STIFFNESS

Stiffness and the Sean Kelly Paradox

Stiffness. You mash down on your pedals. The bike snaps forward, leaping by your structurally deficient competition. Stiff is good, stiffer is better. What more is there to say?

Even though it seems downright simple to almost anyone, I am not convinced. I'd understand it at this point if you thought that I was a bit thick, but please keep in mind that I'm getting paid for this, and at least hear my case.

Stiffness relates the deflection of an mechanical structure to loads that try to distort it. The applied loads must be small enough to avoid permanently altering the structure's shape or we have to start talking about strength. Stiffness and strength are not the same. If the structure is stiff, its shape doesn't change much when the loads are applied. If it is not stiff (flexible), its shape will change quite a bit when loaded. How much "doesn't change much" and "changes quite a bit" actually are depends on the structure and the application; we're talking bikes here, eh? Simple.

A coil spring has stiffness, and it's referred to as it's spring constant if you push in the right direction. Push a little, and the spring squashes down. Push a bit harder and it shortens a bit more. Stop pushing and it returns to its original length. While a bicycle is much more complex than a coil spring, the analogy is useful. To think about the meaning of it all we need to figure out what the loads on the bike are like, what complexities there are in its structure that might spoil the analogy, and how the bike's deflections affect its performance. I'd like to start this rant with a simple basis for distrusting the common version of modern stiffness dogma, the version out on the streets. I'll talk more about the physics later.

First of all, I'm suspicious, in principle, about anything technical that is common knowledge and therefore considered obvious by folks without a technical background (like magazine staffers). I know, I know. I'm a skeptic; what can I say?

Here's a good reason for this. Solid, resilient, lively, dead, responsive, mushy, snappy, supple ... all the adjectives that are commonly used to communicate thoughts about a bicycle's stiffness are imprecise, and are very subjective. This makes it difficult to communicate a connection between the stiffness of a bicycle and performance in a quantifiable way. But this is how most of us have come to respect the attributes of a stiff bicycle. I find it troublesome to trust the opinions that I hear or read (except in MTB Pro of course).

I'm not completely senseless though; I can't ignore sensory inputs. I agree that a very stiff bike can "feel" quicker when you sprint. And I agree that, at first glance, it should accelerate infinitesimally better in principal. While I've felt this, I haven't observed a definite, consistent change in actual sprinting or climbing performance. I know I'm slow, but I'm not just making excuses. A snappy feel is not the same as a real, significant improvement in mechanical efficiency. Sorry for what seems like hair splitting; I hope the difference will become less murky later.

My introductory paragraph, the one that seemed to confirm what you knew all along about a stiff bike, the one that had you nodding contentedly while you were reading it, was rife with bad assumptions. I believe that our perceptions of performance, gotten while riding, are not always very accurate (back to the snappy feel is not the same as real changes in efficiency comment). I picked up this invaluable observation while working on motorcycles. My job was to develop fast motorcycles. I would go out to a track with very skilled riders, world championship caliber riders, and test with parts and tuning combinations in order to determine whether they could go faster. We tried the setups and solicited opinions from the rider, asking him how the bike worked and whether he thought he was going faster than the last time around. Of course, I would measure the rider's performance with a stopwatch as well. So I had two indications of performance, the rider's opinion and the clock's version of the story.

When the results from the two methods were compared, the rider's view and the clock's account often differed. Test configurations that seemed fast to the rider weren't always good according to the clock, and parts that made the rider think he was slow could actually be making fast lap times. The stopwatch earned its keep.

Comparing the rider's opinion of a given part's performance characteristics and what he thought should happen proved interesting as well. If the rider didn't have any hints about the upcoming test, his description of the result often differed considerably from what I actually did to the bike. The riders couldn't always tell what was happening from what they felt as they rode. On he other hand, if the rider knew beforehand what should happen in the test, his report was much more likely to be in agreement with the changes we performed. A good way to get a uniform, consistent, but potentially useless evaluation of a part was to discuss the anticipated result with the test rider before the test. Sound familiar?

