Application of ultraspherical polynomials to non-linear, non-conservative systems subjected to step function excitation

Abstract

Ultraspherical polynomials are employed for the purpose of deriving approximate solutions of non-linear ordinary differential equations that describe the motion of non-conservative systems subjected to step function excitation. The method of analysis explicitly accounts for the effect of damping that is linearly proportional to the velocity, in such a way that the approximate solutions of the non-linear equations reduce in the limit to the exact solutions of the corresponding linear problems as the small parameter of non-linearity tends to zero. Numerical calculations based upon approximate formulas for the amplitude and phase as well as upon the fourth-order Runge-Kutta method of numerical integration were performed, and the results are depicted graphically.