Power and forces
A bicycle is actually a locomotion-machine driven by muscle power; we will look at how man, as the engine of such a machine, can perform. The old unit horsepower is replaced by the Watt (W). One horsepower is ± 740 watts. For a few seconds, a trained athlete can produce more than 2200W (3 hp). Over a prolonged period (eg one hour) the maximum effort, will drop to 300-400W. Healthy, but not really trained, young men provide up to 150-200W for 1 hour. A flat timetrial of 10 minutes in the Tour de France, will be won at ± 53 km/ h; that is 500 - 600 watts. Maintaining the same training scheme, the performance of a 25 year old athlete will be steady, till about the age of 40. After 40 it will be reduced approximately by 1-1.5% per annum. On a windless day, with the rider crouching, we see the following relationship between power and speed: 50W: ± 20 km/h, 150W: ± 30 km/h, 350W: ± 40 km/h, 550W: ± 50 km/h, 1000W: ± 60km/ h This latter poweroutput obviously applies only for a brief sprint. During the 2019 Amstel Gold Race, Mathieu van der Poel rode the last 400m averaging 1200W and a 1400W peak.
The poweroutput (P total) of the rider is the multiplication of the total resistance (R total) and speed (v) divided by the efficiency (η = eta) of the drivetrain; usually about 95% (η = 0,95):
P total = R total * v / η.
The main losses in the drivetrain are chain (2%) and derailleur (2%); together an average of 5%; but bad maintenance can cause 10% loss and more. In well-greased and adjusted bearings of hubs and axles, the total loss of the bearings is less than 1%; it should be noted that in the current generation hubs and axles friction several times higher than in the Campagnolo Record series from the seventies. The use of small cogs on the freewheel (<16) will also be at the expense of efficiency; each tooth less will cost about half a percent.
There are several types of resistance: rolling resistance Rrol, airdrag Rair, slope resistance Rslope and acceleration resistance Raccel.
Together: R total = R air (1) + R rol (2) + R accel (3) + R slope (4)
(1) The air resistance R air = 0,5 * ρ air * Cd * A * v².
In calm weather is v equal to the speed of the cyclist Vc.
The ρ air is the density of air at 1 bar (at sealevel it is about 1,23 kg/ m³; at 1800m about 1 kg/ m³).
Cd is the aerodynamic form-factor, for a clunckerbike 1,1; a deep seated racer 0,9, at very low recumbents even 0,8. The real profit comes from streamlined recumbents: Alleweder 0,45; Quest 0,22; recordmachines < 0,1!
A is the frontal area; for a big guy of 2m and 100kg upright on a clunkerbike A is about 0,75m². A deep seated racer will have a frontal area of 0,3-0,35m²; ( with his hands on top of the brakes 0,5-0,6m²); a low recumbent 0,25m² and an Alleweder velomobile 0,45m². Of course, this are averages!
The power needed to overcome the air-resistance is calculated by multiplying drag and speed v and dividing by the efficiency of the drivetrain.
At 20km/h, for example, the racer provides: Pair = 0,5 * 0,9 * 1,23 * 0,35 * 5.56³ / 0,95 = 35W; on a clunckerbike: 0,5 * 1,23. * 1, 1 * 0,75 * 5,56³ / 0,95 = 91W
Although for velomobiles Cd is less, the speed gain over a racer or a two-wheeled recumbent is very limited, because of the increase of frontal area, weight and rolling resistance. Around an average of 30 km/h benefit starts to arise; but not everybody rides typically at 200 watts or more.
The winddrag of the cyclist is actually a friction force, opposite to the direction of travel. The formulas above are only correct when there is no wind!
When the wind blows, the case is more complicated. R air becomes R wind.
The winds associated with the Beaufortscale (0-12 scale) are measured at 10m height. The speed of the wind in meters per second as an approach to wind power: 1: 0,2 m/s 2: 1 m/s 3: 3 m/s 4: 5 m/s 5: 9 m/s 6: 12 m/s 7: 16 m/s 8: 19 m/s 9: 23 m/s 10: 27 m/s 11: 31 m/s 12:> 35 m/s
This is the wind speed on top of the hill or dike; at cyclists level we are glad this speed is usually less. Therefore we need to correct it; one scale less in open terrain; between houses, hedges or bushes, two scales off. A low recumbent catches even less wind, so one extra scale off. The effect of the crosswind is larger, because it hits a larger area. Wheels, luggage, frame tubes, all offer more resistance (factor area increases from 1 to 1,5). Above a real wind speed of 15m /s cycling in head wind is no fun at all; it is more like wrestling. Wind strength and direction during a trip is hard to measure and will only be constant for a few seconds. To be able to calculate, we will use everages as constants. The wind direction is measured in degrees: 0 is head-on against the wind and 180 is the wind right behind us. This is the angle of the wind and the direction of travel, at standstill !
The wind we feel on the bike, the result of the riders speed VF and wind speed Vw: the apparent wind-speed Vs. This results in an apparent wind force FS; it has an apparent wind direction αS and is resolved in the forward and side component. The lateral force does not count, because it does not use energy. We lean against the wind, and perceive this sideforce typically when a truck comes along and disturbes the equilibrium. αW is the angle of the wind and the direction of travel in degrees (at a standstill!) and αS is the angle of the apparent wind direction and the direction of travel in degrees.
