Industrial parts are increasingly being designed to be more lightweight in modern manufacturing for energy saving and material cost reduction. However, the high-speed motion of flexible systems tends to excite severe residual vibrations that result in positioning accuracy degradation and loss of productivity. This study proposes a closed-form trajectory optimization method for vibration suppression based on trapezoidal velocity profiles, which are most widely used in industrial robots and machines. First, the formulation and minimum time solution under actuator limits of the motion profile are defined. Then, the relationship between the trajectory parameters and the vibration response is investigated. It is shown that residual vibration can be eliminated by properly tuning the acceleration/deceleration switching times according to the natural frequency. Based on the derived vibration suppression conditions, a tuning procedure for time parameters compliant with actuator limits is established to generate fast and precise movement. A main advantage of the proposed method is easy implementation for general machines without requiring extra computational resources or modification to the control system. The effectiveness and practicality of the proposed approach are verified through experiments conducted on a robot. The experimental results show that the optimized trajectory achieves superior residual vibration reduction performance.