It requires 3 dimensions, and it consists of multiple coupled springs.

I agree they are coupled, so that simplifies the problem. Why is the direction perpendicular to the bike frame of relevance? (Even if you really do need to treat it as a coupled, 3D system, that is hardly unusual, let alone fatally complicating.)

Demonstrate a device with multiple springs that has a single input and a different output with no moving parts.

Coupled springs don't simplify the problem, they complicate it. For instance, a model with three uncoupled springs can be reduced to three independent 1 dimensional models. A model with 3 coupled springs cannot, in general, be reduced to simple independent models.

If they are in the same plane and oscillating in phase (due to the coupling), the problem simplifies to a 1-D problem. The coupling in the perpendicular direction can probably be ignored; in fact, the spring in the perpendicular direction itself is likely safely ignored, due to the impossibility of it contributing to the proposed effect.

It is standard practice to start with the simplest model, and add in the complications iff necessary.

This is a problem very familiar to physical chemists, because a polyatomic molecule is a set of atoms connected by bonds, which can be approximated by springs. The molecular vibrations are quantized, but apart from that, it is the same problem (the classical version is easier), and it is one that undergrads routinely learn to solve.

The end points/joints on the bicycle frame are the "atoms", and the tubes that connect them are the "bonds."

In both cases, the bonds can be treated, to excellent approximation, as simple harmonic oscillators (springs). The motions are the normal vibrational modes.

But only a diatomic molecule can be reduced to a single 1-dimensional oscillator, as you were suggesting could be done with a bike frame. And a bike frame is not really modeled well by discrete masses connected by springs -- the mass is distributed and it's bending modes that are of interest.

I didn't say single. I said you could separate out the components, and just treat the active/relevant mode as a single SHO. This is not as complex as problem as you are claiming, and so what if it was?

Have a look at the Feynman Lectures where he treats ammonia (which has 4 atoms).

There's nothing that says you can't model it with a single oscillator. The question is, does that simple model give you any insight into how planing might occur?

That isn't the question and you know it. Now you're just deflecting since you can't explain your objection to the simple question of flex return.

Yeah, I was there a bit ago, but his posts are equally entertaining as they are annoying (as well as wrong) so....
I just wish I could express that more clearly...

You're the other empty mouth that has nothing to say. Put up or shut up - if you know better show how it works.

Because you aren't entertaining. Just rude.

Entertaining thread.

What is absent from my perspective is the concept that rider comfort (as in smoother riding not sitting in the sun drinking fruit margaritas) allows more power transfer when the road isn’t smooth.

No idea how to describe it in engineering terms, but my gravel bike is faster than my skinny tired road bike on rough roads.

I would say that concept is suspension, and a frame can be part of that.

Too much arguing, too many reports, thread is closed.