Evolution of the Nonlinear Normal Truncation Modes with energy in Diatomic Chains with Strong Cubic Nonlinearity

Abstract Truncation resonances exist within the bandgap of finite periodic systems and are often localized at boundaries, enabling prominent applications in wave guiding, vibration attenuation, and flow control. Several papers have investigated the energy transmit 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 truncation resonance mode shape, which quantifies the energy localization, when nonlinearity is introduced. This paper 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 combined with a numerical continuation method to characterize the dependence of the truncation resonance mode shape on energy, 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 system. The paper 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. Results also reveal how the delocalization energy coincides with the transition energy defined elsewhere in the context of nonlinear effects on topologically protected interface modes. This study provides a methodology to analyze nonlinear effects on truncation resonances and emphasizes importance of understanding the mode shape evolution to quantify system characteristics.