This study addresses the design and properties of a sliding-mode neural-network control (SMNNC) system for a nonlinear flexible arm that is driven by a permanent magnet (PM) synchronous servo motor. First, the dynamic model of a flexible arm system with a tip mass is introduced. When the tip mass of the flexible arm is a rigid body, not only bending vibration but also torsional vibration occurr. In this study, the vibration states of the nonlinear system are assumed to be unmeasurable, i.e., only the actuator position can be acquired to feed into a suitable control system for stabilizing the vibration states indirectly. Then, a SMNNC scheme without the feedback of the vibration measure is proposed to control the motor-mechanism coupling system for periodic motion. All adaptive learning algorithms in the SMNNC system are derived in the sense of Lyapunov stability analysis, so that the system-tracking stability can be guaranteed in the closed-loop system. The effectiveness of the proposed control scheme is verified by both numerical simulation and experimental results.