In praise of riding low pressure tyres fast

[JimK]

The Rigida Andra 30 has an "inner well width" of 19mm:

http://www.sjscycles.co.uk/rigida-andra-30-26-(559)-mtb-css-rim-black-32-hole-rohloff-drilling-prod13269/

It sure seems like the rim strength ought to enter the equation too!

The 1% per kg formula also gets me thinking. The tire carries a load equal to the contact patch area times the pressure. So, to keep the contact patch area constant, the pressure should increase proportionally with the load. If the starting load is 50 kg (per wheel),... that would call for a 2% increase for each kg increase in load? Where is my mistake?

Keeping the contact patch area constant is one goal in determining tire pressure. Another is keep the rim elevated, to avoid pinch flats etc. Seems like when one hits a bump, the contact patch area increases which lets the pressure push the bike upwards. So an interesting number is the ratio between the cruising contact patch area and the maximum contact patch area (just before the rim hits ground).

Or much of the time when riding over bumps, the bike doesn't really get pushed up to clear the obstacle, but the tire deformation just takes up the variation in the road surface. This is clearly one major advantage of fat tires. Especially with a heavy load, the bike is just not going to move up and down to get out of the way of rocks or whatever. The extra height of a fat tire can completely absorb small obstacles, and gives the bike more time to move to avoid larger ones.

Yeah, lots to chew on!

[JimK]

still chewing...

The contact patch of a fat tire is both longer and wider than that of a narrow tire. The tire can start pushing the bike up and over an obstacle as soon as the contact patch reaches the obstacle. Thus the longer in space the contact patch, the longer in time the bike has to move upward to avoid the obstacle.

Seems like two sorts of obstacles need to be looked at: sharp obstacles that are smaller than the contact patch, e.g. an sharp step up, and long obstacles, like a sharp step down.

There's a nice problem. Just drop the loaded wheel onto a smooth surface. Will the rim hit the ground?

Let x be the tire width, i.e. the distance from rim to the ground when the wheel is completely unloaded.

Model the tire as a simple spring. The tire pressure becomes the spring's stiffness, k. We'll inflate the tire to achieve some fixed fractional deformation of the tire, i.e. k varies as the inverse of x.

The energy of the dropping bicycle, proportional to v**2, must be taken up by the spring, whose energy is k*(x**2). But since k is inversely proportional to x, the maximum energy the spring can absorb is proportional to x. I.e. the maximum velocity, with which we can allow the bike to drop, is proportional to the square root of the tire width. That velocity is proportional to the height from which the bike is dropped.

Looking at that Schwalbe inflation chart, they do have pressure roughly inversely proportional to tire width. The pressure seems to go up a bit faster as width goes down.... lots of other factors at play here!
 

[Danneaux]

As an afterthought, imagine these calculations if you had a 26" wheeled touring tandem?!?!?

Oh, but I do!  I bought it used from a family where each member weighed well in excess of 375lbs/170kg, so a pair was a serious load.  And they had really ridden it, including at least one Seattle-to-Portland ride of about 325km/202miles.  I repainted it, converted it to a road orientation, and relaced the wheels, truing and tensioning them with the original spokes (they were originally built with uh, "inconsistent" lacing).  I've used it a lot, with heavy stokers and on really rough roads.  The bike alone weighs 20.9kg/46lbs set-up for touring with Arai drum brake, racks, bottle and cages, pump, fenders, etc. as pictured.  My Sherpa is perilously close to that at 18kg/40lbs in a similarly ready state.

When my Dutch pal visited from The Netherlands in 2007, we decided on a couple night's tour up into Oregon's Cascade mountains on a mix of paved and very rough gravel roads.  My tandem runs CR18 Sun Metal rims (25mm outside width) with 36 spokes and Matrix (Trek) 26x1.5 Road Warrior slicks that measure a true 38mm/1.5" in section width, and I ran them at the listed sidewall-listed maximum of 5.86 bar/85psi front and rear.  We never had a problem, and this thing was the very definition of Heavy Metal.

