Asymptotic methods for vibrations of the pure non-integer order oscillator

In this paper oscillators with a restoring force which is the function of a non-integer power exponent of deflection are considered. The oscillatory motion is described by a differential equation with a rational-power term. The equation is first analyzed qualitatively. The new analytical methods are developed for solving the differential equation with a non-integer order term. The methods are based on the assumption that the vibration of the non-integer oscillator has to be close to that of integer order one. The new perturbation method based on variation of the order of the non-linearity is developed. The unperturbed system is the integer order non-linear oscillator. One of the methods uses the perturbation of the amplitude and phase, and the following two techniques introduce the straightforward expansion using the known values for the pure integer order oscillators. The first order approximate solutions are obtained. Their accuracy is checked on several examples. The results obtained are compared with the exact numerical solution, showing good agreement. The vibrations are widely discussed.