Approximations for motion of the oscillators with a non-negative real-power restoring force

Oscillators with a non-negative real-power restoring force are considered in this paper. This type of restoring force is related to systems with a quasi-zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature. Examples of these types of restoring force are grounded in real physical and engineering systems. Periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and Hamilton's variational principle. Then, the approach is extended to non-conservative oscillators by adjusting the elliptic Krylov–Bogoliubov method. The methods proposed for the conservative and non-conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force. Several examples, the majority of which are so far unsolved, are given to illustrate the methods proposed and to demonstrate their generality, which permits unforeseen solutions for motion, containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force. The results obtained are compared with numerical results and have excellent accuracy.