Damping characteristics of the mechanism consisted of a double-cone, travelling on straight and curved rails, have great influence on the hunting motion of railway vehicles, and on the efficiency of wave-powered electrical generators. In this work, damping associated to reciprocating movement of a double-cone travelling on straight or curved divergent-convergent rails, is evaluated. Concretely, rails are attached to the hull of a reduced-scale ship, which, by using a pendulum, is excited to roll along its longitudinal axis. By employing three laser displacement sensors, the rolling angle of the ship and the position of the double-cone on the rails, are simultaneously measured. From the recorded response signals, the damping ratios of the pendulum and double-cone are determined by using the logarithmic decrement method, both in the free and coupled oscillatory movements. Replacing the straight-rails with curved-rails, the total travelling time of the double-cone was largely increased, by reducing the energy loss at the middle point, where the rails change from the divergent to the convergent zone. In order to explain these results, the opening angles of the straight and curved rails, as well as the angles and radii of contact of the double-cone with the rails, are explicitly given. Then, by deriving relations between the damping ratio and the friction coefficient, and also between the damping ratio and the restitution coefficient related to collisions of the double-cone at the middle point of the straight-rails, phenomenological explanations are presented. Compared to the monotonical reduction of the vibration amplitude, known for oscillations under constant damping, a heaving variation of the amplitude was observed in the case of this two-degrees of freedom vibration system, this suggesting a periodically variable damping ratio.