Wednesday, July 29, 2009

effect of road grade on VAM

VAM is an easy to calculate proxy for power-to-mass ratio commonly reported for professional riders in European races, where television coverage plus well documented hill profiles facilitates its extraction. VAM is the rate of altitude gained per unit time, for example in meters per hour. I recently recorded 1402 metters per hour in a break-through ride up Old La Honda, breaking my previous PR. The excellent Science of Sport Blog recently posted the following plot of some VAMs from recent Tour history:


VAMs from the Tour de France, from The Science of Sport.

Michele Ferrari's numerical calculation of the effect of road grade on VAM are often used as a reference, by me included, but an analytic form is perhaps more useful.

I'll again assert a simplified, windless bike power-speed model, which is for "PowerTap power" neglecting drivetrain losses, but it is assumed drivetrain losses are a function of total power:

p = fm m s + fw,

where p is a fixed power, s is bike speed, and fm and fw are coefficients. This model neglects drivetrain losses, but I'll assume they depend primarily on total power p, and thus are fixed.

Differentiating with respect to speed s yields:

dp / dgrade = ∂p/∂s ∂s/∂grade + ∂p/∂grade

which since dp = 0 yields, keeping only non-zero partial derivatives:

0 = (fm m + 3 fw s²) ∂s/∂grade + m s ∂fm/∂grade

or

∂s/∂grade = ‒ m s ∂fm/∂grade / (fm m + 3 fw s²)

Using a definition for VAM with our linearized grade (sine = tangent):

VAM = grade × s

yields:

( grade / VAM ) (d VAM / d grade) = ( grade / VAM ) [ ∂VAM / ∂grade + (∂VAM / ∂s) (∂s / ∂grade) ]
      = 1 ‒ m grade ∂fm/∂grade / (fm m + 3 fw s²)
.

The quantity on the left is the relative increase in VAM with the relative increase in road grade. For example, what is the ratio of the % increase in VAM with the % increase in grade? This can be simplified by defining δVAM as the fractional increase in VAM and δgrade as the fractional increase in grade.

So given the following:

fm = ( grade + Crr) g

then

∂fm/∂grade = g

where g is the gravitational acceleration, and therefore

δVAM / δgrade = 1 ‒ m g grade / (fm m + 3 fw s²).

If f is the fraction of power coming from wind resistance this yields:

δVAM / δgrade = 1 ‒ ( m g grade s / p ) / ( 1 + 2 f ).

This can be simplified by defining αRR = CRR / (CRR + grade), the fraction of mass-proportional power from wind resistance, yielding

δVAM / δgrade = 1 ‒ (1 ‒ αRR) (1 ‒ f) / (1 + 2 f)
      = [ 1 + 2 f ‒ 1 + αRR + f ‒ αRR f) ] / (1 + 2 f)


with a final result:
δVAM / δgrade = [ (1 ‒ f) αRR + 3 f ] / (1 + 2 f)

using the following definitions:
δVAMfractional increase in VAM
δgradefractional increase in road grade
αRRfraction of mass-proportional power from rolling resistance
ffraction of total power from wind resistance


An alternate but equivalent representation would be (I prefer the previous one):

δVAM / δgrade = [ αRR + (3 ‒ αRR) f ] / (1 + 2 f).

Or, if αRR is close to 1, then this more symetric representation may be better, as the function of f on the right is the effect of relative decreases is mass on climbing speed, derived in an earlier post:

δVAM / δgrade = 1 ‒ (1 ‒ αRR) (1 ‒ f) / (1 + 2 f).

Consider the following example:

αRR ≡ 1 / (CRR / grade + 1) = 0.05,

f ≡ fraction of power to wind resistance = 0.12.

Then a 10% relative increase in grade (for example, from 7% to 7.7%) will result in a 3.2% increase in VAM (for example from 1400 m/hr to 1445 m/hr).

There are two limits for this result, each providing a nice check of the derivation. One is where f is essentially zero, in which case it simplifies to αRR. So in this instance the VAM is reduced proportional to the fraction of power going into rolling resistance. In the other limit of αRR → 0, it simplified to 3 f / (1 + 2 f), which represents the decreased power going into wind resistance with steeper grades. There is also the second-order term in the numerator ‒αRR f which has negligible effect on steep relatively smooth roads. To the opposite extreme, where the road is almost flat and therefore f or αRR are close to 1, the result will be close to 1, which implies for very shallow roads dominated by rolling resistance and/or wind resistance a fractional increase in road grade barely affects speed, and so the grade contributes directly to VAM. This is of course correct, so in each of these test limits the formula does the right thing.

Old La Honda PR

Well, racing this year has gone terribly. There's really no other way to put it. In road races, DNF's in Spring Hill and Pescadero. Crashed out of Berkeley Hills. Dropped at Mt Hamilton and Diamond Valley. This isn't what we train for. It'd have been better to focus on double centuries.

I've done better on hillclimbs: Diablo as solid a 5th out of 6 as you can get, and a satisfying 2nd Ross' Epic. Small fields in each case, but the guys who beat me are guys who should be beating me. I rode well in each case.

But besides racing, my goal this year (as last year) was to crack 17 minutes up Old La Honda. My existing PR has been 17:03 from 2002. That was on a sub-16 lb Fuji Team, actually a Teschner hand-built bike for the Mercury team. The frame cracked on me last year, and I replaced it with another Fuji, this time a real Fuji, the SL/1 carbon frame, which I love. Weenied out it weighs in at a very nice 11.35 lb. So with that sort of advantage, I certainly should have been able to crack 17 minutes. But I wanted a bit more: I wanted to further beat my previous time by enough to offset the difference in bike weight.

