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