Stationary points created by resonances in a chain of bilinear oscillators

This study investigates resonances in the symmetrical systems of coupled bilinear oscillators (identical masses connected by identical bilinear springs). The bilinear oscillators are often used to model the dynamics of various discontinuous systems exhibiting different stiffnesses in compression and tension. These natural and engineering systems are normally characterised by multiple degrees of freedom and their behaviour is far from being completely understood. In this study, we present a method for determination of the resonances in a one-dimensional chain of bilinear oscillators. We demonstrate that the resonant frequencies of the system coincide with the resonant frequencies of two basic oscillators: a one-mass oscillator and a two-mass oscillator. In resonance, the system oscillates as independent basic oscillators, effectively decomposing itself into either one-mass oscillators or two-mass oscillators. These basic oscillators are separated by masses, which remain stationary (stationary points). The presence of these stationary points is a unique feature of the symmetrical systems of bilinear oscillators. We also show that the super-harmonic resonances (resonances at frequencies multiple of the main frequency) and sub-harmonic resonances well known in single bilinear oscillators can occur in the chains of the bilinear oscillators.