Different dynamics of a periodic mass-in-mass nonlinear chain during a single mode excitation

Abstract

The multiple scale dynamics of a periodic chain composed of nonlinear mass-in-mass cells is studied. Based on a continuous approach of the one-dimensional chain, dispersion equation is obtained which provides the general form of solutions of the linearized system. Taking into account a single harmonic of the chain around a 1:1 resonance with a targeted mode, fast dynamics of the system leads to the detection of the slow invariant manifold and its stability borders. Slow dynamics permits to predict singularities and equilibrium points leading to frequency responses. The developments predict periodic and non-periodic responses and permit to tune parameters of the chain for the aim of localization of vibrating energy and design of periodic or non-periodic responses.

Design and Experiment of a Biomimetic Duckbill-like Vibration Chain for Physical Weed Control during the Rice Tillering Stage

“…For a suspended chain, its dynamic equation is generally solved using Bessel functions and Neumann functions to obtain its natural frequency of vibration, energy distribution, chain tension, composite vibration, standing wave characteristic points, etc., [ 33 , 34 , 35 ]. As shown in Figure 10 , for a chain with length L and density ρ , suspended along the x -axis, with endpoint A undergoing sinusoidal vibration in the y-direction, the vibration displacement of any point B on the chain at time t is u ( x , t ), and point C at a distance dx ( dx approaching 0) from point B is subject to tension FT ( x ), FT ( x + dx ), and external force dF .…”