An approximate analysis of non-linear, non-conservative systems using orthogonal polynomials

Abstract

Approximate solutions of second-order non-linear ordinary differential equations containing a linear velocity dependent damping term and a small parameter of non-linearity are obtained with the aid of sets of orthogonal polynomials. The method of solution treats the effect of linear damping in such a way that the approximate solutions of the non-linear equations reduce to the exact solution of the corresponding linear problems as the small parameter of non-linearity tends to zero. Ultraspherical polynomials, which depend upon a parameter λ, the value of which is selected at the time of performing numerical calculations, are found to be efficacious for solving problems of engineering significance. Numerical calculations, based upon the approximate method for several values of λ and upon the fourth-order Runge-Kutta method of numerical intergration, for three non-linear equations, were performed and results were compared. It was found that, in these cases, the calculations effected with λ=3/2 most nearly agreed with the Runge-Kutta calculations.