A nonlinear absorber for the reflection of travelling waves in bars

In this work, the use of a nonlinear absorber for the reflection of travelling waves in a slender bar is investigated. The absorber is a mass spring system attached to an infinite bar. The absorber stiffness has a linear and a cubic nonlinear part. The equation that governs the relative absorber motion is first obtained. After proper scaling is applied, this equation takes the form of a weakly forced single degree of freedom system with weak nonlinearity and weak damping even though damping is not present in the system. The method of averaging is applied, and the approximate steady state solution is first determined. The stability of the obtained solution is discussed. Since weak nonlinearity is assumed, the system linear response is altered in only three regimes which are considered. In the primary regime, the nonlinearity influences the transmitted wave and it is shown that the absorber should be under-tuned in the hardening case and over-tuned in the softening case. In the remaining two secondary regimes, while the nonlinearity alters the absorber motion, it has no effect on the transmitted wave. Hence, these regimes are not to be considered for vibration reduction. The verification of all analytical results is achieved by comparison to two numerical solutions. The first is the direct solution of a set of nonlinear ordinary differential equations, and the second is the numerical resolution of the finite element model of a bar with an absorber system. Both methods showed a very good agreement with the analytical solution.