Chaotic dynamics of granules-beam coupled vibration: Route and threshold

TL;DR

This paper investigates the chaotic behavior of granules-beam coupled vibrations, deriving a Melnikov criterion for chaos onset and experimentally confirming the route to chaos, which enhances understanding of granular-structure interactions.

Contribution

It introduces the first Melnikov criterion for chaos in granules-beam systems and validates it through experiments, revealing the chaos route and threshold.

Findings

Chaotic responses occur after symmetry-breaking and period-doubling cascade.

The Melnikov criterion accurately predicts the chaos threshold.

Experimental results support the theoretical chaos evolution pattern.

Abstract

Although granular materials are the second most processed in industry after water, the theoretical study of granules-structure interactions is not as advanced as that of fluid-structure interactions due to the lack of a unified view of the constitutive relation of granular materials. In the previous work, the theoretical model of granules-beam coupled vibration was developed and verified by experiments. However, it was also found that the system exhibits significant stiffness-softening Duffing characteristics even under micro-vibration, which implies that chaotic responses may be reached under certain conditions. To reveal the route and critical conditions for the system to enter chaos, in the present work, the chaotic dynamics of the system are studied. In qualitative analysis, Melnikov method is applied to analyze the instability behavior of the perturbed heteroclinic orbit of the system, thus the critical condition for the system to enter Smale horseshoe chaos, i.e., the Melnikov criterion, is obtained. The validity of the criterion is verified numerically. In experimental studies, the existence of chaotic responses and the route to chaos for granules-beam coupled vibrations is revealed. The experimental results suggest that the system response first experiences symmetry-breaking then undergoes a complete period-doubling cascade, and finally enters single-scroll chaos. In addition, although the Additional Dynamic Load (ADL) generated by granular media is highly complicated, a general and simple evolution pattern of the chaos threshold is found by parametric experiments, which is also supported by the Melnikov criterion. In short, the chaotic dynamics of granules-beam coupled vibration is revealed, which is a contribution to the engineering vibration aspect. On the theoretical side, an impressive result lies in that for the first time, the Melnikov criterion of such fractional-order...