On Bicycles, and.... what else is there?

I previously posted a route profile for the Chanteloup climb near Paris, France. This is a climb with an amazingly rich cycling history, going back to Velocio who used it to demonstrate the superiority of multi-gear bikes to the fixed-gear bikes which dominated professional racing in the early 20th century. Later it became the sight for the "Poly de Chanteloup" event, which was contested by randonneurs and professional racers. The randonneur event unfortunately seems to have died, but the professional race continued as the Trophée des Grimpeurs , a traditional last race of the season until sponsorship was lost in 2010 and it also expired. Jan Heine, in his highly recommended book "René Herse" , has the following quote about the gearing of the winning tandem in the 1949 randonneur contest. I posted this last time but I repeat it here because I find it remarkable: A single chainring was sufficient for the 14% of the climb of Chanteloup, which had to be climbed e

I've been reading Jan Heine's truly excellent book, "René Herse" . Here's a quote, referring to the tandem ridden by the builder's daughter Lili and Rene Prestat in winning the 1949 Poly Chanteloup, a circuit race which included a popular climb outside of Paris: A single chainring was sufficient for the 14% of the climb of Chanteloup, which had to be climbed eight times. In 1950 a report listed the gearing of their tandem: a single chainring with 46 teeth and a 5-speed freewheel with 13-15-18-21-22 teeth. They never used the 22-tooth cog, and they made their attack with the 46-18. I figured 14%? Sure, typical French hyperbole. It's probably 14% on the inside of a switchback. And indeed, the book has a route profile which shows the climb to average only 7%. Riding a 46-18 up a 7% grade is still a considerable gear. I PR'ed Old La Honda Road, which averages 7.3%, riding a 36-18, and the tandem gear was 28% bigger. So I checked Strava. Here's t

I was reminded of some issues in differential calculus when working on the previous blog post. I had an equation for power as a function of speed and wind speed: p(s, s w ) Since I assume that power is constant, it is an implicit equation: it defines contours in (s, s w ) space of constant power. In the case of power, solving s as a function of s w may be a considerable challenge. But The Chain Rule comes to the rescue. With the Chain Rule I can write: dP = (∂ P / ∂ s) ds + (∂P / ∂ s w ) ds w . But I'm assuming constant P, so dP = 0. I can then write: (∂ P / ∂ s) ds = −(∂ P / ∂ s w ) ds w . And with re-arranging terms I get what I want: ds / ds w = −(∂ P / ∂ s w ) / (∂ P / ∂ s) . This was all fairly simple (although that minus sign confused me when I first saw it as an undergraduate). The tricky bit for me was the second derivative. But there's no reason for it to be complicated, either. If I write the first derivative as a function r ≣ ds / ds w : d

I already showed my "simple" model for how an arbitrary wind affects speed. This was based in part on a numerical fit to an implicit power-speed calculation, where I found to decent approximation the logarithm of speed as a function of tailwind/headwind was well fit by a parabola. The formula I used, combining the effect of a tailwind/headwind with the effect of a crosswind, was: s' = exp[ −(s w ' / 3) 2 ] exp[ 2 s wx '/ 3 ], where v 0 ' is the ratio of flat-road speed with the wind to flat-road speed without the wind, s w ' is the ratio of wind speed to flat-road speed without wind, and s wx ' is the component of that in the direction of rider travel. I already showed an analytic derivation of the cross-wind term, which is proportional to the square of s w '. I also did a first-order dependence of s' on s wx '. However, to justify this full model other than numerically requires a second-order dependence of s' on s wx '.

Last time I gave up trying to solve this integral. I wanted to watch Cyclocross Worlds so got lazy and did a numerical solution and fit. But I realized while out running after the racing that I could have done much better. So I'm back for more. (1 / 2π) ∫ dφ exp[ (s w ' / 3) 2 ] exp[ −2 s wx ' / 3 ], where the integral is over the full circle and φ is the angle of the wind relative to the rider (0 = pure tail wind). The key here is to recognize that this can be well-approximated by a Gaussian for s w ' to at least 1. This isn't the solution of the integral, but it's a good approximation, so once I recognize the analytic form of the solution I can get away just matching derivatives with respect to s w ' = the ratio of the wind speed to the zero-wind rider speed. So my solution will be the following, where I must solve for K: exp[ (s w ' / K) 2 ]. I recognize that for every value of positive s wx ', there is a corresponding negative value

Last time, I discussed a simple approximation which comes fairly close to predicting the speed at constant power in an arbitrary wind of reasonable magnitude. The formula can be written as a normalized speed s' (normalized to no wind) and a normalized wind s w ' (normalized to the same scale) as follows: s' = exp[ −(s w ' / 3) 2 ] exp[ 2 s wx '/ 3 ], where s wx ' is the normalized speed of the wind in the direction of the rider (positive for tailwind, negative for headwind). For a typical closed circuit, the rider will head in random directions relative to the wind. If the wind is constant than the average distance-weighted direction is zero: the rider starts where he begins. Last time I calculated a result assuming a square course aligned with the wind. That's an over-simplification, even if it's probably representative. But I'd prefer to consider the more generally applicable case of all directions equally likely. So I assume the rider

The motivation for the preceding analysis was to add wind effects to my heuristic speed model. The philosophy of the heuristic speed model is to not rely on a constant power approximation, but try to model cyclist behavior directly. So how do riders behave when faced with a wind? When I first started riding with a heart rate monitor, an early lesson was my heart rate dropped when I was riding into the wind. This was obviously psychological: the wind was defeating me and I lost the motivation to pedal hard. Then I moved from the San Francisco Bay area to Austin where I lived for three years and winds were a predictable part of every ride, while extended hills were essentially gone. Instead of challenging myself on long hills, I learned to use treat the headwinds as a challenge rather than bad luck. As a result, I began increasing my heartrate, rather than decreasing it, when I encountered headwinds. With groups it's different: the group may be motivated to hammer into the