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

Since thanks to Ron Brunner and Dean Larson, I now have the complete set of Low-Key Hillclimb results (the '90's pages were lost when Giorgio Cosentino, the webmaster at the time, had his account expire, and I wasn't backing up his stuff, and I lost the older stuff when my UNIX account from grad school got flushed). Anyway, a bit of work saved then doing backups resulted in a lot more work today reconstructing results. After typing in data from scans of Dean's printouts, getting some of the HTML from the Wayback machine (and all of 1995 ), regenerating 1996 - 1997 HTML from the data I'd typed in, and mining data from the existing HTML, I now have a nice set of times for every week the series was run. The exception was 1998 where I barely had anything to work with, so generated all fresh HTML using my current scripts, modifying the scoring algorithm to match that year's. The series started in 1995 with the top rider getting 100 points, everyone else a sco...

Caltrain is tasked to be public transportation, not commuter rail. However, weekend service, especially "counter-commute" (SF->SJ AM, return PM) is so marginal to be almost useless. This weekend really reinforced that point. My coach was running a weekend training camp: rides daily at 8:30 am. I couldn't make it. Not Sunday, not Saturday, not even Friday which isn't an official holiday. Not even close. I thought things were bad when service was restored in June 2004 following a two-year down-time due to track work to allow for "Baby Bullet" service, which required passing zones. After a period of debate in which four different scheduling options were considered, the worst of the four, full local service on each hourly train, was chosen. Every time it was suggested various stations would be omitted from the service of some or all of the trains, people from those stations would complain. People from places like Atherton, where the local station i...

It's often suggested that tandems are slower climbing than single bikes. Yet it's rare that one gets to see solid data supporting this. One nice thing about the Low-Key Hillclimbs is they generate a lot of interesting data. One nice thing about this year is we have Brian and Janet. They each did a climb solo, and did another climb as a tandem pair. Brian further ran another climb, but that's another matter. While each climb is different, as long as a climb is continuous and doesn't provide much opportunity for drafting, the ratio of a rider's speed to the median male speed, using single bikes only (I specifically refer to male speed as the number of males tend to be a lot more than the number of females, so there's less random variation in the median male speed), tends to be fairly constant one climb to the next. So I'll use the ratio of climbing speed to median male climbing speed as a measure of general climbing speed. So here's a comparison: Br...

Last time on this subject I came up with the following relation for determining the orientation of the spindle-based accelerometers relative to the desired coordinate axes: tan Δ φ = < a 2 > / < a 1 >  , < a r > > 0, where " <> " refers to a time average. But the time average isn't as straightforward as might initially be thought. Consider the case where the rider pedals a bit then the bike sits for several minutes. The accelerometers are accumulating data associated with the pull of gravity in a particular direction for an extended period of time. This isn't what's wanted. For the average of the transverse component of acceleration to go to zero (the component associated with the pedal around the circumference of its circle), we want to average over the angles of the circle, not really over time. A time average is a convenience, since accelerometers sample the acceleration at specific intervals in time. But if the pedals are...

Last time I described modeling some MetriGear Vector test data . The data were from two accelerometers which appeared to be misaligned relative to the radial and tangential axes of the test rig. What I'll do here is a bit of a restatement of a previous post . Yet since I have real data to chew on here it's worth starting from the beginning. Suppose we have two acceleration components, a 1 and a 2 . We know these accelerometers are oriented at a right angle (90°) relative to each other, and that they're perpendicular to the principal axis of the spindle. However, we aren't sure of the orientation of the spindle with respect to the end of the crank. This orientation can be described by an angle Δ φ . We wish to derive acceleration components a r , a radial acceleration component toward the center of the pedal's orbit, and a t , an acceleration component in the direction of the pedal's motion. Then we can write a relation: a 1  = a r cos Δ φ ‒ a t sin Δ ...

Enough political commentary! Back to science and engineering. In the MetriGear Blog they published data from a test rig in which accelerometers were rotated in a circle, and orthogonal components of acceleration were measured. Here's their plot: Metrigear accelerometer data from their test rig The blue curve represents data from an accelerometer measuring primarily tangential acceleration (in the direction of motion of the pedal as it rotates). The red curve represents data from an accelerometer measuring primarily radial acceleration (perpendicular to the direction of pedal motion through its circular orbit). The radial direction includes components due to gravity (the accelerometer rotates relative to the direction of gravity) and due to the centrifugal acceleration. The tangential acceleration also includes the gravity component, 90 degrees out of phase with the radial gravitational component, plus it has a component proportional to the rate of change of rotation. This appea...

It's been awhile since I mentioned the Metrigear Vector; I've been interested lately in solving the more fundamental issues of how fluctuations in speed, wind speed, and road grade affect power. Indeed, I fear that I may have made an error in my analysis of grade fluctuations. But I'm still checking over these calculations, so back to the Metrigear Vector for now. First, congratulations to Metrigear for getting the Vector nominated as one of candidates for "best new product" in the 2009 CyclingNews reader poll ! And they clearly contributed to the nomination of "Power Meters" in the nomination for "best tech innovation". Now I need to come clean here. I didn't vote for it in either category. Well, for "best new product", it's not a product until the blue tape is retired. And for tech innovation power meters have been around for a long time, and there continues to be incremental improvement. And while the Vector promises ...

