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Research Papers

Limit-Cycle Analysis of Three-Dimensional Flexible Shaft/Rigid Rotor/Autobalancer System With Symmetric Rigid Supports

[+] Author and Article Information
DaeYi Jung

Korea Institute of Research and
Medical Sciences,
75 Nowon-ro Nowon-gu,
Seoul 01812, South Korea
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 January 11, 2015; final manuscript received January 11, 2016; published online April 7, 2016. Assoc. Editor: Paul C.-P. Chao.

J. Vib. Acoust 138(3), 031005 (Apr 07, 2016) Paper No: VIB-15-1016; doi: 10.1115/1.4032718 History: Received January 11, 2015; Revised January 11, 2016

In recent years, there has been much interest in the use of automatic balancing devices (ABD) in rotating machinery. Autobalancers consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance. 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 to 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 positions resulting in increased rotor vibration. To explore this nonsynchronous behavior of ABD, the unstable limit-cycle analysis of three-dimensional (3D) flexible shaft/rigid rotor/ABD/rigid supports described by the modal coordinates has been investigated here. Essentially, this paper presents an approximate harmonic analytical solution to describe the limit-cycle behavior of ABD–rotor system interacting with flexible shaft, which has not been fully considered by ABD researchers. The modal shape of flexible shaft is determined by using well-known fixed–fixed boundary condition due to symmetric rigid supports. Here, the whirl speed of the ABD balancer masses is determined via the solution of a nonlinear characteristic equation. Also, based upon the analytical limit-cycle solutions, the limit-cycle stability of three primary design parameters for ABD is assessed via a perturbation and Floquet analysis: the size of ABD balancer mass, the ABD viscous damping, and the relative axial location of ABD to the imbalance rotor along the shaft. The coexistence of the stable balanced synchronous condition and undesirable nonsynchronous limit-cycle is also studied. It is found that for certain combinations of ABD parameters and rotor speeds, the nonsynchronous limit-cycle can be made unstable, thus guaranteeing asymptotic stability of the synchronous balanced condition at the supercritical shaft speeds between each flexible mode. Finally, the analysis is validated through numerical simulation. The findings in this paper yield important insights for researchers wishing to utilize ABD in flexible shaft/rigid rotor systems and limit-cycle mitigation.

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Figures

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Fig. 1

Three-dimensional flexible shaft/rigid rotor/ABD system with rigid supports

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Fig. 2

Comparison of whirl orbits between the analytical solution (LC) and the numerical ones (FS): (a) in-plane transverse deflection [vr(x,t)vs.wr(x,t)]| x=0.3L and (b) out-plane tilting angle [α(x,t)vs.β(x,t)]| x=0.3L

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Fig. 3

Comparison of shaft responses between the analytical responses (LC) and the numerical ones (FS): ((a) and (b)) the first mode at Ω¯=1.4, ((c) and (d)) the second mode at Ω¯=7, and ((e) and (f)) third mode at Ω¯=16

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Fig. 4

Comparison of whirl speeds and maximum amplitudes between the analytical solution (LC) and the numerical ones (FS) at every operating speed: (a) whirl speeds and (b)–(d) maximum whirl amplitudes

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Fig. 5

Whirl speed of ABD balancer and damped natural frequencies of baseline system (rotor–shaft system): (a) Ld=0.75L and LAB=0.45L and (b) Ld=0.5L and LAB=0.3L

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Fig. 6

(a) Stability influence of the relative mass size mb of ABD to the mass of rotor and shaft msL+md and (b) system stability influence of ABD ball damping coefficient cb

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Fig. 7

Comparison of whirl speeds on time domain between the analytical solutions (LC) and the numerical ones (FS): (a) whirl speed at Ω¯=4 and (b) whirl speed at Ω¯=15

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Fig. 8

(a) The difference between minimum and maximum whirl speed in FS and (b) the absolute magnitude of coefficients in the higher-order harmonics truncated for an approximation

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Fig. 9

Synchronous balanced equilibrium position of ABD balancer: (a) Ld=0.75L and LAB=0.6L, (b) Ld=0.5L and LAB=0.45L, (c) Ld=0.75L and LAB=0.3L, and (d) Ld=0.5L and LAB=0.5L

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Fig. 10

The limit-cycle behavior for the relative location of ABD along with shaft to imbalance source (displayed as a dotted line on each plot): imbalance rotor at 0.75L (a) and 0.5L (b) and the performance of balancing condition for three different amplitude levels: Λ>1, Λ<1, and Λ<0.5 in (c) and (d)

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