Research Papers: Hydrodynamic Lubrication

Bifurcation Analysis of a Rotor Supported by Five-Pad Tilting Pad Journal Bearings Using Numerical Continuation

[+] Author and Article Information
Sitae Kim

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: sitaekim@tamu.edu

Alan B. Palazzolo

Fellow ASME
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
e-mail: a-palazzolo@tamu.edu

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received September 28, 2016; final manuscript received August 8, 2017; published online September 29, 2017. Assoc. Editor: Joichi Sugimura.

J. Tribol 140(2), 021701 (Sep 29, 2017) (12 pages) Paper No: TRIB-16-1300; doi: 10.1115/1.4037699 History: Received September 28, 2016; Revised August 08, 2017

This paper presents analytical bifurcations analysis of a “Jeffcott” type rigid rotor supported by five-pad tilting pad journal bearings (TPJBs). Numerical techniques such as nonautonomous shooting/arc-length continuation, Floquet theory, and Lyapunov exponents are employed along with direct numerical integration (NI) to analyze nonlinear characteristics of the TPJB-rotor system. A rocker pivot type five-pad TPJB is modeled with finite elements to evaluate the fluid pressure distribution on the pads, and the integrated fluid reaction force and moment are utilized to determine coexistent periodic solutions and bifurcations scenarios. The numerical shooting/continuation algorithms demand significant computational workload when applied to a rotor supported by a finite element bearing model. This bearing model may be significantly more accurate than the simplified infinitely short-/long-bearing approximations. Consequently, the use of efficient computation techniques such as deflation and parallel computing methods is applied to reduce the execution time. Loci of bifurcations of the TPJB-rigid rotor are determined with extensive numerical simulations with respect to both rotor spin speed and unbalance force magnitude. The results show that heavily loaded bearings and/or high unbalance force may induce consecutive transference of response in forms of synchronous to subsynchronous, quasi-periodic responses, and chaotic motions. It is revealed that the coexistent responses and their solution manifolds are obtainable and stretch out with selections of pad preload, pivot offset, and lubricant viscosity so that the periodic doubling bifurcations, saddle node bifurcations, and corresponding local stability are reliably determined by searching parameter sets. In case the system undergoes an aperiodic state, the rate of divergence/convergence of the attractor is examined quantitatively by using the maximum Lyapunov exponent (MLE).

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Fig. 1

Five-pad TPJB (load on pad) diagram and the respective coordinates in x–y plane

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Fig. 2

Finite element TPJB model and typical pressure distributions on pads

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Fig. 3

Symmetric TPJB support—rigid rotor system

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Fig. 4

Flowchart of shooting method with deflation and parallel computing

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Fig. 5

Computation time of shooting method for solving a three-multiple root nonlinear equation problem with the computation acceleration techniques. (a) Computation time with 6 cores and (b) computation time with 12 cores.

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Fig. 6

Bifurcation diagrams with regards to rotor mass and imbalance eccentricity—using direct NI

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Fig. 7

Loci of bifurcation diagram—using direct NI (mp = 1/2, α/β = 0.5, and μ = 13.8 mPa·s)

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Fig. 8

Bifurcation diagram and coexistent solutions with respect to rotor revolution speed—shooting/continuation (eimb = 0.3Cb, mp = 1/2, α/β = 0.5, and μ = 13.8 mPa·s)

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Fig. 9

Bifurcation diagram and coexistent solutions with respect to imbalance eccentricity on disk—shooting/continuation (rpm = 16,000, mp = 1/2, α/β = 0.5, and μ = 13.8 mPa·s)

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Fig. 10

Geometry of the pad–pivot parameter sets: (a) case 1, (b) case 2, (c) case 3, and (d) case 4

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Fig. 11

Bifurcation diagrams with respect to pad preloads (mp) and pivot offset (α/β): (a) case 2 (mp = 1/2 and α/β = 0.6), (b) case 3 (mp = 2/3 and α/β = 0.5), and (c) case 4 (mp = 2/3 and α/β = 0.6)

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Fig. 12

Bifurcation diagrams (mp = 1/2 and α/β = 0.5) with lubricant viscosities: (a) μ = 27.0 mPa·s and (b) μ = 10.3 mPa·s

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Fig. 13

(a) Bifurcation diagram (Poincaré sections) and (b) MLE (λmax) plotted versus spin speed (W = 19.6 kN, eimb = 0.3Cb, μ = 13.8 mPa·s, mp = 1/2, and α/β = 0.5)

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Fig. 14

Orbits, Poincare attractors, and frequency spectra: (a) rpm = 14.5 k, (b) rpm = 15.5 k, and (c) rpm = 18.5 k



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