0.653 kcal/(mile∙lb) for walking/running,
• 16.96 kJ/(km∙kg) or 2.93 kcal/(mile∙lb) for swimming.

For many people whose running might be limited by muscle and knee pain, cycling offers comparable outdoor exercise that can be enjoyed by people of a wide range of fitness levels: it is a "no-impact" sport that is easy on the body, especially on the knees, as long as the bike is properly "fit." In addition, since bicycling can also provide convenient transportation, less self-discipline may be required to keep to the activity, since it has a practical purpose. However, because of its efficiency, cycling requires a longer distance, and often greater time, than running to consume the same amount of energy.

The average "in-shape" person can produce about 3 watts/kg for more than an hour (e.g., around 200 watts for a 70 kg rider), with top amateurs producing 5 watts/kg and elite athletes achieving 6 watts/kg for similar lengths of time. Elite track sprinters are able to attain an instantaneous maximum output of around 2,000 watts, or in excess of 25 watts/kg; elite road cyclists may produce 1,600 to 1,700 watts as an instantaneous maximum in their burst to the finish line at the end of a five-hour long road race. Even at moderate speeds, most cycling energy is spent in overcoming aerodynamic drag, which increases with the square of speed; therefore, power needs increase approximately with the cube of speed.

[edit] Typical speeds

Typical speeds for bicycles are 16 to 32 km/h (10 to 20 mph). On a fast racing bicycle, a reasonably fit rider can ride at 50 km/h (30 mph) on flat ground for short periods. The highest speed ever officially recorded for any human-powered vehicle on level ground and with calm winds without external aids (such as motor pacing and wind-blocks) is 130.36 km/h (81.00 mph). That record was set in 2002 by Canadian Sam Whittingham with the Varna Diablo II, a highly aerodynamic recumbent bicycle.

[edit] Weight vs power

There has been major corporate competition to lower the weight of racing bikes through the use of advanced materials and components. Additionally, advanced wheels are available with low-friction bearings and other features to lower road resistance. In measured tests these components have almost no effect on cycling performance. For instance, lowering a bike's weight by 1 lb, a major effort considering they may weigh less than 15 lb to start with, will have the same effect over a 40 km time trial as removing a protrusion into the air the size of a pencil. For this reason more recent designs have concentrated on lowering wind resistance, using aerodynamically shaped tubing, flat spokes on the wheels, and handlebars that allow the rider to bend over into the wind. These changes can impact performance dramatically, cutting minutes off a time trial.

[edit] Kinetic energy

Consider the kinetic energy and "rotating mass" of a bicycle in order to examine the energy impacts of rotating versus non-rotating mass.

The kinetic energy of an object in motion is^[1]:

E = ½mv²

Where E is kinetic energy in joules, m is mass in kg, and v is velocity in meters per second. For a rotating mass (such as a wheel), the formula is:

E = ½Iw². I = mr², w = v/r

Where I is the moment of inertia, w is the angular velocity in radians per second, r is the radius in meters. When a rotating mass is moving down the road, the KE of the mass at speed plus the KE of the rotating mass. Since a wheel is rotating at the same speed the bike is traveling, the KE of this wheel is now:

E = ½mv² + ½Iw²

substituting for I and w, we get: E = ½mv² + ½mr²v²/r²

the r² term cancels, and we finally get: ½mv² + ½mv² = mv²

In other words, twice the KE of a non-rotating mass on the bike. The is the kernel of truth in the old saying that "A pound off the wheels = 2 pounds off the frame."^[2] One other interesting point from this equation is that for a bicycle wheel, rotating at the same speed as the bike is moving, the KE is independent of wheel radius (r^2 cancels). In other words, the advantage of 650C or other smaller wheels is due to their weight savings (less material in a smaller circumference) rather than their smaller diameter, as is often stated. The KE for other rotating masses on the bike is tiny compared to that of the wheels. For example, pedals turn at about 1/5 the speed of wheels, so their KE is about 1/25 (per unit weight) that of a spinning wheel.