It is pretty fair to say we took everything but the kitchen sink with us, including his heavy Dutch Army surplus tank-driver's boots, since that's all he had for hiking at our destination. Towing my homemade trailer loaded with 56kg/125lbs alone, our total weight was in the neighborhood of 272kg/600lbs.

I'm not proud of the weight, but it was a wonderfully fun trip, and a nice introduction to tandem touring and camping for my friend.  We ate like gluttons and lounged in a camp equipped for kings (we even took two tents, albeit small 1-person models).  We had neither tire nor rim problems, though the distance was short.  Maybe I've been extraordinarily lucky all these years, but the fact also remains that despite the weight, these were fairly narrow tires on fairly wide rims, so that may well be a factor.

For my part, I am just tickled by this topic, and look forward to hearing more ideas on the subject.  I'll dig further, too.

Best,

Dan.

[JimK]

When hitting a sharp step up at high speed, the bike cannot respond quickly, and the height of the step that can be managed is simply proportional to the tire width.

Another interesting case is an abrupt change in direction, e.g. the start or end of an inclined plane, or where the angle of inclination suddenly changes. This seems more like a step-down in that the spring of the tire will have time to push the bike in the new direction. The limit must be something like the sine of the angle of changed direction being proportional to the square root of the tire width.

A pot hole is a nice case, i.e. a step down followed by a step up. A very long pot hole, where the bike gets to riding along the bottom before hitting the far edge, is just the simple combination of the two steps. A shorter pot hole will let the bike hit bottom but still be bouncing when the step-up comes - clearly the worst case is when the step-up arrives just as the bike is at the bottom of the first descending compression.

Then another case is when the bike never hits the bottom, it just flies off the step-down and hits the step-up. Since the bike has started to fall, this is like a step-up onto an inclined plane. The step-up eats some of the tire width, and then the rest had better be enough to push the bike in the right direction. If I get insanely desperate I will come up with a formula but maybe I should go for a ride instead!

[Danneaux]

Jim,

I've enjoyed spending time thinking about your posts on this topic; they're well-reasoned and information-dense, and give a lot to think about!  You surely have a good mind.

Your thoughts and calculations appear reasonable to me, but the larger problem we all face is we are not tire engineers at the company in question, so we don't have access to their test data and must rely on our own speculation.  For example, we know the shape of the tire patch has something to do with rolling resistance, but we don't know for sure what size that patch should be to optimize a given parameter.  I wish we did, 'cos I could get a lot deeper into all this; its a great topic!  Like you, I keep turning from the computer to my copies of _Bicycling Science_ and other sources in search of answers.

You wrote a few things that really got my gears turning.  Apologies for taking your quotes out of context...

The contact patch of a fat tire is both longer and wider than that of a narrow tire.

Yes, but as I recall reading, it is the shorter patch length that reduces rolling resistance of a fat tire versus a narrow one for a given inflation pressure.  I'll check my notes, but I think this point emerged from Jan Heine's tests mentioned earlier in this topic thread.  Hmm.

This seems more like a step-down in that the spring of the tire will have time to push the bike in the new direction. The limit must be something like the sine of the angle of changed direction being proportional to the square root of the tire width.

Yes, but remember there is some hysteresis in the equation as well.  Yes, there are some momentary localized pressure spikes, but the tire and tube and rim together comprise an endless closed column that is itself flexible to a degree.  Wouldn't the overall pressure remain fairly constant for a given static pressure regardless of momentary dynamic loading? In other words, if I put a small amount of air in a tire and attached a manometer, I don't think I would see a pressure rise when poking the tire with my thumb (just did it; no change in instrumented pressure).  While I would be compressing the casing/tube locally, if the overall air column is not constricted except by its own flexible casing, it should bulge out an equal amount, distributed about the remainder of the tire/tube circumference.  Moving on from this a bit, I'm not sure any impact-related bulge remains at "ground zero" long enough to alter the contact patch too much apart from any gap that is bridged by the tire. I once saw some fascinating high-speed camera footage of a motorcycle tire as it passed through a pothole, and the resulting casing deformation appeared to take place on the tire *after* the wheel passed the pothole; in other words, the tire had rotated, and the bulge resonated through the tire after impact.

