Abstract
•Used active control to eliminate the vibration of the buckled beam system.•The influences of the quadratic and cubic terms on non-linear dynamic of the buckled beam are studied.•Second-order approximation of the system by the method of multiple time scale perturbation is obtained.•Stability and bifurcation analysis of the system is investigated and period doublings (PD) were obtained.•Different region of the bifurcation parameters were discovered giving stable and unstable steady state, stable periodic.
In this paper, we focus on applying active control to nonlinear dynamical beam system to eliminate its vibration. We analyzed stability using frequency-response equations and bifurcation. The analytical solution of the nonlinear differential equations describing the above system is investigated using multiple time scale method (MTSM). All resonance cases were extracted from second order approximations. Numerical solutions of the system are included. The effects of most system parameters were investigated. The results demonstrated that proposed controller is efficient to suppress the vibrations. Increasing the quadratic stiffness coefficient term vanished the multi-valued solution. Bifurcation diagrams refiled the effects of various system parameters on its stability showing different bifurcation cases. Finally, we conclude that for low values of natural frequencies dynamical system, the controller is more effective. The results show that the analytical solutions of the system are in good agreement with the numerical solutions.