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TECHNICAL PAPERS: Gas Turbines: Structures and Dynamics

On LMI-Based Optimization of Vibration and Stability in Rotor System Design

[+] Author and Article Information
Matthew O. Cole

Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200, Thailandmatt@dome.eng.cmu.ac.th

Theeraphong Wongratanaphisan

Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200, Thailandwong@dome.eng.cmu.ac.th

Patrick S. Keogh

Department of Mechanical Engineering, Faculty of Engineering and Design, University of Bath, Bath, BA2 7AY, UKenspsk@bath.ac.uk

J. Eng. Gas Turbines Power 128(3), 677-684 (Mar 01, 2004) (8 pages) doi:10.1115/1.2135818 History: Received October 01, 2003; Revised March 01, 2004

This paper considers optimization of rotor system design using stability and vibration response criteria. The initial premise of the study is that the effect of certain design changes can be parametrized in a rotor dynamic model through their influence on the system matrices obtained by finite element modeling. A suitable vibration response measure is derived by considering an unknown axial distribution of unbalanced components having bounded magnitude. It is shown that the worst-case unbalanced response is given by an absolute row-sum norm of the system frequency response matrix. The minimization of this norm is treated through the formulation of a set of linear matrix inequalities that can also incorporate design parameter constraints and stability criteria. The formulation can also be extended to cover uncertain or time-varying system dynamics arising, for example, due to speed-dependent bearing coefficients or gyroscopic effects. Numerical solution of the matrix inequalities is tackled using an iterative method that involves standard convex optimization routines. The method is applied in a case study that considers the optimal selection of bearing support stiffness and damping levels to minimize the worst-case vibration of a flexible rotor over a finite speed range. The main restriction in the application of the method is found to be the slow convergence of the numerical routines that occurs with high-order models and/or high problem complexity.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Block diagram of augmented system with output weighting and overall transfer function T̃(s)=W(s)T(s)

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Figure 2

Schematic of rotor test rig considered in case study

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Figure 3

Vibration response bound γf(Ω) for γ=1

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Figure 4

Rotor-dynamic model, showing selected dimensions

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Figure 5

Optimization surface showing contours of constant γmin and optimization path for case 1

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Figure 6

Optimization of γmin in case 1

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Figure 7

Optimization surface showing contours of constant γmin and optimization paths for cases 2A and 2B with different initial values of cB and kB. The region of permissible values is indicated by - - - boundary.

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Figure 8

Worst-case vibration response for optimized and unoptimized bearing coefficients from case 2A. The final bound (scaled by γopt) is also shown.

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Figure 9

Worst-case vibration response with optimized bearing coefficients from case 3. The optimized bound (scaled by γopt) is also shown.

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