[edit] Convert to calories

Assuming that a rotating wheel can be treated as the mass of rim and tire and 2/3 of the mass of the spokes, all at the center of the rim/tire. For a 180 lb rider on an 18 lb bike (90 kg total) at 25 mph (11.2 m/s), the KE is 5625 joules for the bike/rider plus 94 joules for a rotating wheel (combined 1.5 kg of rims/tires/spokes). Converting joules to calories (multiply by 0.0002389) gives 1.4 calories.

That 1.4 calories is the energy necessary to accelerate from a standstill, or the heat to be dissipated by the brakes to stop the bike. These are kilocalories, so 1.4 calories will heat 1 kg of water 1.4 degrees Celsius. Since aluminum's heat capacity is 21% of water, this amount of energy would heat 800 g of alloy rims 8 °C (15 °F) in a rapid stop. Rims do not get very hot from stopping on flat ground. To get the rider's energy expenditure, consider the 24% efficiency factor to get 5.8 calories—accelerating a bike/rider to 25 mph requires about 0.5% of the energy required to ride at 25 mph for an hour. This energy expenditure would take place in about 15 seconds, at a rate of roughly 0.4 calories per second, while steady state riding at 25 mph requires 0.3 calories per second.

[edit] Advantages

So, what is the advantage of light bikes, and particularly light wheels, from a KE standpoint? KE only comes into play when speed changes, and there are certainly two cases where lighter wheels should have an advantage: sprints, and corner jumps in a criterium.^[3]

In a 250 m sprint from 36 to 47 km/h to (22 to 29 mph), a 90 kg bike/rider with 1.75 kg of rims/tires/spokes increases KE by 6,360 joules (6.4 calories burned). Shaving 500 g from the rims/tires/spokes reduces this KE by 35 joules (0.04 calories = 0.01 watt-hour). The impact of this weight savings on speed or distance is rather difficult to calculate, and requires assumptions about rider power output and sprint distance. The Analytic Cycling web site (www.analyticcycling.com)^[4] allows this calculation, and gives a time/distance advantage of 0.16 s/ 188 cm for a sprinter who shaves 500 g off their wheels. If that weight went to make an aero wheel that was worth 0.03 mph at 25 mph, the weight savings would be canceled by the aerodynamic advantage. For reference, the best aero bicycle wheels are worth about 0.4 mph at 25, and so in this sprint would handily beat a set of wheels weighing 500 g less.

In a criterium race, a rider is often jumping out of every corner. If the rider has to brake entering each corner (no coasting to slow down), then the KE that is added in each jump is wasted as heat in braking. For a flat crit at 40 km/h, 1 km circuit, 4 corners per lap, 10 km/h speed loss at each corner, one hour duration, 80 kg rider/6.5 kg bike/1.75 kg rims/tires/spokes, there would be 160 corner jumps. This effort adds 387 calories to the 1100 calories required for the same ride at steady speed. Removing 500 g from the wheels, reduces the total body energy requirement by 4.4 calories. If the extra 500 g in the wheels had resulted in a 0.3% reduction in aerodynamic drag factor (worth a 0.02 mph speed increase at 25 mph), the caloric cost of the added weight effect would be canceled by the reduced work to overcome the wind.

Another place where light wheels are claimed to have great advantage is in climbing. Though one may hear expressions such as "these wheels were worth 1-2 mph," etc. The formula for power suggests that 1 lb. saved is worth 0.06 mph on a 7% grade, and even a 4 lb savings is worth only 0.25 mph for a light rider. So, where is the big savings in wheel weight reduction coming from? One argument is that there is no such improvement; that it is "placebo effect". But it has been proposed that the speed variation with each pedal stroke when riding up a hill explains such an advantage. However the energy of speed variation is conserved; during the power phase of pedaling the bike speeds up slightly, which stores KE, and in the "dead spot" at the top of the pedal stroke the bike slows down, which recovers that KE. Thus increased rotating mass may slightly reduce speed variations, but it does not add energy requirement beyond that of the same non-rotating mass.