There's a nice problem. Just drop the loaded wheel onto a smooth surface. Will the rim hit the ground?

Well, not unless something goes terribly wrong!  All kidding aside, this is a fascinating problem.  I once cut an old rim in two and did the same for a tire and tube, then carefully mounted one inside the other and then cyclically loaded the assembly so I could better visualize how pinch flats ("snakebite punctures") were caused.  This was a crude little experiment, but it appeared pretty clear that with insufficient inflation, the tire beads would bow inward under vertical load, causing them to be unseated from the rim's bead-retaining hooks.  Even on my little model, the tube was quick to try and fill the gap and became caught between the tire bead and seat when the momentary vertical load was eased.  Because the tire was loaded vertically and equally, both beads bowed inward, and trapped the bulging tube equally, which is why those sorts of punctures are nearly always paired.  I realize you were postulating hypothetically, but in reality, I do think the rim could  hit the ground if the impact was sufficient to cause a pinch flat (or if the tube was underinflated).  By the way, a sew-up tubular tire never suffers from pinch flats 'cos there are no beads to trap the tube.

Model the tire as a simple spring. The tire pressure becomes the spring's stiffness

I agree the inflated tire will act as a spring, but I am not at all sure that spring action is linear.  That could be devilishly hard to model, given the dynamic operation of the spring-as-tire coupled with variations in carcass construction, tread, and overall design.  It would be interesting to build an instrumented test rig for this, to augment the calculations.  I know intuitively that rough roads have the effect of increasing rolling resistance, but the obstacles encountered will surely have a greater effect than dynamic changes in contact patch size, shape, and orientation, though each of those are affected by impact.  A large-diameter rotating drum or billiard-table smooth surface would make empirical calculations of pressure and rolling resistance much easier.  In the past, I had fun inking various tire treads then loading them vertically through the axle on a dummy fork to see how load and pressure affected contact patch shape and size.  I wish I had also tried doing this with the same tire mounted on rims of varying width.

It sure seems like the rim strength ought to enter the equation too!

It surely does!  We already know rim width makes a difference, and it is worth noting it forms the rest of the cylinder containing air and therefore plays a part in determining overall air volume.  Rim width also plays a role (though less important) in determining casing shape and therefore determining contact patch shape, size, and orientation as well.  The same nominally-sized tired will plant itself differently depending on whether it is mounted on a narrow, medium, or wide rim.  handling and rolling resistance are affected as well.

I registered for an account at Schwalbe, their email contact link is broken (could they have seen me coming?), so I guess I will need to phone them to inquire further.  That will be tomorrow or the next day, 'cos in between working and perusing this Forum, I've been out riding.    Some really cold weather is coming, so I've been trying to cram in as many shakedown rides as I can to get the Sherpa sorted for my needs (and to play with tire pressure, of course!  Got the motion-detecting alarm sorted as well; it is fun and will be useful at night when wild-camping solo).  

Best,

Dan.

[JimK]

Here is a simple analysis of the shape of the contact patch:


r is the radius of the tire, and R is the radius of the wheel. Again, I assume we'll inflate to the point where the load compresses the thickness of the tire by 15%.

The contact patch grows in both directions as the width of the tire increases. But the width of the contact patch increases more quickly: linearly with the tire width. The length of the contact patch only increases as the square root of the tire width.

Unfortunately my copy of Bicycling Science is stuffed in my storage unit ten miles away, probably buried in a heap of boxes. Not an ideal situation! But it is fun to try to think through this stuff!

