Wow. OK. Cool. This will be a welcome challenge... lol. This isn't a subject I know much about or that I've given much thought to, although I have read that even the major factories chase their tails over it. I will tell you what I know and try to do some research on it to form some discussion. From what I understand of it though, it is something of a black art.
In a nutshell, although a perfectly stiff frame/chassis would seem to be desirable, ultimately, unlike a race car, there is such a thing as "too stiff" for the motorcycle chassis. As I understand it, part of this fact is due to the shortcomings of the telescopic front fork suspension. And moreover the inability of a motorcycle suspension in general to operate efficiently while leaned over, ie. it can't react to bumps in the road when it is sideways. Obviously, the details of where and why the chassis needs to flex, ie. near the headstock and swingarm pivot, as opposed to the engine cradle or spars which need to be stiffer, or how to brace the swingarm for torsional rigidity while allowing some lateral or horizontal flex is a huge subject far beyond my own experience or knowledge at the moment. However...
If I recall correctly, chassis flex first became a big issue in the seventies when big motors (like the Kawasaki Z1, 900/1000/1100 or the H1/H2 two-stroke triples) became so powerful that they literally twisted the old tubular frame bikes up in knots. Chain pull would cause the swing arm to actually twist in its pivot and the frame with it. Sometimes the chassis would actually bend and stay bent, but, more often, it would wind itself up like a spring and release that energy in an uncontrolled fashion. (Just ask the guys who raced them... like Eddie Lawson, Wes Cooley or... ahem.. Keith Code). So, stiffer frame/chassis became the goal. However, frame technology, materials and design have come so far that today's chassis/frames can be so light/stiff that they actually start to vibrate or oscillate allegedly causing front end "chatter" or perhaps masking feedback of what is happening at the pavement. Or whatever else riders complain about before they begin removing engine mounting bolts from their frames... like Colin Edwards allegedly did his last year of WSB.
Using Google, I found some interesting articles like this blog:
http://firstsynn.blogspot.com/2007/04/moto...ame-primer.html
Here is an interesting "engineering oriented resource":
http://www.dim.unipd.it/lot/HTML%20flexmot...%20Madrid1.html
Thanks for taking a stab at it! This topic is obviously pretty complicated and I'm curious how one begins learning about these things without the benefits of an engineering degree or the desire to get one for the sake of a hobby. I found another resource for this in case you were interested. I'm not sure about the credentials of the guy who wrote the book because he throws out some really wild ideas but it seems a pretty good place to start.
The link to his book is as follows: http://www.tonyfoale.com/reviews.htm
I would be happy to hear any additional comments if other people care to chime in. The subject just plain fascinates me.