Zvonko Rakaric

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.

A non-simultaneous variational approach for the oscillators with fractional-order power nonlinearities

Abstract This paper is concerned with a class of conservative oscillators the restitution force of which is of a power form which includes positive non-integer exponents. It is shown how an approximate Lagrangian and Hamilton’s variational principle can be used to obtain a second-order approximate solution for their free vibrations. Due to the fact that, in a general case, when the restoring force is multi-term, the period cannot be obtained from the energy conservation law in a closed form, the problem is formulated as a one-point boundary-value problem, and a non-simultaneous variation is introduced. The explicit expressions for the amplitudes and frequency of oscillations are derived, in which there are no restrictions on the values of the non-integer powers. The analytically obtained results are compared with numerical results as well as with some approximate analytical results from the literature.