Originally Posted by AlanO

First of all, a bicycle frame is not a spring, it is a complex system made up of many different pieces, each with its own properties. More importantly, none of the models that you learn in your undergrad physics or engineering classes are true, they are all approximations. There is no such thing as an ideal spring, springs have mass, they do not have linear restoring forces. If you enclose a real spring in a vacuum and launch it into space it won’t continue to vibrate forever. It may oscillate for a long time but not forever. A real beam which is deflected will not continue to oscillate, it’s not an ideal beam. In fact, a beam is much further from ideal than a simple spring. The more complicated a system becomes the less it behaves as ideal. Every piece of a bicycle frame interacts with every other piece. None of the pieces are ideal, they have properties which contribute to the behavior. That’s not to say that ideal models aren’t useful as a first approximation in a simple scenario but they are inadequate for describing real physical phenomena as you depart further from the ideal.