The nonholonomic redundancy of second-order nonholonomic mechanical systems is investigated. It has been verified that the self-motion can be implemented demonstrably by some nonholonomic mechanical systems such as the underactuated redundant manipulators. An exponentially stabilization control method is proposed for a class of underactuated manipulators, of which the number of actuated joints is no less than that of the passive joints. It has been shown that this class of underactuated manipulators are completely controllable when the dynamic coupling of the underactuated manipulators is non-degenerated and the up-boundary of the inputs is large enough. By the proposed control method, we exhibit this class manipulators with zero weight can realize the "self-motion" as a full-actuated redundant one. As a typical application, the problem of path tracking with suppressing vibration is investigated for the underactuated redundant manipulators. It is revealed that the vibration of the underactuated redundant manipulator can be converted into an internal resonance that is compatible with the "self-motion", while it leads to no vibration at the end-effector of the manipulator. Some numerical simulations by a planar four-DOF underactuated manipulator with two actuated joints and two passive joints show the effectiveness of the accurate trajectory control method and the value of the self-motion compatible internal resonance.