Propagation characteristics of an elastic bar coupled with a discrete snap-through element

Abstract

Wave propagation characteristics of an elastic bar coupled at one end with a single degree of freedom, bi-stable, essentially nonlinear snap-through element are considered. The free end of the bar is subjected to sinusoidal excitations. A novel approach based on multiple time scales and harmonic balance method has been proposed to analytically investigate the reflected wave from the nonlinear interface and the dynamic response of the snap-through element. A unified approach to the non-dimensional representation of the governing equations of motion, boundary conditions and system parameters, which is consistent across all the externally applied excitation frequencies and excitation amplitudes, has been developed. Through Taylor series expansion of the non-autonomous forcing functions arising in the governing differential equations and natural boundary condition about an initial stable configuration of the system and the proposed asymptotic method, approximate closed-form analytical solutions have been derived for sufficiently small amplitudes of the excitation pulse. Numerical results obtained through a finite difference algorithm validate the asymptotic model for the same small amplitudes of the excitation pulse. A stability analysis has been subsequently performed for the discrete snap-through element by using the extended Floquet theory for sufficiently large amplitudes of the excitation pulse by approximating the displacement at the nonlinear interface as a sinusoidal function of time, and the Mathieu plot of the excitation frequency vs the excitation amplitude showing the stable and unstable regions for the motion of the snap-through element has been generated. The expressions derived here give the most comprehensive and consistent description of the wave propagation characteristics and the motion of the snap-through element, which can be directly used in finite difference analysis over a wide range of parameter values of the excitation pulse.