Lighter bikes are easier to get up hills, but the cost of "rotating mass" is only an issue during a rapid acceleration, and it is small even then.

[edit] Explanations

Possible technical explanations for the widely claimed benefits of light components in general, and light wheels in particular, is as follows:

1. Light weight wins races with significant climbing because the heavier bike can't make up the gap on descents or on the flats: the rider on the lighter bike just drafts. Alternatively, if the identical riders of heavier and lighter bikes simultaneously reach the bottom of a climb to the finish, all of the advantage goes to the lighter bike. This is not the case in a hilly time trials (or riding solo), where the advantage of heavier, but more aerodynamic wheels would easily make up the distance lost in climbs. In climbing, lighter wheels offer no particular advantage vs. a lighter frame, because there is no net loss of KE.

2. Light weight wins sprints because it accelerates more easily. But note that heavier aerodynamic wheels gain significant advantage as speed increases, and for a good part of a sprint a rider is doing little accelerating but is working hard against a high-speed wind. So many sprint situations may favor heavier but more aerodynamic wheels.

3. Light weight wins in criteriums because of the constant acceleration out of every corner. Heavier but more aerodynamic wheels offer little advantage because the riders are in a group most of the time. The energy savings from lighter wheels is minimal, but it may be more significant that the leg muscles have to put out just that bit of extra effort at each jam.

There are two "non-technical" explanations for the effects of light weight. First is the placebo effect. Since the rider feels that they are on better (lighter) equipment, they push themselves harder and therefore go faster. It's not the equipment that increases speed so much as the rider's belief and resulting higher power output. The second non-technical explanation is the triumph of hope over experience—the rider is not much faster due to lightweight equipment but thinks they are faster. Sometimes this is due to lack of real data, as when a rider took two hours to do a climb on their old bike and on their new bike did it in 1:50. No accounting for how fit the rider was during these two climbs, how hot or windy it was, which way the wind was blowing, how the rider felt that day, etc.

Another explanation, of course, may be marketing benefits associated with selling weight reductions.

In the end, the "incremental muscle power requirement" argument is the only one that can support the claimed advantages of light wheels in "jump" situations. This argument would state that: if the rider is already at the limit on each jump or each stroke of the pedals, then the small amount of extra power required for the extra weight would be a significant physiologic burden. Whether this is true is not clear, but it is the only explanation for the claimed advantage of wheel weight savings (compared to saving weight from the rest of the bike). For these accelerations, it makes no difference whether 1 lb is taken off the wheels or 2 lb off the bike/rider. The miracle of light wheels (compared to saving weight anywhere else in the bike/rider system) is hard to see.

[edit] Aerodynamics vs power

Heated debates over the relative importance of weight savings and aerodynamics are a fixture in cycling. This is an attempt to at least get the equation-based parts of the debate clarified. There will always be those who argue that "experience trumps mathematics" on this issue, so this will attempt to highlight those areas where experience might disagree with the math. From this, perhaps further discussion can focus on the topics of dispute rather than questioning known physics. To be as clear as possible, this will cover 1) the power requirements for moving a bike/rider 2) the energy cost of acceleration, and then 3) why experience and the math might disagree.

[edit] Power required

There is a well known equation that gives the power required to push a bike/rider through the air and to overcome the friction of the drive train:

P = (Vg*W*(K1+G) + K2*(Va)^3)/375

Where P is in horsepower, Vg is ground speed (mph), W is bike/rider weight in pounds, G is the grade, and Va is the rider's speed through the air (mph). Grade is feet or altitude gain per foot of horizontal distance, and while often expressed in per cent, in this equation is used as a decimal (a 6% grade is 0.06). K1 is a lumped constant for all frictional losses (tires, bearings, chain) and units conversion, and is generally reported with a value of 0.0053. K2 is a lumped constant for aerodynamic drag and is generally reported with a value of 0.0083. Note that power to overcome friction and gravity is proportional only to rider weight and ground speed. Power to overcome wind drag is proportional to the cube of the air speed.

