Evolution of the Nonlinear Normal Truncation Modes With Energy in Diatomic Chains With Strong Cubic Nonlinearity

Abstract Truncation resonances reside within the bandgap of periodic systems as a result of a finite-length system being truncated from its infinite chain. Because they exist in the band gap, truncation resonance modes are often localized on boundaries, and as such, they have prominent practical applications such as wave guiding, vibration attenuation, and flow control. Several papers have investigated the energy transmission rate, energy localization, and existence conditions of truncation resonances, and recent work has shown how nonlinearity affects topologically protected modes, which are a specific type of truncation resonance. However, current work does not explicitly study the characteristics of the truncation resonance mode shape, which quantifies the energy localization, when nonlinearity is introduced. This article bridges this gap by investigating the evolution of the mode shape of truncation resonances in a grounded diatomic chain with hardening and softening nonlinear springs. We use nonlinear normal mode analysis to characterize the energy dependence of the truncation resonance mode shape, and then present a mathematical functional fit of the mode shape evolution as a function of energy to quantitatively identify major energy-dependent characteristics of the nonlinear system. This article shows how the delocalization energy determined through mode shape analysis depends on system parameters, and specifically, it correlates linearly with the bandwidth between the truncation resonance and nearest propagating band edge in hardening nonlinear systems. This study provides a methodology to analyze nonlinear effects on truncation resonances and emphasizes the importance of understanding the mode shape evolution to quantify system characteristics.