It's rare -- very rare, in fact, I get to try for a "real time" up Old La Honda. Normally I've ridden hard Saturday and Sunday. I go into Tue still not fully recovered, then do some brutal interval or sprint session. On Wednesday, I'm hardly firing on all cylinders. Then if I do the climb, I'm carrying a pump and tool bag, probably with a full waterbottle. These are not the ways to get a PR. To really nail the climb you need to strip down all extra weight, come in with fresh legs, get an optimal warm up riding close to threshold but not over, and bring your best equipment. The last time I'd done this had been 2006, when I scored a tepid 17:43 at the Low-Key Hillclimb.


2006 Low-Key up OLH (Sheri Greenspan photo)

I am lighter now then I was then, and have been training smarter, so expected more. Indeed, my recent times have been close to this just on the Noon Ride, riding my steel Ritchey Breakaway with boat-anchor Powertap rear clincher wheel and a typically heavy front clincher wheel.

Yesterday I realized I had my chance. After a solid climbing workout last Wednesday, I'd had good preparation: a long massage + light ride on Thursday, no ride (travel) on Friday, a steady ride @ Diamond Valley on Saturday, a steady climb of Carson Pass on Sunday, then recovery on Monday. Freshness without staleness. Plus, the air temperature was essentially perfect: warm but not hot. The only downer was a 12 mph westerly wind. Old La Honda's eastern slope is well sheltered from westerly winds, but even an undetectable breeze can be worth 10 seconds.

So anyway, with my bike fully weenied (if I'd gotten a flat, it would have been a long walk back to the office...), I set off on the Noon Ride. They were doing the Tue Double Alpine route, which was perfect as it reduced the pace on Alpine to the Portola intersection. They continued on up Alpine, while I bailed on Portola for a moderate spin to Old La Honda Road.

During the spin, I selected on my strategy: put it in the 36/18 and keep that gear spinning. In lieu of a power meter, which I'd left at home in favor of the considerable mass savings of my Mt Washington rear wheel, I'd go old school: pace using gears and cadence. Old La Honda's grade varies considerably on a short-scale, but over longer length scales it's remarkably consistent. So a constant cadence in a constant gear means a relatively constant rate of altitude gain and a relatively constant power output.

The pace felt way too easy at the bottom. But I knew from experience with the Powertap that even what feels easy at the bottom can result in struggling in the final kilometers, and it was important to hold my power the full distance. Unlike when I pace with the power meter, however, I resisted the temptation to upshift when the grade reduced. I was married to that 18 cog. In the final kilometer, I'd reconsider.

As the climb progressed, it became harder to keep that gear spinning. But never too bad. When I hit the "3 mile" marker, with 500 meters to go, I knew it was time to ramp it up. Here I switched to alternating between the 17 and 18 cog, following the grade allowed, finally jamming it in the last 200 meters. I hit my watch, not really expecting that I'd achieved my goal.

But there it was: 16:49.44. I'd done it, at long last, and with time to spare.

So what about the weight? I'd had much lighter shoes in 2002 (custom 4-hole Rocket 7's versus my heavy Sidi Megas with Speedplay cleat adaptor plates, a total difference of close to 200 grams), but my helmet was around 70 grams heavier then. My bike was the big difference: around 4.5 lb lighter. So let's call the net difference around 4.1 lb. I weighed 124.4 lb yesterday morning, then add in 3.1 lb for clothing, shoes, and helmet. The net result is 4.1 lb would be around 2.9% of total mass. Given my calculations I've recently posted here, this would increase my time by around 1.8%, 18.05 seconds, bringing me up to 17:07.5, 4 seconds away from my 2002 time. Still, were I to add in the headwind, the two efforts are probably similar.

One potential advantage is tires. I'd have used some ultra-light Michelin clinchers in 2002 with ultra-light butyl tubes. This time: Veloflex Record tubulars, which have exceptionally low rolling resistance. And of course the state of the pavement varies. There's a lot of patching on the road today, but I think it's better than in 2002.

So I probably climbed better in 2002. But let's be real: I'm 7 years older, so that's not unexpected.

Maybe I'll have a chance to knock a bit more time off this time this year. Certainly my climbing power has been increasing over the past month. I'll definitely give it at least one more shot.

Some calculated numbers (with certain assumptions):

VAM: 1402 meters/hour (Contador was 1860 up Verbier)
wind resistance power: 33 watts
rolling resistance power: 16 watts
climbing power: 240 watts
Powertap power: 290 watts
drivetrain loss: 9 watts
SRM power: 299 watts = 5.31 W/kg (Contador was 6.7 W/kg up Verbier)

These numbers are good for me. My previous best power up Old La Honda on the Powertap was 289 watts, on a ride last April 8. But of course there's a lot of uncertainty in these calculations.

Tuesday, July 28, 2009

Diamond Valley Road Race

I came, I saw, it kicked my ...

The Diamond Valley Road Race hit me on Saturday like a punch in the gut. Our well-tempered pack of not much more than 30 riders started peacefully enough: we rode civilly from starting area to begin the first of five laps of the 11.4 mile course. My goal: stay near the front to have a good position on the subsequent climb. This is something I'd failed to do each of my previous two rides here, and in each case it cost me added anaerobic stress during the beginning of the two-part climb. In each case I'd managed to survive these efforts, move up during the race, and exploit general attrition to place (for me) fairly well. But it's always easier climbing from the front than the back: position on a climb this short is money in the bank worth more than any weight-weenie gram shaving.

I'd arrived with Cara to our campground late Friday afternoon. A solid night's rest camping at Grover Hot Springs, then a relaxed breakfast, and I still arrived to the race two full hours before my 11:20 am start. I knew the altitude was an issue, and so had focused particularly on hydration since my arrival in camp. After registering and dealing with a brake pad issue (use Campy, not Shimano-compatible pads on Zero Gravity brakes...), I had plenty of time for a leisurely tour of the course before my start.

