Generalized Krylov-Bogoliubov Method for Solving Strong Nonlinear Vibration

In this paper an analytic procedure for solving strong nonlinear differential equation of the vibration is developed. The Krylov–Bogoliubov method, introduced for weak nonlinear equations, is generalized for the strong nonlinear ones with power order nonlinearity. Based on the exact solution of the truly nonlinear differential equation (without linear term), given in the form of the Ateb function, the approximate solution of the perturbed version of vibration equation is obtained. In addition, the method is applied for strong nonlinear oscillator with time variable parameter. As the special case, motion of the strong nonlinear oscillator with slow variable mass is investigated. In the paper the steady state motion of the forced strong nonlinear one-degree-of-freedom oscillator is also analysed. Using the Melnikov criteria the condition for existence of deterministic chaos in excited strong nonlinear oscillator with linear damping is determined. For suppressing of the chaos the delayed feedback control, i.e., the ‘Pyragas method’ based on the idea of the stabilization of unstable periodic orbits embedded within a strange attractor is suggested.