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

Breaking the mass law for broadband sound insulation through strongly nonlinear interactions

“…Recently, nonlinear acoustic metamaterials (NAMs) [25][26][27], with embedded nonlinear local resonators, have attracted increasing attention owing to their outstanding features that are absent in their linear counterparts. NAMs' bandgaps are amplitude-dependent [25,[28][29][30][31][32][33], which is essentially dominated by the band degeneration process [34]. NAM [35,36] can induce nonreciprocal wave transmission [37], i.e.…”

Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

“…This approach leads to closed-form expressions of the bifurcations and sensitivity analyze. In [41] a multiple-scale approach was presented to capture internally resonant wave interactions in weakly nonlinear lattices and metamaterials, while in [42] this method was employed to study analytically the dispersion properties characterizing the free propagation of harmonic waves in pantographic metamaterials. The standing and traveling waves in undamped periodic systems possessing cubic nonlinearities were studied asymptotically both with or without 3:1 internal resonances in [43,44] and, more recently, in [45] where it was shown that the method of multiple scales provides more general results than Lindstedt-Poincaré in the case of wave-wave interactions.…”

“…Lie series operators and Hamiltonian perturbation theory: a short overviewPerturbation methods are flexible and efficient mathematical tools to determine analytical-asymptotically approximate-solutions for nonlinear dynamic problems. Within the context of wave propagation in periodic materials and structures, different perturbation techniques have been employed to study the nonlinear dispersion properties of harmonic waves, including the modified harmonic balance (with ordering of the harmonic amplitudes)[53], Lindstedt-Poincaré[55,56] and the method of multiple scales[22,42,43,45,46].…”

Wave propagation in viscoelastic metamaterials via added-state formulation

“…According to the former approach, the eigenproblem governing the free wave propagation can be stated, solved and -in principle -inverted to analytically assess the microstructural parameters satisfying desired spectral requirements, like the existence and position of a given harmonic spectral component or the amplitude and centerfrequency of certain pass or stop spectral bands [24,25]. Naturally, the analytical solution of the direct and inverse eigenproblem tends to be infeasible in presence of high model dimensions, large parameter spaces or important nonlinearities, leaving space to consistent mathematical approximations, like asymptotic perturbation-based solutions [26,27]. According to the latter approach, the optimal solution in the multidimensional space of the design parameters is numerically identified by minimizing or maximizing a suited objective or multi-objective function [28][29][30][31].…”