Saturday, April 30, 2011

SB910: supporters and opponents

The Senate has posted an analysis of SB910, the bill which would impose a 3 foot minimum passing distance for motor vehicles passing cyclists. There is now on AroundTheCapitol an analysis, dated Thursday 28 April.

First, I'll jump to the end:

SUPPORT: Office of the Mayor, City of Los Angeles (co-sponsor)
California Bicycle Coalition (co-sponsor)
Amgen Cycling Club
Channel Islands Bicycle Club
Humboldt Bay Bicycle Commuters Association
Sacramento Area Bicycle Advocates
Santa Cruz County Cycling Club
Silicon Valley Bicycle Coalition
47 individuals

OPPOSED: AAA Northern California
Automobile Club of Southern California


AAAJust in case any cyclists out there are members of AAA, please quit now. Try Better World, for example. AAA members are supporting the AAA's car-centric legislative agenda, opposing every attempt to improve the rights of cyclists on the roads. It is incomprehensible to me that any member of any bike coalition would be also a member of AAA. It's like chasing down your own teammates in a bike race.

This happens time and time again: every initiative to improve cyclist rights, cyclist safety, is opposed by AAA. Recently they fought to get Federal transportation infrastructure dollars more focused on highways, less on trains, less on cycling infrastructure. Every dollar sent to them is supporting their pro-car agenda. AAA is, like it or not, a political lobbying organization, and if you sign on to their free maps or insurance or whatever then you're signing on to their agenda.

On the other hand, hats off the the organizations supporting this legislation. In particular, I'm glad to see Silicon Valley Bicycle Coalition there, and disappointed to not see San Francisco Bicycle Coalition. SFBC, to be fair, is focused on city issues. However, I would have liked to see them step up and show support for such an important state-wide effort.

Friday, April 29, 2011

SB910 gutted?

Yesterday I hit refresh on my screen yesterday morning, and I saw this ( link )... the text of SB910, the bill supposedly requiring a 3 foot passing buffer when drivers pass cyclists, was revised.

Changed from:
21750.1. (a) (1) The driver of a motor vehicle overtaking a bicycle proceeding in the same direction shall pass to the left at a safe distance, at a minimum clearance of three feet, at a speed not exceeding 15 miles per hour faster than the speed of the bicycle, without interfering with the safe operation of the overtaken bicycle.

To:
21750. 1. (a) The driver of a motor vehicle overtaking a bicycle proceeding in the same direction shall pass to the left at a safe distance, at a minimum clearance of three feet or at a speed not exceeding 15 miles per hour faster than the speed of the bicycle, without interfering with the safe operation of the overtaken bicycle.

Wow -- talk about flip flop from replacing a comma with an or. So it's now considered safe to blow by a cyclist riding 20 mph going 45 mph within one foot of the rider? Now it's clearly too weak.

Not to mention the penalty has been substantially reduced, from felony to a $220 fine. We're not talking petty infraction here: the key text is "causes great bodily injury".

(b) If a person operates a motor vehicle in violation of subdivision (a) and that conduct proximately causes great bodily injury, as defined in subdivision (f) of Section 12022.7 of the Penal Code, or death to the bicycle operator, the person driving the motor vehicle, upon conviction, shall be punished by imprisonment in a county jail or in the state prison. (b) A violation of subdivision (a) is an infraction punishable by a fine of two hundred twenty dollars ($220).

One plus: Joe Simitian used to run a "It outta be a law" contest in which citizens could submit proposed laws to him which, if it won the contest, he would file as a bill. Mine, allowing cars to cross double-yellows up to 3 feet if there was sufficient visibility when passing "non-vehicular road users" (cyclists and pedestrians), lost out to a "mandatory running of windshield wipers" bill (which passed). A similar provision was added here, at least for cyclists:

(applying to double yellow lines)
(2) As provided in Section 21460.5. (c) (1) Either of the markings as specified in subdivision (a) or (b) does not prohibit a driver to whom any of the following applies from crossing the marking (A) The driver is on a substandard width lane, passing a person riding a bicycle or operating a pedicab in the same direction, and it is safe to do so.

So the 3-foot passing bill is now too weak: the penalty would too small and the 15 mph exception is too broad. If we're going 35 mph, it would okay to have a car pull next to me @ 0 mph delta and pull within 1 foot? Silly. But I'm glad to see the proposed rule allowing cars to cross the double yellow when passing cyclists. I (when I drive) and 98% of drivers do that already.

The whole speed differential thing really should go, though. I'd be willing to support it if it said the car was going no faster than 15 mph, period, no differential, to handle the turbulent conditions near intersections, but even then something a bit slower would be better.

Wednesday, April 27, 2011

boat puzzle: still confused

Back to the boat puzzle...

I freely admit this is perhaps among the most boring topics ever posted to a blog in this history of the Intenet. But I feel the irrational need to put some sort of closure on this.

I proposed that all solutions to the arbitrary boat puzzle could be reduced to the simple example of Mac-Nadia-Flannel.

It was first proposed to me that if there is at least one member not a part of a maximal independent set incompatible with no more than two members of the maximal independent set, the problem can always be solved. This is true: just leave everyone else on the boat and solve the simple Mac-Nadia-Flannel-cans problem with the one remaining seat.