The wind was where I wanted it: a headwind on the descent, a tailwind at the finish. This basically neutralized the descent: the big gravity-assisted power-maniacs couldn't cause any damage there, while the approach to the finish with its combination of gentle upslope and wind from the tail would favor a longer attack rather than the late sprint which a headwind rewards. If I had a chance for a good result, for example top ten would be nice, then these were the conditions which would maximize it. I was 11th in 2007, so top ten was realistic, I thought.

Soon after the start we began the descent of Carson River Road. Chris Phipps almost immediately moved up on the right side. That was the way it should be done. Flash-back to last year when Mike Hutchison moved up across the center-line on the left: he subsequently attacked with Chris Wire and Mike Taylor, staying away the rest of the race. Honestly I hoped Chris would also get away today. He can probably sustain around a half-watt-per-kilogram more than me and is training for the pro race at the Tour of Utah. The sooner the Chrises Phipps and Wire (back this year) were gone, the more likely the pace would suit my fitness and ability.

Sure enough, the headwind was a big factor, compressing the small pack on the gradual descent. Hanging in the back row was easy enough, but not the right thing to do. Nevertheless my inherent distrust of what I can't see (and riders with whom I've not ridden) did me in. I wasn't willing to do what needed to be done to move. To do what Chris had done. "I belong at the front, so that's where I'm going." Instead, as usual, I was more worried about my dermis staying epi than about my pre-stated goal.


The view down the first climb, from SteepHill.tv

Whatever I expected from the climb, it wasn't what happened. The peace absolutely exploded the moment we'd turned onto Diamond Valley Road. From nervous and easy to near-maximal effort in the reduced air, I was in trouble. Take it easy, I told myself, pace yourself and you'll be fine. But I wasn't fine. The group had gapped me, and the gap was growing. I was left with a few stragglers, struggling up this harmless-looking hill in the dry heat.

I was despirited, to put it lightly. A few riders caught me from behind on the false flat following the second part of the main climb just as I overtook another. We were four. There was still hope: we could regain the pack which would unwilling and unable to sustain the hard effort to the end of the now gradual climb to the feed zone not far past the finish line. One rider pulled, then it was my turn. A quick pull, not too much, just enough to keep it moving, then I moved aside. But then the next guy in line drilled it, catching me too depleted to respond. I was gapped.

They hovered in front of me for awhile. If it hadn't been for that pointless acceleration, I'd have been able to follow. But then this gap also grew. Eventually, when I had a clean line of sight, I saw the main pack ahead, but no sign of the chasing three. It seemed they'd recaptured the main group. I, on the other hand, had little chance to do anything but pick off stragglers. I settled in for the long haul. Only 50 miles.

But it proved to be a long 50 miles. In the heat, with the combination of the headwind on the descent and the grind of the long gradual climb back to the feed zone, the vast majority exposed to the bright sun, it was seemed further. Thank heaven I had Cara in the feed zone. Her super-smooth hand-offs of chilled carbohydrate solution kept me going. And also thanks to Greg's wife Sam who cheered me as she walk-ran a counter-race circuit of the course. Dropping out simply wasn't an option.

I'd hoped to maybe hold off the non-championship E3-E4-E5 race, but no luck there. It had shattered into three groups, each of which passed me between laps 4 and 5. Fortunately I was never lapped by the 30+ or 35+ packs, which had started 10 minutes and 5 minutes before me. So I was riding steadily if not particularly fast.

Finally I finished, probably 10 minutes or more behind winner Chris Phills, far more depleted than I though justified for the sub-60 mile distance of the race. I'd logged close to 70 total for the day, but with the heat and altitude, it clearly felt like more. When the results were posted, numbers only, I was 20th of maybe 23 finishers. Persistence had paid off in a result higher than I probably deserved.

But despite not meeting my goal of top 10, the weekend was still a winner. Camping at Grover Hot Springs with Cara was a blast. Even if we couldn't hike or ride together because of her slowly recovering knee injury, even with the too-frequent engine noise from the many vehicles at the park, how can one go wrong in a tent under the tall trees in the pleasantly cool night? I slept from dusk to dawn, as we were designed to do, at least an hour more than I usually get in our world of artificial light and the constant temptation of the internet.

On Sunday, after we'd packed and soaked in the hot spring, we drove the four miles down to Markleeville for a wonderful salad lunch at the deli. Then I unloaded my bike again and recalled the Death Ride by riding the 21 miles to the Kit Carson summit. The last 15 miles or so are a long grind in the windy heat (1:13 from Woodfords to the pass) with far more traffic than I'd prefer, some even unfriendly. But who can deny the magic of a Sierra pass? Still blissfully unaware of the results of the Tour the day before, I bonded with racers in France who'd ridden the Ventoux: similar distance, similar conditions, even if Ventoux is considerably steeper.

Thursday, July 23, 2009

effect of body mass change on climbing (with wind resistance)

I calculated the effect of body mass on wind resistance, and previously the effect of wind resistance on climbing speed, and before that the effect of total mass on climbing speed, so these three analyses can be combined to calculate the total effect of body mass change on climbing speed.

Assuming body mass changes are relatively small, the effect of a body mass change on wind resistance can be linearized. Assuming the wind resistance associated with the body, initially a fraction of total wind resistance βw. Then a fractional change εw of the body's wind resistance will have an effect on total wind resistance = εw βw. Assuming the change in the body's wind resistance is due to mass change at a fixed height, this fractional wind resistance change εw is to first order half the fractional change in body mass εm (as a result of the square root). So, if δw is the fractional change in total wind resistance, it can be calculated:

δw = εm βw / 2.

If f is the fraction of a bike's power due to wind resistance (as opposed to the mass-proportional power component), then from the previous result, the fractional change in speed δs can be calculated:

δs = ( εm βw / 2 ) f / ( 1 + 2f ) + ... .

