Originally Posted by
unterhausen
One experiment with $20000 worth of test equipment is all that is required to prove your intuition wrong in this case. Below the first resonant frequency of a structure, you will not get any vibration in the structure itself. It acts as a rigid body -- you're just shaking the frame back and forth with the fork/wheel. So changing the tubing would have no effect until you made it into a wet noodle. The wet noodle wouldn't ride well enough to even consider as a bike.
What you say about the first resonant frequency of a structure is true, but you can change that resonant frequency by changing the stiffness or mass of the structure. As a very simple example, imagine a vibrating string. If you put the string under more tension (analogous to a stiffer frame), that will raise the first resonant frequency. If you increase the mass of the string (analogous to putting a load on the bike), that will lower the first resonant frequency. A bicycle is a much more complex system than a string, but I am unaware of any system where the first resonant frequency is not increased by increasing stiffness and decreasing mass.
I suspect that when you load a bicycle, you can lower the first resonant frequency to the point where it can be excited by the rocking motion when you ride out of the saddle. If you build the frame out of a stiffer material, you can raise that resonant frequency back above that range.