Occurrence of gradual resonance in a finite-length granular chain driven by harmonic vibration

This study presents numerical simulations of the resonance of a finite-length granular chain of dissipative grains driven by a harmonically vibrated tube. Multiple gradual resonant modes, namely non-resonance mode, partial-resonance mode, and complete-resonance mode, are identified. With a fixed vibration frequency, increasing vibration acceleration leads to a one-one-one increase in the number of grains participating in resonance, which is equal to the number of grain-wall collisions in a vibration period. Compared with the characteristic time of the grain–grain and the grain–wall collisions, the time of free flight plays a dominant role in grain motion. This situation results in the occurrence of large opening separation gaps between the grains and independent grain–grain and grain–wall collisions. A general master equation that describes the dependence of the system energy on the length of the granular chain and the number of grain–wall collisions is established, and it is in good agreement with the simulation results. We observe a gradual step-jump increase in system energy when the vibration acceleration is continuously increased, which is dedicated to an individual energy injection. The phase diagrams in the spaces of packing density and vibration acceleration, chain length and vibration acceleration show that shorter granular chain and larger packing density favor the occurrence of complete-resonance mode.