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

Abstract

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 semi-infinite 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.

Appendix: Numerical Solution for 1-D Problems

As reference for the HBM solution, the governing equations for the 1-D systems (Eq. (16.6) with Eqs. (16.7) and (16.8), or with Eqs. (16.8) and (16.11)) are solved numerically. Here, we pay special attention to the incorporation of the non-reflective boundary condition (Eq. (16.11)) in the solution scheme (for the half-space problem). First of all, the range of integration is limited to $\tau =[0,t]$ accounting for the Heaviside function in C(t) and the fact that the excitation starts only at $t=0$ in the numerical solution. Furthermore, we apply a finite-difference discretization in time to the response quantities at the boundary:

$$\begin{aligned} u_{,z}(z_0,t_j) = \sum _{i=1}^{i=j-1} \frac{u(z_0,t_{i+1})-u(z_0,t_i)}{\Delta t} \int \limits _{t_i}^{t_{i+1}} C(t_j - \tau ) \mathrm{d}\tau , \end{aligned}$$

(16.34)

where $\Delta t$ is the time step, $t_1 = 0$, and the integral of the weakly singular memory function is known analytically:

$$\begin{aligned} \int _{t_i}^{t_{i+1}} C(t_j - \tau ) \mathrm {d}\tau = \sqrt{\frac{\rho }{G_0}} \left[ \text {erf}{\left( \sqrt{\tfrac{G_0}{\eta }(t_j-t_i)}\right) }-\text {erf}{\left( \sqrt{\tfrac{G_0}{\eta }(t_j-t_{i+1})}\right) } \right] , \end{aligned}$$

(16.35)

where $t_{i+1} \le t_j$. Now, the non-reflective boundary condition is no longer an integral equation, but an ordinary differential equation in z, in which the response at $t_j$ is unknown and can be solved for together with the other boundary condition and the equation of motion.

For both 1-D problems, the system of equations is solved using a standard finite-difference discretization of the spatial domain, while the time integration is done using a fourth-order Runge-Kutta scheme. To save computational time, we employ the stationary responses computed using the HBM as initial conditions, which makes that the simulated behaviours reach the stationary states immediately (provided the HBM solution is converged).