A New Spatial and Temporal Incremental Harmonic Balance Method for Obtaining Steady-State Responses of a One-Dimensional Continuous System

A new spatial and temporal incremental harmonic balanced (STIHB) method is developed for obtaining steady-state responses of a one-dimensional continuous system. In the STIHB method, Galerkin procedure for a governing partial differential equation (PDE) in the spatial coordinate to obtain a set of ordinary differential equations (ODEs) and the harmonic balance procedure for the set of ODEs in the temporal coordinate to obtain the harmonic balanced residual are combined to be Galerkin procedures for the PDE in the spatial and temporal coordinates to simultaneously obtain the spatial and temporal harmonic balanced residual, and integrations in Galerkin procedures are replaced by the fast discrete sine transform (DST) or fast discrete cosine transform (DCT) in the spatial coordinate and the fast Fouriour transform (FFT) in the temporal coordinate, which is referred to as a DST-FFT or DCT-FFT procedure. The harmonic balanced residual for an arbitrary second-order PDE can be automatically and efficiently obtained by a computer program when the expression of the PDE is given, where numbers of basis functions in the spatial and temporal coordinates can be arbitrarily selected and no more extra derivations are needed. There are two versions of the STIHB method. In the simple version, the DST-FFT or DCT-FFT procedure to calculate the harmonic balanced residual and Broyden’s method that is a quasi-Newton method are combined to find solutions that make the residual vanish, which can be used to construct steady-state solutions of the PDE. In the complex version, the exact Jacobian matrix is derived and used in Newton-Raphson method to achieve faster convergence. While its derivation is complex, the exact Jacobian matrix for the arbitrary PDE can be automatically and efficiently obtained by following a calculation routine when the linearized expression of the PDE is given, and it can be easily implemented by a computer program. The exact Jacobian matrix can also be used to study stability of steady-state responses, where no more extra derivations are needed. The STIHB method is demonstrated by studying the transverse vibration of a string with geometric nonlinearity; its frequency-response curves with weak and strong nonlinearities and different numbers of trial functions are calculated, and stability of solutions on the curves is studied.