On the Effect of Multiple Incident Waves on the Reflected Waves in a Semi-infinite Rod with a Nonlinear Boundary Stiffness

Abstract

AbstractThis paper concerns a potential application of vibration control using nonlinearity. In this paper, a nonlinear boundary problem for a thin rod is considered. An incident wave propagates along the rod at frequency \(\omega\) giving rise to an infinite number of reflected waves with frequencies \(n\omega\). This paper concerns the reflected waves produced by multiple waves at frequencies \(n\omega\) incident on the boundary of a semi-infinite rod with linear and cubic nonlinear stiffnesses. Equations are truncated including the third harmonic and solved using the harmonic balance method for the case with two reflected and two incident waves. The effects of different parameters on the magnitudes of the reflected waves are studied. It is shown that the phase difference between the incident waves affects the reflected waves’ behaviour. Numerical examples are presented to find the conditions at which the magnitude of the reflected wave of the 1st harmonic is minimum and the maximum energy leaks from the 1st harmonic to the 3rd harmonic. The results show that the presence of the second incident wave can decrease the magnitude of the reflected wave of the 1st harmonic. KeywordsReflection coefficientMultiple incident wavesNonlinearityNonlinear boundaryHarmonic balance method