Wave Propagation in a Strongly Nonlinear Mass-in-mass Chain with Soft Springs and Stability and Bifurcation Analyses of Periodic Solutions

A modified incremental harmonic balance (IHB) method is used to obtain a periodic solution for wave propagation in a strongly nonlinear mass-in-mass chain with soft springs. To analyze its stability and detect bifurcations of the solution, a method based on the Hill's method is employed. This work firstly reveals that the solution is in a hyperplane with two dimensions. Comprehensive analyses are conducted to explore relationships among the amplitude, the frequency, and system parameters. Notably, superharmonic resonances and fusion of optical and acoustic branches are identified, which is absent in the chain with hard springs. The results of the modified IHB method are compared with thosof the Lindstedt-Poincaré (LP) method. While the results from the two methods have the same trend at a low amplitude, the LP method fails to capture complex nonlinear characteristics at a high amplitude, such as bifurcations and turning points. Finally, attenuation zones for optical and acoustic branches are determined.