But there's a more general approach, in which each elemnt of the "Mac-Nadia-Flannel-cans" puzzle is represented by a group rather than an individual. First, identify which animals which are going to go across on the first trip (the rest are a maximal independent set). Then bring the rest across. Then decide which of those from the first trip will stay on the second shore, and which will stay on the boat for the duration of the entire solution. Next, make trips to shuttle across those in the independent set compatible with those left on the second shore. Then shuttle across a first group not compatible with those on the second shore. Swap these animals with those left on the second shore, bringing those back. Then swap these for the last of the animals left on the first shore. Finally, head back, pick up everyone else, and shuttle them across.

For this to work, the number of animals originally left on shore 1 who are incompatible with those originally left on shore 2 must number no more than twice as many as those originally left on shore 2.

Anyway, that's a lot of detail. I wrote a program to carry out this solution. I first generated either all possible puzzles (for up to 8 animals) or a set of 8192 random puzzles (for up to 16 animals here ... 3, 9, and 13-16 had no failures). The fraction of puzzles which could not be solved is as follows. You see this captures most cases.

plot


So you can see the algorithm does pretty well.

However, I felt the solution was too general. It was making too restrictive assumptions about the individual groups. So I checked my code for simple rejected cases and identified the following as a solvable puzzle. The unique maximal independent set is outlined:

tricky puzzle


This problem can be solved.

In considering the solution of this problem, we observed that after swapping 1 and 2, swapping 2 and 3, and swapping 4 and 5 the reverse solution ("playing the video backwards") is the same as the forward solution. That's curious. There's an underlying symmetry here which is being exploited. Indeed, I think it can be shown that the optical solution of any puzzle is also an optimal solution when played backwards. In the case of a symmetric puzzle an optimal solution exists where the puzzle played in reverse is the same as the forward solution after swapping symmetric pairs.

But I was never very good at abstract math...

Tuesday, April 26, 2011

SB910: 3 foot passing zone

The California State Senate is considering SB910, the latest in a long-running series of bills requiring a minimum passing margin for cyclists. The text of the bill is here.

The key text:

21750.1. (a) (1) The driver of a motor vehicle overtaking a bicycle proceeding in the same direction shall pass to the left at a safe distance, at a minimum clearance of three feet, at a speed not exceeding 15 miles per hour faster than the speed of the bicycle, without interfering with the safe operation of the overtaken bicycle.

I'm a huge fan of a 3 foot passing margin. It really changes nothing, since any rational observer would conclude passing with less than a three foot margin is unsafe, and it's illegal to pass at an unsafe distance. But this codifies that less than 3 feet is de facto unsafe, removing the burden to prove it. For example, presently a driver could pass with a two-foot margin, hit a cyclist, then claim it was the cyclist's fault for veering to the left. The 3-foot margin gives the cyclist some breathing room.

Of course, it could be argued if a 3-foot margin is necessary to deal with unforseen circumstances, like the cyclist hitting an object on the road and swerving two feet to the left, then the law would actually require an anticipated 6-foot margin, to give the driver a 3 foot margin to avoid the 3-foot margin and therefore a violation. But this is perhaps being picky.

However, what I don't like about this bill is the 15 mph speed difference. It's simply impractical. Bikes are allowed on certain shoulders of I280, for example. Shall all passing traffic slow to 25 mph to accommodate a 10 mph cyclist? The fatal flaw is that the safe passing speed is actually a function of the gap. Perhaps with a 3 foot margin, a 15 mph differential is recommended. But then with a 4 foot gap, the speed can be higher, and with a 5 foot gap, higher still. The law doesn't recognize this: it simply places a 15 mph speed differential limit, even if the car is all the way into the opposite lane.

So I view this presently as bad legislation. I have been told by a member of the California Bicycle Coalition that an attempt will be made to clean this up. Nobody seems to know, I was told, how that 15 mph differential item got into the bill. Given that it's been there for several months, I don't know what's taken so long. It's so obviously flawed.

I really hope this one gets cleaned up and sent through. The arguments against it, that a 3-foot passing requirement will unduly slow traffic, are absurd because they imply, without explicitly stating, that it's acceptable to pass a cyclist closer. As it stands, it's simply too hard to prove a pass was "unsafe". With this bill you'll simply need to prove the pass was closer than the 3-foot margin.

Saturday, April 23, 2011

"aero road frame" versus front fender

I'm working my way backwards through Bicycle Quarterly and am currently at the Fall 2007 edition. There's a very interesting article there: Jan Heine went to the wind tunnel to test his Alex Singer Randonneuring bike with various options. One of the tests was the effect of a front fender.

Bicycle Quarterly


As he expected, the front fender appeared to reduce wind resistance. He used a telescoping fender to test the effect of length. I plot the result here: "N" corresponds to no fender, "S" for short, "M" for medium length, "L" for long, "XL" for extra-long, and "MF" for a mud-flap on an extra-long fender.

plot


There appears to be an optimal length: the fender blocks the wind from entering the brake area, then going longer than that is counter-productive.

Jan dismisses the effect as relatively small. However, ignoring his "repeatability" tests to establish error bars, I'll plot the naked numbers of the difference in CdA from the fenders to the difference in CdA reported by Scott for "aero" mass-start bike frames relative to a "round-tube" frame, my guess the long-head-tubed Scott CR1. Scott's tests were with a dummy mounted on the bike. Jan's tests were with a rendered outline on a display against which is aligned an image of his body for repeatability when conducting the wind tunnel tests. I use the zero-yaw data from the Scott test since that is how the fender test was done.

plot


Curiously, the short front fender seems to offer a greater advantage than the Cervelo and Felt, and not much less than the Scott Foil. Velo Orange fenders cost around $60. The Scott Foil frame is probably around $3500.

