Fig. 6: Experimental validation of the optimal nonlinear and linear chains, compared with simulation.

The optimal nonlinear chain corresponds to that identified in Fig. 4a. a–d Spatiotemporal evolution of kinetic energy in the system (normalized by the input kinetic energy, or initial total energy). The nonlinear chain is shown in a (experiment, nonlinear trial 1) and b (simulation), while the linear case is shown in c (experiment, linear trial 1) and d (simulation). The experimental spatiotemporal plots include a smoothing of displacement values to assist with noise induced by differentiating the discrete time camera data. The spatiotemporal plots for other trials are included in Supplementary Information Note 9. Time histories of the experimentally measured (solid lines) and simulated (dashed lines) velocities for the fifth and last unit cell in the nonlinear (e) and linear (f) chains. g A ratio of maximum kinetic energy (linear/nonlinear) seen at each unit cell for both experiment and simulation. A value greater than 1 (denoted by the horizontal dashed black line) indicates superior performance of the nonlinear chain as compared to the linear. The experiment values are the average taken from three experimental trials. The X marks the average experimental value recorded by the vibrometer (see Supplementary Information Note 9 for the full data sets), which collected data from only the last unit cell. h The simulated sensitivity of the KE ratio (truncated at 1) to impact conditions, wherein the simulation was run with the coefficients that were found experimentally from the nonlinear spring in Fig. 5D (i.e., the physically achieved coefficients). Source data are provided as a Source Data file.