Influences of time-delays on the performance of a controller based on the saturation phenomenon

Abstract The nonlinear vibrations control of a vertically supported Jeffcott-rotor system is investigated within this article. A modified version of the controllers based on the saturation phenomenon is connected to the system via quadratic coupling nonlinearity. Time-delays in the control loop are included in the considered model to explore their influences on the controller performance and stability chart. Asymptotic analyses are sought to obtain an approximate solution to the time-delayed differential equations simulating the whole system dynamics. Bifurcation control is conducted by means of bifurcation-diagrams. The analyses illustrated that the proposed controller can suppress the system vibrations effectively, but the system may lose its stability via Hopf bifurcations if time-delays exceed a specific value. The limits of the time-delays at which the system solution remains stable have been determined. Then, the acquired results are simulated numerically by utilizing the ODE45 and DDE23 Matlab solvers. The nature of the system motion is analyzed by constructing Poincare-map, frequency-spectrum, and whirling-orbits. The numerical results showed that the system has chaotic and quasi-periodic motions in addition to the periodic ones. Finally, the optimal control parameters are reported and a comparison with recently published articles is included.

[1]  Richard David Neilson,et al.  Experimental verification of Jeffcott rotor model with preloaded snubber ring , 2006 .

[2]  Jian Xu,et al.  An improved time-delay saturation controller for suppression of nonlinear beam vibration , 2015 .

[3]  Vimal Singh,et al.  Perturbation methods , 1991 .

[4]  A. H. Nayfeh,et al.  Adaptive Control of Flexible Structures Using a Nonlinear Vibration Absorber , 2002 .

[5]  Tsuyoshi Inoue,et al.  Vibration Suppression of Nonlinear Rotor Systems Using a Dynamic Damper , 2007 .

[6]  N. A. Saeed,et al.  Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system , 2017 .

[7]  M. Eissa,et al.  Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations , 2013 .

[8]  Youngbae Kim,et al.  A MULTIPLE HARMONIC BALANCE METHOD FOR THE INTERNAL RESONANT VIBRATION OF A NON-LINEAR JEFFCOTT ROTOR , 1997 .

[9]  Jun Jiang,et al.  Stability Analysis of Sliding Whirl in a Nonlinear Jeffcott Rotor with Cross-Coupling Stiffness Coefficients , 2001 .

[10]  N. Saeed,et al.  Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system , 2016 .

[11]  M. Eissa,et al.  Saturation-based active controller for vibration suppression of a four-degree-of-freedom rotor–AMB system , 2014 .

[12]  Tsuyoshi Inoue,et al.  Nonlinear theoretical analysis of contacting forward whirling vibration of a rotating shaft supported by a repulsive magnetic bearing , 2013 .

[13]  Toshio Yamamoto,et al.  On the Vibrations of a Shaft supported by Bearings having Radial Clearances , 1955 .

[14]  M. Eissa,et al.  Nonlinear vibration control of a horizontally supported Jeffcott-rotor system , 2018 .

[15]  A. Nayfeh,et al.  NONLINEAR COUPLING OF PITCH AND ROLL MODES IN SHIP MOTIONS , 1973 .

[16]  Y. Ishida,et al.  Theoretical discussions on vibrations of a rotating shaft with nonlinear spring characteristics , 1977 .

[17]  Tsuyoshi Inoue,et al.  Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor , 2011 .

[18]  H. Yabuno,et al.  Nonlinear Analysis of Rotor Dynamics by Using the Method of Multiple Scales , 2007 .

[19]  M. Wiercigroch,et al.  Bifurcation analysis of a preloaded Jeffcott rotor , 2003 .

[20]  M. Wiercigroch,et al.  Non-linear dynamic interactions of a Jeffcott rotor with preloaded snubber ring , 2004 .

[21]  F. F. Ehrich,et al.  High Order Subharmonic Response of High Speed Rotors in Bearing Clearance , 1988 .

[22]  R. Ganesan.  Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances , 1996 .

[23]  L. Shampine,et al.  Solving DDEs in MATLAB , 2001 .

[24]  Livija Cveticanin.  Free vibration of a Jeffcott rotor with pure cubic non-linear elastic property of the shaft , 2005 .

[25]  S. T. Noah,et al.  QUASI-PERIODIC RESPONSE AND STABILITY ANALYSIS FOR A NON-LINEAR JEFFCOTT ROTOR , 1996 .

[26]  N. Saeed,et al.  On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force , 2017 .

[27]  Jian Xu,et al.  Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system , 2012 .