Variable Speed Gearboxes?

[jkelleyrtp]

This concept is an actual thing, however it does not typically involve gears or the exact mechanic you proposed. Some hybrid cars make use of what's known as continusouly variable transmission (CVT) which utilizes belts and conical pulleys to change the effective ratio (r1 /r2). Belts are more resilient in this application than gears. Check out variomatics if you are curious on how to apply it on a smaller scale. I could see some FRC use but the required pulley sizes for the HTD profile might not be feasible in a compact "gearbox."