I know my methods were imperfect in many ways; we were under time constraints. But I believe it supports my point that even a skilled rider's perceptions are not always reliable and can be influenced by what they think before the test. Objectivity is tough to come by, maybe impossible, and I think that this is more or less the case when it comes to evaluating the relative merits of stiffness while riding, especially when the opinions come from a magazine writer or acquaintance.

Of course, our image of the off road bicycle is not often affected by the unbiased and insensitive (with respect to designer's, advertiser's or rider's feelings) criticism of the stop watch. But until it is sorted out, you know where I'm putting my money. Of course, if anyone has the money and desire to sort it out, you know where to find me.

And whenever the subject comes up in a discussion, I'm haunted by images of Sean Kelly in his prime, shooting by a handful of thick legged Italian and Belgian sprinters on his wimpy glued aluminum Vitus, to win a sprint in a classic. If ever there was a contrary example, this is it, and it sticks with me.

Detractors will quickly point out that Kelly (while not a sprint specialist himself) was simply much stronger than all of the sprinters that day. The fact that the frames involved in the sprint could have differed in stiffness by a factor of two or more, to Kelly's disadvantage, would just be one more noteworthy aspect of Kelly's remarkable physical superiority. I revere Kelly, but I don't buy it.

So, at a minimum, I should have called the popular dogma on the subject of stiffness into question in your minds. Think of Kelly's wet noodle of a Vitus frantically carving out a drunken sinusoidal path through the pack whenever you waiver. In coming months I'll discuss how stiffness can affect performance, a very crude method that you can use to evaluate how the stiffness of individual parts can affect the stiffness of the bicycle overall and what is likely to happen due to stiffness variations in the most important specific load cases on a bike.

Until then have fun in the mud; I'll be thinking of you from sunny Santa Cruz.

Stiffness 2 - More ranting about stiffness . . . if you haven't already had enough.

STIFFNESS #2

Welcome back. Last month I tried to convince you that there were good reasons to distrust the common wisdom on stiffness, without even invoking arguments based on physics. While I'm sure that my reasoning wasn't Rumpolian in its persuasiveness, I thought it was okay for a hired Yank. Now I'm going to cover some simple physics and then we can get on with applying this to some bike parts. I'm going to sound like I actually care about stiffness a bit here and there, but I assure you it is only because it's hard to talk about this stuff any other way. My original skepticism is unwavering as I hope yours is.

Remember the coil spring we talked about last time? A spring undergoing a constant force, and the associated math, looks like this:

F = kx

Where F is the force applied to the spring

K is the stiffness of the spring, the spring constant expressed in force/distance

and x is the distance the spring deflects

When you press down slowly on the spring the force that the support feels is the same as what the spring feels, according to Mr. Newton. The spring doesn't steal any force. While springs do this when a load is applied to them slowly, they can do some really strange things when they are hit really hard but for short time intervals (like if you hit it with a hammer). As long as you haven't taken hammering too literally, we should be all right in our assumptions.

It's worth while to think about one slightly more complicated, but extremely useful, spring example. What happens when you attach a bunch of springs together end to end and push on them? Think about a system of springs each with a different stiffness, lined up in a series like this:

The math goes something like this:

Springs that are arranged like this combine to form an overall stiffness that is not obvious to many people. So what's so important about that? I'm getting to that. Actually this is some really good stuff here; bound to make you a much wiser technoperson. Look at what happens if we have two springs, one of which is 10 times stiffer than the other:

The resulting stiffness is about 90% of the softest spring! The 2 spring system behaves like the soft spring; you can almost completely ignore the stiffer spring in the system.

So how does a series of springs like this have anything to do with bicycles? Get ready; what follows is a very long winded, tedious, but momentous concept:

You can't easily estimate the effects of a given component's stiffness on the overall stiffness (and potential performance) of a bicycle. You need to know a lot about how the component is attached to the bicycle. You need to know how stiff the other parts in the load path are. Sorry, that was a mouthful. Now I'll try to sort it out.

