Some years ago, when I still lived in New York City, I tried building a stabilizing camera mount for a bicycle. I used to have videos of some of the contraptions I built, but I don’t think I still have them. I’ll look through the old backups when I have a chance.
What I can tell you is that I seriously doubt that any system based on springs and levers is likely to work for a lightweight camera. There’s simply not enough inertia. But what might work if you want to play around is a system based on magnets. That was the one that came closest to working for me.
I created a camera mount out of 1/2 inch PVC pipe that rode up and down on two vertical wooden dowels, suspended by magnetic fields at the top and bottom. In other words, the magnet on the movable mount had magnets both above and below it, each of which opposed the movable magnet and kept it suspended.
That actually worked better than anything else I’d tried. But it was crude and kept hanging up on the dowels. With refinement, I think it would have worked. But before I could actually refine it, the old lady and I split up; and I hurried up and moved before she changed her mind.
That’s actually true, by the way. I always wanted to live in the country, but my ex hated rural life. She was the only reason I still lived in the city; so as soon as she left, I sold the tech support part of my business and started looking for a place in the country. I can do Web development anywhere I can get the Interwebs, after all.