Experimental Demonstration of Amplitude-Dependent Band Structure and Plane Wave Stability in Nonlinear Lattices

This study experimentally investigates wave propagation in nonlinear lattices, extracting amplitude-dependent band structures from direct measurements of travelling waves and observing amplitude-dependent plane wave stability. Such findings provide experimental validation of analytical predictions previously derived from a multiple scales perturbation technique. Both one-dimensional (1-D) and two-dimensional (2-D) monatomic shear lattices are explored. Heavy magnetic inclusions periodically spaced in a matrix of nylon wire form the discrete shear lattices, which are arranged in both 1-D (single column) and 2-D (multiple column) configurations. Such systems are subject to geometric stiffness nonlinearity primarily due to the non-constant tension that develops in the wires at sufficiently high amplitudes. In the experiments performed, an elastodynamic shaker excites plane waves in the system, while laser Doppler vibrometery is employed to measure the subsequent waveform at multiple points in the lattice. The experimental findings in this work demonstrate the viability of fabricating wave-based devices that exploit the amplitude-dependent filtering and stability characteristics of nonlinear plane waves. 

For band structure measurements, the shaker (B&K 4809) gradually introduces harmonic loading to the chain’s boundary. As the wave propagates, two single point laser Doppler vibrometers (Polytec PDV 100) record the velocity time histories at neighboring unit cells after the shaking mass. Time windowing the data to study a region free of reflections from the boundary provides two time-delayed velocity signals characteristic to travelling waves. Identifying the phase difference of the plane wave’s fundamental frequency component provides direct measurements of wavenumber that, when paired with the shaker’s sending frequency, yield the lattice’s band structure. A lifting of the passband occurs when re-measuring the band structure for successively higher plane wave amplitudes, confirming the nonlinear hardening behavior predicted by multiple scales. Plane wave stability is experimentally observed by using a high-stroke shaker (LDS V408) to launch wave packets into the lattices and collect velocity measurements at multiple locations via a scanning laser Doppler vibrometer (Polytec PSV 400). In the 1-D chain, time-windowed measurements of the wave packet arrivals at the unit cells confirm the amplitude-dependent stability of plane waves. Low amplitude wave packets retain their spectral content almost exclusively at the fundamental frequency and higher harmonics (integer multiples of the fundamental). By contrast, the instabilities that develop for high amplitude wave packets are characterized by nontrivial generation of broadband incommensurate frequency content. Stability exhibits its expected direction dependence in the 2-D lattice for symmetric (i.e. uniform) stiffness distributions. Wave propagation along the lattice direction is inherently less stable than inclined directions consistent with the multiple scales analysis.