This is why I brought up the example about the series of springs in the first place believe it or not. Remember the series of springs? The parts of a bicycle are connected together in a way that resembles a series of springs. Anytime the force acting on the bike goes along a path through the assembled structure, through a series of connected parts, the individual parts along the path behave like a series of springs. They combine in roughly the same way we showed above. Pedaling loads, terrain induced loads, inertial loads, they always end up connecting through a series of components arranged roughly (schematically anyway) end to end, along the load path.

So lets put this to work. If one brand of left hand crank arm is 10% stiffer than your current part, how much less will the pedal deflect under your awesome pedaling forces should you chose to buy it for lots of money?

First of all, you should never trust anyone to tell you the answer of this sort of question if they own one of the parts in question.

From the example above we know that the resulting change in the stiffness you feel while pedaling the bike could be much smaller than the 10% your paying for if the cranks are stiff compared to some other component in the series - like the bottom bracket spindle. This may seem a bit exaggerated, but imagine that the torsional stiffness of your ultra trick titanium bottom bracket spindle is roughly 10% of the bending stiffness of the crank arms (I know, a torsional stiffness and a bending stiffness are like apples and oranges. If you're that clever, you can work out the difficulties for this particular case). Now lets say you pop for the cranks and race home to install them. The net change in stiffness that you (can't) feel at the pedals ends up at about 0.75% . You got a bad deal if you were after the big change in stiffness.

The lesson? I'm not picking on left hand crank arms here; there are instances of this sort of thing all over a bicycle. A bike is a jumbled up mess of stiffness once all the parts are assembled together, and a big increase in a given component's stiffness here or there may not have a meaningful effect on the overall performance of the thing (assuming it could in the first place - remember my skepticism on this point). You have to know how everything fits together to sort it out at all, and this is pretty hard to do precisely.

But you can do it pretty well without a testing lab and a team of scientists. Fool around with your bike. Push on things and get someone to watch what happens. Trade places. Let her push while you watch. Think about the stacked springs while you do it. Soft, skinny or long parts on the bike are flexible. Bars, seat posts, steerers, saddles, tires, axles, spindles, and human flesh and bones dominate the flexibility of your bike. With a little bit of this you can probably figure out whether a claim, based on an increase in stiffness, is potentially valid or not. Many aren't.

Of course, you shouldn't be all that worried about how stiff a part is if you bought my argument last month. But I've assumed that many of you followed too well. That is, you accepted the correctness of being skeptical about this sort of thing, and then you immediately assumed a skeptical position about everything including my skimpy arguments. So I've given you this means of qualitatively evaluating stiffness so you can prove it to yourself. See if you can't get a feeling for how much the handlebar, stem, crank arm, bottom bracket spindle, and frame each contribute to the flex in that system, and then figure out where you'd spend your time if you wanted to stiffen it up a bit. I did.

Next month we'll talk about some more bicycle related details. Sorry for all the messing about, but I had to lay some groundwork, and the series of springs model really works wonders in sorting things out sometimes. In the meantime see how many different bikes you can take for a spin (sprint). Think about what's happening as you ride and see if you can feel what I'm talking about. If you think I'm off the mark, wait a couple of months (so the publishers don't care anymore) and tell me what you really think.

STIFFNESS 3

The Energy Efficiency Thing

The big deal about a stiff bike is how efficient it is right? It leaps forward at every pedal stroke, converting all of your precious effort into forward thrust, and so on. Ask brother Gary about it; he'll tell you.

How does this work though?

In the simplest case, not at all. Think about springs again. As a spring is compressed, you do work on it. Never mind the equations this time. This work is stored as potential energy in the spring. Work is returned to the surroundings when the spring is released. The bicycle analog: you push down on the pedal and yank up on the bars; everything in between flexes. Now you let up a bit, and the bike relaxes back a bit and you're right back to where you started. 100% efficient!

