Effect of geometrically nonlinear damping on systems controlled by negative derivative feedback under harmonic excitation

This study investigates the effectiveness of a Negative Derivative Feedback (NDF) controller in suppressing vibrations in a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The system incorporates geometrically nonlinear damping, which introduces complex dynamic behavior. To analyze the system’s response, the averaging method is employed to derive approximate analytical solutions, while the fourth-order Runge–Kutta method is used for numerical integration. Particular attention is given to the case of simultaneous resonance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , a critical condition in nonlinear dynamics. Comparative analysis between the numerical simulations and the perturbation-based solutions confirms the high accuracy and reliability of the proposed control strategy. Furthermore, the impact of key system parameters on vibration attenuation is examined. The results highlight the NDF controller’s potential in enhancing the performance of nonlinear vibration mitigation devices, with direct applications to systems such as Centrifugal Pendulum Vibration Absorbers.