In the triangle formed by VS, V F and parallel to V W the cosinus-rule can be used ; NB.: cos (180-αw) = cos αw VS² = VF² + FW² + 2 * VF * Vw * cos αw. So we can derive VS . The cosinus-rule, used in the triangle with αS enables us to determine cos αS; cos αS = (VS² + VF² - FW²) / (2 * V F * VS )
The strength of the apparent wind FS = 0,5 * ρ air * C w * A * VS². The resistance of the wind head-on is : R wind = Fs * cos αS . The power needed for air + wind resistance, i.e. the energy required to overcome the air resistance + wind, is: Pwind = R wind * VF / 0,95. The wind direction in the computer must be entered in degrees, 0 against wind and 180 wind right behind. This is the wind direction at a standstill!
(2) The rolling resistance R rol = m * g * Cr. The power consumption of rolling resistance, can be calculated if we multiply Rrol with velocity v in m/s, and divide it by the efficiency of the drivetrain.
The rolling coëfficiënt Cr for the following tyres on smooth concrete (eg a gym or racetrack) is:
Clement Seta tubular with 8 bar = 0.002 ; Racing tyre 23 mm 7 bar = 0.003 ; Touring tyre 32 mm with 5 bar = 0.005 ; ATB tyre rough profile 47 mm 3 bar = 0.009 ; Anti-puncture tyre 37 mm with 5 bar = 0.007 ; Anti-puncture tyre 37 mm 3 bar = 0.01
NB: especially in a tough anti-puncture tyre, the tyre-pressure is hard to estimate by squeezing it with your hand! If we do not use smooth concrete, but rough asphalt or poor road surface, rolling resistance will be much higher), on good pavement the value for 23mm racing tire is 0.005 (so + 0.002) .
(3) The acceleration resistance R accel = m * a is mass (m) multiplied by acceleration (a). This variable (3) is actually only included for completeness. If we ride at a constant speed, this value is zero. If there is more power produced, then used at the moment, the cyclist will accelerate. At a higher speed restistance increases (especially the airdrag). When the available, and used power are equal, we riding at a constant speed again.
Rotating parts use extra energy as we accelerate, especially if the radius of the movement is large, wich is the case with wheels and tires. (This extra energy, about 2%, is neglected in the calculations below).
(4) The slope resistance is: R slope = m * g * "%". Here, m is the mass in kg, g gravity 9.81 N / kg, and "%" is the percentage slope. The use of the inclination is not entirely correct. This is because we should use the tangent of the angle of inclination. Actually, we are using the sinus, but the difference will only be noticeable above 21% (tan12 = 0.213 and sin12 = 0.208 : so both 21%). A negative inclination is downhill!
The calculation: This program can be run if you have a spreadsheet like Excel on your computer! You can download power&speed.xlsx at the end of this page . A free spreadsheet program is available through Open Office; there is one in Google apps too. The effect of the wind on our power-out (P total) can be compared wind-less weather (Pcalm). NB. At the bottom of the calculation is a field in which your poweroutput can be converted into an expected speed; it's a cube root. This calculation is complicated, see J.Lieh at “Info on the web”. The wind and acceleration are not included in this calculation.
An example: a rider and his bike weigh 75 kg; they climb the Mont Ventoux. The climb is 18 km long and is completed in one hour (5 m / s); the inclination is 9%. The required power for climbing is about 331 W. But our hero has also got an air resistance of 31W and a rolling resistance of 18W to overcome. In total about 380W. Unfortunately, there is still 5% loss in the drivetrain, so he must deliver 400W.
If his gearing used was 42-21 and the tire had a circumference of 2.1m, one revolution of the crank is (42/ 21) * 2.1 = 4.2 m; this is 238 revolutions per kilometer and about 71 rpm. When he is sixtie and 25 kilos heavier, he want to do the climb again, but he only delivers 150W. How long will it take? What gearing should he use, when he wants to ride at 60 rpm (1 per second) ?
If the power output is low, almost all power is used for climbing. Roughly Pslope is about 135W = m * g * "%" * V / η Now we can calculate his speed V; this is 1.5 m / s = 5.4 km / h. The climb will take approximately 3:20 h. This is 200 minutes or 12,000 revolutions. The distance traveled per revolution is 18000m: 12000 = 1,50 m. If we take an ATB-crankset, with a small sprocket 24, used in combination with a 34 as largest sprocket on the cassette, we find a development of 1.48 m per revolution of the crank. So the gearing 24-34 would be a good choice.
