Research Papers

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

[+] Author and Article Information
Jianzhe Huang

Department of Energy and Power Engineering,
Harbin Engineering University,
Harbin 150001, China

Albert C. J. Luo

Department of Mechanical and
Industrial Engineering,
Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1805
e-mail: aluo@siue.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 1, 2016; final manuscript received April 1, 2017; published online September 7, 2017. Assoc. Editor: Tomasz Kapitaniak.

J. Comput. Nonlinear Dynam 12(6), 061014 (Sep 07, 2017) (11 pages) Paper No: CND-16-1595; doi: 10.1115/1.4036518 History: Received December 01, 2016; Revised April 01, 2017

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

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Fig. 1

The constrained polygon domain Ω1 for a discontinuous dynamical system

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Fig. 3

A bouncing flow at the boundary ∂Ω1∞(1)

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Fig. 4

The bouncing flow from x(α1) to x(5) at the corner Σ1∞(5)

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Fig. 2

Boundaries and corners in the constrained single domain Ω1

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Fig. 5

Global mappings: (a) P1 switching on boundary ∂Ω1∞(1), (b) P2 switching on boundary ∂Ω1∞(2), (c) P3 switching on boundary ∂Ω1∞(3), and (d) P4 switching on boundary ∂Ω1∞(4)

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Fig. 6

Global mappings: (a) P5 switching at corner ∑1∞(14), (b) P6 switching at corner ∑1∞(12), (c) P7 switching at boundary ∑1∞(23), and (d) P8 switching at corner ∑1∞(34)

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Fig. 7

Eight generic local mappings

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Fig. 8

Parameter map (Ω, F0(1)) with other system parameters in Eq. (31)

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Fig. 9

Bifurcation scenario varying excitation frequency Ω at F0(1)=3.00 with system parameters in Eq. (31): (a) switching state xk and (b) switching phase mod(Ωtk, 2π)

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Fig. 10

A global view for analytical prediction varying excitation frequency Ω at F0(1)=3.00 with system parameters in Eq. (31): (a) switching state, (b) switching phase, (c) real parts of eigenvalues, and (d) magnitudes of eigenvalues

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Fig. 11

Analytical prediction varying excitation frequency Ω at F0(1)=3.00 with system parameters in Eq. (31). Zoom-1 view 1: (a) Switching state and (b) switching phase. Zoom-2 view: (c) switching state and (d) switching phase.

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Fig. 12

Periodic motion relative to mapping P(21)22346 at F0(1)=3.00 with parameters in Eq. (31) for Ω=4.1: (a) trajectory, (b) state response, (c) zoomed view around corner ∑1∞(8) with entering that corner, and (d) zoomed view around corner ∑1∞(6) with entering that corner. The initial conditions are (xi,x˙i,Ωti)≈(−0.0151,−0.9835,1.7053).

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Fig. 13

Periodic flow relative to mapping P23 at F0(1)=3.00 withparameters in Eq. (31) for Ω=5.1: (a) trajectory and (b) state response. The initial condition is (xi,x˙i,Ωti)≈(−0.1719,0.8281,0.7018).

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Fig. 14

Periodic flow of mapping P621,234 with system parameters in Eq. (32) for Ω=1.15: (a) trajectory and (b) state variable x. The initial condition is (xi,x˙i,Ωti)≈(3.6670,-1.3331,1.8576).

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Fig. 15

Periodic flow relative to mapping P61(21)37(32)81461(21)37(32)914 with system parameters in Eq. (33) for Ω=5 : (a) trajectory and (b)state response x. The initial conditions are (xi,x˙i,Ωti)≈(1.4576,-3.5425,1.7326).



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