We employ a phase-field lattice Boltzmann model to numerically investigate the complex dynamics of droplets on a mechanically excited, horizontally oriented fiber. Famously, the droplet can wet the fiber in two distinct states: the barrel state and the clamshell state. Initially, we characterize the natural oscillatory behavior of a droplet sitting on a stationary fiber, finding that the clamshell state exhibits a significantly higher natural frequency compared to the barrel state. By systematically varying the vibration amplitude and frequency, we uncover a diverse range of droplet response modes. For vertical fiber vibration, these modes include the quasi-static motion, rigid motion, regular pumping, up-and-down pumping, and subharmonic pumping modes. Furthermore, at sufficiently high acceleration amplitudes, droplet detachment occurs. Additionally, a swinging motion can be induced by an initial disturbance within a specific frequency range. For horizontal fiber vibration, the observed modes encompass regular swinging, combined swinging and pumping, pure side-to-side pumping, and a slinging mode, where the droplet orbits the fiber. We present comprehensive regime maps that delineate the boundaries for each response mode in the parameter space of frequency and acceleration amplitude. Our findings provide valuable insights into the intricate interplay between a droplet and a vibrating fiber, with potential applications in fog harvesting, microfluidics, and droplet-based technologies.