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

TL;DR

This paper investigates the resonance phenomena in finite granular chains under harmonic vibration, revealing multiple resonance modes, the role of collision times, and establishing a master equation correlating energy with chain length and collisions.

Contribution

It introduces a comprehensive numerical analysis of resonance modes in granular chains and develops a master equation linking energy, chain length, and collisions.

Findings

Identification of multiple resonance modes

Energy increases stepwise with vibration acceleration

Dominance of free-flight time in grain motion

Abstract

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, increased vibration acceleration leads to a one-by-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 condition results in the occurrence of large opening gaps between the grains and independent grain-grain and gain-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. Moreover, two typical phase diagrams are discussed in the spaces of $\phi-\it{\Gamma}$ and $N-\it\Gamma$.