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Journal of Vibration and Acoustics | Volume 134 | Issue 2 | Research Paper
Research Papers

Force Feedback Control for Active Stabilization of Synchronous Whirl Orbits in Rotor Systems With Nonlinear Stiffness Elements

[+] Author and Article Information
M. O. T. Cole

C. Chamroon, P. Ngamprapasom

Department of Mechanical Engineering,  Chiang Mai University, Chiang Mai 50200, Thailand

J. Vib. Acoust 134(2), 021018 (Jan 26, 2012) (10 pages) doi:10.1115/1.4005021 History: Received July 05, 2010; Accepted August 16, 2011; Published January 26, 2012; Online January 26, 2012
Article
References
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Citing Articles

Synchronous vibration in rotor systems having bearings, seals, or other elements with nonlinear stiffness characteristics is prone to amplitude jump when operating close to critical speeds as there may be two or more possible whirl motions for a given unbalance condition. This paper describes research on how active control techniques may eliminate this potentially undesirable behavior. A control scheme based on feedback of rotor-stator interaction forces is considered. Model-based conditions for stability of low amplitude whirl, derived using Lyapunov’s direct method, are used to synthesize controller gains. Subsidiary requirements for existence of a static feedback control law that can achieve stabilization are also explained. An experimental validation is undertaken on a flexible rotor test rig where nonlinear rotor-stator contact interaction can occur across a small radial clearance in one transverse plane. A single radial active magnetic bearing is used to apply control forces in a separate transverse plane. The experiments confirm the conditions under which static feedback of the measured interaction force can prevent degenerate whirl responses such that a low amplitude contact-free orbit is the only possible steady-state response. The gain synthesis method leads to controllers that are physically realizable and can eliminate amplitude jump over a range of running speeds.
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References

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Muszynska, A., and Goldman, P., 1993, “Chaotic Vibrations of Rotor/Bearing/Stator Systems With Looseness or Rubs,” Nonlinear Vibrations, 54 , pp. 187–194.
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Kim, Y. B., and Noah, S. T., 1990, “Bifurcation Analysis for a Modified Jeffcott Rotor With Bearing Clearances,” Nonlinear Dyn., 1 , pp. 221–241.  [CrossRef]
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Yu, J. J., Goldman, P., Bently, D. E., and Muzynska, A., 2002, “Rotor/Seal Experimental and Analytical Study of Full Annular Rub,” ASME J. Eng. Gas Turbines Power, 124 (2), pp. 340–350.  [CrossRef]
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Bartha, A. R., 2000, “Dry Friction Backward Whirl of Rotors,” Ph.D. dissertation, ETH, Zurich.
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Williams, R. J., 2004, “Parametric Characterization of Rub Induced Whirl Instability Using an Instrumented Rotordynamic Test Rig,” "Proceedings of the Tenth International Conference on Vibrations in Rotating Machinery", Swansea, Paper C623/012/04, pp. 651–659.
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Childs, D. W., and Bhattacharya, A., 2007, “Prediction of Dry-Friction Whirl and Whip Between a Rotor and a Stator,” ASME J. Vibr. Acoust., 129 , pp. 355–362.  [CrossRef]
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Villa, C., Sinou, F., and Thoverez, F., 2008, “Stability and Vibration Analysis of a Complex Flexible Rotor Bearing System,” Commun. Nonlinear Sci. Numer. Simul., 13 , pp. 804–821.  [CrossRef]
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Ehrich, F. F., 1988, “High Order Sub-harmonic Response of High Speed Rotors in Bearing Clearance,” ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, 110 , pp. 9–16.
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Wegener, G., MarketR., and Pothmann, K., 1998, “Steady-State Analysis of a Multi-Disk or Continuous Rotor With One Retainer Bearing,” "Proceedings of the Fifth International Conference on Rotor Dynamics", Darmstadt, pp. 816–828.
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Lawen, J. L., and Flowers, G. T., 1997, “Synchronous Dynamics of a Coupled Shaft/Bearing/Housing System With Auxiliary Support From a Clearance Bearing: Analysis and Experiment,” ASME J. Eng. Gas Turbines Power, 119 , pp. 431–435.  [CrossRef]
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Cole, M. O. T., and Keogh, P. S., 2003, “Rotor Vibration With Auxiliary Bearing Contact in Magnetic Bearing Systems, Part 2: Robust Synchronous Control for Rotor Position Recovery,” J. Mech. Eng. Sci., 217 (4), pp. 393–409.  [CrossRef]
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Cade, I. S., Sahinkaya, M. N., Burrows, C. R., and Keogh, P. S., 2009, “On the Use of Actively Controlled Auxiliary Bearings in Magnetic Bearing Systems,” ASME J. Eng. Gas Turbines Power, 131 (2), p. 022507.  [CrossRef]
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Ginzinger, L. B., and Ulbrich, H., 2008, “Simulation-Based Controller Design for an Active Auxiliary Bearing,” "Proceedings of the Eleventh International Symposium on Magnetic Bearings", Nara, Paper No. 532.
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Ginzinger, L. B., 2009, “Control of a Rubbing Rotor using an Active Auxiliary Bearing,” Ph.D. dissertation, Technical University Munich.
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Cole, M. O. T., 2009, “Model-Free Control of Touchdowns Involving Circular Whirl in Rotor-Magnetic Bearing Systems,” JSME J. Syst. Des. Dyn., 3 (3), pp. 584–595.  [CrossRef]
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Figures