I'm afraid I played a bit fast and loose with my assumptions in the last post. Here's the deal: two ways to calculate work done are to either integrate power with respect to time (which I did) or to integrate force with respect to distance. So what I did: first I considered the effect of bike speed fluctuations on average power. Bike speed varies over time with a particular average value, this yields fluctuations in power as a function of time, and these fluctuations in power average to a larger value than the power which would result from a constant speed equal to the average speed. But if average speed over time is the same, then distance covered is the same, which is the desired constraint of the calculation. Then I considered the effect of grade fluctuations on speed. But the speed must average the same over time, while grade was analyzed over distance. All good if the speed is constant, in which case distance and time are proportional. But it's not. If the grade i...

To summarize from last time, the physics behind the effect of grade variations on bike speed is that at constant power grade variations lead to speed variations, and speed variations lead to variations in wind resistance. When the bike moves faster than average, wind resistance is higher, but when it moves slower than average, wind resistance is less. To first order, these cancel, but to second order, the increase in wind resistance from riding above average speed is less than the reduction in wind resistance from riding at below average speed. So the greater the grade fluctuations the speed fluctuations and the higher the average power, even if the average speed is the same. We've already analyzed the effect of speed fluctuations on average power. That result, assuming no wind resistance, is: Δ p ≈ 3 f p 0 < Δs ² > / s 0 ², where Δ p is the change in the average power, p is the average power calculated without modeling the fluctuations, f is the fraction of power f...

The last post looked at the effect of wind speed and bike speed variations on climbing speed. However, often it may be difficult to estimate what these quantities are. However, even without metrology, if you have good profile data for a climb, you can at least estimate the variations in the road grade. These may provide an estimate for the road speed variations. Grade variations on Old La Honda Road (Lucas Pereira) One can imagine two simplifying cases for how a non-uniform grade may be approached. One is to ride it at constant speed. In this case, estimating the effect of wind resistance on power is simple: in still air (or a constant relative wind) and with a fixed riding position ride wind resistance is constant. Obviously power fluctuates, perhaps wildly as the grade changes. Interestingly, from a pure physics standpoint this is the most efficient way to climb the hill. For a given speed, it minimizes energy used. However, physiologically it may not be efficient. There is...

Last time on this topic , I ended with the following equation: < p w > ≈ f w [ s 0 ( s 0 ‒ s w0 )² + ( 3 s 0 ‒ 2 s w0 ) < Δs ² > + s 0 < Δs w ² > + ( 2 s w0 ‒ 4 s 0 ) < Δs Δs w > ] . Up to this point, there's nothing climbing-specific about this analysis: it applies as much to descending or riding on the flats as it does to climbing. But the goal of this post is to simplify it a bit, using the result of the last analysis, the effect of headwinds on bike speed . In that analysis, which was corrected on 31 Jan 2013, it was derived that for an absolute wind speed which is relatively small in comparison to bike speed, and for a relatively small (say 20% or less) of the power going into wind resistance, then a change in wind speed change d s w (positive = tail wind, negative = head wind) changes bike speed by d s ≈ 2 d s w f , where f is the fraction of power going into wind resistance. Here's the first "climb approximation...

Recall in the last post there was a term in the result representing the correlation between wind speed and bike speed. To estimate this term, a relationship between wind speed and bike speed is needed. This is a relationship interesting on its own merit, anyway. For this I start, as always, with a bike speed-power relationship, in this case without the usual assumption of zero wind. There are two components to power considered here: one from wind resistance and independent of mass, the other mass-proportional and due to either rolling resistance or to altitude change. Drivetrain frictional losses are assumed proportional to these other two components, and thus don't enter into this analysis. First, the wind resistance component is: p w = f w ( s ‒ s w )² s , where p w is the power due to wind resistance, s is the bike speed, and s w is the speed of the wind in the direction of the bike (a head wind would be negative). Then, we have a mass-proportional component,...

Okay, last time I discussed the problem of why climbing power depends not just on average bike speed and average wind speed, but also on fluctuations in these parameters. So let's break down climbing speed and wind speed into two components each, a constant component and a variable component, the variable component having a zero time-average: s = s 0 + Δs , s w = s w0 + Δs w , where, for < s > signifying the average of s , etc: < s > = s 0 + < Δs > = s 0 , and similarly: < s w > = s w0 + < Δs w > = s w0 . So these can be plugged into the wind power calculation: p w = f w ( s ‒ s w )² s . First, a word of warning: this equation only applies if s w ≤ s , otherwise you need to flip the sign. If s w > s , then it could mean a tailwind is pushing the rider, or else the rider is moving in the negative direction. These sign issue is a big clue that this equation is not grounded in fundamental physics, and its not, it's an approxima...

Even when climbing steep hills, wind resistance is a considerable contributor to total power. Wind resistance force, in the absence of a cross-wind, is typically modeled as proportional to the square of the relative speed of the wind relative to the bike. To get power, you multiply this force by the bike speed. The standard model, neglecting drivetrain efficiency (we assume drivetrain losses are proportional to transmitted power): p w = ( C D A ρ / 2 ) ( s ‒ s w )² s , where s is the bike speed, s w is the speed of the wind (positive for headwind) relative to the Earth's surface at the typical rider height (not 10 meter height, as is typically reported), C D A is the coefficient of drag times effective cross-sectional area of the rider and bike, and ρ is the air density. Too much detail: all we care about is there's a coefficient for wind resistance, which we assume is held constant, f w : p w = f w ( s ‒ s w )² s Now, assuming we know the average value of s and...