To this we need to add the effect of total mass on speed. Given a relative change εm in body mass, and given initially that a fraction βm of total mass is associated with the body, then the fractional change in total mass δm = βm εm. So this yields:

δs = εm [ ( βw / 2 ) f + (1 - f) βm ] / (1 + 2 f).

So that's it. Some typical numbers, estimating for my case (bike weight includes clothing, shoes, helmet, etc), climbing Old La Honda:
βm = 86%
βw = 2/3
f = 12%
.

Then:
δs = 64% εm

So a 1% loss in body mass with the same power increases speed by 0.64%. The improvement in wind resistance increases speed by 0.03%, while the reduction in total mass increases speed by 0.61%. A 0.80% loss in power would be sufficient to cancel the benefit of this 1% loss in mass.

effect of body mass on wind resistance

I already considered the effect of mass on climbing speed, but body mass (as opposed to equipment mass) also affects wind resistance, which is still important when climbing.

This has been estimated empirically by Heil, European Journal of Applied Physiology, V 93 N 5-6; March, 2005. However, I never really liked Heil's analysis, because it looks at body mass without also considering body height. Obviously if you consider gaining mass without height versus gaining mass in proportion to height it is reasonable to expect different results. I'm reminded of a physics lecture I attended as an undergrad: "First, we model the human body as a sphere." The crowd chuckled. It was a sphere of salt water, BTW, useful for electrostatic calculations, but of course the human body is not a sphere. So shape matters.

So first the usual wind resistance model, in the absence of external wind:

Pw = fw

fw = ½ ρ CD A
,

where ρ is the mass density of air, CD is the wind drag coefficient, and A is the cross-sectional area normal to the direction of motion. I'll assume CD A is due to two independent parts: one from the body and one from the bike:

CD A = (CD A)bike + (CD A)body.

Obviously the body affects the drag on bike components, but I'll assume this effect for a given bike is independent of the mass or height of the rider. But obviously a taller rider tends to ride a larger bike, although the wind resistance may not scale in strict proportion, since the wheels and fork blades, as well as the components fail to scale with rider size. So I'll assume a linear model:

Abike = A⁰bike + Weff × height

where Weff has units length and describes how the area of the bike increases with rider height. In contrast, Heil estimates the effective cross-sectional area of a road bike is around 0.11 to 0.13 m², with a time trial bike around 0.066 m².

More important is the body. It's been estimated the body accounts for around 70% of the total wind resistance. Heil claims an empirical formula for cross-sectional area which can be written as 0.306 m² (mass / 100 kg)^(0.762). This formula is useful for establishing a relative magnitude, but the analytic form, as I noted, is less than satisfying.

For this analysis, assume CD is fixed (Heil claims this may not be a good assumption). Then assume area is proportional to width × height, but not depth. It's well known an elongated tube has less wind resistance, up to a 3:1 ratio of depth to width, than a cylindrical tube, for wind striking normal to the principal axis of the tube. In certain cases "depth" may even contribute to a propulsive force via the "sail effect". So depth can either increase resistance (due to skin friction) or decrease it (due to reduced turbulance) depending on specifics of the shape relative to the wind direction and the direction of motion.

So the model I'll use is:

Abody = width × height.

I assume we know height. Assuming mass is proportional to width × height × depth, and height and depth increase in proportion, then height and width are each proportional to sqrt(mass / height). So:

Abody = KA × sqrt(mass / height) × height
    = KA × sqrt(mass × height)


where KA is a constant.

So the total CD A is then:

CD A = CD,bike × ( A⁰bike + Weff × height ) + CD,body × KA × sqrt(mass × height).

Since height is closely correlated with mass, but less strongly correlated with body mass index (BMI) = mass / height², it's useful to cast this equiation in terms of BMI:

CD A = CD,bike × ( A⁰bike + Weff × height ) + CD,body × KA × sqrt(BMI × height³).

Note that if BMI is considered relatively constant, then height is proportional to sqrt(mass), and the body's cross-sectional area is proportional to mass^(3/4), which is quite close to Heil's exponent of 0.762.

This model allows for a refinement of the effect of body mass (whether due to BMI or height) on climbing speed. I'll leave that for later.

Wednesday, July 22, 2009

effect of wind resistance on climbing speed

To continue the previous analysis: what about the effect of wind resistance on climbing?

Well, the result should be easily derived from the last two analyses. But for completeness, starting again from the simple power-speed model, assuming zero wind:

p = fm m s + fw

we can then calculate the derivative of the dependence of wind resistance on subsequent speed s, where we're interested in fractional rather than absolute changes of each:

(fw / s) d s / d fw =

(fw / s) (∂ p / ∂ fw) / (∂ p / ∂ s)


where:

∂ p / ∂ fw = s³,

∂ p / ∂ s = fm m + 3 fw.

So:

(fw / s) ∂ s / ∂ fw =

(fw / s) s³ / ( fm m + 3 fw s² ) =

fw s² / ( fm m + 3 fw s² )


which can be expressed as:

r / ( 1 + 3 r )

where r is the ratio of wind resistance power to mass-proportional power. If you know instead the fraction of total power (neglecting drivetrain losses) due to wind resistance, then substitute r = f / (1 - f), where f is this fraction:

f / ( 1 + 2 f ).

So back to our previous example, if I reduce wind resistance by 1%, and if wind resistance is 10% of total power, then my speed is increased by 0.083%. For example, suppose I'm riding on Old La Honda and have the choice whether to draft a rider or ride at my own pace without the draft: what's the potential benefit from that drafting? Well we know that drafting can be worth 30% of wind power on the flats, implying the same reduction in wind drag force (since the comparison is done at a given bike speed). Suppose only a 10% reduction in wind drag force at the speeds involved in climbing (around half the speed on the flats). Then given wind drag is responsible for 10% of the total power to start with, the increase in speed from drafting would be 0.83% = 8.5 seconds out of 17 minutes. Using the previous analysis of the effect of mass change on speed, this is similar to a 1.1% reduction in mass, or for rider + bike + equipment = 64 kg, 710 grams saved. So drafting is a big deal, even on a climb like Old La Honda.

