A new method for approximate analytical solutions to nonlinear oscillations of nonnatural systems

This paper deals with nonlinear oscillations of a conservative, nonnatural, single-degree-of-freedom system with odd nonlinearity. By combining the linearization of the governing equation with the method of harmonic balance, we establish approximate analytical solutions for the nonlinear oscillations of the system. Unlike the classical harmonic balance method, the linearization is performed prior to proceeding with harmonic balancing thus resulting in linear algebraic equations instead of nonlinear algebraic equations. Hence, we are able to establish the approximate analytical formulas for the exact period and periodic solution. These approximate solutions are valid for small as well as large amplitudes of oscillation. Two examples are presented to illustrate that the proposed formulas can give excellent approximate results.