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Research Papers

Neimark-Sacker Bifurcations Near Degenerate Grazing Point in a Two Degree-of-Freedom Impact Oscillator

[+] Author and Article Information
Shan Yin

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
Hunan University,
Changsha 410082, Hunan, China;
School of Mechanical and
Mechatronic Engineering,
FEIT,
University of Technology Sydney,
PO Box 123,
Broadway NSW 2007, Australia

Jinchen Ji

School of Mechanical and
Mechatronic Engineering,
FEIT,
University of Technology Sydney,
PO Box 123
Broadway NSW 2007, Australia

Shuning Deng

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
Hunan University,
Changsha 410082, Hunan, China

Guilin Wen

State Key Laboratory of Advanced Design
and Manufacture for Vehicle Body,
Hunan University,
Changsha 410082, Hunan, China;
School of Mechanical and Electric Engineering,
Guangzhou University,
Guangzhou 510006, Guangdong, China
e-mail: glwen@gzhu.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 26, 2018; final manuscript received August 12, 2018; published online September 12, 2018. Assoc. Editor: Katrin Ellermann.

J. Comput. Nonlinear Dynam 13(11), 111007 (Sep 12, 2018) (8 pages) Paper No: CND-18-1126; doi: 10.1115/1.4041236 History: Received March 26, 2018; Revised August 12, 2018

Saddle-node or period-doubling bifurcations of the near-grazing impact periodic motions have been extensively studied in the impact oscillators, but the near-grazing Neimark-Sacker bifurcations have not been discussed yet. For the first time, this paper uncovers the novel dynamic behavior of Neimark-Sacker bifurcations, which can appear in a small neighborhood of the degenerate grazing point in a two degree-of-freedom impact oscillator. The higher order discontinuity mapping technique is used to determine the degenerate grazing point. Then, shooting method is applied to obtain the one-parameter continuation of the elementary impact periodic motion near degenerate grazing point and the peculiar phenomena of Neimark-Sacker bifurcations are revealed consequently. A two-parameter continuation is presented to illustrate the relationship between the observed Neimark-Sacker bifurcations and degenerate grazing point. New features that differ from the reported situations in literature can be found. Finally, the observed Neimark-Sacker bifurcation is verified by checking the existence and stability conditions in line with the generic theory of Neimark-Sacker bifurcation. The unstable bifurcating quasi-periodic motion is numerically demonstrated on the Poincaré section.

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Figures

Grahic Jump Location
Fig. 1

Schematic model of the two degree-of-freedom mechanical oscillator with a clearance [27]

Grahic Jump Location
Fig. 2

Determination of the degenerate grazing point in a two degree-of-freedom impact oscillator. The black cross indicates for the chosen degenerate grazing point.

Grahic Jump Location
Fig. 3

Neimark-Sacker bifurcation at ω=ωde−0.0005: (a) one-parameter continuation obtained by shooting method. Here, ‘NS’ denotes Neimark-Sacker, (b) evolution of the eigenvalues of the two branches of 1/1 motions when d>d*, (c) partial enlarged part of (b), and (d) the conjugate eigenvalues of the lower branched 1/1 motion near the Neimark-Sacker bifurcation point.

Grahic Jump Location
Fig. 4

Neimark-Sacker bifurcation at ω=ωde+0.0006. The two main eigenvalues in modulus collide to generate the required conjugate eigenvalues. ‘NS’ denotes for Neimark-Sacker.

Grahic Jump Location
Fig. 5

Transition of the critical eigenvalues of the 1/1 motion near the degenerate grazing point. The blue circle in (b) indicates the intersection point between the two lines.

Grahic Jump Location
Fig. 6

The unstable quasi-periodic invariant set on the Poincaré section Π2: (a) Unstable invariant set is inside the chaos at d−d*=−6×10−7, (b) unstable invariant set is outside the stable fixed point of 1/1 motion at d−d*=−6×10−7, (c) overlap of the two subplots (a-b) to show the relative locations between the chaos, unstable invariant set and 1/1 motion. A slight change of the initial value can lead to different attractors, and (d) Three isolated invariant sets at different bifurcation parameters and drawn by continuous lines. Blue line at d−d*=−6×10−7, black line at d−d*=−5.2×10−7, and red line at d−d*=−4×10−7.

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