Using Pseudotime Solution Framework in Harmonic Balance Methods for Aperiodic Problems

Abstract

Nomenclature D = time-domain spectral time-derivative matrix Ej = error in jth set of equations resulting from approximation made in envelope harmonic balance method F = vector of f sampled over one period of ω; ff x t0 ; f x t1 ; : : : ; f x t2N g F = amplitude of forcing in Van der Pol equation f = vector of nonlinear functions in system of ordinary differential equations F0,Fk1, Fk2 = Fourier-series coefficients for f G = system of harmonic balance equations in frequency domain g = vector of Fourier coefficients for f ; fF0;F1; : : : ;F2Ng ~ G = time-domain pseudospectral equations k2, k1 = indices used to enumerate Fourier coefficients m = dimension of system of ordinary differential equations N = number of terms in Fourier series t = time ti = sample points in time for time-domain pseudospectral method; i2π∕ω 2N 1 x = system states X0, Xk1, Xk2 = Fourier-series coefficients for x X = vector of Fourier coefficients for x; fX0;X1; : : : ;X2Ng Y = vector of systems states sampled over one period of ω; fx t0 ; x t1 ; : : : ; x t2Ng α = parameter in Van der Pol equation Δt = time step used in numerical integration σ = ω1 − ω0 τ = pseudocontinuation parameter (pseudotime) φ = Fourier basis functions; f1; sin ωt ; cos ωt ; : : : ; cos Nωt g φ, ψ = phase angles Ω = Fourier spectral differentiation matrix ω = Fundamental frequency ω0 = frequency of free oscillation in Van der Pol equation ω1 = forcing frequency in Van der Pol equation