Tuesday, July 21, 2009

effect of power on climbing speed

Similar to the last post, the next question is: what is the effect of power on climbing speed? Answering this is a fairly trivial extension of the previous calculation.

Again consider a power-speed model which, simplified and neglecting wind speed and acceleration, is expressed as:

p = fm m s + fw

where terms are as described last time. Then we can calculate the effect of fractional changes in speed on fractional changes in power required:

(s / p) (∂ p / ∂ s) =

(s / p) ( fm m + 3 fw s² )
.

What we want, however, is the inverse of this: the effect of fractional changes in power on fractional changes in speed:

(p / s) / ( fm m + 3 fw s² ).

Expanding p / s yields:

( fm m + fw s² ) / ( fm m + 3 fw s² )

which can be simplified in terms of the ratio of wind resistance to mass-proportional power used in the last blog entry:

(1 + r) / (1 + 3r).

Again, if what we know isn't the ratio of wind resistance power to mass-proportional power, but rather the ratio of wind resistance power to total power, using r = f / (1 - f) and multiplying numerator and denominator by 1 - f:

1 / (1 + 2 f).

In contrast to the previous analysis, in this one the f form is the simpler one. So if wind resistance constitutes 10% of total power, and I put out 1% additional power on a climb, I get 1% / 1.2 = 0.83% more speed on the hill. Compare this to the case for saving 1% of mass, which increased speed by 0.75%.

Consider, then, the case where you gain 1% of body mass, starting from your bike + equipment weighing 12% of body mass and climbing with 10% of power going into wind resistance. How much power do you need to gain to keep climbing speed the same? Well, if you're addicted to "watts/kg", the answer is obvious: 1%. But that's not quite right, assuming wind resistance was unaffected by the mass gain. Total mass increase = 1% / 1.12 = 0.89%, decreasing speed by 0.67%, requiring a 0.80% increase in power to compensate. Of course there is typically some increase in cross-sectional area with any mass increase, so the actual number may be a bit higher.

Monday, July 20, 2009

effect of mass reduction on climbing speed

A frequent question in cycling speed-power calculations is: "If I save a certain percent of weight, then what is the effect on my speed?". The same ananalysis applies to other mass-proportional power terms: the effect of grade and/or rolling resistance changes on speed. This has tangible significance when considering how much it's worth to invest in some über-light weight-weenie widget, or when considering the trade-off between aerodynamic versus mass in wheels or frames.

Consider a power-speed model which, simplified and neglecting wind speed and acceleration, is expressed as:

p = fm m s + fw

where fm is the mass-proportional coefficient and fw is the wind resistance coefficient. In the standard linearized model, fm = g (grade + Crr) / α, where g is the acceleration of gravity, grade is the road grade, Crr is the coefficient of rolling resistance, and α is the drivetrain efficiency (characterizing the drivetrain as a power-proportional loss term may be a poor approximation, but it's the one which is typically made). The wind-proportional term is ½ ρ CDA / α, where ρ is the mass density of air (typically around 1.1 kg/m³), CD is the coefficient of drag (typically around 0.7), and A is the effective cross-sectional area (typically around 0.5 m² for a racing position). The details of these coefficients aren't terribly important for this analysis, however. All that's important is the ratio of fm and fw.

We can then calculate the derivative of the dependence of mass m on subsequent speed s:

(m / s) d s / d m =

(m / s) (∂ p / ∂ m) / (∂ p / ∂ s)


where:

∂ p / ∂ m = fm s,

∂ p / ∂ s = fm m + 3 fw.

So:

(m / s) ∂ s / ∂ m =

(m / s) fm s / [fm m + 3 fw s²] =

fm m / (fm m + 3 fw s²)


which can be expressed as:

1 / (1 + 3 r)

where r is the ratio of wind resistance power to mass-proportional power. If you know instead the fraction of total power (neglecting drivetrain losses) due to wind resistance, then substitute r = f / (1 - f), where f is this fraction:

(1 - f) / (1 + 2 f)

If f is sufficiently small, this can be slightly simplied to second order as:

1 / (1 + 3 f + 3 f²)

So there it is. If you're riding such that, for example, 10% of the power is from wind resistance, and I reduce mass (or equivalently grade + Crr) by for example 1%, then the effect on speed will be 1% / [1 + 3 × 0.1 / (1 - 0.1)] = 0.75%. This shows that the frequent assumption that a 1% reduction in weight will increase speed by 1% is a significant overestimate of benefit: wind resistance, even if it's a relatively small fraction of total power, is still marginally important when balancing weight and wind resistance on climbs.

So how much is it worth to save a second up Old La Honda? If I run these numbers for my stats and assuming I'm within one second of my 17-minute goal, that second up Old La Honda requires 86.3 grams saved. A regression I did around a year ago on SRAM component prices (SRAM is a good choice, as they have the same design of each of their groups other than weight savings) is weight reduction came at a cost of $3.35/gram. I tend to stick with this number when contemplating other upgrade choices, as well. So it would appear for me saving a second up Old La Honda is worth at least $290. If not, I'd be using Rival on my race bike.

Sunday, July 19, 2009

Contador on Verbier (and on Old La Honda)

Note: I had to revise the numbers on this page because mapmyride does a relatively poor job with distance, as I determined by considering its profile of Old La Honda Road. Since I did the original calculation, an official km-by-km climb profile was posted to The Science of Sport.

