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The Brachistochrone problem

Brachistochrone

I was struck by this GiF recently, at:

http://imgur.com/5t32VJU

 

This little video shows three possible paths – the straight line (the shortest), a path that descends quickly and then becomes flat as it gets to the destination, and one in between, looking more elliptical* in shape.

Ever wondered what downhill profile of road will get you most quickly to the bottom of the hill freewheeling from a given point T (the top) to a given point P (the pub)? Now you know – it’s a cycloid, appropriately enough. Now I just need to think about the best shape of road from the pub to the top for the fastest ascent for a given a) wattage, and b) for a given pedal pressure…

…The classical problem is well-known as the difficult to solve Brachistochrone problem. See http://www.math.utk.edu/…/m231f08/m231f08brachistochrone.pdf for a solution – minimising a quantity over a famiły of functions, or http://www.osaka-ue.ac.jp/ze…/nishiyama/math2010/cycloid.pdf which is perhaps a little more accessible and talks a little more about the cycloid (the resolution of the problem).

Even more arcanely (I was going to say more interestingly, but I’m realistic about reading endurance!) the cycloid, as the second article says, is isochronous. It means that even if you start freewheeling from a rest position R half way down the hill, it still takes the same time to get to the pub. Why? Briefly, because starting higher up, at the intermediate point R you are moving, whereas starting from R you are at rest. Weird, huh? It’s the same for any point R, no matter how close to the pub.

*the elliptical path is nearly as fast as the cycloid – but not quite!!

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The tacx concept of “Virtual speed”

I have another interesting quirk to discuss with you relating to my Tacx Flow turbo, or indeed any other Tacx turbo that has a relatively low maximum slope simulation capability (i.e. other than the Neo, i-Genius or Bushido) such as the Flow at 6% max or the Vortex at 7% max. This is from the Tacx support pages http://forum.tacx.com/viewtopic.php?f=213&t=25258:

“When a trainer cannot apply enough resistance, the measured speed does not comply with your exercise (training program) anymore.
To solve this the speed is calculated (Virtual Speed) based on the power you are producing and the exercise data (current slope).

“e.g. You are riding an exercise which consists of a single 10 % slope on a trainer which can only apply enough resistance to simulate a 7 % slope.

“Using measured speed you will finish the exercise much sooner compared to actually riding the same 10 % slope in real life, because the trainer cannot apply enough resistance causing you to go faster resulting in the exercise costing you less effort.

“Using virtual speed your exercise will be much closer to actually riding the same 10 % slope in real life, because it calculates how fast you would be going on a 10 % slope based on the effort (power) you are producing on the trainer

“Keep in mind speed shown in TCA is also NOT the raw speed from the trainer. It’s always calculated with the same virtual speed calculations but not flagged as virtual speed cause it’s less than 10% deviation from the RAW speed of the trainer.

“Measured speed from a sensor is treated as raw speed.”

So if you use a speed/cadence sensor with your turbo bike, you will get raw speed recorded, allegedly. I have certainly noticed downhill speeds being greater then I would have thought, and uphill speeds possibly slower (but then that may be my issue!)

On the retailer Wiggle’s site I see it asserted that apparently Zwift have partnered with Strava to allow virtual rides to count towards Challenges if you use such a sensor (as I am only riding turbo at the moment, I have zero distance and elevation in Strava Challenges for January, and probably December too). My personal targets are as normal. See:

http://guides.wiggle.co.uk/zwift-indoor-cycling-guide

Never mind the potholes on a real road; on a turbo, it’s a minefield out there!

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Zwift’s “difficulty setting”

Here’s a very interesting article with, for me, some initially counter-intuitive outcomes – until I thought about it more.

For context, I’m using the Smart Tacx Flow as an alternative to my non-Smart i-Genius trainer (a contradiction in terms…).. The Flow goes up to 6% incline and 750W. No motor for downhill, but it DOES reduce the flat riding resistance which can be felt to about 3% downhill, and then it stays the same. No motor means if I stop pedalling it eventually stops whether up or downhill. For my money it’s great value. I reckon it’s swings and roundabouts on 6% max and no motor compared to I-Genius. I work harder on big downhills but less hard on steeper hills. The Zwift “difficulty setting” offers a way to make the 6% hills and above feel tougher…

The article at http://zwiftblog.com/using-the-trainer-difficulty-setting-in-zwift/ describes resetting the resistance levels when using Zwift to increase or reduce the multiple of training effort for any given slope, WITH NO CHANGE TO THE WATTS REQUIRED. How can that be, one might ask? Surely, if you increase the resistance multiple for any hill, thereby increasing pedal pressure (the intention of the change (in my case) to circumvent the Flow’s 6% max slope simulation) then isn’t the power (pedal force X speed = work rate) increased, or do they reckon because you are likely to reduce cadence (for a given gear) as a result, the speed diminishes in proportion with the increase in pedal pressure?

Having a PhD in Theoretical Physics means I’m a little ashamed of myself even to have a momentarily different intuition initially! But people of my age who did Applied Maths at school ought to remember the old Horsepower formula HP = Pv/550 (in appropriate units). Power is proportional to the multiple of force P and velocity v. It’s why some sports car manufacturers used to be so fond of highlighting their cars’ horsepower, because by making their engines rev at very high revs (enhancing engine wear no end) they could advertise very high horse-power. Very UNusable power, given how high you had to rev in all the low gears to get performance. The American way was to make engines bigger and bigger (427 cubic inch (7 litre) V8 preferably) so that TORQUE (turning force) was very high and the engines were just ticking over at quite high speeds, generating power at much more usable revs. Even smaller diesels (whose fuel characteristic is different from petrol) produce high torque at low revs too, compared to lighter, faster revving petrol engines.

