Study of Hopf curves in the time delayed active control of a 2DOF nonlinear dynamical system
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Study of Hopf curves in the time delayed active control of a 2DOF nonlinear dynamical system Ali Kandil1 Received: 27 July 2020 / Accepted: 30 September 2020 © Springer Nature Switzerland AG 2020
Abstract Within this paper, we are focusing on the time delay effects on the improved positive position feedback (PPF) controller for the oscillations of a 2DOF nonlinear dynamical system which represents a thin-walled pre-twisted rotating blade. Herein, we are improving the performance of the traditional positive position feedback controller (PPF) that suffers from the existence of two high amplitude peaks on both sides of its minimum amplitude point. The improvement involves coupling double nonlinear saturation controllers to the main system to notch down the high peaks. As an active control process, the time delay is inherent in the process which produces a locus of Hopf bifurcation points separating between the stable and unstable regions of operation. Variation of different parameters is studied to relate their effects with time delay on the system operation. To ensure the validity of our proposed work, a comparison between the approached multiple scales analytical solutions and the computed Rung-Kutta numerical solutions is done at the end of this paper. Keywords Positive position feedback controller · Nonlinear saturation controller · Multiple time scales method · Time delay · Hopf curves List of symbols ̈ p, ̇ p Horizontal acceleration, velocity p, and displacement of the blade cross-section ̈ q, ̇ q Vertical acceleration, velocity q, and displacement of the blade cross-section ̈ x, ̇ x Acceleration, velocity and displacex, ment of the PPF controller ̈ y, ̇ y Acceleration, velocity and displacey, ment of the first NSC controller z̈ , z,̇ z Acceleration, velocity and displacement of the second NSC controller μ, μ1, μ2, μ3 Damping parameters of the blade and controllers ω, ω1, ω2, ω3 Natural frequencies of the blade and controllers
β11, β21, β13, β25, β5 Coupling factors between the blade vibrational directions β5 Cubic nonlinearity parameter of the blade β14, β24 Parametric excitation parameters f0 , f Force excitation amplitudes Ω Excitation frequency c1, c2 Gains of control signals c3, c4 Gains of feedback signals σ1, σ2, σ3, σ4 Detuning parameters 𝜏1 , 𝜏2 Time delays
1 Introduction Thin-walled pre-twisted blades are of the most important structures to be controlled in the modern time due to their participation in many industrial applications. They
* Ali Kandil, alikandil21@el‑eng.menofia.edu.eg; [email protected] | 1Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt. SN Applied Sciences
(2020) 2:1924
| https://doi.org/10.1007/s42452-020-03614-0
Vol.:(0123456789)
Research Article
SN Applied Sciences
(2020) 2:1924
are built-in components in industrial applications like helicopter blades, robot manipulators and rotating compressor blades that are our case of study in this work. Rotating compressor blades m
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