Friday, April 22, 2011

boat puzzle: solving the simple cases

Previously I described the boat puzzle and how it's provided some mental exercise for my train commutes.

The traditional example is the Mac-Flannel-Nadia case:



The solution, as I described, is to identify a maximum independent set, in this case Mac and Nadia. Then you need a number of seats on the boats sufficient to transport the rest, in this case Flannel. So:

1. Flannel taken from shore A to shore B.
2. return with empty boat.

Next step doesn't matter, as Mac and Nadia are identical in this problem. I'll pick Mac:
3. Mac taken from shore A to shore B.

Now Mac and Flannel are together. I need to return with Flannel on the boat, as I cannot leave the two, or Mac will turd.

4. Flannel taken from shore B to shore A (leave Mac on B).

But then....

5. Take Nadia from A to B (leave Flannel alone). Nadia is now with Mac, and they lick each others faces, so that's fine.
6. Return with empty boat.
7. Bring Flannel from A to B.

When I add in the cans, it's only slightly more complex. I circle the maximum independent set, the cats which need to be left behind on the first boat trip:

Mac-Nadia-Flannel-cans

Here I add steps between steps 2 and 3 in the above:

2.1 bring cans from shore A to shore B. Leave those with Flannel, who can't open them.
2.2 return with empty boat.

Then continue with step 3.

Here is a problem which cannot be solved in a nontrivial number of seats on the boat. Recall if I have more than the number of seats sufficient to isolate the maximum independent set, then a solution becomes trivial, so I am only interested in problems with non-trivial solutions.

Another problem: I add Joshua to Mac, Nadia, and Flannel. Joshua is very friendly, but Flannel is a pest, and due to certain issues, Joshua can't defend himself as well as he might. So I don't want to leave Joshua with Flannel, either. The maximal independent set is obviously Joshua, Nadia, and Mac. I therefore want to be able to solve it with only one seat on the boat:

Mac-Nadia-Flannel-Joshua


Out of luck here: I know of no solution.

So I conjectured that all problems can be reduced to either Mac-Nadia-Flannel-cans or Mac-Nadia-Flannel-Joshua. If the former, I can solve them. If the latter, I cannot. That was my conjecture, but I no longer believe it to be true.

Tuesday, April 19, 2011

Golden Gate Bridge proposed 10 mph speed limit

Jym Dyer sent notice of this gem along the SF Bike mailing list:

APRIL 21, 2011 BUILDING & OPERTATING COMMITTEE TO DISCUSS 10 MPH SPEED LIMIT FOR BICYCLES (sic)

SF Examiner story

Here was my response, which I emailed to the Golden Gate Bridge Transportation District:

I have read the consultant’s report and the corresponding proposal to introduce a 10 mph speed limit on the bridge.

The statistics do not justify such an action. Collision rates (per rider) have been decreasing over time, even with an enormous increase in the number of inexperienced cyclists (renting from Blazing Saddles and other agencies) crossing the bridge. And only a minority (39%) of collisions are labeled with “speed as a factor”. This is in spite of the fact that virtually all experienced cyclists ride in excess of 10 mph across the bridge (there are plenty of data available on Strava.com). Checking my own data, 25 kph is a typical speed even when riding alone in a relaxed, noncompetitive fashion.

There isn’t even a case presented that safety is a problem on the bridge. You cite 164 collisions over 10 years on a bridge where “up to 6000 cyclists” used the bridge in a day. If you assume 15 thousand cyclists per week (an average of only 2 thousand per day), that’s roughly 750 thousand cyclists per year, which corresponds to 1.3 million cycling miles per year at one crossing per cyclist (I assume the report counts round-trips as “two cyclists”). Over ten years this corresponds to one collision per 81 thousand miles ridden. This is likely lower than typical urban motor vehicle collision rates.

Even if a speed limit is imposed, 10 mph is too slow, especially on the western path from which pedestrians are forbidden. The report justifies imposing a speed limit on both sides as follows:

“Bicyclists accustomed to riding unimpeded at a relatively quick speed over the Bridge on weekend rides may have difficulty adjusting to sharing the path with slower moving pedestrians. Pedestrians may feel intimidated by large groups of bicyclists passing by.”

With this logic, auto speeds should be limited by the safety concerns of the slowest nearby streets. For example, the motor vehicle limit on the bridge should be no more than 25 mph, as motorists accustomed to going faster may have difficulty adjusting to conditions on the Bridgeway. This line of reasoning is inconsistent with standard motor vehicle policy where the “safest speed” is judged on a road-by-road basis.

If there is a safety issue on the western path, and the data hardly indicate a safety concern given the traffic volume, then that issue is with pedestrians: people standing by their bikes taking photos, often moving without care. Pedestrians are forbidden on the western path, and that implicitly applies to people standing next to their bicycles who are legally pedestrians:

California Vehicle Code 467. (a) A “pedestrian” is a person who is afoot or ….

