I'm neither a physics person or an engineer, but I think I can respond in a way that makes sense:
Two reasons why the anti-spring argument doesn't pass muster.
1) At least some portion of the energy used to flex the frame in one direction is indeed "returned as useful" as the springing back accentuates the pedal strong in the opposite direction. But "some portion" implies loss, so see next part.
2) The frame flexes in response to the force applied to it. As the rider exerts downward force on the pedals, some of the force is orthogonal to the plane of the frame. How much the frame deflects depends on the force applied and the rigidity of the frame. For a given force, "soft" frames deflect more than relatively more rigid frames. The key here is "for a given force."
In the spring example, put the same force on different springs. They may compress by different amounts while the force to compress them is the same. The stiffer spring compressed less, but it still took the the same force to compress it even though it compressed by a relatively lesser amount.
As I've said before, riding the "flexible frame" may waste energy from the rider in terms of body motion (but not from applying force to the pedals). But a frame with too little flexiblilty doesn't accentuate the riders pedal stroke.
For analytical types, my guess is that one could express optimal rigidity as a concave function of rider-body-motion and pedal-stroke-efficiency.
Mike Kone in Boulder CO
> Well, without going to the archives, and without getting drawn into a
\r?\n> big discussion, the argument about that energy from frame flex is
\r?\n> conserved because bicycles are springs is rubbish.
\r?\n> Yes, deflecting a frame causes the frame to store energy. Yes, when the
\r?\n> frame springs back, it releases the energy. However, this energy does
\r?\n> not go back up the rider's legs, nor does it move the bicycle forward.
\r?\n> It is just wasted movement.
\r?\n> It often appears to me that many posters on this list seem to be unaware
\r?\n> of basic principles of physics. Maybe we should start a physics mailing
\r?\n> Louis Schulman
\r?\n> Tampa, FL
\r?\n> firstname.lastname@example.org wrote:
\r?\n> > Without getting into the testing methodology, the conclusions on the web site
\r?\n> cited below are in my opinion useless.
\r?\n> > The author claims there is no downside to a frame that is too stiff. That in
\r?\n> my opinion is just plain wrong. Again, a Raleigh 3sp is super stiff, but dead.
\r?\n> A frame must be neither too stiff or too flexible. It must have flex
\r?\n> characteristics that enhance the riders pedal stroke without causing a
\r?\n> sub-optimal amount of additional body motion.
\r?\n> > Without getting drawn back into discussion that is in the archives - remember
\r?\n> that frames DO NOT absorb significant amounts of energy - they are big springs.
\r?\n> > The author of the frame flex tests and discussion cited below seems to have
\r?\n> missed this important point.
\r?\n> > Mike Kone in Boulder CO
\r?\n> > -------------- Original message --------------
\r?\n> >>You might want to start at Sheldon's archive of Damon Rinard's bike
\r?\n> >>tech site.
\r?\n> >>Almost all the tests include frames from the pre-'83 time period or at
\r?\n> >>least a "KOF" frame of some kind.
\r?\n> >>Vancouver, B.C.
\r?\n> >>On Wednesday, Sep 28, 2005, at 13:20 US/Pacific, charles nighbor wrote:
\r?\n> >>>IN RIDE Issue Number 29 WINTER 2005 Page 139 there is a very
\r?\n> >>>interesting article on Frame Flex Testing with a Jig. Than on the
\r?\n> >>>following pages five frames are tested (Yes Modern new frames way past
\r?\n> >>>1983 cut off but still tests results worth noting) and worth studying.
\r?\n> >>>When we discussed brazing materials silver versus brass and testing of
\r?\n> >>>them it brought this article to my mind. If you get a copy decide which
\r?\n> >>>one you prefer over the others. Than decide how it would affect ride.
\r?\n> >>>Then read how it did ride. See if you can go from test numbers to a
\r?\n> >>>correct interruption of results. I wonder just how much testing was
\r?\n> >>>done on steel lugged frames we all love and admire?
\r?\n> >>>Charles Nighbor
\r?\n> >>>Walnut Creek, CA