What forces do cyclists apply? The main forces are exerted on the pedals. Here is the formula: the average effective force is the average power, divided by the pedaling speed (rotationspeed in m/s) : Favg = Pavg / vpedal
If Pavg is 300W at 60 rpm, and the cranklengte 175mm (d = 0.35 m), vpedal is: pi * 0.35 M * 1/s = 1.1 m/s ; Favg is: 300 Nm/s / 1.1 m/s = 273N
Here we see actual forces exerted by a real cyclist on the pedal. The direction of the force is more or less vertical. We will decompose the force into two forces: one component perpendicular to the crank and a force in the longitudinal direction. The portion that is converted into labor, is the component perpendicular to the crank arm. This is converted through the axle into chaintension. The component in the longitudinal direction is absorbed by the crank. This is similar to the reaction force of the floor. There is no work and no loss of power!
As you can see, the rider looses power, because he lets his pushing leg, bring up the other leg. So the actual force on the pedal, is higher than the effective force. At the dead spots ( 6h clock), no power is developed! In all steep slopes (> 15%) you will notice this effect. It is also felt very well on exercise bikes without flywheel. Normally a good circular motion is maintained on the basis of the kinetic energy in the system.
Trained cyclists use their strength better than this rider. They will push later and stop earlier. In addition, they pull their clipless pedals in the upward movement; it is small contribution, a few percent, but it assists and does not cost energy.
The peak force on the pedals will be higher than the average effective force, about 1.3-1.5 as high. In a sprint even 2-3 times as high. The rider can no longer remain seated, because he exerts forces greater than its weight. He will get out of the saddle and combine rotation, muscle strength and weight. This is called "jacking". It is similar to the run of a high jumper, wherein energy forward, is converted into speed in the vertical direction. Powerpeaks above 2000 N have been measured!
In efforts ± 250W, the force will be around 20% of body weight (this is inter alia dependent on the resistance). Our riding force is not a pure torque, because we only use one leg at a time to push. Twice per revolution there is a pulse.
If the top position of the pedal is called 0 degrees and the lower position 180, we generate about 90% of our power between 50 and 140 degrees. Moment (or torque) is force multiplied by lever-arm: M = F * r
With a locked shoe we can excert a pulling action (max.5%) on the lifting pedal. The torque of trained cyclists is never really zero, but the minimum is not more than a few percent.
Now what is the tension in the chain? Say a touringcyclist with luggage rides a 20% hill; the maximum force exerted is 500N, the cranklengte is 175mm and it is used for a triple crankset with a 28 sprocket (a radius of 57mm). The maximum time is 500N * 0,175m = 87.5 Nm; it also passes through the gear: F = M / r = 87.5Nm: 0,057m = 1535N. This is the peak of the tension in the chain.
At a gearratio of 42 teeth in front and 21 teeth behind, the speed of the rear axle doubled compared with the cranks and the torque halved! If the wind is head-on, we need to shift down; as pedaling and step rate will remain the same, take speed and speed of the rear wheel off. The torque is increased to compensate for the increased resistance. The power remains the same: it is linked times crankshaft speed.
The pedaling frequencies for tourists are usually between 60-80 rpm. For racers 90-120 rpm is common ( good trackracers, even more than 200 rpm). High speeds are not really efficient; the rider goes for maximum power, not maximum efficiency.
What force F does a cyclist need to ride a long 17% slope? If the total weight of rider + bike 1000N, then: F = 1000N * 0.17 = 170N.
This force should be delivered to the street by a wheel with 670mm diameter; the torque to be delivered : M = 170N * 0,335m = 57Nm.
This torque also passes through spokes and hubs. Set our flange has a radius of 0.025m; then the force in the hub flange: 57Nm / 0.025m = 2280N. If there are 36 spokes present, it is at least 63N per spoke.
This is the case when the spoke is perpendicular to the axis of the flange ; at 3x crossed spoke-pattern the angle to the axis is 60° , so we must divide by sin 60° ; this gives a value of 73N per spoke.
Is this rider able to climb that hill, if he has a defect rearderailleur, and just one gear 42-14? Say the maximum average pedaling force he can devellop is 500N and the cranks are 170mm. Then his maximum torque will be (0,170m * 500N) / (42/ 14) = 28.3 Nm! Not at all, he needs twice as much: 57 Nm ; he should use 42- 28 at least!
Low pedaling frequencies (<40) can harm the knee joints. How much power he will have to produce, if we keep 40 as a minimum?
If he does 40 rpm (= 0.67 per sec) the wheel circumference = pi * D = 2, 1m; the gear ratio is 42 / 28 = 1,5.
The speed (velocity) v : 2,1m * 0,67 1/s * 1.5 = 2m/s (= 7,2km /h). The required climbing power P slope = 1000N * 0.17 * 2 m/s = 340W. If we estimate 40W for rolling resistance , air resistance, and efficiency- losses, we need 380W total output.
INFO ON THE WEB:
Wikipedia on Bicycle performance: https://en.wikipedia.org/wiki/Bicycle_performance
For people who to like caculate, a link: www.analyticcycling.com
The mathematical background of the calculation from power to speed comes from:
Junghsen Lieh: Determination of Cycling Speed Using a Closed-form solution from Nonlinear Dynamic Equotations
A good German book M. Gressmann: Fahrradphysik und Biomechanik; ISBN: 978-3-7688-5222-7
A cyclist moves three times faster with the same effort, as a walker; the bicycle is the perfection of the locomotion of man
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