The bike paths on the bridge are as legitimate transportation infrastructure as the motor vehicle lanes. Regular users of the bridge, such as I, should be able to cross the bridge in a timely manner, subject to the constraints of safety. Yet under general conditions, the maximum safe speed on the bridge is more like 20 mph than 10 mph. Under conditions of exceptional congestion the speed needs to be reduced, as it must be reduced on the roadway when there is heavy traffic. But you do not set roadway limits under the worst-case traffic scenario, and you should not set cycling speeds under worst-case traffic scenarios.

There is a precedent for what constitutes a safe and reasonable speed for cycling infrastructure: Table 1003.1 of the California Highway Design Manual specifies the design speed for Class I bikeways where mopeds are forbidden to be 40 kph. It is clearly recognized that speeds of up to 40 kph are reasonable for bicycle use under appropriate conditions. And those conditions often exist on the bridge between the supports in early mornings and in winter. There is no need for a blanket regulation.

I am thus strongly opposed to this proposed limit.

Monday, April 18, 2011

boat puzzle:: the first trip and maximal independent sets

I'll now continue my discussion of the boat puzzle in which the goal is to get cats across a river in a boat, where there are fewer seats on the boat than cats, and certain cats must not be left with each other unsupervised.

For the first trip across the goal is simple: to leave no two incompatible cats back on the first short. For example, consider the following problem, with cats Mac, Nadia, and Flannel, where I spiced it up slightly with the requirement to also transport a supply of cat food cans:

Mac-Flannel-Nadia-Cans


I want to find a group which has no mutual connections to leave back on the shore. Ideally, my boat should require no more seats than the number of cats + objects to be transported in that first trip. Thus I want this group without mutual connections to be as large as possible. The largest group without connections is called the maximal independent set.

Here is the maximal independent set for the graph shown above:

maximal independent set


The only element not in the maximal independent set is Flannel, so Flannel goes in the boat first, and I need at least one seat in the boat to accomplish my task (which is obviously the minimum for any non-zero number of cats + objects).

If I have two seats, there is a trivial (perhaps nonoptimal) solution. I keep Flannel in one seat, while I use the other seat to transport cats across one-by-one. For example, I take Flannel + Mac, drop off Mac, go back with Flannel, pick up Nadia, drop Nadia off with Mac (leaving Flannel on the boat), go back with Flannel, pick up the cans, then exit the boat with Flannel and the cans: all across. The interesting question is if I can solve it with only enough seats to isolate the maximal independent set.

Here is another graph I described last time: Flannel, Mac, and Molly: none of the two cats can be left together unsupervised.

Molly puzzle


Here there are three candidates for the maximal independent set, each with only one member:

maximal independent set
maximal independent set
maximal independent set


So I need at least two seats to solve this problem. And I can solve it with two seats: for example, I pick Molly as my independent set, and put Mac and Flannel in the boat. I drop off Flannel, and take Mac back. Then I pick up Molly and take Mac and Molly and I'm done.

The whole trick is in finding a maximal independent set which allows the problem to be solved with just enough seats on the boat to carry the others across on that first trip. Here is a randomly generated graph. The Perl code which generated this plot isn't very creative, so used numbers for the elements of the graph, as opposed to cat names:

random graph


There are three solutions for the maximal independent set here; here is an arbitrary one:

maximal independent set (1/3)


You can see I would need five seats on the boat to make the first trip. So the question is: is this enough? As I described, there is a trivial solution if there are six seats on the boat, so I want to solve it with only five.

Sunday, April 17, 2011

comparing Scott Foil windtunnel data to VeloNews and Tour results

A few more plots on the Scott Foil comparative windtunnel data...

First VeloNews. Both Cervelo S and Felt AR frames were tested by both VeloNews and Scott. VeloNews tested the Masi Competizione, which they claim is similar to a Trek Madone. Scott tested the Trek Madone, so I include the results from these tests as well. VeloNews' tests were with Zipp 404 wheels except for the Masi which was tested with Fulcrums. However, I adjusted the VeloNews data based on the (considerable) improvement the Zipps had on the Ridley and Cervelo in their test, which were each tested with both the Fulcrums and the Zipps.

VeloNews tested at positive and negative yaw, while Scott reported only wind from one side of the bike, so I averaged positive and negative data from VeloNews. In each case, both for the North Carolina windtunnel used in the VeloNews test, and in the Mercedes windtunnel used by Scott, I assumed an air density of 1.2 kg/m³ to convert from force (VeloNews) or power (Scott) to CdA (the coefficient-of-drag-normalzed cross-sectional area for wind resistance).

The Scott test was "bike only with similar wheelset". VeloNews tested complete "stock" bikes with both "stock" and Zipp 404 wheels.

Here's the comparison:



First, the Scott numbers are obviously lower. Other than that, the absolute difference between the Felt and the control bike is rather similar in the two cases. The difference between the Cervelo and the control bike is larger in the VeloNews test. As a result, the Cervelo does better than the Felt in the VeloNews test, while the two are much closer in the Scott test.

Curious is the shape of the Madone curve versus the aero bikes. The Madone has a steeper CdA-versus-yaw slope. In the VeloNews test, all bikes, after adjusting for the same wheels, had similar slopes. I wonder if the Scott test was done with Firecrest wheels on the Madone, or like in the VeloNews test, there is an undisclosed exception made for the "control" bikes. The Scott test had three control bikes, and all three followed similar slopes, while all three aero frames followed a reduced slope.