[JimK]

I'm not sure any impact-related bulge remains at "ground zero" long enough to alter the contact patch too much

I agree that the tire pressure really isn't changing dynamically in any significant way. Sure, sound waves must be bouncing around inside the tire, but I don't think there is any significant effect. The total change in volume due to loading and compressing the tire is not very significant either.

But the bike is getting accelerated up and down on our bumpy road. Gravity is constant, so the force of the road, transmitted through the tire, has to be changing. That force is just pressure times contact area. So the contact area must be fluctuating just in parallel with the variations in vertical acceleration.

[JimK]

I know intuitively that rough roads have the effect of increasing rolling resistance, but the obstacles encountered will surely have a greater effect than dynamic changes in contact patch size, shape, and orientation, though each of those are affected by impact.

I am not addressing rolling resistance at all. I remember there was a lot of analysis in Bicycling Science of rolling resistance... and I remember getting rather lost in those formulas! It is surely an interesting question, how does rolling resistance change with load and with tire width.

It is also true that a rough road is tough slogging, not just because of rolling resistance, but also because the load is not rigid but soaks up energy from vibrations. The same sort of thing could happen on a perfectly smooth road, if one were pushing hard on the pedals and then letting up - the back and forth accelerations could get cargo sloshing around and absorbing energy.

But I am not looking at efficiency here. I am just thinking that a key criteria for a tire to carry a load is that you really don't want the rim to hit the ground. That can bend the rim, or give one a pinch flat, and sure it rattles the rider too! So I am just trying to see how a wider tire is more capable of keeping the rim from hitting.

[Danneaux]

A quick look at www.schwalbe.co.uk shows that site has all their technical information bundled into a single tidy PDF file, and it is more complete than the technical information listed on Schwalbe's US website.  

The UK PDF makes for interesting reading relevant to our present topic and is here:
http://www.schwalbe.co.uk/_webedit/uploaded-files/All%20Files/Technical%20Info.pdf

Page 10 indicates, "A dense carcass is important for low rolling resistance".

Pages 16-17 are devoted to a discussion of rolling resistance and contact patch shape and size, as well as energy requirements for two tires at various pressures, with the following quote:  "At 2 bar a 60mm wide tire rolls as well as a 37 mm tire at 4 bar".

Schwalbe's tire life estimates are intriguing as well, and they indicate a tire can be ridden so long as there is rubber present on the tread.

Of particular interest to me was the news that a Schwalbe tire can be stored for up to five years with no problem, and perhaps longer if stored properly.  I stored some spare tires for the tandem some 18 years ago, and came across them last Spring.  They mounted with no problem and aired up just fine.  No cracks or issues of any kind, but I had stored them in air-evacuated plastic bags in a cool, dry place in the garage.

Best,

Dan.

[JimK]

That PDF is very nice - thanks for posting that link!

Fascinating, the table of rim vs. tire width is different in that pdf, compared to http://www.schwalbetires.com/tech_info/tire_dimensions - the crucial difference I see is that the pdf says a 19mm rim can handle up to a 60 mm tire, while the usa website says the 19mm rim is limited to a 50 mm tire width. The Rigida Andra 30 rim has a 19mm interior width, so that is a significant difference!

The discussion of contact patch shape in the pdf is quite interesting. Note that when they say that a narrow tire has a longer contact patch, that's assuming that the pressure is the same in both tires. If, instead, the pressure is kept inversely proportional to tire width, then the wider tire will have a larger contact patch...

ah, it seems my contact patch shape calculation, in a previous post, must be wrong. The shape must be roughly elliptical and the area must be proportional to the product of the length and the width. But I have the patch width increasing proportional to the tire width and the patch length proportional to the square root of the tire width. This would have the area increasing with the 3/2 power of the tire width. But if the pressure is inversely proportional, the area must be proportional linearly. Rats, back to the drawing board!