There are some problems with that model (or I'm quickly running out of a topic). When you look beyond the high school physics level you find that a "real" spring doesn't work this way. It stores almost all of the work done on it, and then gives almost all of it back when it returns to its original shape. It consumes a bit of power each time it deflects, and this shows up as heat in the spring. This is a fundamental property of all materials and is called material damping. So, in order to figure out how much power is at stake due to the stiffness (or lack of it) of a bicycle, one of the things we might do is determine how much the bike heats up as it's ridden. It would make it easy to estimate power losses if we could measure the change in the temperature of some of the parts of a bike after a hard sprint. We'd make the measurement after a sprint because the it's the worst case; losses during normal pedaling would be much smaller proportionately. But before you go out and singe your fingers on your crank arm after one of your notorious knobby shredding sprints, let's get a feel for how hot a part might get. Remember, safety first!

Imagine that all of your horsepower is consumed in the torsional flexing of your titanium bottom bracket spindle (unrealistic) and that it happens so fast that all of the heat stays in the spindle (also unrealistic, but necessary). How hot would it get after a sprint?

1/2 horsepower for ten seconds or so is a good sprint, so:

0.5 Hp to watts; 372 watts. For 10 seconds; 3720 joules. Into (gram)calories; 888 calories. Into 100 grams of titanium with a specific heat of 0.124; the change in temperature is 71 degrees C. It would get hot, eh?

But not all of your power ends up in the spindle. The power consumed in all of the stressed material of the bike is a small fraction of the rider's output, and only a fractional amount of the total is consumed in the bottom bracket spindle. Let's guess that 1% is lost there for now. What would we have to do to try to measure this? The part will heat up 0.7 degrees C. This temperature change is too small to measure with the sort of scientific equipment that is easily obtainable in the bicycle industry (like bits of string, old radios, and empty beer cans). So we can't do it this way.

But, there's another way. There are bright, friendly people in other industries, industries that are typically financially dependent on an ambitious or paranoid government, who are well paid to find out about these things and write what they find down and publish it all where everybody can get to it if they try. Since bicycles are made of the same sort of stuff that missiles and jet fighters are, and you need to know this sort of thing to predict what happens after all the buttons are pushed, there has been plenty of money around to make sure that this work is done well. Wouldn't want the rockets breaking down. I keep up on this stuff when I can and I found some that applies in this case. We might as well put it to good use since its paid for it and just lying around otherwise.

The study I found went something like this: Suppose you caused a bunch of beams, each made of a different material, to vibrate one at a time by hitting them with some sort of periodic force. Now suppose you suddenly stop beating on the poor things. How long would it take for each of the beams to stop vibrating? As material is deflected it heats up - material damping - remember?. Heating the material robs power, reducing the kinetic energy of the vibrating system, so the vibration in the beams will die off. If a beam of a given material stops fast it will have converted more of the kinetic energy into heat in a given period of time than if the vibration dies out slowly.

Thinking about it another way, the material that heats up fast as it is driven by the periodic force will require more power to deflect at a given rate since it will lose more power to heat whenever it is deflected, everything else being equal. That's not a perfect explanation, but it'll work for us.

It turns out that this is a pretty hard thing to measure for most metals because they lose this energy very slowly! They make good bells but bad heaters. Despite these difficulties, it turns out that the scientist types persevered (spent money on better apparatus) and put the answer to this question into fairly simple, though not necessarily inexpensive, terms. They invented a quantity called "specific damping capacity", symbolized it with a pretentious Greek letter, and then measured it for quite a few materials. This quantity represents the fractional difference in the kinetic energy of the beam between one cycle and the next. That is, it is the power lost to heating the vibrating material during one cycle of a decaying vibration.

Got that?

Maybe an example will help. If the specific damping capacity for a vibrating beam is 1%, and it starts off with 20 calories of work done on it, it will lose 0.2 calories between the first and second cycles of its vibration and so on until it stops (actually this can't really work - it will never stop if it loses energy at a rate expressed as a fraction of the energy state of the previous cycle - it's just close). Simple huh?

Specific damping turns out to be a very useful measure of a material's behavior in the case we are interested in. The deflection a bike undergoes during a pedal stroke is like half of a vibration. The bike is pushed out of shape and it stores this work as potential energy. It returns to its original shape after the forces are released, does some work back on the rider in the process, and ends up heated up a bit. This heat will be equal to the energy stored in one cycle of each of the bike's components along the pedaling load path multiplied by the specific damping of each of these components.