One issue with windtunnel tests is you need to subtract off the "tare": the wind resistance due to the set-up and not due to the bikes. So it's possible the tare subtraction differs in the two tests. I wouldn't necessarily expect the two tunnels to yield the same result for the exact same bike. Then there's the issue of air density. I don't know what the relative air density is is in the two tunnels. But there's countless other factors which could differ in the two tests, starting with frame size. I simply don't know much without photos.

Then there's the Tour Magazine tests. These were done with a dummy mounted on the bike. Results with the dummy are more variable than those without the dummy, at least to my eye. Here is a comparison of the Felt, Cervelo, and Cannondale tested by Tour to the Felt, Cervelo, "round-tube" control, and Scott Foil tested by Scott.



The difference here is just huge. The Scott numbers, around a CdA of 0.22, would be about the best you get from a time trialist with a super-optimized position like Levi Leipheimer or Dave Zabriskie. The Tour values, around 0.31, are much more typical of a mass-start position. Tour showed a photograph of their dummy set-up, while Scott did not. Maybe the Scott dummy had no arms. That would be my guess. Such a simplification would facilitate set-up.

Looking at the Tour result, that low-yaw Cervelo result is just weird. I see nothing so anomalous in the Scott test.

In the Scott case, the Scott suddenly does substantially better than the Cervelo and Felt with the dummy on the bike. Why? Looking at images of the bikes, I can't see a good reason for this. You might think the Cervelo and Felt have some aerodynamic feature, like an aero post, which helps them much more without a rider than with the dummy mounted. But all three bikes have aerodynamic seat posts.

What do I conclude from all of this? Basically I think it's clear that the aero frames do improve things, and significantly, albeit less than other factors like body position (or perhaps if the rider has arms or not). It also appears the Scott is a legitimate aero frame, in the same class as the Felt and Cervelo, despite being much lighter. In the Tour test other so-called aero frames, like the Canyon and Stevens, did not do so well in the wind tunnel relative to their Cannondale control.

But it's obvious wind tunnel tests are hard to do. Make different assumptions, get a different result. Looking at end-results only, without details about the assumptions and the set-up, is of limited value. This is especially true when you test with a dummy rider, which in principle seems the right thing to do, but then you need to make sure you're meticulous in setting up the dummy in exactly the same position, and that fitting the dummy to the bike in this position doesn't introduce any biases for one frame versus the other since different frames fit different riders differently (especially a dummy whose position can't be tweaked even a little without confounding tests). Hats off to Scott for showing results both with and without the dummy rider. They could have shown only the dummy results, and claimed unambiguous superiority.

added 13 Jun 2011: after writing this I learned Scott tested the bikes with two water bottles and cages, while the other tests were done without water bottles. This provides an advantage to Scott (as it would to Litespeed, for example) for a better interface between the truncated down tubes and the bottle shape. Second, it would award more optimized placement of the water bottle holes. Bottles may either increase or decrease wind resistance relative to no bottles, but the much lower CdA's apparently evident in the Scott test relative to the Tour test, for example, still puzzles me.

Saturday, April 16, 2011

Scott Foil wind tunnel test of aero mass-start frames

Scott just released a new bike, the "Foil", their entry into the "mass-start aero frame" category.

This bike has been around in the pro peloton for a year now. Mark Cavendish famously rode it in the first stage of last year's Tour de France, where he had a disappointing result. When he switched back to his previous bike, a beefed-out Scott Addict (the consumer Addict is under 800 grams, Mark's custom bikes are reportedly close to 1200 grams with a Pinarello-like excess of carbon fiber), he started winning races again. But the rest of his HTC Columbia team continued to ride the bike, including Mark Renshaw and the rest of the lead-out train on which Mark relies so heavily for his wins.

Interestingly, until this consumer release it has been called the F01. Very clever: the F01 becomes the F01L, or the "FOIL". Scott wins the bike naming game against Specialized with the latter's recently-released Venge.

The differentiating factor with the Scott is its "Kamm Tail" design. You can see Scott's website for plenty of info: basically they cut the tail off an aerodynamic shape, and since the flow of the wind has already been established by the tail, the air flows almost as smoothly over the shape than it would have with the tail still in place. The advantage is you save mass by leaving the tail off, and additionally attain UCI compliance. The UCI forbids frames to have more than a 3:1 aspect ratio (length to width) on any of the main tubes. (Apparently the ratio does not apply to quick release levers, for example). Here is their styled representation of the concept:



Their video shows their engineers apparently inventing the concept, but Trek has been using it for several years on their time trial frames.

Scott puts wind tunnel data up on their site to justify their claim that the frame is at least as good as the competition. So far in the tests I have reviewed, one by Tour Magazine and the other by VeloNews, the Felt AR and the Cervelo S have been compared. In the Tour test the Cervelo and Felt were fairly close, with the Felt better at zero yaw. This test was done with a dummy on the bike, but no cables installed. Then VeloNews published a test in which the Cervelo came out clearly ahead. This had the bikes fully built up except for pedals without a rider on board.