Grahic Jump Location
+Nyquist plot for the linear part of the model (case 2 low ks)
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Nyquist plot for the linear part of the model (case 2 low ks)
Figure 4

Nyquist plot for the linear part of the model (case 2 low ks)

Grahic Jump Location
+Rotor-stator system dynamics under control with force-feedback gain matrix H
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Rotor-stator system dynamics under control with force-feedback gain matrix H
Figure 5

Rotor-stator system dynamics under control with force-feedback gain matrix H

Grahic Jump Location
+Region for stable low-amplitude whirl under force-feedback control (case 1). Results with controller gain optimized for two different frequencies are shown.
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Region for stable low-amplitude whirl under force-feedback control (case 1). Results with controller gain optimized for two different frequencies are shown.
Figure 9

Region for stable low-amplitude whirl under force-feedback control (case 1). Results with controller gain optimized for two different frequencies are shown.

Grahic Jump Location
+Region for stable low-amplitude whirl under force-feedback control (case 2). Results with controller gain optimized for two different frequencies are shown.
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Region for stable low-amplitude whirl under force-feedback control (case 2). Results with controller gain optimized for two different frequencies are shown.
Figure 10

Region for stable low-amplitude whirl under force-feedback control (case 2). Results with controller gain optimized for two different frequencies are shown.

Grahic Jump Location
+Flexible rotor-AMB test rig: (a) photograph and (b) CAD model
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Flexible rotor-AMB test rig: (a) photograph and (b) CAD model
Figure 11

Flexible rotor-AMB test rig: (a) photograph and (b) CAD model

Grahic Jump Location
+Measured rotor vibration response without control: (a) orbit amplitudes and (b) selected orbit plots
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Measured rotor vibration response without control: (a) orbit amplitudes and (b) selected orbit plots
Figure 12

Measured rotor vibration response without control: (a) orbit amplitudes and (b) selected orbit plots

Grahic Jump Location
+Measured rotor vibration response with force-feedback control
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Measured rotor vibration response with force-feedback control
Figure 13

Measured rotor vibration response with force-feedback control

Grahic Jump Location
+Rotor-stator dynamic model with nonlinear interaction: (a) subsystems and (b) combined model with nonlinear feedback element
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Rotor-stator dynamic model with nonlinear interaction: (a) subsystems and (b) combined model with nonlinear feedback element
Figure 1

Rotor-stator dynamic model with nonlinear interaction: (a) subsystems and (b) combined model with nonlinear feedback element

Grahic Jump Location
+Rotor-stator model involving a single nonlinear stiffness element
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Rotor-stator model involving a single nonlinear stiffness element
Figure 2

Rotor-stator model involving a single nonlinear stiffness element

Grahic Jump Location
+Multiplicity of possible synchronous whirl responses can be determined from a Nyquist plot for the linear part of the model (case 1 high ks)
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Multiplicity of possible synchronous whirl responses can be determined from a Nyquist plot for the linear part of the model (case 1 high ks)
Figure 3

Multiplicity of possible synchronous whirl responses can be determined from a Nyquist plot for the linear part of the model (case 1 high ks)

Grahic Jump Location
+Calculated unbalance response showing effect of local feedback of measured rotor-stator interaction force u=-0.7f: (a) with positive interaction stiffness k=0.02 MN/m and (b) with negative interaction stiffness k=0.01 MN/m
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Calculated unbalance response showing effect of local feedback of measured rotor-stator interaction force u=-0.7f: (a) with positive interaction stiffness k=0.02 MN/m and (b) with negative interaction stiffness k=0.01 MN/m
Figure 6

Calculated unbalance response showing effect of local feedback of measured rotor-stator interaction force u=-0.7f: (a) with positive interaction stiffness k=0.02 MN/m and (b) with negative interaction stiffness k=0.01 MN/m

Grahic Jump Location
+Region for stable low-amplitude whirl: comparison of Nyquist and Lyapunov boundaries (case 1)
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Region for stable low-amplitude whirl: comparison of Nyquist and Lyapunov boundaries (case 1)
Figure 7

Region for stable low-amplitude whirl: comparison of Nyquist and Lyapunov boundaries (case 1)

Grahic Jump Location
+Nyquist plot for actively controlled system (case 1) with force feedback. Results with controller gain optimized for two different frequencies are shown.
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Nyquist plot for actively controlled system (case 1) with force feedback. Results with controller gain optimized for two different frequencies are shown.
Figure 8

Nyquist plot for actively controlled system (case 1) with force feedback. Results with controller gain optimized for two different frequencies are shown.

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