On $$\frac{1}{2}$$-DOF active dampers to suppress multistability vibration of a $$2$$-DOF rotor model subjected to simultaneous multiparametric and external harmonic excitations
Nasser A. Saeed, Jan Awrejcewicz, Randa A. Elashmawey, W. A. El-Ganaini, Lei Hou, Mohamed Sharaf
Abstract
Abstract This article addresses the bifurcation characteristics and vibration reduction of a $$2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> -DOF dynamical system simulating the nonlinear oscillation of an asymmetric rotor model subjected to simultaneous multiparametric and external excitations. To suppress the system's vibrations, two $$1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> -DOF active dampers are attached to the system in linear and cubic nonlinear forms via a magnetic coupling actuator. The closed-loop system model is derived as two differential equations with multi-control terms, including cubic, quantic, and septic, coupled nonlinearly to two first-order systems. Applying perturbation theory, the system model is solved, and the autonomous system describing the closed-loop slow-flow dynamics is obtained. Through numerical algorithms, the motion bifurcation is analyzed using various tools such as 2D and 3D bifurcation diagrams, two-parameter stability charts, basins of attraction, orbit plots, and time response profiles. The analytical investigations confirm that the uncontrolled model behaves like a hardening Duffing oscillator with multistability characteristics, displaying simultaneous mono-stable, bi-stable, tri-stable, or quadri-stable periodic oscillations depending on both the asymmetric nonlinearities and angular velocity. Subsequently, the influence of different control parameters is analyzed to determine the threshold between mono and multi-stability conditions. Finally, optimal control parameters are designed to eliminate multistability characteristics and achieve minimum and safe vibration levels.