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

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 ...

Back in July 2009 I did a series of first-order calculations on the effect of various parameters on riding speed. Calculating speed from power is difficult to do explicitly, but to first order the calculations become straightforward. First-order analysis is where much of the intuition is, anyway. One of these calculations was the effect of wind resistance on riding speed . Then in November 2009 I extended that analysis to the effect of wind speed on riding speed . To my dismay, I found an error in that result. I'd even rationalized the wrong result with incorrect arguments. I had to track the consequences of that error through the following two blog posts. I think I fixed everything. The corrected result was: d s / d s w = 2f / [ 2f + 1 ‒ s w / s ] where s is the rider speed, s w is the tailwind speed, and f is the initial fraction of retarding force due to wind resistance. There appears to be a singularity issue for strong tail-winds (see the denominator) but then ...

So much for my 6.74 degree estimate.... Theory: Experiment: Not that I'd want an aero frame in those conditions... like the guy on the Cervelo SLC who can barely keep his bike from going airborne.

Here's my fourth post in my series where I try to derive a probability distribution for yaw angle. I ask myself why I spent so much time on it, but it became something of a silly obsession, and I suppose as far as temporary obsessions go it isn't so bad. I could have been doing Sudoko or doing crosswords, for example, or playing games on my new "smart" phone. Or worse, responding to work emails and studying manuals or reading literature. Anyway, here's the equation I derived: P(yaw) ∝ 2 cos yaw / sqrt [ γ² ‒ sin² yaw ] First I want to integrate this to derive an analytic normalization constant. For that I cheated and used the Wolfram Integrator : 2 arctan [ sin yaw / sqrt ( γ² ‒ sin² yaw ) ] which integrated across allowed values of yaw yields 2 π. In retrospect this is obvious, as this is the range of headings I'm considering, and I'm normalizing over all headings. So I can write the normalized result: P(yaw) = cos yaw / (π sqrt [ γ² ‒ s...

Last time I began the problem of predicting the probability distribution of yaw angle if wind speed (|v|) is fixed but if the rider's heading relative to the wind (Φ) is unnbiasedly random. I calculated the derivative of yaw with respect to heading. Rider speed is s. The probability distribution of yaw is inversely proportional to the magnitude of the derivative at the given value of yaw. P(yaw) ∝ | ( (|v|² + s²) + 2 s |v| cos Φ ) / [ |v| ( |v| + s cos Φ ) ] | Special case: cos Φ = ‒1 P(yaw) ∝ | ( (|v|² + s²) ‒ 2 s |v|) / [ |v| ( |v| ‒ s ) ] | = | (|v| ‒ s)² / [ |v| ( |v| ‒ s ) ] | = | (|v| ‒ s) / |v| | = | 1 ‒ s/|v| | which goes to zero for s = |v|. But I'll look more at that case later. But in general, for this to be useful, cos Φ needs to be eliminated from the equation: tan yaw = |v| sin Φ / ( |v| cos Φ + s ) ( |v| cos Φ + s ) tan yaw = |v| sin Φ ( |v| cos Φ + s )² tan² yaw = |v|² ( 1 ‒ cos² Φ ) ( |v|² cos² Φ + 2 s |v| cos Φ + s² ) tan² yaw = ...

The last time I came up with a proposed distribution of wind speeds at the bike's level. This was based on an assumed relation between wind speed and height, which is for urbanized areas. It doesn't apply to open coastlines, but then it doesn't apply to dense forests, either: just an attempt to model truly "typical" conditions. But next I need to consider that the bike may be moving in any direction relative to the wind. I assume all headings relative to the wind are equally likely. Each contributes equally to the final net yaw probability distribution. I'll start by assuming the bike is moving at a constant speed and constant direction, and the wind is also at a constant speed and constant direction. I'll worry about probability functions later. To go from wind speed |v| to yaw, I add a parameter Φ, the relative angle between the rider's heading (at speed s) and the wind direction. Then I get two components of the relative wind, one parall...

First Tour magazine , and now VeloNews have published wind tunnel data on "aerodynamic" mass‒start bikes. Each measured the bikes over yaw angles from 0 to 20 degrees, and each averaged over measured angles. Tour measured at 0, 5, 10, and 20 degrees, which implicitly gave uniform weight to the ‒12.5 to +12.5 degree yaw range, then a reduced weight to yaws out to at least ‒20 and +20 degrees. VeloNews tested at uniform intervals across the range from ‒20 to 20 degrees. Zipp reports, via NY Velocity , that the measured yaw distribution has peaks at around +/‒ 13 degrees, with peak extending (depending on conditions and rider speed) from 7 to 15 degrees, as measured at the head tube. So does this make sense? I thought so, but I wanted to do an analytic analysis of the problem. If wind is varying in speed, and the rider is varying in his heading, averaged over time what relative yaw angle is expected? Worth a quick calculation, I figured. Unfortunately my "quick...

Last two times, I discussed an article in Bicycle Quarterly which described how low-trail bikes should perform better in cross-winds. I showed that this is true for two effects from the sidewind, one that a bike leaning into a side-wind will tend to steer into the wind if it has positive trail, and the other that a sudden gust will cause a reaction force on the front wheel that will tend to rotate the wheel. Actually, these two effects are really the same, one the steady-state effect, the other a short-term effect. But what the article missed was that the sudden gust is also going to tend to blow the bike over in the direction of the wind, so an effect which causes the bike to suddenly steer away from the wind can compensate for this. But there's another effect. Everyone knows that a deep-dish rim is often less stable than a shallow rim, or even that bladed spokes may be less stable than round ones. This is due to the direct force of the wind on the front wheel, although t...

The description I gave last time was only part of the story described in Bicycle Quarterly and Tony Foale . Suppose I'm riding along when a cross-wind strikes. It will apply a force to the rider, to the frame, to the rear wheel, and to the front wheel. The last time I dealt with the force as a whole, which causes the bike to lean, and the lean induces the front wheel to turn proportionate to the lean. But there's an additional factor. Okay, several, but there's only space to describe two of them in this post. In the last analysis, I assumed a steady wind. But assume instead the wind strikes suddenly from the side. In this case, the bike is starting vertical: not leaned over. But the wind hits and suddenly it's pushing against the rider + bike. Without friction, the bike would accelerate sideways in response to this force, wheels and all. But there is friction: the tires grip the road, the and consistent with Newton's Third, the ground pushes ...

Moving backwards through the four copies of Bicycle Quarterly I recently received, I came across a very interesting article in the Spring 2010 edition. It addressed a topic I'd been puzzled about before: the relationship between trail and cross-wind stability. It's well established that trail contributes to bicycle stability. For example, this reference describes trail as a generally positive thing for stabiltiy: more trail = more stable. So if I'm riding in a cross-wind with deep-dish wheels, I should want a lot of trail, obviously. Yet of my two bikes, a Ritchey Breakaway (somewhat slack 72.5° head tube angle) and my Fuji SL/1 (super-slack 71° head tube angle), the Ritchey seems to do better in cross-winds. What's up with that? Everything I'd read says trail is what makes bikes move in a straight line. That should apply to cross-winds as well as to other conditions, right? Well, not quite. What trail does is cause the bike to steer when it leans. ...