Wave Propagation in a Strongly Nonlinear Mass-in-mass Chain with Hard Springs and Stability and Bifurcation Analyses

A modified incremental harmonic balance (IHB) method is employed to obtain periodic solutions for wave propagation in a strongly nonlinear mass-in-mass chain with hard springs. To analyze the stability and bifurcation of the solutions, a method based on Hill's method is utilized. This study first reveals possible types of solutions and verifies that the solutions are in a hyperplane with two dimensions. Furthermore, relationships among the amplitude, the frequency, and system parameters are thoroughly analyzed and numerous bifurcations are identified. Two frequency formulae for wave propagation in a weakly and strongly nonlinear mass-in-mass chain with hard springs are derived using a fitting method and the modified IHB method. Notably, the formulae for weak nonlinearity are consistent with the formula obtained via the Lindstedt–Poincaré (LP) method. Finally, attenuation zones for optical and acoustic branches are identified.