Here's Scott's data for its bike compared to the Cervelo, the Felt, and to three "non-aero" bikes, the Specialized SL2, the Trek Madone, and a "round-tubed bike" which I'd guess is the Scott Addict, or even more likely a Scott CR1 with its longer headtube. I first show the results with no dummy on the bike. I converted their data, described as power at 45 kph, to a CdA value assuming an air density at the Mercedes Wind Tunnel of 1.2 kg/m³.

no dummy


This is exactly the sort of thing you might expect. The Cervelo and the Felt, designed with more traditional aerodynamic optimization, come out slightly better at low yaw than the Scott. At higher yaw the Scott does better. When the wind hits from more of a side angle the elongated aerodynamic tubes may provide more surface area for wind resistance.

All three finish well ahead of the Specialized SL2 and the Trek Madone, but interesting is that the Specialized does so much better than the Trek (and the "Round Tube Bike"). Here's a comparison of these frames:

Trek Madone
Scott CR1
Specialized Tarmac SL2


I'm not sure I see why the Specialized did better than the others. Were the frames tested with seat posts? If not, maybe the swoopier top tube-seat stay transition helped. But since the Foil has an aerodynamic seat post, I suspect they were tested with seat posts. Did they test the Madone "performance" or "pro" fit? If the "performance", then maybe it's an issue of head tube length: the Madone and the Scott both having longer head tubes than the Specialized. I'm really not sure.

Here's the aero frames tested:
Cervelo S
Felt AR
Scott Foil


They also tested the bikes with "dummy" riders. I'm not sure if the frames had components installed or, if so, what components. Here is the results, which again I converted to CdA using the same assumed air density of 1.2 kg/m³:

dummy


These results are very different from the tests of the bare frames. The Scott Foil does clearly better than the Cervelo and the Felt. How is that possible? Without seeing photos of the dummy mounted on the bikes, I worry about how closely matched the rider positions were, and how the riders were placed in that position. So it would be difficult to speculate on this result.

This is getting long, so I'll save more more for later...

Thursday, April 14, 2011

boat puzzle

Another puzzle... actually I spent quite a few train commutes thinking about this one.

There's a classic puzzle where a farmer needs to get a fox, a sheep, and a cabbage across a river on a boat. I don't have much experience with transporting fox or sheep, and cabbage aren't very exciting, so I'll recast the problem in terms of cats.

Consider I want to transport three cats across a river on a boat. I can watch over the cats while they're on the boat, but when they're on one shore or the other, they need to get along. The names of the three cats are Mac, Flannel, and Nadia.

If I can watch them they're fine, but if Nadia is left unsupervised with Flannel, Flannel will harrass Nadia. who needs her rest. And if Flannel is left unsupervised with Mac, Mac will turd, so we need to be careful not to let that happen. On the other hand, if Mac and Nada are left with each other, they'll just lick each others faces, so that's fine.

The challenge is there is only one extra spot on the boat for cats. For example, I can transport Nadia, but if Nadia's on the boat and Mac steps on, Mac will sink the boat.

The goal is to get the three across the river without leaving Nadia and Flannel unsupervised or Flannel and Mac unsupervised. How to proceed?

Well, I can represent the relationships between the cats with a graph. Each cat is a node in the graph, and lines connect cats which cannot be left together without supervision. Here is the graph for this problem:

Mac-Flannel-Nadia


I can make the problem slightly harder by adding in a tray of cat food cans which must be transported with the cats. If I carry the cans, that's all I can carry. Cats can't open cans, so the cans can be left with any of the cats. The diagram becomes:

Mac-Flannel-Nadia-cans


So far, not much new. Now suppose I consider another problem. Instead of carrying Nadia, I want to carry Molly. Molly will rip to shreds any other cats if she's not supervised, so the diagram (without cans) now becomes:

Mac-Flannel-Molly


I can solve this one trivially with two or three seats on the boat, but with one seat, is it possible?

In general, I can construct arbitrary graphs with an arbitrary number of connections. The question posed to me during my commute was: how many seats does the boat need to have to transport everything across?

Wednesday, April 13, 2011

Paris-Roubaix head-tube length record?

This year Cervelo overhauled its road bike geometry, merging the previous R and RS series into a single unified R. I plot head tube angle versus reach for 2010 and 2011 series: the 2011 R series goes from R-like at small sizes up to RS-like in large sizes.

2010 vs 2011 Cervelo geometry


On Sunday, Garmin-Cervelo's Johan Van Summeren produced an epic win at Paris-Roubaix riding his modified "R3". Johan's a tall guy: reportedly 2 meters in height, and rides the largest-size Cervelo. From the plot you can see this bike has a 22.5 cm head tube, 2 cm longer than the comparable bike from last year. And you can see every cm of this length in photos of Johan's bike, borrowed from VeloNews:

VeloNews photo


VeloNews photo


Impressive. Johan is at the limit of the frame's geometry with a slammed -17 degree stem (of course he could have used a retrograde stem to get even lower, but this is generally considered undesirable). This seems to support Cervelo's geometry.

But it got me thinking: is this the longest head tube ever to win in Paris-Roubaix's prestigious 106-year history?

The first Paris-Roubaix was held in 1896, won by Joseph Fischer on a bike which would look very much at place in the streets of San Francisco (Wikipedia photo):



Lacking any web pages with geometry charts for his bike, I need to estimate the head tube length myself. For that I need a length reference. Usually for this purpose I use wheel diameter in the direction of interest. The isuse is, however, that wheel diameter was not yet standardized in 1896.

