NUMERICAL-SOLUTIONS OF 2ND-ORDER IMPLICIT NONLINEAR ORDINARY DIFFERENTIAL-EQUATIONS

The existing literature usually assumes that second order ordinary dif ferential equations can be put in first order form, and this assumptio n is the starting point of most treatments of ordinary differential eq uations. This paper examines numerical schemes for solving second orde r implicit non-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them ar e based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differ ential equation is transformed into a set of algebraic ones that are s olved iteratively. The third method is based on a fourth order (Houbol t's scheme) and an eighth order backward finite difference method (FDM ). Each method is presented first, and then applied to specific exampl es. It is shown: (1) that the Picard method is not valid for solving i mplicit equations containing large non-linear inertial terms; (2) that the IHB method yields accurate periodic solutions, together with the frequency of oscillation, and the dynamical stability of the system ma y be assessed very easily; and (3) that both Houbolt's and the eighth order scheme can be used to compute time histories of initial value pr oblems, if the time step is properly chosen. Finally, it is also illus trated how the combination of IHB and FDM can be a powerful tool for t he analysis of non-linear vibration problems defined by implicit diffe rential equations (including also explicit ones), since bifurcation di agrams of stable and unstable periodic solutions can be computed easil y with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM. (C) 1996 Academic Press Limited