Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms

Abstract

This paper is focused on solving the generalized second-order strongly nonlinear differential equation \({\ddot{x}+\sum_i {c_i^2 }x \left| x \right|^{i-1}=0}\) which describes the motion of a conservative oscillator with restoring force of series type with integer and noninteger displacement functions. The approximate analytical solution procedures are modified versions of the simple solution approach, the energy balance method, and the frequency–amplitude formulation including the Petrov–Galerkin approach. For the case where the linear term is dominant in comparison with the other series terms of the restoring force, the perturbation method based on the solution of the linear differential equation is applied. If the dominant term is nonlinear and the additional terms in the restoring force are small, the perturbation method based on the approximate solution of the pure nonlinear differential equation is introduced. Using the aforementioned methods, the frequency–amplitude relations in the first approximation are obtained. The suggested solution methods are compared and their advantages and disadvantages discussed. A numerical example is considered, where the restoring force of the oscillator contains a linear and also a noninteger order term (i = 5/3). The analytically obtained results are compared with numerical results as well as with some approximate analytical results for special cases from the literature.