Wave dispersion in a damped beam supported by cubic nonlinear springs: A multiscale freewave approach

Researchers are drawn to exploring wave dispersion in nonlinear systems because of the amplitude-dependent tunability of the band gap. This paper investigates the amplitude-dependent wave dispersion in continuous beam structures supported periodically by nonlinear springs. Additionally, it examines the influence of inherent beam damping on wave dispersion. The analytical framework consists of homogenization of the unit cell and the method of multiple scales with two distinct time scales to derive the wave dispersion equation. The proposed analytical approach for nonlinear wave propagation is validated through numerical finite element simulations. It is observed that the frequency shift is positive for hardening and negative for softening supports. Following this, the dispersion shift over time in the damped systems is examined by considering viscous and strain rate--dependent damping. The sensitivity of strain rate damping to propagation constant and the independence of viscous damping from propagation constant are thoroughly investigated. In a damped system, the frequency shift diminishes over time as the amplitude decreases reducing the effect of nonlinearity. This study opens up avenues for controlling or filtering vibrations through the tunable band gap of continuous nonlinear metamaterials.