Hopf-Hopf Interactions of Surge and Rotating Stall

[+] Author and Article Information
B. D. Coller

Department of Mechanical Engineering, Northern Illinois University, DeKalb, IL 60115

J. Comput. Nonlinear Dynam 1(4), 320-327 (Jun 06, 2006) (8 pages) doi:10.1115/1.2338324 History: Received November 15, 2005; Revised June 06, 2006

In this paper, we examine the interaction of two instabilities that occur in axial compressors using the Hopf-Hopf normal form. As a result, we illuminate some gaps in understanding the dynamics of standard compressor models. We find a possible dynamic mechanism, which explains certain “curious” behavior observed in experiments, and are able to predict and explain failure mechanisms in previously proposed control strategies.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Schematic diagram of compression system

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Figure 2

(a) Compressor and throttle characteristics. (b) Creating an “effective” throttle characteristic via feedback.

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Figure 3

Stability of the nominal operating point

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Figure 4

Partial parameter space for the Hopf-Hopf normal form 9. Dotted lines separate different degenerate behaviors (solid letters). Solid lines separate nondegenerate classes (open Roman numerals). Dashed lines serve the dual purpose.

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Figure 5

Phase portraits for two interacting Hopf bifurcations. Sixty-six additional cases are obtained by reversing the sense of the arrows.

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Figure 6

Compressor characteristic and throttle curves for a variety of settings. As one continuously varies the throttle setting, one traverses a one-dimensional curve through a normal form parameter space.

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Figure 7

Sequence of phase portraits for the normal form as one varies the throttle setting

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Figure 8

Branches of rotating stall amplitudes

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Figure 9

Simulation of Moore-Greitzer model with KT=0.873. Variables x1 and y1 are the sine and cosine components of the first Fourier mode of the stall cell. Two slightly different initial conditions lead to dramatically different results.

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Figure 10

(a) Simulation of Moore-Greitzer model with and without feedback control. With β, small surge dynamics are highly damped and the Liaw-Abed-type controller works. For larger β as in (b), however, the controller fails.

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Figure 11

Eveker ’s (4)Φ̇ feedback successfully controls the system with larger β

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Figure 12

Portraits corresponding to normal form dynamics of controlled systems

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Figure 13

Phase portraits showing an example of a tertiary Hopf bifurcation

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Figure 14

Simulation depicting three-frequency behavior in compression system




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