As for cycling, and using the Zwift “difficulty setting” to change the perceived resistance, I guess, as the article says, just like the automobile analogy, it’s really a cadence control. You push harder on the pedals, say, but at lower revs, so the power is the same. Watch those quads grow!

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Some Tacx, Zwift and turbo training learnings

Having used the Tacx non-smart i-Genius turbo trainer for a while, I have also been using my Tacx smart Flow trainer a lot recently.

The Tacx Flow is great value, the lowest price Tacx trainer that is fully smart, i.e. ANT FE-C and Bluetooth bidirectional. Thus the trainer software (Tacx films or 3rd party like Zwift, on laptop or handheld device where applicable) will control the Flow brake, and the Flow sends all speed, power and cadence data to the training software.

You can also control a smart trainer like the Tacx Flow with the Garmin x20 series, i.e. such as the 520 or 820 that have the right data protocols.

At this low end of the Tacx range, you are limited to 6% uphill simulation and 750W power, but I don’t find this too much of an issue. Because it doesn’t have a motor to help simulate downhill freewheeling (like the I-Genius I also have) you work harder downhill to make up for it being easier at 7% uphill and above because the resistance is limited to simulating 6% max.

The Tacx Vortex does 7% and 950W but not worth quite a bit extra price in my opinion. I have published quite a bit on all this on Facebook to my cycling Club I have also described how Zwift offers the option of increasing what they call “difficulty” from the default standard 50% to something higher. This increases resistance on hills, lowering cadence and speed and leaving power the same.

Tacx have something called “virtual speed” which decreases reported speed uphill and increases it downhill to compensate for the maximum simulated uphill and downhills on trainers like the Flow and Vortex.

The much more expensive Tacx Neo, Genius and Bushido don’t need that.

DCRainmaker always offers good information, especially his “all you wanted to know…” page which is at www.dcrainmaker.com/2016/07/everything-you-ever-wanted-to-know-about-ant-fe-c-and-bike-trainers.html
32 minutes ago

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A visit to the Westerham Cyclery

Having seen this excellent blog post on Facebook about the Westerham Cyclery,

Westerham Cyclery at the Green
Westerham Cyclery at the Green

and linked to it from there – https://summit2ride.com/2016/10/05/the-cafe-ride-westerham-cyclery/ I thought I should add my own pictures and very positive comments, having been hosted into hosted by their very friendly staff!img_6246 I went there to meet up with a cycling Glasgow Green Cycle Club friend, Susie Goodwin, and then to ride one of the climbs back over the North Downs.

img_6247
The cake options, with second piece of bread pudding!

The highlight of my visit, I have to own up to, was the (two pieces of!) bread pudding, but they have all sorts of cakes and drinks, plus a well stocked bike shop, a very safe place in the courtyard to park (and lock if desired) the bike, and tables and seats in that courtyard for

My first helping of bread pudding
My first helping of bread pudding

a quiet al fresco rest.

img_6244I am based near Glasgow in Scotland, but now, added to the Spitfire café in Biggin Hill, Westerham Cyclery will be at the top of list for my next ride south of London when visiting family. The café garden is a great place to meet people and relax.

img_6336From a riding point of view, there are quite a few climbs back over the North Downs, including Chalkpit Lane which I rode on another visit. My first return was via the signposted route from Westerham to Croydon, which is a more gentle climb that anyone can manage, so go ahead and make a visit!

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Relating Steady State to Dark Energy and Dark Matter

After a little fall from my bike (ouch!) I have some time to do some reading! See

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This week’s Strava rides

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Latest Strava rides

Here are my latest bike rides on Strava – click on one for more details…

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North over the Crow and south over the Tak

A very good 50 mile ride yesterday with Douglas Beattie. We took the Drymen road to Carbeth, down the Cuilt Brae, and over to Lennoxtown to go over the Crow Road to Fintry. Then the horrible surface of the Carron Valley Road to Carron Bridge, up the Tak (through a flood at the bottom) and down to the Boathouse in Kilsyth for well deserved scones and cream tea! A gentle ride from there to Twechar and thence via Torrance to Bearsden completing our 50 miles. Along the way we met some interesting people – Paul at Carbeth walking with his lovely, friendly bull mastiff Zenden, and also his hawk Ruby who was eventually seen on top of a telegraph pole waiting for us to leave so that she could return to Paul’s glove. Lots of chat about dogs and how to drink whisky! Then, at the top of the Crow, we met Alec, 88 years young, who had run quite a few marathons (he says New York is the best, by the way, with great goodie bags!) and lots of other running. His athletic life showed in his healthy and fit demeanour! The only surprise of the day was not meeting Alex, Natalie or Andy of West Coast Velo at the Tak ma Doon car park! Altogether a great day out on bikes!
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Tak ma Doon climb from Carron Bridge to viewpoint

This is the easier side of the Tak, but there are some very steep little ramps, limited to 10-11%, which is a critical 2% or so less than the north side, where the steeper parts are also longer. On this side, however, there is plenty of less steep road where a rest can be taken, back in the saddle. I have been up a few times on my geared bike, and also single speed but only on 46/18, 69″ gear. i think that is a sensible maximum for me! There is a ford about half way (surprisingly there is a loch, not at the bottom but also half way up) and at the top the views from the car park viewpoint are all the way over to Grangemouth, the Forth Bridge and the new crossing. Very much worth the climb!!