• A wave approach is introduced to study forced vibration of a rod with nonlinear boundary stiffness. • Energy leakage occurs between the fundamental and higher harmonic reflected waves. • An iterative wave-based method is proposed for efficient steady-state forced response analysis. • The system shows nonlinear hardening behavior with jumps and multiple steady-state solutions. • Parametric analysis identifies regions of multiple solutions and nonlinear response features. A wave approach is used to study the forced vibrations of a rod with a nonlinear boundary stiffness under time-harmonic excitation. Waves with frequencies multiples of the fundamental frequency are incident on the boundary. This will produce a set of reflected waves whose amplitudes depend on the linear and nonlinear stiffnesses of the system together with amplitudes of the incident waves. It is seen that leakage of energy exists between the fundamental incident wave and higher harmonic reflected waves. Under the assumption of cubic nonlinearity, the maximum energy leakage is shown. Then, a wave approach is introduced to find a numerical iterative solution for the forced response of the rod. The suggested method can directly calculate the steady-state forced response of the system from the reflection of axial waves, reducing computational cost compared to methods based on coupling elements. The hardening behaviour is shown accompanied by jumps and multiple solutions, as is common in nonlinear systems. The effects of different parameters on the response of the rod are studied. It is indicated that as the force magnitude or nonlinear stiffness increases, the system becomes almost linear with one solution. A numerical study is performed to determine the forced response, together with the region in which there are multiple solutions. Various approximate solutions are developed to give insight into the underlying behaviour. Numerical examples indicate that as the harmonic force magnitude or nonlinearity increases, the frequencies of the peak response of the system asymptote to a constant value which depends only on the rod properties.
Abdi et al. (Fri,) studied this question.