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Research Papers

# Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

[+] 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
Professor
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77840
email: a-palazzolo@tamu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 12, 2015; final manuscript received August 30, 2016; published online December 5, 2016. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 12(3), 031003 (Dec 05, 2016) (15 pages) Paper No: CND-15-1288; doi: 10.1115/1.4034733 History: Received September 12, 2015; Revised August 30, 2016

## Abstract

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

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## Figures

Fig. 1

Bifurcation scenarios based on Floquet theory: (a) (+1,0)→symmetry breaking or pitchfork or saddle node bifurcation, (b) (−1,0)→periodic doubling bifurcation, and (c) cross the unit circle→Secondary Hopf or N-S bifurcation

Fig. 2

FRB middle plane and its coordinate system

Fig. 3

Layouts of mesh and boundary conditions of finite FRB model: (a) inner film and (b) outer film

Fig. 4

Rigid rotor supported by floating ring bearing (FRB)

Fig. 5

Bifurcation diagram (L/D = 0.2, ε = 0.0): (a) Revolution speed versus max/min yj and (b) revolution speed versus period ratio (τ/τs)

Fig. 6

Identified possible responses using the shooting method at 80,000 rpm (L/D = 0.2): (a) limit cycle #1 (Stable*), (b) limit cycle #2 (Unstable*), and (c) limit cycle #3 (Stable*). *The stability of the responses is evaluated by Floquet theory.

Fig. 7

Repelling motion (journal related to housing) of the unstable orbit (LC#2): (a) LC#2 → LC#1 and (b) LC#2 → LC#3

Fig. 8

Stable periodic solutions (journal orbit related to ring) among N-S bifurcations: (a) PS #1 (50 krpm), (b) PS #2 (105 krpm), (c) PS #3 (150 krpm), (d) PS #4 (176 krpm), and (e) PS #5 (250 krpm)

Fig. 9

Bifurcation diagram (L/D = 0.5, ε = 0.0): (a) Revolution speed versus max/min yj and (b) revolution speed versus period ratio (τ/τs)

Fig. 10

Schematics support excitation to FRB housings

Fig. 11

Jump phenomenon between two limit cycles due to bump from FRB base at 80,000 rpm (L/D = 0.2, ε = 0.0): (a) before: limit cycle #1 (blue) → after: limit cycle #3 (red), (b) before: limit cycle #3 (blue) → after: limit cycle #1 (red) (see figure online for color)

Fig. 12

Bifurcation (xj-poincaré versus journal revolution speed) and corresponding maximum Lyapunov exponent diagrams (L/D = 0.2, ε = 0.4Ci)

Fig. 13

Nonlinear response evaluation at 10,000 rpm (L/D = 0.2, ε = 0.4Ci): (a–d) for orbits and ring speed, (e–g) for Poincaré maps, (h) for frequency spectrum, (i) for Lyapunov exponents, and (j) for maximum Lyapunov exponent

Fig. 14

Nonlinear response evaluation at 16,000 rpm (L/D = 0.2, ε = 0.4Ci): (a–d) for orbits and ring speed, (e–g) for Poincaré maps, (h) for frequency spectrum, (i) for Lyapunov exponents, and (j) for maximum Lyapunov exponent

Fig. 15

Nonlinear response evaluation at 20,000 rpm (L/D = 0.2, ε = 0.4Ci): (a–d) for orbits and ring speed, (e–g) for Poincaré maps, (h) for frequency spectrum, (i) for Lyapunov exponents, and (j) for maximum Lyapunov exponent

Fig. 16

Nonlinear response evaluation at 40,000 rpm (L/D = 0.2, ε = 0.4Ci): (a–d) for orbits and ring speed, (e–g) for Poincaré maps, (h) for frequency spectrum, (i) for Lyapunov exponents, and (j) for maximum Lyapunov exponent

Fig. 17

Response with added unbalance on the disk at 40,000 rpm: (a) response with no unbalance (ε = 0.0), (b) response with ε = 0.2Ci, (c) response with ε = 0.4Ci, (d) response with ε = 0.6Ci

Fig. 18

Bifurcation diagrams and associated waterfall diagrams obtained using finite bearing method versus short bearing approximate (L/D = 0.2, ε = 0.4Ci): (a) Bifurcation diagram (finite bearing method), (b) bifurcation diagram (short bearing approximate), (c) waterfall diagram (finite bearing method), and (d) waterfall diagram (short bearing approximate)

Fig. 19

Comparison of orbits and ring speed ratio, which are obtained by finite element method (blue) and short bearing approximation (red) at (a) 10,000 rpm, (b) 16,000 rpm, (c) 20,000 rpm, (d) 35,000 rpm, and (e) 50,000 rpm in case of L/D = 0.2 (see figure online for color)

Fig. 20

Bifurcation diagrams and associated waterfall diagrams obtained using finite bearing method versus short bearing approximate (L/D = 0.5, ε = 0.4Ci): (a) Bifurcation diagram (finite bearing method), (b) bifurcation diagram (short bearing approximate), (c) waterfall diagram (finite bearing method), and (d) waterfall diagram (short bearing approximate)

Fig. 21

Orbits and frequency spectrum at 60,000 rpm from finite and short bearing methods (L/D = 0.5, ε = 0.4Ci)

Fig. 22

Orbits and frequency spectrum at 120,000 rpm from finite and short bearing methods (L/D = 0.5, ε = 0.4Ci)

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