Abstract
This paper studies the nonlinear vibrating behaviour of a nonlinear cantilever beam system (primary system) using a nonlinear absorber (the secondary system). The nonlinear vibrating behavior for the present dynamical system is considered with the effect of the external force. The one controller type, nonlinear saturation controller (NSC), is introduced to decrease the vibration of this system. Perturbation method treatment is produced to get the mathematical solution of the equations for the dynamical modeling with NSC. The perturbation technique is used to obtain the approximate solution of the dynamical system. This research focuses on resonance case with primary and 1:2 internal resonance. Time histories of the primary system and the controller are shown to demonstrate the reaction with and without control. The time-history response, as well as the impacts of the parameters on the system and controller, are simulated numerically using the MATLAB program. Routh-Hurwitz criterion is used to examine the stability of the system under primary resonance. A numerical simulation, using the MATLAB program, is obtained to show the time history response, the effect of the parameters on the system and the controller. The effects of system parameters on the performance of the primary system and the controller are investigated. A comparison between all the obtained solutions made to confirm the results. Validation curves are provided to show how closely the perturbation and numerical solutions are related. A comparison is made with recently released papers.