During high-speed handling of large-sized thin-walled structural components by industrial robots, flexible payloads undergo significant vibrations due to the flexibility. To describe and address the vibration issues in multi-degree-of-freedom industrial robots carrying flexible payloads, this paper established a dynamics model and developed an online modal parameters identification method with multi-mode vibration suppression methods. First, through the assumed modes method and Lagrange equation in modal coordinates, we establish the dynamics model for robot-payload systems where robot motions serve as excitation and modal coordinates describe response. Then, an online identification method is designed using the Levenberg-Marquardt algorithm to identify natural frequencies and damping ratios of multiple payload modes. Finally, based on the identified parameters, a joint-space multi-mode input shaper is developed and its capability to achieve approximate zero vibration during dynamic robot-load handling processes is theoretically demonstrated. Experiments confirmed that the proposed method effectively suppressed the vibration of regular and irregular payloads handled by a six degree-of-freedom industrial robot.