An aircraft turbomachine rotor has a rotor disc extending transversely with respect to a longitudinal axis X, a plurality of blades, and a damping device. The damping device having: a support ring arranged transversely with respect to the longitudinal axis X and mounted on the outer periphery of the rotor disc, and has a plurality of damping members that are attached to the support ring and project upstream from the support ring, each damping member being configured to extend at least under a platform of a blade so as to exert a radially outward force to dampen the radial movement of the blade when the turbomachine is in operation.