Question for engineers and potentially computer scientists who are so familiar with complexity problems they solve it while they crap:

Velomobile is a fully covered tricycle. It's much more aerodynamic than a bicycle but weighs at least twice more. I'm trying to quantify the overall performance comparison between touring with a bicycle and a velo, and as posted on this thread, I'll bring in only the mathematical part:

---

A stock 520 weighs 11.5kg. Add front rack and panniers, you get 13kg. Let's make the velomobile 33kg. Although it weighs 2.5x more, you have to consider the total working weight, which is 105 for 520 and 125kg for velo. Now it only weighs 19% more. Now I'll throw in some random number. The ratio of increasing effort to increasing weight on a vehicle on wheels, accounting only bearing, rolling resistance and tyre-road friction on flat terrain (let's call this drag A), is 1% for every 1% weight increased. That is, if you add 1kg more onto a 520, it will take you 99km instead of 100km in the day. Since velo package weighs 19% more, we'll say it's 20% more effort to get through drag A.

Now, take a typical long distance tour, cross USA, and the most efficient direction, west to east. This way, you'll start off with lots of hills, and gradually move into flatter terrain. Your legs take the most punishment in the beginning, having to climb hills on something that is 20% more effort to climb on. Let's say (again pulling numbers out of my ass) 5% grade takes 10% more effort, and 15% grade 40%. Let's say the average climbing grade in the beginning is 10%, descending in the same rate after the peak, and finally flat for 4 times the distance. We'll make effort to climb 10% something high, like 30%. We also need to know how much more effort is required for every percentage of weight increase when pulling it up a grade. So let's just some everything up to assume that 20% more effort on drag (A) translates to 40% more effort to climb in a velo.

Velo looks very bad so far. But we have something Velo shines on - aerodynamic efficiency. Since our cyclist is going from west to east, she'll expect nice tailwinds most of the time. The average winds are 15kph SW (depending on where it is), which makes 11Kph W effective (15 x cos(45)). However, from my experience for every 3 days of tailwinds, we can expect 2 days of headwinds. At 11Kph, on a usual cruise speed of 25Kph, a headwind slows me to 21Kph but propels me to 32Kph as tailwind. Assuming a velo is 40% more efficient, with the same effort headwind slows her to 31Kph ((21-25)/25 * (1-.4) * 35) but propels her to 47Kph ((1 + (32-25)/25) * 35 = 45; add 2 just for fun) as tailwind. During the 5 days she cycled 6 hours on tailwind days and 3 hours on headwind days (she hates headwinds), covering 768km. On a velo, she would've covered 1032km. That makes the Velo 34% better at winds than a conventional bike, on flat terrain.

And, the descent. Assuming a similar 12% grade, she'll cycle up to 28Kph, but on a velo it reaches 39Kph (28/25 * 35). Here we see that velo is 39% more efficient on descents.

Finally, we'll add this all up. Ms. Tourist sets out on a journey that has 6 parts: 1 ascent, 1 descent, 4 flat. There are no winds during the ascent/descent stage, 11Kph average and using the 3 tailwind/2 headwind model on flat. We add some time weight in here to make it 2 : 0.8 : 4, the rational being you're half your usual speed climbing 12% grade. This ratio translates to 1 month climbing, 2 weeks descending, and 2 months on flat. So, velo:normal bike will be:

((1-0.4 [climb effort] ) ^ 2) * (1.39 [descent efficiency] ^ 0.8) * (1.34 [flat efficiency] ^ 4)

= 1.29

That is, it's actually 29% more efficient to cycle on a velo given the above sets of conditions. Plotted on a graph, you will see this:

where the red area is 29% larger than the green.

-------------

The more I think about it, the more I think it's bad. How would I go about improving it? How would you compare them? I would need to know more about the relationship between energy and drag, but that's waaaay beyond me.