As luck has it, I have Jan Heine's excellent The History of the Competition Bicycle and that has geometry measurements for an 1895-1896 Humber bicycle. Here's a poor photo of the diagram of the bike:

Jan Heine schematic of 1895 Humber


The front wheel is slightly larger than the rear on this one, something not evident in the photo of Josef's rig. I have a few candidate reference lengths I can use: for example the bottom bracket height or the chain stays. The result? Fairly close: it could go either way. I think Josef's head tube is longer than Johan's, but I'm not sure.

Certainly by the era of Louis Trousselier, the 1905 winner, riders had begun to adapt more aggressive positions and head tubes were shrinking.

Teousselier, 1905 winner


Riders tend to run bars lower than they did in the mid-20th century, since riders previously used the drops more, the hoods less, than standard practice today. There was also a tendency to ride with the arms further forward (more stretched out). So if there was a rider of Johan's height who's won it before, that rider may well win the "prize".

Saturday, April 9, 2011

VeloNews vs Tour, Cervelo vs Felt

Both the VeloNews aero bike test and the Tour aero bike test (my discussion of that test is here) examined the Cervelo S and the Tour AR. In the case of VeloNews, the bikes were tested "stock" with the exception both used Zipp wheels. Cables were installed and there was no rider. Tour on the other hand used a dummy rider (the only way to get a reasonably consistent position) and Mavic wheels. The Tour article is in German, so the finer details are beyond my comprehension. But they didn't use bar tape, while VeloNews used whatever bar tape the bikes had installed. Same deal with the saddle: the VeloNews bike tested bikes with "stock" saddles, while in the Tour test, the saddle was less important due to the use of a "rider".

Despite differences you might think conclusions about which bike was faster would be fairly consistent. But comparing the only two bikes tested by both magazines, this is not the case:

comparison of results

I averaged the positive and negative yaw results from the VeloNews test, since Tour reported values for only one side of the bike.

If you read Tour, and you think low yaw angles are the most important, you rush out and buy a Felt. On the other hand if you read VeloNews you definitely get the Cervelo.

Neither test result was without surprises. The Cervelo at zero yaw was slower even than the Cannondale control bike in that test. Nobody's ever looked at a Cannondale System 6 and walked away with the impression of aerodynamic elegance. In contrast, the Cervelo has a beautifully tapered head tube.

True, the Tour test overlooked the advantage of internally routed cables. But I can't help wonder how consistently the dummy was fit on the bike in each case in the Tour test.

So this sort of bike-to-bike comparison testing is very challenging. Far different than looking at weight difference, where you pop that sucker up on the scale and photo the result. Sure, there's subtleties, like what sizes to compare, and whether to compare naked frames or "modules", but wind tunnel testing is just in a different league.

Thursday, April 7, 2011

VeloNews aero bike test: playing with the numbers

I've been spending a lot of train time playing around with puzzles, doing "work-work", and even reading the newspaper following recent world events. Time to get back to those VeloNews frame data.

VeloNews plots its drag force in grams. Grams is not a standard unit of force. A "gram" of force is defined as the force exerted by a standard gravity on a one-gram mass. But with wind resistance, we're probably most interested in power. And to get watts from force we want the standard metric unit of force, which is Newtons. Newtons multiplied by meters/second equals watts. So I converted their data to Newtons.

But even then, Newtons are good only for the relative wind speed at which they did the test. Actual force depends on relative wind speed and air density. Wind resistance force is generally proportional to the product of air density and the square of the relative wind speed. I don't know the air density, but I can guess it was close to 1.2 kg/m³, since they tested close to sea level (in North Carolina) as opposed to, for example, a wind tunnel in Boulder CO, which is where the VeloNews offices are. Air density falls approximately 1% each 100 meters gained in altitude, at a given air temperature. So being in Boulder at 1600 meter altitude would make a big difference.

Furthermore, I of course assumed wind force is proportional to the square of wind speed. This is often quoted as an exact result, but it is not: it is based on simplifying assumptions. But it works quite well. In any case, if CdA is considered a weak function of speed, then the CdA's I derive here are assumed to apply best at near 30 mph.

So here's VeloNews' data, where each bike was tested in a "stock" configuration:

stock wheels


I don't place much value in these data alone: the wheels are a huge component in the result, and it's a lot easier to swap wheels than it is to swap frames. But VeloNews did a good thing on these tests: they also tested the frames with Zipp 404 Firecrest wheels, which are probably the best non-HED clincher wheels sold from an aerodynamic perspective for their rim depth. The exception was the Masi "control" bike (the Masi not designed for aerodynamics), which was tested with Fulcrum wheels both times. I can try to fix that error, however.

Anyway, here's the comparison of the two tests, the increased resistance of the stock wheels versus the Zipp 404's:

Zipp 404 vs stock wheels


The Zipps are not so surprisingly far superior to the Fulcrums, especially at large yaw, but perhaps more surprising is the substantial advantage at large yaw of the Zipps over the Mavic Cosmics.

Note two bikes offered the Fulcrum vs. Zipp comparison: the Cervelo and the Ridley. The two were improved by the Fulcrum-Zipp swap remarkably similarly: the Cervelo benefited a bit more at every measured yaw, but the two improvement curves track closely together. For a lack of anything better I took the average of the two results (shown in the plot) to apply to the Masi to predict how the Masi would have performed with Zipp 404's, since all other bikes were tested with 404's in addition to their stock wheels. I plot that result versus the actual results from VeloNews for the other bikes with the Zipps:

all bikes Zipp 404


If you compare this to the plot from the article, in the article the advantage of the aero frames versus the Masi at high yaw was profound. Here you see the high-yaw boost is mostly from the wheels. The curves here track fairly parallel.