[Danneaux]

Schwalbe's 2012 UK _product_catalog_ also has some information relevant to this discussion.  It is available here:
http://www.schwalbe.co.uk/_webedit/uploaded-files/All%20Files/Bike2012-English.pdf

There is a nice photographic comparison of the entire line, starting on page 41 for the "Marathon Concept".  The Evolution series is presented as their "Best" and includes Racer, Supreme, Dureme, and Mondial (XR replacement).  The next tier down is the Plus series, which includes the Plus and Plus Tour, the latter with a more aggressive tread. The Original series includes only the Marathon with Green Guard puncture protection.  The Evolution series has the least rolling resistance and lightest weight, while the Plus has the greatest flat protection and durability.  Their chart shows the original Marathon is like the Evo series, but heavier and less easy-rolling.

Page 44 indicates...

The Big Apple started off the Balloonbike trend ten years ago: Comfortable cycling without using complicated technology! Air cushion tires are used as natural suspension. Inflated to around 2 Bar a Balloonbike rolls really easily and with a full suspension effect. A normal tire with a width of 37 mm must be inflated to a rock-hard 4 bar in order to roll similarly well.

This means at the same pressure, the narrower tire will be at a disadvantage, so Schwalbe is using the same pressure for their rolling-resistance comparison rather than adjusting it proportionally.

The Energizer series (page 42) is optimized for low rolling resistance and high grip.  I presume this comes at some expense of overall longevity and flat resistance.  If one needed really grippy, fast-rolling tires, these might fill the bill for, say, commuting.  Probably not so good near the curb as in the traffic lane.  

Yeah, Jim, there's quite a few differences in Schwalbe's UK info vs the US.  I'm still pondering it all, and plan to check out their sites in other countries as well, though my limited foreign-language skills will make it a tough slog (Google Translate doesn't work on PDFs).  In your last post, you observed...

Note that when they say that a narrow tire has a longer contact patch, that's assuming that the pressure is the same in both tires. If, instead, the pressure is kept inversely proportional to tire width, then the wider tire will have a larger contact patch...

<nods>  Yes, I think that is the basis for their argument that fat tires can roll more easily than skinny ones; they're using the same pressure in their comparison (see above).  When I read that, I had one of those "head-slap" moments; well, duh!

Still wrapping my own mind around all this,

Dan.

[JimK]

Aha... if you use my contact patch formulas...



L is the constant load, P is the pressure. The pressure should be proportional to the -3/2 power of tire width. It's a nice fit to the Schwalbe recommended pressure:

[JimK]

While I am thinking about it, I wanted to clean up my math a bit:



I decreased my tire pressure a bit before my ride today - to 50 psi in the rear and 45 in the front. Felt just fine, though that was only about 5 psi less than what I had - I couldn't really swear that I noticed any difference!

[Danneaux]

Jim,

You may well enjoy seeing this:  
http://books.google.com/books?id=uDQEAAAAYAAJ&pg=PA1007&lpg=PA1007&dq=How+a+pneumatic+tire+supports+its+load&source=bl&ots=N9LCRVitB6&sig=fljjpXQy-7IfG20L3zwo5f5qsfo&hl=en&ei=UzR-TOCgLMKqlAeh6a3tCw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CB0Q6AEwAjgK#v=onepage&q&f=false

It is a Google Books excerpt from vol. 26 of the 11th Edition (1910-1911) _Encyclopedia Britannica_, page 1007.  The article envisions the tire as a spring and shows their calculations for determining the ellipse of the contact patch, among other things.  It really is a nifty little article on tire design, construction, and science for anyone interested in bicycle tires.  It also addresses our discussion of rolling resistance through a thumbnail presentation of the work of a Professor Osborne Reynolds.  Reynolds was a pretty interesting fellow himself -- an Irish mathematician who contributed greatly to the field of fluid dynamics.  See: http://en.wikipedia.org/wiki/Osborne_Reynolds  His research helped me a great deal some years ago when I examined boat propeller design and the effects of cavitation on propulsion efficiency.  I remembered his work on fluid dynamics and managed to track down this reference pertinent to our discussion of rolling resistance and tire design.

