Non-perturbative analysis of bifurcation and chaotic dynamics of a cantilever beam under triple time-delayed feedback control

In this work, we aimed to improve vibration suppression in a flexible CB structure through the application of time-delays nonlinear feedback control, surpassing conventional linear control under external excitation. To improve the effectiveness and stability of vibration lessening in flexible constructions exposed to external perturbations, three time-delayed nonlinear feedback controllers provided enhanced attenuation of the fundamental CB responses compared to conventional linear or instantaneous control. The controllers employed linear/nonlinear procedures of position, velocity, and acceleration. The core methodology was based on the non-perturbative approach, mainly derived from He's frequency formula, which converted a weakly nonlinear oscillator of an ordinary differential equation into an equivalent linear one. The resulting parametric ODE was validated using the Mathematica Software. The stability/instability analyses were explored under multiple scenarios. Moreover, two objective functions were developed to optimize control gains and delay parameters for effective vibration suppression and bifurcation control. Furthermore, the method was grounded in first principles, diminished analytical complexity, and maintained high numerical accuracy in treating nonlinear parametric systems. To explore the nonlinear dynamics of the CB, several tools were employed, including bifurcation diagrams, Poincaré sections, phase portraits, and the largest Lyapunov exponents.