The Felt is rather striking in its relatively poor low-yaw result. This seems to contradict the Tour magazine result where the Felt AR did quite well at low yaw. In the VeloNews it is conjectured the thick bar tape on that bike is in part responsible. Wider handlebars would also contribute to poor low-yaw drag.

I take the difference in yaw between the Masi and each bike, using the Zipp-corrected Masi numbers. Each 0.01 m² difference in CdA translates to an approximate 1% difference in speed at a given power for wind-resistance-limited riding (recall Tour magazine measured approximately 0.32 m² total for a dummy rider + bike, and it takes a 3% change in power to change speed 1% in the wind-resistance limit).

Masi comparison: all bikes Zipp 404


These are nice results from VeloNews, although perhaps slightly overstated due to the lack of a rider and any water bottle on the bike, and the differences in handlebars, tape, saddles, and cables make it difficult to put too much importance on small differences between bikes. But I hope VeloNews continues with their testing: it'e great to get numbers to play with.

Sunday, April 3, 2011

Strava, running versus cycling

Strava has really gained traction among the San Francisco Bay area cycling community. In some group rides, in particular competitive non-racer groups like Mission Cycling and SF2G, typically 3/4 of the riders will upload rides, all the more remarkable considering it requires an expensive GPS-enabled cycle computer (although the price of entry gets lower with each new generation of Garmins, as the users dump their old units).

Here's the Marincello trail page for cycling (which includes both mountain biking and road biking). As I write this, 99 users have uploaded rides which include this segment, ridden a total of 567 times. The top time, by David Beldon, was with a VAM of 1194 meters/hour, quite impressive for a gravelly dirt fire road. I'm ranked 24th.

But Strava isn't just about cycling. Basically any activity which involves transport by human power is compatible. Strava activities can be categorized in a number of areas, and of course running is one of these. The Marin Headlands trails, every weekend day are full of trail runners. This is an absolute hotbed of trail running in the United States, and the Marin Headlands are a favorite for both training and racing.

So after running in the headlands today, I created a segment for the Marincello trail. "Run 1 time by 1 person". I'm ranked first. And only. What's up with that?



Come on people, get on the program. I'm not sure what website the trail runners are uploading their GPS too, but it's obviously not Strava. What am I missing?

Saturday, April 2, 2011

prisoner puzzle: big hint

Previously I described a puzzle of prisoners and hats. Each of 100 prisoners had a 50-50 chance to pick his number from his 50 picks from hats. For any prisoner to survive (puzzles tend to be violent; it's true), every one of the 100 prisoners needed to succeed with his 50-50 chance. Simple probability says the chance of the prisoners surviving is essentially zero: you can't flip a fair coin 100 times and get heads every time. The key, it turns out, is to use a rigged coin.... Go to that link now if you don't want to see a really big hint.

Move below the cute kitten photos to proceed...

kitten
kitten
kitten
kitten
kitten


Okay, enough of that. Here's that big hint. For simplicity, I'll assume 8 (not 100) prisoners and 4 (not 50) hats. I show two sequences: one which wins, the other which loses.

The winning sequence:
4, 6, 3, 0, 2, 1, 5, 7



The losing sequence:
4, 2, 3, 0, 1, 6, 5, 7



These sequences are permutations: various orderings of the same elements. The key insight is that these permutations descibe loops. The probability distribution of the length of a loop associated with a random element is unifrom from one to the number of elements in the set. So for 100 prisoners, if I pick a given hat, then follow the sequence as shown, I will end up back at that hat in from one to 100 steps, each with equal probability. Complete the loop and you've found your number. If your loop has a length no more than half the number of elements, you win, or else you lose. If you win, because your loop is short enough, then everyone else in the same loop will win. And if you lose because your loop is too long, then everyone else in the loop will lose.

This is why the solution is like a weighted coin: you don't know if it's rigged to come up heads or tails, so the first flip is 50-50 chance of a win. But if you commit to picking either heads or tails every time, if you luck out and guess right on the first flip, every successive flip is in your favor. Same with the prisoners. If the loops are all length less than half the total, then every prisoner wins. If any loop is more than half, then every prisoner on that long loop will fail. It's impossible that less than half the prisoners will fail except that no prisoner fails.

So what's the probability of success? Consider the limit of a large number of prisoners. The first prisoner has the probability 50% of success. If he succeeds, then the length of his loop has any value from 0 to 50% the number of prisoners with equal probability. Eventually a prisoner will pick a hat which is in a different loop. The probability of success for that loop is better, since there's fewer remaining hats after the first loop, and the maximum length of the loop is still half of all of the hats together. Simple analysis shows the second loop has a 75% chance of success.

My fellow Caltrain passenger explains the theoretical chance of failure, in the limit of an infinite number of prisoners, is the natural logarithm of two, or 69.3%. That leaves a 30.7% chance for success. My 100 thousand trials had had a 29.919% success rate, with a standard error of 0.45%.

So I didn't explicitly describe the solution, but it should be fairly obvious from the diagrams.