0
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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

The constrained polygon domain Ω1 for a discontinuous dynamical system

Grahic Jump Location
Fig. 2

Boundaries and corners in the constrained single domain Ω1

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
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)

Grahic Jump Location
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)

Grahic Jump Location
Fig. 7

Eight generic local mappings

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
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)

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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).

Grahic Jump Location
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).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In