Three control techniques for suppressing nonlinear vibrations in a 2-DOF dynamical system subjected to combined external and multi-parametric harmonic excitations with 1:1 internal resonance

This article addresses the vibration control of a nonlinear two-pole rotor generator system through the application of three distinct control strategies. The equations of motion governing the system are formulated as a nonlinear two-degree-of-freedom model, subjected to a combination of external, linear parametric, and nonlinear parametric excitations. A closed-loop control system is then developed, incorporating the rotor model alongside the proposed control approaches, resulting in six coupled nonlinear differential equations. The proposed control strategies explored are Positive Position Feedback Control ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> ), Integral Resonant Control ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> ), and the hybrid approach <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> + <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . Utilizing perturbation theory, an accurate analytical solution is derived for the closed-loop system. The bifurcation characteristics of the rotor model are examined, and the vibration reduction effectiveness of the proposed control methods is evaluated. The findings from the analytical investigations reveal that both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> and the hybrid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> not only fail to effectively mitigate the vibrations of the rotor under the combined excitations but may also destabilize the system at specific angular velocities. This instability manifests quasi-periodic or unbounded oscillations, which could have potentially destructive consequences. In contrast, the tuned <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> combination successfully suppresses the rotor vibrations, reducing them to near zero. However, a loss of tuning conditions in this approach can critically compromise the stability of the closed-loop system. While the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> strategy is less effective at vibration suppression compared to the tuned <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> <mml:mi>F</mml:mi> <mml:mi>C</mml:mi> <mml:mo>+</mml:mo> <mml:mi>I</mml:mi> <mml:mi>R</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> , it is more robust against instability, simpler, and more reliable, making it the optimal control strategy for such systems. Finally, numerical simulations of all derived analytical results demonstrated the accuracy of the analytical solution despite the complexity of the introduced mathematical model.