Reynolds inverted the problem we're dealing with and made his initial calculations upon an iron roller resting upon a sheet of India rubber.  He postulated rolling resistance is the result of slippage and distortion at the contact interface, and will vary due to hysteresis.  The upshot of it is his steel roller had the least rolling resistance due to the least hysteresis (deformation) caused by vertical compression (as a solid, the steel was uncompressible within any measurable limits for the test), followed by a pneumatic tire pumped hard and then an underinflated pneumatic tire.  

What really made me raise my brows was his statement saying a solid-rubber tire had the greatest rolling resistance of all!  Professor Reynolds says...

...If the vertical compression cd of the tire be denoted by y, the energy lost may be said to be proportional to Hy. Comparing three tires of steel, solid rubber and air respectively rolling on a smooth, hard surface, H (hysteresis) is probably smallest for steel and largest for rubber, y is least for steel, greater for a pneumatic tire pumped hard, greater still for solid rubber and for a pneumatic tire insufficiently inflated.  The rolling resistance of the steel tire will therefore be least; next in order come the pneumatic tire inflated hard, and the pneumatic tire inflated soft, while the solid rubber tire has the greatest resistance.

Whoa!  

Intuitively, the solid rubber tire "should" behave like a pneumatic tire inflated rock-hard so it has essentially no vertical compliance beyond the natural hysteresis provided by compression of the rubber tread, since any suspension effect of the casing would be essentially "locked out".  However, according to Osborne's thesis a solid-rubber tire is instead more akin to an underinflated pneumatic tire in terms of rolling resistance because of higher energy loss due to greater hysteresis for the same vertical compression (which I take to be deformation as a function of vertical load).  I need to ponder this a bit more.  

Jim, with regard to your concerns about the rim bottoming out (I'm thinking aloud here), the air inside a tire provides the opposing tension that supports the load.  The air in a tire serves mostly to keep the casing intact on the rim everywhere and/but actually supports the load directly only at the contact patch, where deformation of the casing transfers the load upward from the contact patch and on to the tire sidewalls and then on to the rim and centrally-supported axle and thence to the frame or fork.  If the air pushes equally outward all around the tire casing, then the only place where it can actually support the load _is_ at (or through) the contact patch...the overall pressure in our tire remains the same, despite the local deformation in shape.  The outward force of the air at the contact patch must equal the weight load in order to support that load to some degree.  Since pressure in the tire remains constant, the sidewalls must necessarily bulge outward when the center of the tread is deformed flat under load.  That is what transfers the load...the sidewalls bulge as the tire deforms under load and the vertical air pressure support (or tension) of the casing is reduced at the bulge; it pushes out instead (same overall pressure).  This is not so different than the way tensioned bicycle spokes support an axle load through a wheel, only in this case the air is providing the tension on the tire casing.  The tire tread at the contact patch isn't compressing...it just has less vertical tension that it would with no load.  Another way of putting it is the load _hangs_ rather than _stands_.  It just hangs less at the contact patch.    The _area_ of the tire contact patch is a function and result of both pressure and load.  The _shape_ of the contact patch is a product of tire design (i.e. conventional bias ply versus the radial-ply tires that came OEM on my '89 Miyata 1000LT), width, diameter, and pressure.  It is important to remember the inflated pneumatic tire cannot support a load separate from the wheel it is mounted on.  In the case of a bicycle, we're talking about two tensioned structures supporting the central axle load.

Whew!  My head hurts.  I'm putting this one to bed tonight, and then myself.

Best,

Dan,

[Danneaux]

For those still following the discussion, there is more to chew on here:

http://www.sheldonbrown.com/tires.html

Best,

Dan.