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

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.