Effects of time delay on the dynamical behavior of nonlinear beam on elastic foundation under periodic loadings: Chaotic detection and it control

In this paper, we analyze the effects of the time-delay feedback position on the dynamical behavior of the nonlinear beam on elastic foundation under periodic external force and the chaos control. Firstly, by using the formulation of Lagrange as well as geometric analysis, the nonlinear equation of the system with time-delay is established. In this case, the system rests on a Winkler-type elastic foundation soil that acts on its bottom interface (soil-beam). Secondly, the equilibrium points are found, their stability studied and the condition of Hopf bifurcation established. Thirdly, by using the Melnikov process, the analytical constraints necessary to have chaos or not as a behavior of the system are sought, which is confirmed by numerical investigations, with plotting of the time series, Lyapunov exponent and the bifurcation diagram, which are used to study the effects of time-delay on dynamical behavior of the system. This time-delay being used as a control parameter. The results show that when the value of time delay is small, the ergodic tori and resonance cycles with different rotation numbers on the torus can be observed. When the value of time-delay increases the chaos behaviors take place in the system. Taking into account the fact that chaotic situations are considered disturbance and are harmful for operation, the control method based on the time-delay parameter is used to quench and also to prevent chaotic behaviors.