Harmonic Balance Method for the Stationary Response of Finite and Semi-infinite Nonlinear Dissipative Continua: Three Canonical Problems

The Harmonic Balance Method (HBM) is often used to determine the stationary response of nonlinear discrete systems to harmonic loading. The HBM has also been applied to nonlinear continuous systems, but in many cases the nonlinearity consists of discrete nonlinear elements. This chapter demonstrates the application of the HBM to dissipative continua with distributed nonlinearity by analysing three canonical problems: (a) 1-D layer with a free surface and rigid base (interfering upward and downward propagating shear waves), (b) 1-D half-space with a rigid base (vertically propagating shear waves), and (c) 2-D axially symmetric semiinfinite medium with a circular cavity (radially propagating compressional waves), all of them subject to harmonic excitation at a boundary. Results show that systems (a) and (c) exhibit softening behaviour and super-harmonic resonances, while only the former displays multiple response amplitudes for certain excitation frequencies; the unique frequency-amplitude relationship of system (c) is due to the strong damping (i.e., radiation damping and internal dissipation). Furthermore, although system (b) essentially does not resonate, the third-harmonic component exhibits a maximum caused by the interplay between the dissipative and nonlinear effects, a phenomenon that also occurs in system (c). Finally, the considered systems have applications in earthquake and geotechnical engineering, among others, but the presented methodology is generic.