Time-delayed control to suppress a nonlinear system vibration utilizing the multiple scales homotopy approach
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O R I G I NA L
N. A. Saeed · G. M. Moatimid · F. M. F. Elsabaa · Y. Y. Ellabban
Time-delayed control to suppress a nonlinear system vibration utilizing the multiple scales homotopy approach
Received: 8 January 2020 / Accepted: 12 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Nonlinear vibrations · Resonance cases · Position-velocity controller · Time-delay · Quasiperiodic motion · Frequency spectrum
List of symbols q, q, ˙ q¨ μ ω α1 , α 2 , α 3 f Ω β1 , β2 τ
Non-dimensional displacement, velocity, and acceleration, of the cantilever beam system, respectively Non-dimensional linear damping coefficients of a cantilever beam system Non-dimensional linear natural frequencies of a cantilever beam system Non-dimensional cubic nonlinearity coefficients of a cantilever beam system The amplitude of the excitation force Frequency of the excitation force Non-dimensional control signal gains Control loop time-delay
N. A. Saeed Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt G. M. Moatimid · F. M. F.Elsabaa · Y. Y. Ellabban (B) Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt E-mail: [email protected]
N. A. Saeed et al.
1 Introduction Cantilever beam system is often used to simulate the nonlinear vibrations of many engineering structures such as helicopter rotor blades, wind-turbine blades, aircraft wings, large space structures, robot manipulators, arm-type positioning mechanisms, and links of slider-crank mechanisms... etc. The nonlinear vibrations that occur in such engineering structures have many different causes such as the geometrical nonlinearity of the structure, nonlinear properties of the system materials, and the external excitation forces. In some cases, the negligible of the system nonlinearities is possible, but when these systems are excited away from their resonance conditions with small excitation amplitudes. However
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