Back to the ti spindle to apply this. Let's guess that the rider's peak forces under his left foot are about three times his 167 pound body weight, around 500 pounds (this is not far off what a strong rider would put out starting up an incline from a stop, so it's on the high side for a sprint). The work done on the deflected spindle will be about 10 calories at its maximum deflection. The specific damping factor for ti is about .03% in this stress range, so the work lost to the spindle is about 0.3 calories per left foot stroke.

Then the rider would lose 3 calories to heating the spindle over the ten second interval at 150 rpm. Given a total output of 888 calories in that interval the rider's loss to material damping would be about 0.3 percent or a temperature change of about 0.2 degrees C. The spindle doesn't heat up much.

This may not be exactly right; we made a few assumptions along the way, though none of them are grossly out of range in my book. But you get the idea. We turned up another one of those miserable "fraction of a percent" answers. They're not enough to get excited about but not enough to ignore either (unless you are just in it for the fun of course).

I'll take this further next month - it's been raining here and I like the mud so I'm going for a ride now. I think that's one good way to keep your mind from overrevving on this sort of thing - go for a good mud ride. You've got so much to worry about just keeping on the trail and avoiding disaster and humiliation that small matters like stiffness, efficiency, and so on become trivial. And, for the most part, that's where they are anyway.

Stiffness 4 - And more!

STIFFNESS 4

Stiffness - once again anybody?

What about energy you say? The main advantage of a stiff bike is all of the power it saves you. Things that don't flex are efficient, right? As a spring is compressed, you do work on it. In the simplest case this work you do is stored as potential energy in the spring and comes back to you when you let up on it. You push down on the spring; it pushes back up on you and nothing's lost. So in the simplest case the spring gives all of the work back and efficiency is not an issue.

But that's not all there is to it. When you look a little closer you find that a "real" spring stores the work it takes to compress it, and gives almost all of it back when it returns to its original shape. It consumes a bit of power each time it moves, and as it does it heats up. This is called material damping and is a fundamental property of all materials. So, we have to figure out how much the spring (the bike remember?) heats up as it sproings along.

This problem is a bit more difficult to solve than anything else we've done so far and really is one of the questions when it comes to stiffness and energy efficiency. All parts will heat up as you ride and flexible parts will heat up the most all else being equal. Now I can't say, based solely on my riding experiences, whether the heat lost to the environment due to these losses is significant, and I doubt that anyone else can either.

But can we measure it? It would make it easy to figure power losses if we could measure the change in the temperature of the bike after a hard sprint. But before you go out and singe your fingers on your frame after one of your famous "after burner" sprints, let's try to get a feel for how hot a part would get if the power losses were a substantial fraction of your output. Imagine that all of your horsepower is consumed in the uniform torsional flexing of your brand new titanium bottom bracket spindle. Lets also assume that it happens so fast that all of the heat stays in the spindle. How hot would it get after a hard sprint?

Let's see, 1/2 horsepower for ten seconds or so would win almost any sprint, so:

0.5 Hp to watts; 372 watts. For 10 seconds; 3720 joules. Into (gram)calories; 888 calories. Into 100 grams of titanium with a specific heat of 0.124; the change in temperature is 71 degrees C.

But we assumed that ALL of your power went into the spindle. We can guess that the power consumed in the material is a small fraction of the rider's output, say 1%. Then the temperature change changes by the same proportion and the part heats up 0.7 degrees.

Based on this we can figure that the thermal changes are very small, way too small to measure with easily obtainable equipment (like strings and coffee cans).

But, there may be another way to do this which, while less satisfying than a good experiment, will help out quite a bit. Now we need a bit more physics.

There are people in other industries who are well paid to care about this sort of thing, and who actually find out more or less what really happens (whatever it may be) rather than letting an unsuspecting public go on wondering about it for decades and an equally unsuspecting media go on writing artsy lines about it in technical articles for an equally absurd amount of time. They've solved a similar problem and written about it. It goes like this:

Suppose you caused a beam (long skinny chunk) of material to vibrate by driving it with some sort of force that makes it vibrate. It might ring like a bell or thunk like a pillow. Now suppose you stopped whacking on the poor thing. How long would it take for the beam to stop vibrating? If it stops fast it will convert more energy to heat than if the vibration dies out more slowly. And the internal mechanism that causes the vibration to die will cost you energy when you make it flex.

