Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type

In this paper, an approximate analytic procedure is developed for solving the strong non-linear differential equations of the Duffing type with complex-valued function which describes the dynamical behavior of many real systems. The method is based on the elliptic-Krylov–Bogolubov procedure where the solutions are the Jacobi elliptic functions. As an example, the self-excited vibrations of the rotor with variable shaft rigidity are considered. The analytical results of this example are compared with numerical ones and excellent agreement is found between them.