Monday, December 13, 2010

Techy Stuff

"In order to find the best compromise between performance and cost, the round tube may be sized such that its bending stiffness is identical to that of a square tube with the same wall thickness.  Using the first order findings of Eq. 2.12 and setting equal results in:  

wrd = cuberoot(16/3*Pi) * wsq 

Plugging the diameter wrd found in Eq. 2.18 into Eq 2.13, the torsional stiffness of a round tube can be found to be larger by a factor of 4/3 (33%) while being lighter by 7%."  

I may never become a commercial success at frame-building (or perhaps any pursuit) because of my resolute tendency to cry BS at anything that tickles me as marketing tekno-babble rather than real engineering.  And today, in any successful commercial endeavor, there there seems to be a lot of marketing tekno-babble.  No where is this more true today, than in the bicycle industry; where so much that is discussed, advertised, and written regarding design and bicycle attributes is just so much nonsense.  

Recently, I was perusing: Principles of Rapid Machine Design by Eberhard Bamberg, M.Sc., Advanced Manufacturing Systems Brunel University, 1993 Dipl.-Ing, Maschinenbau Universit├Ąt Stuttgart, 1996  SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT OF THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY, June 2000 © 2000 Massachusetts Institute of Technology All rights reserved.  This is the source of the opening quote.

Let's begin by acknowledging that my academic career stayed far away from the hard sciences and engineering, thus making my tastes in light reading that much more unusual.  And no, the goal of this read was not to learn anything about designing or building bicycles.  However, while I've known for quite some time that a round shape is the most efficient shape of tubing for resisting both bending and torsional forces; I haven't had a scholarly reference or mathematical formula to back up this understanding.  So, running across this quote, in the midst of a larger discussion of analyzing the properties of tubes, seemed like a bit of an early holiday present.

All of the large bike brands make much of the shapes of their tubes, suggesting that that they have found miraculous new ways to: 1) improve stiffness; 2) reduce weight; 3) improve ride qualities.  I'm calling BS right here, right now.  Stiffness is increased by increasing the diameter of tubes and/or increasing the wall thickness.  Weight is reduced by cutting down the amount of material in the tubes, either by using smaller diameter tubes, and/or decreased wall thickness.  Ride comfort, however, is a different question - and it is difficult to argue that tubing diameter or wall thickness is directly related to ride comfort.

Among stock phrases appearing in bicycle reviews, "... greater lateral stiffness combined with improved vertical compliance..." is probably the one most often ridiculed by the cognoscenti of bicycle design and building.  And this ridicule is truly well earned.  The side-view of a bike frame is essentially based on a Warren Truss, which is a very efficient structure for resisting the weight carried by a bridge.  In other words, it focuses on resisting vertical forces.

Viewed from the front, however, there is no structure to resist lateral forces except the form of the tubes themselves. 

Both the shape of a tube, and the form of a Warren Truss service to harness the opposing forces of tension and compression.  Both also gain stiffness by physically separating those opposing forces as far apart as possible (hence the greater stiffness of a larger diameter tube).  But the distances within tubes and between tubes vary by greater than an order of magnitude.  So while a round tube's inherent stiffness is equal in the vertical and horizontal planes, the Warren Turss does not offer this direction-independent equality of stiffness.

What does this mean?  A bicycle frame is more apt to bend laterally than vertically, regardless of what magazine writers or marketing specialists say. 

So maybe that's true, but how about a top tube that is flattened?  Surely its shape allows more vertical compliance and lateral stiffness than a round tube, because most of the material is oriented horizontally, right?  And this beneficially tunes ride characteristics?  Wrong Kemosabi!

Think of an I-beam: there is a vertical web connecting two horizontal plates.  Fundamentally, its the plates doing the hard work.  Why is this?  Well, the greater effective distance a portion of a beam is stretched or compressed, the more work that portion of the beam does in terms of preventing bending.  Get a compass, and draw three concentric circles.  The inner represents the bending of the lower plate of the I-beam, the middle circle represents the bending of the web, and the the outer circle represents the bending of the upper plate.  Clearly, the lower plate is compressed relative to the web and upper plate.  The upper plate is stretched relative to the web and lower plate.  And the web is merely bent.  Now this is a slight generalization of the forces are work, but it makes the point. 

A tube is more like a box beam, where there are two webs, one on each side, and two plates.  A beam is made more stiff by increasing the distance between the plates - thus increasing the differential between the compression and tension.  That squashed top tube causes the sides of the tube to be further apart from each other, and the top/bottom to be closer together.  So.... while it appears as if it should be stiffer laterally, and more flexible vertically, it is in fact the opposite. 

It's hard to beat round tubes for a bicycle frame.  Where there is a benefit to ovalizing them, it is generally to give a larger area with which to bond (using a weld/braze/glue/carbon wrap/or structural lug) to the adjoining tube - and not to tune the ride or stiffness.  And, that's the way it is.

Having beaten this point to death, my hope is that a few more folks will now recognize that: a) shaped tubing is just a marketing gimmick; b) frames flex laterally (or in torsion) rather than vertically.  And thereby they will be better informed consumers.