Research Papers

Nonlinear Vibration of Saturated Water Journal Bearing and Bifurcation Analysis

[+] Author and Article Information
Shoyama Tadayoshi

Appliances Company, Panasonic Corporation,
Yagumo-Nakamachi 3-1-1,
Moriguchi 570-8501, Osaka, Japan;
Department of Aeronautics and Astronautics,
School of Engineering,
The University of Tokyo,
Bunkyo, Tokyo 113-8654, Japan
e-mail: shoyama.tadayoshi@jp.panasonic.com

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 1, 2017; final manuscript received November 14, 2018; published online January 16, 2019. Assoc. Editor: Costin Untaroiu.

J. Vib. Acoust 141(2), 021016 (Jan 16, 2019) (10 pages) Paper No: VIB-17-1521; doi: 10.1115/1.4042041 History: Received December 01, 2017; Revised November 14, 2018

We developed a turbo compressor that has water-lubricated bearings driven at 30,000 rpm in a saturation condition, where the ambient pressure is at the saturation point of the discharged lubricant water. The bearings are supported with nonlinear elastomeric O-rings. At rotational speed exceeding 15,000 rpm, the rotor showed many subharmonic vibrations that are nonlinear phenomena unpredictable from a linear equation of motion. Instead, a stability analysis with a bifurcation diagram is an effective method to tackle these problems. In this paper, we investigated these rotor vibrations by bifurcation diagrams of the vibrations measured in experiments of saturated water journal bearings. The angular velocity was used as a bifurcation parameter. The bifurcations among synchronous, subharmonic, and chaotic vibrations were shown. Next, the nonlinear dynamics of the rotating rigid shaft were analyzed numerically with the nonlinear stiffness obtained by a commercial code that utilizes the two-dimensional (2D) Reynolds equation. The dynamic properties of the supporting structure were modeled with a complex stiffness coefficient. As a result, a Hopf bifurcation was found and a subharmonic limit cycle appeared spontaneously as observed in the experiments. The parametric studies revealed the influences of the dynamic properties of the structural components, especially the sensitive effect of the damping of the bearing support on the onset frequency and the amplitude of these vibrations. Furthermore, linear eigenvalue analysis of the motion equations clarified the mechanism of the sensitive effects.

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Fig. 1

Bearing and supporting structure

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Fig. 2

Photograph of the compressor and the bearing with O-ring support

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Fig. 3

Schematic of experimental turbo compressor and displacement sensor locations

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Fig. 4

Waterfall plot of vibration data (HSY)

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Fig. 5

Two-dimensional torus and Poincaré section

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Fig. 6

Bifurcation diagram of measured vibration data (HSY)

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Fig. 7

Periodic orbits in bearing cross section

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Fig. 8

(Top) Enlargement of region 2 of bifurcation diagram. Some dense regions (303 Hz, 354 Hz, etc.) are frequencies at which rotational speed was kept constant for a long time; they have thus no special implications. (Bottom) Frequencies of periodic motion and ratio to RRW frequency, corresponding to each window in bifurcation diagram.

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Fig. 9

Vibration analysis model, only half of which is considered for symmetry

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Fig. 10

Equilibrium positions for different speeds. Vertical downward load F is varied on each line at constant speed.

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Fig. 11

Nonlinear stiffness force matrix of bearing film

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Fig. 12

Waterfall plot of nonlinear dynamic analysis

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Fig. 13

Bifurcation diagram for the result of nonlinear dynamic analysis (k = 1.1, Dsfd = 2000 (N s/m)). Densely filled region indicates that orbit is not periodic but chaotic.

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Fig. 14

Effect of damping of supporter on bifurcation frequency (k = 1.0)

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Fig. 15

Influence of unbalance of rotor on the maximum amplitude of each component (frot = 367 (Hz), k = 1.0, Dsfd = 1800 (N s/m))

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Fig. 16

Effects on the bifurcation frequencies of: (a) O-ring stiffness, (b) bearing stiffness factor, and (c) bearing film damping. (Dsfd = 1800 (N s/m)).

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Fig. 17

Comparison of real part of eigenvalues and instability of RRW appeared in nonlinear analysis. The far smaller negative eigenvalues were not shown.

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Fig. 18

Comparison of imaginary part of the eigenvalue having maximum real part and RRW frequency in nonlinear analysis

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Fig. 19

Influence of bearing support stiffness on real part of eigenvalues

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Fig. 20

Influence of bearing support damping on real part of eigenvalues



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