Analysis of non-linear autonomous conservative oscillators by a time transformation method

The problem considered is that of the free vibration of the class of strongly non-linear oscillators ü + mu + αf(u) = 0, where m = 1, 0 or -1, f(u) is a non-linear function of the displacement u( t), and α is not small. Through a continuous stretching of the time t a new time T is found in whose domain the response is simple harmonic. This time transformation method allows the response u( t) and period τ to be calculated to any desired degree of accuracy; however, the primary objective is to develop reasonably simple, accurate approximations for the oscillators considered.