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Research Papers: Gas Turbines: Structures and Dynamics

Instability and Chaos of a Flexible Rotor Ball Bearing System: An Investigation on the Influence of Rotating Imbalance and Bearing Clearance

[+] Author and Article Information
T. C. Gupta

Department of Mechanical Engineering, National Institute of Technology, Jaipur 302017, Indiatcgmnit@gmail.com

K. Gupta

Department of Mechanical Engineering, Indian Institute of Technology, Delhi 110016, Indiakgupta@mech.iitd.ac.in

D. K. Sehgal

Department of Applied Mechanics, Indian Institute of Technology, Delhi 110016, Indiadks@am.iitd.ac.in

J. Eng. Gas Turbines Power 133(8), 082501 (Apr 05, 2011) (11 pages) doi:10.1115/1.4002657 History: Received April 14, 2010; Revised May 01, 2010; Published April 05, 2011; Online April 05, 2011

In this paper, a horizontal flexible rotor supported on two deep groove ball bearings is theoretically investigated for instability and chaos. The system is biperiodically excited. The two sources of excitation are rotating imbalance and self excitation due to varying compliance effect of ball bearing. A generalized Timoshenko beam finite element (FE) formulation, which can be used for both flexible and rigid rotor systems with equal effectiveness, is developed. The novel scheme proposed in the literature to analyze quasiperiodic response is coupled with the existing nonautonomous shooting method and is thus modified; the shooting method is used to obtain a steady state quasiperiodic solution. The eigenvalues of monodromy matrix provide information about stability and nature of bifurcation of the quasiperiodic solution. The maximum value of the Lyapunov exponent is used for quantitative measure of chaos in the dynamic response. The effect of three parameters, viz., rotating unbalance, bearing clearance, and rotor flexibility, on an unstable and chaotic behavior of a horizontal flexible rotor is studied. Interactive effects between the three parameters are examined in detail in respect of rotor system instability and chaos, and finally the range of parameters is established for the same.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

A beam segment and the analogous motion planes

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Figure 2

Deformation in balls and races

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Figure 3

Vibration wave forms at low rotor speeds

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Figure 4

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic; for e=10 μm, γ0=20 μm, λ= (a) 2.49×10−4, (b) 2.0×10−3, (c) 6.77×10−3, and (d) 5.43×10−2

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Figure 5

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic; for γ0=20 μm, λ=2.49×10−4, e= (a) 0.0 μm, (b) 10 μm, (c) 30 μm, and (d) 60 μm

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Figure 6

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic; for γ0=20 μm, λ=6.77×10−3, e= (a) 0.0 μm, (b) 10 μm, (c) 30 μm, and (d) 60 μm

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Figure 7

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic); for γ0=20 μm, λ=5.43×10−2, e= (a) 0.0 μm, (b) 10 μm, and (c) 30 μm

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Figure 8

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic; for e=10 μm, λ=6.77×10−3, γ0= (a) 20 μm, (b) 10 μm, and (c) 5 μm

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Figure 9

((i) and (ii)) Frequency response curves; (iii) instability map: ▲ unstable and △ stable; and (iv) chaos map: ● chaotic and ○ nonchaotic; for e=10 μm, λ=5.43×10−2, γ0= (a) 20 μm, (b) 10 μm, and (c) 5 μm

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