It turns out that this is a pretty hard thing to do for many metals because they dampen this vibration very slowly! They make good bells. But isn't this kind of like the answer we were looking for? It turns out that these fine folks have even gone to the trouble of putting the answer into fairly simple terms. They invented a quantity called specific damping capacity, gave it a Greek letter for a symbol like they always do (I won't tease the typesetters with it), and then went out and measured it for quite a few materials.

This quantity represents the difference in the energy between one cycle of a vibrating material and the next cycle. Got that? Maybe an example will help. If the specific damping capacity for a vibrating beam is 1%, and it starts off with 20 calories of work done on it to cause it to distort, it will lose 0.2 calories between the first and second cycles of its vibration after you let it go. The next cycle's maximum amplitude will be smaller by the amount necessary to account for the energy dissipated in the material during this first cycle.

Specific damping turns out to be a very useful measure of a material's behavior in the case we are interested in. The deflection a bike undergoes during a pedal stroke is like half of a driven vibration. A pedal stroke pushes the bike out of shape so it stores some work as potential energy. It returns to its original shape after the forces are released, does some work back on the rider, and ends up heated up a bit - if what I'm spouting is true. This heat will be equal to the energy stored in one cycle of each of the bike's springy components along the pedaling load path multiplied by the specific damping of each of these components.

Back to the ti spindle. Let's guess that the rider's peak forces under his left foot are about three times his 167 pound body weight, around 500 pounds. The spindle will deflect torsionally and the work stored in this part will be about 10 calories at its maximum deflection. (1.5" pedal shaft deflection, k=333, U=.5(kx^2) The specific damping factor for ti is about .03% in this stress range, so the work lost to the spindle is about 0.3 calories per left foot stroke at this load. The bike doesn't heat up much. It is not unreasonable that a national class rider will produce this sort of force at a power output and for an interval like the one mentioned above. If that were the case the rider would lose 5 calories to the spindle. Given a total output of 888 calories in that interval the rider's loss to material damping is on the order of 0.5 percent of his energy expenditure during that interval. Not very much.

Stiffness 5 - And more!

STIFFNESS 5

Stiffness in 3D

Engineers live in another reality. Their physical concepts are often crafted to demonstrate basic principles. This is generally because whatever it is they are talking about had to fit originally on a printed piece of paper and in a young brain. The reality we ride in doesn't though, and to that extent, it is often different than the engineer's.

Stiffness is one of these concepts. The basics of springs are simple and instructive, but they are not the whole story. I showed last time that, as the work you do on a bike causes it to flex, some heat is lost to material damping. That's the simple stuff, but how can we get this more real? We need to get past thinking about springs to do this. So this is the last spring installment. I promise.

I'll warn you now, I'm not going to conclude this article by saying anything like "So now you can see that stiffness really does (not) matter and you can do (you pick the scenario) if you want to without thinking about stiffness anymore". Back of the napkin calculations can't solve the thing; I think we need to make some actual measurements. But instead of solving it, we can define the problem more clearly for now. That always helps one way or another.

Consider the pedaling forces a strong rider exerts in a sprint. When she sprints at the end of a race she concentrates on making the biggest forces she can - in a hurry. Form is not a primary issue, though it is developed in practice. With the race on the line a sprint won't be decided by style points or efficiency, only raw power output. Even if some component of her effort is wasted due to frame flex or other sources of inefficiency, it's okay as long as she goes faster for that brief period. Of course, she may be faster if she rides a bike that minimizes the material damping losses, as long as it doesn't slow her down elsewhere. There are also some other possible sources of inefficiency that we haven't talked about yet that may be involved. But the basic idea is true. A sprinter just cranks the pedals as hard as she can.

