How does the landing gear damp vibrations?
By having a spring element and a damping element - some small planes use use leaf or tubular legs bending to act as springs, other use rubber bungy cords, others use, like bigger aircraft commonly do, a gas strut damped by liquid viscosity. The damping in the puddle jumpers is the scrub of the tyres moving laterally on the runway as the suspension moves up and down. Shimmy damping in the nose gears can be either by elements providing viscous damping or coulomb/friction damping.