Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

Abstract

The reflection of time-harmonic waves in a waveguide with a nonlinear boundary stiffness is considered with applications to rods and beams. Incident waves at frequencies that are multiples of a fundamental frequency give rise to reflected propagating and nearfield waves at the same frequencies. An infinite set of equations is developed for the reflection coefficients, which depend on the amplitudes and phases of the incident waves. Nonlinear boundary conditions are applied, and equations is truncated by using the harmonic balance method and solved numerically. The case of zero linear boundary stiffness, i.e. essential nonlinearity, is studied. First, the case where there is only one incident wave is considered. An approximate solution is found when retaining only two reflected waves. Numerical examples are presented, energy being seen to leak into the higher harmonics. The minimum magnitudes of the reflection coefficients of axial and flexural vibrational waves at the fundamental frequency and the maximum energy that can leak into the higher harmonics are determined. Accuracy and convergence when retaining different numbers of reflected harmonics are illustrated. The case of two incident waves is then considered. Multiple incident waves affect the leakage of energy to higher harmonics and can have a significant effect on the reflection coefficient for the fundamental harmonic. With some parameters, a much lower reflection coefficient is obtained for the wave at the fundamental frequency as compared to the case of one incident wave. It is seen that with two incident flexural waves, the reflection coefficients can be multi-valued for certain values of the system parameters. A numerical study is performed to show the region of multiple solutions.