On the effects of a nonlinear boundary with cubic stiffness on the reflection coefficients of time harmonic flexural waves in an Euler-Bernoulli beam

Abstract

The reflection of time-harmonic flexural waves in a thin Euler-Bernoulli beam incident on a nonlinear boundary stiffness is studied. A set of waves at frequencies an integer multiple of a fundamental frequency is incident on the boundary, giving rise to an infinite set of reflected propagating and nearfield waves whose amplitudes are defined in terms of reflection coefficients. Equations for the wave amplitudes are truncated and solved numerically using the harmonic balance method. The boundary nonlinearity is assumed to be cubic with no quadratic term. Emphasis is placed on essential nonlinearity for which the linear stiffness is zero. This case is of interest due to stronger nonlinear effects being produced. The case with one incident propagating wave and two reflected propagating and nearfield waves is considered. It is seen that vibrational energy leaks from the 1st harmonic to the 3rd harmonic. A boundary configuration comprising linear springs is suggested, and the stiffness is calculated for this case and when the maximum leakage of energy occurs from the 1st harmonic to the 3rd harmonic, i.e., for the minimum reflection coefficient of the 1st harmonic. It is seen that the reflection coefficient of the incident wave at the fundamental frequency can be significantly less than 1.