About the Free Parameter Vibrations of the Mechanical Systems

Abstract

The oscillation movement of a mechanical non-linear system is not easy to solve exactly in an analytical way. The approximate solutions are based on different methods and give different values with different approximation degree. In this paper it is shown that for such differential equations, such as m x - c × tk × x = 0 which describe free parametric vibrations of some elastic systems, there can be found analytical or approximate solutions. Using the substitution x = x × y , y=y(t), the differential equation becomes a Riccati special equation for which, using the Bessel functions, we obtain analytical or approximate solutions.