For reference, 1 hp-hr = 641 "calories" delivered to the pedals, 1 hp = 746 watts, 1 calorie = 4.186 kJ. Here, all calories are kg-calories, "big" calories, or "food calories." The human body runs at about 24% efficiency for a relatively fit athlete, so to deliver 1 hp (746 watts) to the pedals requires the body to consume about 2700 calories, more than 4 times the 641 "calories" delivered to the pedals.

Obviously, both of the lumped constants in this equation depend on many variables, such drive train efficiency, the rider's position and drag area, aerodynamic equipment, tire pressure, road surface, etc. Also, recognize that air speed is not constant in speed or direction nor easily measured. It's certainly reasonable that the aerodynamic lumped constant would be different in cross winds or tail winds than in direct head winds, as the profile the bike/rider presents to the wind is different in each situation. Also, wind speed as seen by the bike/rider is not uniform except in zero wind conditions. "Weather report" wind speed is measured at some distance above the ground in free air with no obstructing trees or buildings nearby. Yet, by definition, the wind speed is always zero right at the road surface. Assuming a single wind velocity and a single lumped drag constant are just two of the dreaded "simplifying assumptions" of this equation. You should know that many high powered Computational Fluid Dynamics modelers have looked at the "bicycle problem" and pronounced it "really hard." In layman's terms, this means that much more sophisticated models can be developed, but they will still have simplifying assumptions.

Given this simplified equation, however, one can calculate some values of interest. For example, assuming zero wind, one gets the following results for calories required and power delivered to the pedals (watts):

- 630 calories per hr (174 watt-hrs) for a 200 lb. bike + rider to go 20 mph on the flats (76% of effort to overcome aerodynamic drag), or 5.7 mph on a 7% grade (3% of effort to overcome aerodynamic drag).

- 1125 calories per hr (310 watt-hrs) for a 200 lb. bike + rider at 25 mph on the flats (83% of effort to overcome aerodynamic drag) or 9.8 mph on a 7% grade (7% of effort to overcome aerodynamic drag).

- 585 calories per hr (161 watt-hrs) for a 140 lb. bike + rider to go 20 mph on the flats (82% of effort to overcome aerodynamic drag), or 7.4 mph on a 7% grade (3% of effort to overcome aerodynamic drag).

- 1070 calories per hr (295 watt-hrs) for a 140 lb. bike + rider at 25 mph on the flats (87% of effort to overcome aerodynamic drag) or 12.6 mph on a 7% grade (7% of effort to overcome aerodynamic drag).

Shaving 1 lb. off the weight of the bike/rider would save 0.01 mph at 20 mph on the flats (1 second in a 25 mph, 25 mile TT). Losing 1 lb. on a 7% grade would be worth 0.04 (200 lb. bike + rider) to 0.07 mph (140 lb bike + rider). If one climbed for 1 hour, saving 1 lb. would gain between 225 and 350 feet - less effect for the heavier bike + rider combination (e.g. 0.04 mph * 1 hr * 5280 ft/mile = 225 ft.). For reference, the big climbs in the TdF have the following average grades: Tourmalet = 7%, Galibier = 7.5%, L"Alpe D'Huez = 6.8%, Mont Ventoux = 6.7%. You can confirm the accuracy of this equation as follows: calculate "level ground power" with the equation, then add the climbing power (raising the weight of the bike/rider so many vertical feet per hour). You get the same result as the equation delivers with the grade calculation included.

[edit] Energy cost of acceleration

[edit] Why experience and the math might disagree

[edit] Notes

• 1 "Bicycle Technology", S.S. Wilson, Scientific American, March 1973
• 2 "Johns Hopkins Gazette", Aug.30, 1999
• 3 See Chapter 9 of "Bicycling Science" (Reference, below) for details of transmission efficiency.
• 4 Physics-bases simulation of bicycle race performance

[edit] References

[edit] See also

• Bicycle
• Bicycle and motorcycle dynamics