The Tour today ended with a relatively steep, short climb on a stage on which riders had relatively fresh legs: a perfect opportunity for some truly impressive rates of climbing to be demonstrated. And sure enough, they didn't disappoint.

Various numbers have been tossed about for the stats on Verbier. Originally I used elevation data from MapMyRide's map of the stage: 630 meters climbed over 7.576 kilometers starting from the sharp left from Route de Verbier at km 199.1 in the stage, an average gradient of 8.32%. However, since then official data were posted in comments to The Science of Sport: 638 meters gained over 8.8 km, an average grade of 7.25%. They also reported reports that Contador did the climb in 20:36, a solid Old La Honda time for sure, but on a climb of this length an number as amazing as we've come to expect in the Tour.

Climbing rate is normally reported in a vertical rate of ascent, or "VAM", typically in meters/hour. The steeper the road the higher the number, assuming adequate gearing and traction: obviously on a dead flat road VAM will be zero, no matter how much power you produce. Using climbing times from The Science of Sport:
rider
time
VAM
Alberto Contador
20:36
1858
Andy Schleck
21:19
1796
Carlos Sastre
21:42
1764
Lance Armstrong
22:11
1726

So who cares about the silly Tour, right? Old La Honda is the climb which really counts.

Lucas Pereira calculated Old La Honda gains 393 meters @ 7.3%. This is essentially equivalent to the grade of Verbier, sparing me the need to adjust for wind and rolling resistance. But Old La Honda is also shorter. Assuming Contador has an anaerobic work capacity to critical power ratio of 60 seconds, as a first iteration assuming climbing time is proportional to altitude gained, one can adjust for the distance using te critical power model: power ∝ 1 + AWC/CP / duration. This yields 2.9% higher power for Old La Honda Contador is dissipating around 69 watts in wind resistance (around 15% total), so added power doesn't all go into speed and therefore into VAM. Too lazy to do a formal calculation, I'll claim this is around 2.6% improvement in VAM, taking him to 1906 meters/hour up a well-paved Old La Honda (dream on!).

So then for a well-paved Old La Honda, Contador would have climbed it in.... drumroll... 12:22.5. Lance would have crawled up to the stop sign in a crippling 13:19.3, nevertheless winning the 35+ 1-2-3. The similar VAM numbers validate the assumption of time proportional to altitude gained. Comments on facebook suggest Eric Wohlberg has the record with 13:50, beating Greg Drake's best time, where Greg had cracked Eric Heiden's previous best of 14:15.

But what about that pavement? Old La Honda's in much better shape these days than it was in it's low-point in the early part of the millenium. A CRR = 0.6% might be about right. Let's say Verbier is around 0.2% lower. Then this yields a relative difference in mass-proportional power of 2.7%. Adjusting for wind resistance this would take Contador's time up to 12:35, with Lance at 13:32.

In any case, whatever fun we have with numbers, one thing is clear. These guys can ride.

Friday, July 17, 2009

Hipp memorial ride: Sunday 9am Palo Alto Form Fitness

A recovery week for me this week, a time to reflect on my fitness, my training, and my priorities. A chance to avoid digging myself into a metabolic hole, something I've done too many times before, and something which so many cyclists invariably do in the endless season which is San Francisco Bay area cycling. Hammering is an addiction, as strong as any, as potentially self-destructive as most.

This time for recovery was deflected from its canonical trajectory by the news of Chris Hipp's sudden death. It even cracked the New York Times, and not because of his cycling prowess. Yet despite his fame, of which I was unaware, Chris who was always a mystery to me. My most recent impression of Chris is from the Noon Ride, when I was intrigued by his Leopard frame.

Cyclists can generally be binned into two groups: those who obsess about equipment and those who don't. I'm definitely among the former. When I see an interesting frame, I want to know everything. How does the head tube angle and fork race correlate with handling? How does the stiffness affect comfort? How much does the weight of the paint affect an Old La Honda time?

However, Chris wasn't interested in any of these things. When asked what he thought of the bike, he responded only "it's good", or something equally unsatisfying. Subtext: it's the rider which really matters, and if you understood that, maybe you'd be faster.

Well, I won't get into that discussion here. But in this case it was clearly the rider who did matter. Frames fracture, handlebars snap, spokes break. In this case it was the body which broke. A sobering perspective.

Chris may have been a bit opaque to those who met him casually, like on a Noon ride or Spectrum ride or Morning Ride or Valley Ride or Dave's Ride or one of the oh-so-many rides on which he was a regular. But his sense of humor shined through, at least to me, most in his on-line videos. Landis on Palomar. The finish of Pescadero RR.

Only after his death did I learn about his engineering accomplishment. He basically led the revolution of blade computing: modular low-cost low-profile CPUs mounted in massive server networks, a backbone of so many web sites. But like so many techies, professional success clearly wasn't enough for Chris. The tech world isn't what it's made out to be. Too cynical, too political. He only truly found himself on the bike.

Even though I barely knew him, I'll miss him. And I'll certainly be there Sunday morning, 9am, for the memorial ride in Palo Alto. Unlike too many memorial rides, this won't be an exercise in rage at a system which places so little importance on human safety, but rather a sobering introspection at the fragility of human life.

Monday, July 13, 2009

Fremont Peak Hillclimb

So, the word is that Fremont Peak will get the Mt Tam spot on the calendar this year, as Mt Tam had resistance due to the threat of park closure due to the California State "budget crisis".

Okay, political commentary. Skip to next paragraph. I wouldn't be surprised if shutting the state parks for a few weeks actually cost more money than it saved. The parks are already fairly poorly funded. It's obviously political posturing to dangle some potential for pain in front of the People so that when the pain is withheld, the "mild discomfort" which remains is better tolerated. Okay, end of political commentary.


