Numerical Solutions of Second Order Implicit Non-Linear Ordinary Differential Equations

Author： Semler C. Gentleman W.C. Paidoussis M.P.

ISSN： 0022-460X

Source： Journal of Sound and Vibration, Vol.195, Iss.4, 1996-08, pp. : 553-574

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Abstract

The existing literature usually assumes that second order ordinary differential equations can be put in first order form, and this assumption is the starting point of most treatments of ordinary differential equations. This paper examines numerical schemes for solving second order implicit non-linear differential equations. Based on the literature review, three specific methods have been selected here. Two of them are based on series expansions: Picard iteration using Chebyshev series and Incremental Harmonic Balance (IHB), in which the non-linear differential equation is transformed into a set of algebraic ones that are solved iteratively. The third method is based on a fourth order (Houbolt's scheme) and an eighth order backward finite difference method (FDM). Each method is presented first, and then applied to specific examples. It is shown: (1) that the Picard method is not valid for solving implicit 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 may 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 problems, if the time step is properly chosen. Finally, it is also illustrated how the combination of IHB and FDM can be a powerful tool for the analysis of non-linear vibration problems defined by implicit differential equations (including also explicit ones), since bifurcation diagrams of stable and unstable periodic solutions can be computed easily with IHB, while periodic and non-periodic stable oscillations may be obtained with FDM.