Nonlinear dispersion properties of one-dimensional mechanical metamaterials with inertia amplification

Architected metamaterials offering superior dynamic performances can be conceived by inducing local mechanisms of inertia amplification in the periodic microstructure. A one-dimensional cellular lattice characterized a pantograph mechanism tetra-atomic cell is proposed as minimal physical realization inertially amplified metamaterial. discrete model formulated to describe undamped free dynamics The ordinary differential equations motion feature quadratic and cubic inertial nonlinearities, induced indeformability arms connecting principal atoms with secondary atoms, serving amplifiers. An asymptotic approach employed analytically determine dispersion properties governing propagation harmonic waves pantographic First, linear wavefrequencies waveforms are obtained solving eigenproblem lowest order. invariant parametric form achieved for pass stop band structure, corresponding attenuation branches spectrum plane complex wavenumbers. major effects due mass ratio amplifiers discussed. Particularly, existence conditions, amplitude centerfrequency gap separating acoustic optical bands determined analytically. Second, nonlinear hierarchical problems higher orders. Analytical, although asymptotically approximate, functions waveforms, which show dependence on oscillation amplitudes. mechanical conditions softening/hardening behaviour different topological manifolds associated Finally, numerical simulations provided validate analytical results.