View from Fremont Peak (Virtual Tourist)

But with the calendar spot taken by a Fremont Peak climb, although those of us in San Francisco and northern environs are inconvenienced by the additional distance, on a climb-for-climb basis I'd have to say Fremont Peak wins. I've never been up Fremont Peak, but thanks to the incredible data resource which is MotionBased, a route profile is just a few mouse clicks away.... okay, and a bunch of keystrokes.


Fremont Peak from San Juan Baptista (Motionbased:mooseman)

Nice: 9.5% sustained for much of the 792 vertical net meters of climbing, plus a lower section hovering around 2.5% to test a rider's pacing strategy. A most glorious climb.

Thanks to Greg Bloom and anyone else involved with organizing this extremely exciting event!

Sunday, July 12, 2009

Tour de France stage 9 gap analysis

I love it when the break stays away. And what a classic sprint! A regression of the trend to 5 km to go was a 3 second gap at the finish, but then Caisse d'Epargne basically gave up the chase.

Saturday, July 11, 2009

House of Puncture Vine

I exited the BART in Walnut Creek to find Anthony getting off the same train. I always have difficulty finding my bearings outside the Walnut Creek station, so welcomed having someone to follow. We were both there for the same purpose: the House of Pain.

Well, almost.... he was doing "House of Pain Lite", a less maniacal version of the ride. Or should that be "The House Of Mild Discomfort"? Maybe another time for me. Today I wanted to throw myself at the mercy of the real deal. Ben's spoken highly of it: an East Bay hammerfest, a chance to acquire some heat adaptation not likely available from Roasters.

We rode together to Danville Peet's, arriving around 8:40 am. A decent group was there for the Lite ride, and they were soon gone. I waited basically alone for a group to congeal for the main ride, analogous to the group which picks up Spectrum from Los Altos Peet's, but nothing really formed. I guess most of the riders go to the Park and Ride, or meet it along Danville before the ride turns onto Railroad where Peet's lives.

The ride finally arrived around 9:20 am. I joined in near the front. No pain yet -- it was easy spinning through downtown Danville. What awaited? I would soon find out, I suspected.

But I suspected wrong. Less than a mile after starting, my rear wheel went suddenly flat. I pulled over to the side of the road, causing some distress in riders behind I'm afraid, to inspect. The ride moved on, I got off my Ritchey, and looked at the tire.

Almost immediately I found the thorn. The most and justifiably feared puncture vine. But with such a localized puncture I'd be able to patch it almost as quickly as I could replace it. But wait -- not far away I found another. Then another. My tire was full of them. One by one I pulled them from the rubber, only to find more. I realized then that this wasn't looking good for my front tire either. Not surprisingly, it was also flat. Only one spare tube, and these tubes were going to be beyond any sane patch job. So it was back to Peet's for me, grab the vest (needed for the ride to 16th St BART) and the journal (needed for the BART ride itself) I'd stashed in one of their cabinets, then find a bike shop for new tubes.

Which I easily found: Pegasus Bicycle Works had just two weeks before opened their new shop next door. To salvage something from the day I found my way to South Gate Road, rode the Diablo summit, then home via North Gate and Pleasant Hill BART. A decent ride, after all. Maybe I'll be back next week and hope for better luck.

I never did figure out from where I picked up all those thorns.

Friday, July 10, 2009

good week

Tuesday: 20-50 second sprints on Potrero Hill. I love this sort of workout. Quality pain time. But for much of it I was still chilly in my arm warmers, knee warmers, and vest. A typical summer evening in San Francisco.

Wednesday
: Finally, progress! 274 PowerTap watts up Old La Honda. Only 0.6% lower W/kg than my 2008 best power up the same hill. Given the leg-thrashing I experienced the previous evening, I think I have a few % more in me. My time of 17:40 was very solid considering I was on my steel Ritchey Breakaway with heavy wheels: there's still hope for my sub-17 minute goal. Indeed, calculations show were I on my weenied SL/1, I would have done it today. But calculations don't count. Really, I'd like to break 17 by a sufficient margin for calculations shown I would also have done so on my 16 lb Al Fuji Team I rode in 2002. Given running this past offseason left me at a lower base-line body mass than I've had for years, I just need to keep improving power, just a bit. The key is stay on top of my recovery, not get sick.


Pacing comparison of some OLH climbs. My best of last year was exceptional in that there was very little fade in my average. I started a bit slower this week than in my previous effort, but held a much higher average to the finish. Especially after recent high-intensity intervals have increased comfort in the higher power zones, a key is to keep enthusiasm in check at the bottom where Z5 still feels easy.


Thursday: I rode in with the SF2G crew, a nice route eastward over Guadalupe Canyon Road then onto the canonical Bay Way (with a Foster City "shortcut"). I split off on Marsh to ride Middlefield to California Ave in Palo Alto. I scored a prized "SF2G" sticker in the deal. Sweet.


Jeremy Denk

Thursday Night: San Francisco Symphony does Ludwig. A little overture, then Jeremy Denk played a spectacular 5th Piano Concerto. Capping it off was the 5th Symphony. Cara and I were in the front row, within spitting distance of the piano. Great for the concerto, not as good for the symphony. But a very cool perspective of the violins and the cellos, anyway.

Sunday, July 5, 2009

Wisconsin to Long Beach

Fellow Mouse PPH was driving us to Leesville on Friday. Despite our 5:05 am departure, it was getting warm in the car as we passed through Vacaville on I-80. I checked in the back for my bag, in which I'd stored my water bottles for the day, jammed in there admist my ride clothing, shoes, and food. No bag. Whoops. I'd left it at his place when I'd gone in to use his toilet. I'd be doing support duties for this race... Which turned out to be a lot of fun, and useful on a course where flats are common and the feed zone is essential. If nothing else, being in the feed zone meant I could hand-pick which of the reused discarded bottles the neutral feed was handing out I could give to my teammates. Maybe save them from swine flu. No complaints about the neutral feed, BTW: nobody forces any rider to take neutral water. Beggers can't be choosers.

