Time-delayed control to suppress a nonlinear system vibration utilizing the multiple scales homotopy approach

The nonlinear transversal oscillations of a cantilever beam system at primary, superharmonic, and subharmonic resonance cases are investigated within this work. Time-delayed position-velocity controller is proposed to suppress the considered system nonlinear vibrations. The multiple scales homotopy approach is employed to analyze the controlled nonlinear model. The amplitude-phase modulating equations that govern the system dynamics at the different resonance cases are extracted. The stability charts of the loop-delay are obtained. The influence of the different controller parameters on the system vibration behaviors is explored. The acquired analytical results revealed that the loop-delay has a great influence on the controller efficiency. Accordingly, the optimal values of the loop-delay are reported and utilized to enhance the applied controller performance. Finally, numerical validations of the accomplished analytical results are performed, which illustrated an excellent agreement with the obtained analytical ones.