Time-delayed control of a nonlinear self-excited structure driven by simultaneous primary and 1:1 internal resonance: analytical and numerical investigation

Main objective of this research to eliminate the resonant vibrations and stabilize the unstable motion of a self-excited structure through the implementation of an innovative active control strategy. The control strategy coupling the self-excited structure with a second-order filter, which feedback gain $\lambda$ and control gain $\beta$, as well as a first-order filter, which feedback gain $\delta$ and control gain $\gamma$. The coupling of the second-order filter to establish an energy bridge between the structure and the filter to pump out the structure's vibration energy to the filter. In contrast, the primary purpose of coupling the first-order filter to stabilize the closed loop by adjusting the damping of the system using the control keys $\delta$ and $\gamma$. Accordingly, the mathematical model of the proposed control system formulated, incorporating the closed-loop time delay $\tau$. An analytical solution for the system model obtained, and a nonlinear algebraic system for the steady-state dynamics of the controlled structure extracted. The system's bifurcation characteristics analyzed in the form of stability charts and response curves. Additionally, the system's full response simulated numerically. Findings the high performance of the introduced controller in eliminating the structure's resonant vibrations and stabilizing non-resonant unstable motion. In addition, analytical and numerical investigations revealed that the frequency band within which the second-order filter can absorb the structure's resonant oscillation relies on the algebraic product of $\beta$ and $\lambda$. Furthermore, it was found that the equivalent damping of the system depends on the algebraic product of $\gamma$ and $\delta$, which can be employed to stabilize the negatively damped self-excited systems. Finally, it reported that although the loop delay can potentially degrade vibration control performance, the time-delay stability margin is nonlinearly proportional to the product of $\gamma$ and $\delta$. This finding that increasing the value of $\gamma \times \delta$ can compensate for the adverse effects of loop delay on both system stability and vibration suppression efficiency.