So while the day was productive, it didn't do anything for my fitness, obviously. Unless sweating in a hot feedzone induces adaptations to the heat. If it does, I squandered it the next two days in fog-plagued San Francisco and Marin. My goal for the weekend was to get in some solid miles: make up for the lack of riding at Friday's race. And I succeeded: 87 miles on Saturday, 83 on Sunday, with plenty of Z5-Z6 climbing efforts in the 4-5 minute range. I managed to toss in a few hard sprints, as well.

As I returned from today's ride I crossed paths with two riders loaded down with panniers and gear. I asked where they started. "Wisconsin," one replied. Wow! They'd started there only three weeks ago, doing around 100 miles per day. They wanted to know the way to San Francisco.

I led them through Marin Bike Route 20, which Henry Kingman calls the "Bicycle Highway". We stopped in Larkspur at the park off Magnolia. Along the way I'd asked them where in San Francisco they'd be staying. "We're going to San Jose" was the response, "Do you know the way there?"

Wow -- San Francisco to San Jose is far from trivial. When we stopped, I drew it on the map, a route which dances around and sometimes on CA-280 to Highway 92, then Canada Road, Foothill Expressway, and eventually Saratoga Road. Too many details.

Luckily one of them had an iPhone. I pointed him to SF2G, which describes the San Francisco to Google bike commute routes. This substantially facilitated them following the complex route.

And so I hope they made it to San Jose. From there: eventually Long Beach, then eastward. Really cool.

Thursday, July 2, 2009

filling in the curve

I've been doing a lot of intervals lately... 10 second, 15 second, 30 second, 5 minute. As a result, the left side of my maximal power curve has filled in nicely compared to last year.


Maximal power curve through 04 June, curve for June alone, and curve for surviving data from 2008

You can really see from the curve that the curve using June data is notably improved from earlier in the year: approaching the 2008 numbers. The exception is sprints, but the sprint session in question was late May, virtually June, so is part of the same trend of improvement.

An aside: one thing I've seen with my meter is that if the hill requires a lot of shifting, if the grade varies a lot, average power is going to be lower. It's just harder to sustain a constant effort if the road itself isn't sufficiently constant. The obvious extreme of this is an intermediate descent on a hill. Even a small one is a kiss of death for posting up a good point on the maximal power curve. I've really not done nearly as many of the 5 minute efforts this year as last. Part of the difference may be the total lack of sessions at Huddart Park or Kings Mountain Road. These provide a continuous effort which is good for getting everything I have to offer. So I'm not worried about my 5 minute number.


Something I'll never see on my Powertap (Uhl Albert)


In the 45 second to 4.5 minute range last year also has an advantage. But last year had more intervals of mixed duration encountered on hard group rides: I simply haven't done maximal efforts this year in that time range. If I can go hard at 5 minutes and hard at 30 seconds, nothing magical happens physiological at efforts in between. I'm very confident I could bring this year's curve up in this time range if I tried.

The significant difference from last year is still at longer times: in particular that magic time range near 18 minutes associated with Old La Honda. I just haven't been able to put together a solid climb of that hill yet this year. I've tried, for sure, but it never seems I'm fresh when I do Old La Honda. that's not much different from last year, however. So it's still a fair comparison. Even on my "fatigued" days last year I was putting up better numbers than my best so far in 2009.

In the range of around 30 minutes up to 5 hours the big difference this year is a lack of race data. For example, I did Climb to Kaiser last year with my PowerTap. It's a lot of weight to carry over 13 thousand feet of climbing, but very useful for assessing pacing over the 154 mile event. This year I skipped Kaiser, however. So I don't worry much about what's going on out there.

It's the OLH data I need to improve. One of these week's I'm going to make a run at the hill. Come in fresh, use optimal gears, and see if I can make a further dent in that curve.

Wednesday, July 1, 2009

Middle Road

I started with the Nooner yesterday, then split off @ Golden Oak to do 30 second sprints. First I did four on various hills around there, then I headed up Alpine, planning on doing more on the climb towards the gate where I'd turn around. A small deviation on Alpine Trail led to some hiking around the numerous railroad ties, but soon I was back on Alpine, riding past Willowbrook.

As I gently climbed I was intrigued by the open house sign at the entrance to a private road. Cool: a chance to check out what's hidden behind the tree cover. But the road, which I'd assumed was a short access route to a few oversized domiciles, instead went on and on and on. I climbed slowly, except every 6 minutes or so doing another 30 second sprint. On my third sprint, I finally reached the "open house". I didn't see much of a house, only a driveway, but instead of investigating that further, I continued down the road. Then I came across the colossal compound which is the Fogarty Winery, a few vines growing amidst the tennis courts, swimming pools, and statues.


View Larger Map

A grounds keeper was driving a cart towards me. I asked him where the road went. Open space, he said, 2.5 miles to Skyline. So I moved on from the opulence of the vineyard to a grass-covered but smooth fire road, crossing a gate with "Private! No Tresspassing!" signs on the opposing side. This gradually climbed, with scenic views, crossing a few trail heads, until I reached another gate: Skyline Blvd.

There was a ranger there, who kindly informed me the road from which I'd emerged, "Middle Road", was closed to bicycles. Strange: it was a perfectly good fire road. I explained I'd gotten lost, and that was good enough.

From there, it wasn't far to Old La Honda then to 84, which I descended eastward, my spirits buoyed by the adrenalin rush of a previously unexplored route to Skyline. I managed to get in the rest of my sprints on my ride back to work from there. A bit of a longer lunch than I'd planned.

Power numbers from the day were, for a change, encouraging. More pn power next time.