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.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Thearle, E. L. , 1950, “ Automatic Dynamic Balancers (Part 2––Ring, Pendulum, Ball Balancers),” Mach. Des., 22(10), pp. 103–106.
Kubo, S. , Jinnouchi, Y. , Araki, Y. , and Inoue, J. , 1986, “ Automatic Balancer: Pendulum Balancer,” Bull. JSME, 29(249), pp. 924–928. [CrossRef]
Bovik, P. , and Hogfords, C. , 1986, “ Autobalancing of Rotors,” J. Sound Vib., 111(3), pp. 429–440. [CrossRef]
Majewski, T. , 1998, “ Position Error Occurrence in Self Balancers Used on Rigid Rotors of Rotating Machinery,” Mech. Mach. Theory, 23(1), pp. 71–78.
Jinnouchi, Y. , Araki, Y. , Inoue, J. , Ohtsuka, Y. , and Tan, C. , 1993, “ Automatic Balancer (Static Balancing and Transient Response of a Multi-Ball Balancer),” Trans. Jpn. Soc. Mech. Eng., Part C 59(557), pp. 79–84. [CrossRef]
Lindell, H. , 1996, “ Vibration Reduction on Hand-Held Grinders by Automatic Balancers,” Cent. Eur. J. Public Health, 4(1), pp. 43–45. [PubMed]
Hwang, C. H. , and Chung, J. , 1999, “ Dynamic Analysis of an Automatic Ball Balancer With Double Races,” JSME Int. J., 42(2), pp. 265–272.
Kim, W. , Lee, D. J. , and Chung, J. , 2005, “ Three-Dimensional Modeling and Dynamic Analysis of an Automatic Ball Balancer in an Optical Disk Drive,” J. Sound Vib., 285(3), pp. 547–569. [CrossRef]
Rajalingham, C. , and Bhat, R. B. , 2006, “ Complete Balancing of a Disk Mounted on Vertical Cantilever Shaft Using a Two Ball Automatic Balancer,” J. Sound Vib., 290, pp. 161–191. [CrossRef]
DeSmidt, H. A. , 2009, “ Imbalance Vibration Suppression of a Supercritical Shaft Via an Automatic Balancing Device,” ASME J. Vib. Acoust., 131(4), p. 041001. [CrossRef]
Green, K. , Champneys, A. R. , and Lieven, N. J. , 2006, “ Bifurcation Analysis of an Automatic Dynamics Balancing Mechanism for Eccentric Rotors,” J. Sound Vib., 291, pp. 861–881. [CrossRef]
Green, K. , Champneys, A. R. , and Friswell, M. I. , 2006, “ Analysis of the Transient Response of an Automatic Dynamic Balancer for Eccentric Rotors,” Int. J. Mech. Sci., 48(3), pp. 274–293. [CrossRef]
Jung, D. , and DeSmidt, H. A. , “ Limit-Cycle Analysis of Planar Rotor/Autobalancer System Supported on Hydrodynamic Journal Bearing,” ASME Paper No. DETC2011-48723.
Inoue, T. , Ishida, Y. , and Niimi, H. , 2012, “ Vibration Analysis of a Self-Excited Vibration in a Rotor System Caused by a Ball Balancer,” ASME J. Vib. Acoust., 134(2), p. 021006.
Vance, J. M. , 1988, Rotordynamics of Turbomachinery, Wiley, New York.
Rugh, W. J. , 1996, Linear System Theory, Prentice Hall, Upper Saddle River, NJ.
Issac, F. , 1984, “ Orthogonal Trajectory Accession to Nonlinear Equilibrium Curve,” Comput. Methods Appl. Mech. Eng., 47(3), pp. 283–297. [CrossRef]
Wempner, G. A. , 1971, “ Discrete Approximations Related to Nonlinear Theories of Solids,” Int. J. Solid Struct. 7(11), pp. 1581–1599. [CrossRef]
Desoer, C. A. , 1969, “ Slowly Varying System,” IEEE Trans. Automat. Control, 14, pp. 780–781. [CrossRef]
Frulla, G. , 2000, “ Rigid Rotor Dynamic Stability Using Floquet Theory,” Eur. J. Mech., 19(1), pp. 139–150. [CrossRef]


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
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

Grahic Jump Location
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)

Grahic Jump Location
Fig. 10

Nine simulation results for Fig. 7(b)

Grahic Jump Location
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

Grahic Jump Location
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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In