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Research Papers

Limit-Cycle Analysis of Planar Rotor/Autobalancer System Influenced by Alford's Force

[+] Author and Article Information
DaeYi Jung

Mechanical Aerospace and
Biomedical Engineering Department,
University of Tennessee,
Knoxville, TN 37996-2210
e-mail: dyjung@kirams.re.kr

H. A. DeSmidt

Associate Professor
Mechanical Aerospace and
Biomedical Engineering Department,
University of Tennessee,
234 Dougherty Engineering Building,
Knoxville, TN 37996-2210
e-mail: hdesmidt@utk.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 3, 2013; final manuscript received January 11, 2016; published online February 12, 2016. Assoc. Editor: Yukio Ishida.

J. Vib. Acoust 138(2), 021018 (Feb 12, 2016) (14 pages) Paper No: VIB-13-1344; doi: 10.1115/1.4032511 History: Received October 03, 2013; Revised January 11, 2016

In recent years, there has been much interest in the use of so-called automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancers” consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steady-state. This “automatic balancing” phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor, wherein the balancer masses naturally synchronize with the rotor with appropriate phase and cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steady-state positions resulting in increased rotor vibration. In this paper, an approximate analytical harmonic solution for the limit cycles is obtained for the special case of symmetric support stiffness together with the so-called Alford's force cross-coupling term. The limit-cycle stability is assessed via Floquet analysis with a perturbation. It is found that the stable balanced synchronous conditions coexist with undesirable nonsynchronous limit cycles. For certain combinations of bearing parameters and operating speeds, the nonsynchronous limit-cycle can be made unstable guaranteeing global asymptotic stability of the synchronous balanced condition. Additionally, the analytical bifurcation of the coexistence zone and the pure balanced synchronous condition is derived. Finally, the analysis is validated through numerical time- and frequency-domain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.

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References

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Figures

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Fig. 1

Rigid planar rotor–ABD system supported by a generalized linear bearing holding asymmetric damping and stiffness force including cross-coupling one

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Fig. 2

Stability contour plots (Ω¯  versus K¯XY) under the influence of Alford's force: CXX=0.01KXX (left) and CXX=0.015KXX (right), with mim=mb

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Fig. 3

Stability contour plots (Ω¯  versus K¯XY) under the influence of Alford's force: cb=1×10−4 (left) and cb=1×10−3 (right), with mim=mb

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Fig. 4

Stability contour plots (Ω¯  versus K¯XY) under the influence of Alford's force: mb=0.007mt (left) and mb=0.014mt (right) with cb=1×10−3

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Fig. 5

Bearing stiffness and damping domain for the critical limit cycle frequency (α=0.01 (left) and α=0.1 (right); mb2=0.1305  cbmt is commonly used)

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Fig. 6

Comparison between full numerical solutions (FS) and assumed limit-cycle solutions (LC): (a) KA⋅XY=0.007KXX,  Ω¯=2.5; (b) KA.XY=0.035KXX; (c) and (d) KA⋅XY=0.025KXX,  Ω¯=2.0 with mim=6mb,  mim=mb,Ωim=15deg

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Fig. 7

Comparison between full numerical solutions (FS) and assumed limit-cycle solutions (LC) (KA⋅XY=0.02KXX)

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Fig. 8

Comparison between full numerical solutions (FS) and assumed limit-cycle solutions (LC): (a) KA.XY=0.03KXX,  Ω¯=2; (b), (c), and (d) KA.XY=0.035 KXX

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Fig. 9

Undesirable coexistence of subsynchronous limit-cycle responses and synchronous response (three plots in the first column for KA.XY=0.015KXX and three plots in the second column for KA.XY=0.025KXX)

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Fig. 10

Nine simulation results for Fig. 7(b)

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Fig. 11

max1≤i≤6|λi(x(T))| in Eq. (35) for two cases—KA⋅XY=0.015KXX (left) and KA⋅XY=0.025KXX (right)—of Fig. 9

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Fig. 12

max1≤i≤6Re[si] in Eq. (39) for two cases—KA⋅XY=0.015KXX (left) and KA⋅XY=0.025KXX (right)—of Fig. 9

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