Streamliner Physics |
by Nick Demma - Updated Feb 23, 2006 |
Purpose: The purpose of this analysis is to show some of the practical consequences of the physics associated with riding streamliners. In other words, what's it like to ride one of those things? There are
many good sources of information about bicycle physics, like this: Procedure: Use the information from the web site (above) to figure out the equations which describe the power Vs speed for three different bikes. The first is the upright bike; the second is the recumbent with the full foam fairing; the third is the recumbent with the full hard shell fairing. Simulate a series of hills by assembling them out of a bunch of hyperbolic tangent functions, then give the bikes an initial speed and send them coasting down the hill. Also show what happens when streamliners try to ride with upright bikes. Data: The web site gives the speed at 250 Watts of power output and the power at
40 Km/Hr. From these two points we can find an equation that describes the power as a function of the speed. Let's work through this for the upright bike. All calculations are done using the meter-kilogram-second (i.e. metric) because our system is utterly idiotic. In deference to those who want to understand the results, the answers are converted to familiar units. The drag force due to rolling resistance is proportional to the speed and the drag force due to wind resistance is proportional to the square of the speed. Since the power is the force times the speed, the power needed to overcome the rolling resistance is proportional to the square of the speed and the power needed to overcome the wind resistances is proportional to the cube of the speed. Let's let x = the speed squared and let's let y = the speed cubed. The power as a function of speed is therefore: Power = a * x + b * y = z We know the power for two different speeds, so we have two equations and two unknowns: a
* (8.0556)^2 + b * (8.0556)^3 = 250 Solving this system of equations gives us the coefficients a and b. We now have an equation that gives us the power at any speed. If you want to see it, here it is (Indeed, it is here even if you don't want to see it.): Upright Bike Power = 3.3912 * speed +0.42598*speed^3
Figure 2 shows the speed as a function of the location. The upright bike is shown in black; the foam streamliner is shown in blue; the best streamliner is shown in red. The best streamliner is going so fast that the second hill is little more than a speed bump, but the other streamliner barely makes it over the second hill. The upright bike does not make it to the second hill. Since the point of this analysis is to point out what riding streamliners is like, we should take a close look at the performance (and I use the term loosely) of the upright bike. Notice that its speed drops to a lower level after each of the hills that constitute the main hill. In other words, the rider of the upright bike will not experience the large hill as one hill, but rather as a series of four hills that are separated by plateaus. The streamliners are so efficient that they carry the kinetic energy from the previous hill right into the next hill, so they experience the descent as one big hill.
Figure 3 shows the distance as a function of the time. The upright bike goes 1.23 miles and the best streamliner goes 2.54 miles. It is interesting to note that the presence of the second hill (i.e. the speed bump) actually increases the distance that the streamliners go. The reason why this happens is the going fast wastes energy even in a streamliner. Remember that brief period in American History when people cared about the future enough to impose a 55 MPH speed limit? That saved fuel. The second hill converts kinetic energy into potential energy causing the streamliners to get out of that high-speed energy-wasting mode. Once they have traveled farther and slowed down a bit, the hill gives them back the energy and they travel farther than they would have without the hill. This is not to suggest that going fast down the big hill is a waste of energy. It is better to store the energy from the hill as kinetic energy by going fast than to waste the energy in air resistance and have nothing left shortly after reaching the bottom of the hill. Figure 3: This is a plot of the location as a function of time.
Figure 4: This is a plot of the speed as a function of time. If the bike's weights are the same, then they all derive the same amount of energy from the Earth's gravitational field when they go down the hill. However, power is energy per second and the streamliners go faster, so they derive more power from the hill. They derive more power from the hill because they are going faster and this power causes them to go faster, which, of course, gets them even more power from the hill, so they go even faster. This feedback mechanism is what causes streamliners to go so fast down hills, and it doesn't take much of a hill. There are some bike trails that are built on old railroad beds where there are often very long hills that have very little slope. Sometimes the slope is so small and the area is so wooded that you can't really see the hill. The rider of a streamliner thinks "Hmmm, why am I going 30 MPH? Must be a hill around here somewhere". For a final comparison, let's see what happens when somebody in a streamliner tries to ride with somebody on an upright bike. This is not realistic because riders of streamliners get bored out of their minds when going that slow so they quit riding with people on mountain bikes. Let's suppose that each rider is going to put out 200 Watts when pedaling. The rider of the upright bike will have to pedal all the time to maintain any kind of reasonable speed, which will actually be 16.62 MPH. To stay with the upright bike, the rider of the streamliner will pedal when the speed drops to 1 MPH slower than the other bike and then coast when the speed is 1 MPH greater. Figure 6 shows the speed as a function of time. The red trace is for the streamliner and the black line is for the upright bike. Obviously the recumbent rider doesn't pedal very much. For a better view of just how much, Figure 7 shows the power as a function of time. A closer look reveals that the rider is pedaling 16.5 % of the time. Since both riders put out 200 Watts when pedaling, the total energy used in the ride is proportional to the duty cycle. In other words, after a 100 mile ride, the rider of the upright bike will feel like he has ridden 100 miles that the recumbent rider will feel like he has ridden 16.5 miles. Figure 6: This is a plot of the speed as a function of time.
If the recumbent rider were riding alone and putting out 200 Watts all the time (as did the upright rider), then his speed would be 38.8 MPH and he would finish the 100 mile ride in 2 hours and 35 minutes instead of 6 hours for the upright rider. The recumbent rider would still use only 43% as much energy as the upright rider. Another aspect of the plots in Figure 7 that is interesting is that the coasting periods are 16.5 seconds long. I once rode with a group around Elm Creek Park Reserve and there was a guy on a trike following me. Because he was so low, he could see my feet at the lower part of the pedal stroke because my streamliner is open on the bottom. After the ride, he said "Man, you were hardly ever pedaling!". |