Internal resonance of non-linear autonomous vibrating systems with two degrees of freedom

Free oscillations of a non-linear system having two weakly coupled degrees of freedom in which an internal resonance may occur are treated with a modified version of the K-B-M averaging method. The method presented permits one to obtain a first approximation of the solution very easily in comparison with the perturbation techniques and with the method of multiple scales that have been used to date to solve the same problem. As an example the free oscillations of a system having quadratic non-linearities are treated. This system represents a large number of physical phenomena as mentioned in the paper and the references[1, 2].