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Research Papers: Hydrodynamic Lubrication

Nonlinear Dynamics of Flexible Rotors Supported on Journal Bearings—Part II: Numerical Bearing Model

[+] Author and Article Information
Mohammad Miraskari

Mechanical Engineering Department,
University of British Columbia,
2054-6250 Applied Science Lane,
Vancouver, BC V6T 1Z4, Canada
e-mail: m.miraskari@alumni.ubc.ca

Farzad Hemmati

Mechanical Engineering Department,
University of British Columbia,
Vancouver, BC V6T 1Z4, Canada
e-mail: farhemmati@alumni.ubc.ca

Mohamed S. Gadala

Mechanical Engineering Department,
University of British Columbia,
Vancouver, BC V6T 1Z4, Canada;
Mechanical Engineering Department,
Abu Dhabi University,
Abu Dhabi, United Arab Emirates
e-mail: gadala@mech.ubc.ca

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received January 17, 2017; final manuscript received August 14, 2017; published online October 4, 2017. Assoc. Editor: Alan Palazzolo.

J. Tribol 140(2), 021705 (Oct 04, 2017) (19 pages) Paper No: TRIB-17-1024; doi: 10.1115/1.4037731 History: Received January 17, 2017; Revised August 14, 2017

The nonlinear stability of a flexible rotor-bearing system supported on finite length journal bearings is addressed. A perturbation method of the Reynolds lubrication equation is presented to calculate the bearing nonlinear dynamic coefficients, a treatment that is suitable to any bearing geometry. A mathematical model, nonlinear coefficient-based model, is proposed for the flexible rotor-bearing system for which the journal forces are represented through linear and nonlinear dynamic coefficients. The proposed model is then used for nonlinear stability analysis in the system. A shooting method is implemented to find the periodic solutions due to Hopf bifurcations. Monodromy matrix associated to the periodic solution is found at any operating point as a by-product of the shooting method. The eigenvalue analysis of the Monodromy matrix is then carried out to assess the bifurcation types and directions due to Hopf bifurcation in the system for speeds beyond the threshold speed of instability. Results show that models with finite coefficients have remarkably better agreement with experiments in identifying the boundary between bifurcation regions. Unbalance trajectories of the nonlinear system are shown to be capable of capturing sub- and super-harmonics which are absent in the linear model trajectories.

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Figures

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Fig. 1

The schematic of a journal inside bearing with the adopted coordinate systems

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Fig. 2

Schematic of a flexible rotor supported on journal bearings (a) and the coordinate system (b). Oj and OM correspond to the geometric center of the journal and the central disk, respectively.

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Fig. 3

The attitudes angles as defined by Childs [10,11]

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Fig. 4

Journal bearing steady pressure P¯0 (a), first-order pressure gradients P¯x, P¯y, P¯x˙, P¯y˙ ((b)–(e)), and second-order pressure gradients P¯xx, P¯yy, P¯xy, P¯xx˙, P¯xy˙, P¯yx˙,  P¯yy˙ ((f)–(l)); with L/D=1 at ϵ=0.8

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Fig. 5

Linear stiffness coefficients (a) and damping coefficients (b) for a finite journal bearing with L/D=1

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Fig. 6

Nonlinear second-order stiffness coefficients ((a) and (b)) and damping coefficients ((c) and (d)) for a finite journal bearing with L/D=1

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Fig. 7

Comparison of linear and nonlinear dynamic coefficients for a journal bearing with L/D=1 calculated based on analytic short bearing coefficients of Ref. [24] and proposed finite coefficients of this paper

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Fig. 8

Stability bounds of the flexible rotor-bearing system for a range of nondimensional shaft stiffness values (a) and comparison to experimental results of Wang and Khonsari [19] (b)

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Fig. 9

Selected operating points for a fully balanced flexible rotor-bearing system with k=CK/W=2 in stable and unstable regions for time integration

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Fig. 10

Journal bearing trajectories of a balanced rotor-bearing system at different operating points: point I ((a)–(d)), point II: ((e)–(h)), point III: ((i)–(l)), and point IV: ((m)–(p)) for the case models of Sec. 4: linear coefficient-based model ((a), (e), (i), and (m)); nonlinear coefficient-based model ((b), (f), (j), and (n)); short bearing force-based model ((c), (g), (k), and (o)); and finite bearing force-based model ((d), (h), (l), and (p))

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Fig. 11

Journal bearing trajectories for the unbalanced rotor-bearing system, the linear coefficient-based model (((a)–(d)), nonlinear coefficient-based model (((e)–(h)), short bearing force-based model (((i)–(l)), and finite bearing force-based model (((m)–(p)) at different stability parameters: γ=3.0 ((a), (e), (i), and (m)), γ=4.40 (((b), (f), (j), and (n)), γ=5.53 (((c), (g), (k), and (o)), and γ=5.60 (((d), (h), (l), and (p))

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Fig. 12

The periodic and quasi-periodic attractors (limit cycles) for the unbalanced rotor-bearing system based on the linear coefficient-based model (case 1a—solid), nonlinear coefficient-based model (case 1b—dot dashed), short bearing force-base model (case 2—dotted), and finite bearing force-based model (case 3—dashed), obtained at different stability parameters: γ=3.0 (a), γ=4.40 (b), γ=5.53 (c), and γ=5.60 (d)

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Fig. 13

Waterfall plots of the frequency content of the journal bearing unbalance response in x-direction for linear coefficient-based model (a) and for the nonlinear coefficient-based model (b), short bearing force-based model (c), and the finite bearing force-based model (d)

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Fig. 14

Contour plots of the frequency content of the journal bearing unbalance response in x-direction for linear coefficient-based model (a) and for the nonlinear coefficient-based model (b), short bearing force-based model (c), and the finite bearing force-based model (d)

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