Contrast this with the efficient form that is regarded as good cycling technique. Good pedaling form is referred to as "pedaling circles". That is the way trainers describe it, and it means applying forces to the pedals that are always tangential to the circle the crank sweeps out as it rotates. Even though it has been shown that no one actually does this, the basic idea is clearly right. The forces are directed as often as possible toward providing propulsion. As little as possible is wasted. The bike is not being heated up needlessly, though the rider would be better of if this quantity could be reduced as long as there aren't other costs in doing this. Pedaling circles works.

These are the extremes of cycling from the rider's point of view. Each of these involves the stiffness of the bike, though only indirectly. It becomes an issue because, by minimizing material damping losses in the bike, you might go faster. And this advantage is only real if the additional effects of this change are not detrimental to the overall performance of the bike in some other way. Otherwise the discussion of the two extremes are, for the most part, physiological issues, or how the rider can train or ride to go faster.

From what we know we can say that it doesn't matter whether the work done on a bike would eventually result in propulsion or whether it is wasted and just flexes the bike around. The heat losses still occur. This is simple really. The power is lost forever. So if a bike is built to reduce this type of loss it would be more efficient in this respect. Material selection would matter. For a given material, stiffer is better.

But I think that the issue is still not simple. What if I could find a way that a bike could flex that leads to an increase in power output compared to a stiffer bike, and the increase was potentially larger than the consequent material damping losses? An increase in stiffness may not be warranted in this case.

Back to the example of the sprinter again. As she pushes down on the pedal on one side the frame around the bottom bracket swings over to the opposite side. You can see this happen if you put the pedal at its lowest point and push down on it with the bike at rest. The frame flexes down and also swings over to the side, away from the side of the bike you are pushing on. This deflection can change a rider's power output at the rear wheel. This is because the angle between the force she is applying to the pedal and the bottom bracket spindle orientation changes as the flex occurs. The power output is best when the force and optimal pedal trajectory are parallel. It goes down from there.

And now think about how the bike is tossed back and forth in a sprint. The rider leans the bike over from side to side. I believe that this movement is advantageous for a number of reasons, and that it is made necessary by the rider's natural tendency to center her upper body over the pedal that she is pushing on at the time. She pushes with her right foot, leans the bike to the left and gets her torso and shoulders over the pedal she is pushing on. She is able to exert a larger force in this way and is also balanced better during the sprint. If you don't buy this, try sprinting without leaning the bike this way.

This leaning motion would also hurt power output in the same way that the frame flex described above does. If she pushed straight down while she leaned the bike over, and nothing else happened, her force would no longer be aligned with the optimal trajectory of the pedal. Some of it would be wasted by pointing the pedal trajectory away from the direction her force is applied by leaning the bike over.

But both of these occur in a sprint. As the bike is leaned over it flexes in response to the pedaling force. The two combine in a way that tends to correct the misalignment! She leans the bike over cyclically in phase with her pedaling forces when she sprints because it works better for her (and all humans). The bike flexes in response to her pedaling forces. The deflection of the bike and the angle to which it is leaned happen to combine in a way that the pedaling force tend to align better with the trajectory of the pedal. It is possible that increasing the stiffness of the bike in this common situation could reduce the rider's power output. Fun eh?

The information that would resolve this is requires measurements. The losses due to the misalignment between the rider's pedaling force and the optimal pedal trajectory can be calculated directly from the angle between them. These are likely to be on the order of a percent or two. If the short term power at the rear wheel was plotted as a function of stiffness, the optimum may not be found to be at infinite frame stiffness. If the material damping losses are small (any bets?), then a certain amount of flexibility may actually contribute to efficiency in this case.

I can hear you laughing now. No, I haven't lost my mind as far as I am aware (does one ever really know?). I am not saying that flexible bikes are better in a sprint than stiff ones. My point is to illustrate, with an example that isn't too much of a stretch, how complicated things can get when you think about the importance of stiffness.

And I'll have to leave it that way for now.

Next I will discuss where this lost work comes from and how, believe it or not, suspension can fix this problem, sort of. You knew I was going somewhere trendy with this thing eventually didn't you. I'm sure the editors were hoping so, or at least were lining someone up to take over for me if I didn't get around to it soon.

Stiffness 6...  Nah!