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Research Papers

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

[+] Author and Article Information
Wonmo Kang, Bryan Wilcox, Harry Dankowicz

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Phanikrishna Thota

Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UK

SLIDECONT contains an extensive functionality for the bifurcation analysis of equilibria in Fillipov systems, including continuation of pseudo-equilibria and connecting orbits between pseudo-equilibria. TCˆ, on the other hand, is restricted to the continuation of periodic system trajectories.

J. Comput. Nonlinear Dynam 4(1), 011009 (Nov 12, 2008) (8 pages) doi:10.1115/1.3007975 History: Received September 18, 2007; Revised February 19, 2008; Published November 12, 2008

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of a sample hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, period-doubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

A schematic of the impact microactuator, in which the impactor is shown in its equilibrium position in the absence of excitation and the electrostatic excitation is represented by the voltage amplitude V. Figure modified from Ref. 8 and reproduced with permission from Institute of Physics Publishing.

Grahic Jump Location
Figure 2

The connectivity graph corresponding to the impact microactuator. Each segment of a trajectory of the corresponding hybrid dynamical system is represented by one of the index vectors shown here.

Grahic Jump Location
Figure 3

(Upper panel) A branch of periodic solution trajectories with base unit signature {J5} obtained from a one-parameter continuation with varying V for ω≈0.9189. (Lower panel) The grazing periodic trajectory corresponding to G-1 obtained for V≈0.5876 in the one-parameter continuation. The terminal point of the J5 segment is given by x≈(…00.506.264)T. Here, and in the later figures, ‖x‖2 represents a signature-dependent solution norm implemented in Auto 97.

Grahic Jump Location
Figure 4

Period-doubling (dotted) and grazing bifurcation (solid) curves obtained in a two-parameter continuation with signature {J5}. The initial periodic solutions for these continuations are obtained from the one-parameter continuation shown in the upper panel of Fig. 3.

Grahic Jump Location
Figure 5

(Upper panel) A branch of periodic solution trajectories with base unit signature {J1,J8} obtained from a one-parameter continuation with varying V for ω≈0.9189. (Lower panel) The grazing periodic trajectory corresponding to G-2 obtained for V≈0.7305 in the one-parameter continuation. The terminal point of the J1 segment is given by x≈(…00.50.6743.846)T.

Grahic Jump Location
Figure 6

(Upper panel) A branch of periodic solution trajectories with base unit signature {J1,J8,J1,J8} obtained from a one-parameter continuation with varying V for ω≈0.9189. (Lower panel) The grazing periodic trajectory corresponding to G-3 obtained for V≈0.8095 in the one-parameter continuation. The terminal point of the longest J1 segment is given by x≈(…00.50.7942773.509)T.

Grahic Jump Location
Figure 7

(Upper panel) The grazing periodic trajectory obtained for V≈0.8559 in the one-parameter continuation of a six-segment solution with base unit signature {J1,J8,J1,J8,J1,J8}. (Lower panel) The grazing periodic trajectory obtained for V≈0.8819 in the one-parameter continuation of an eight-segment solution with base unit signature {J1,J8,J1,J8,J1,J8,J1,J8}.

Grahic Jump Location
Figure 8

Bifurcation diagram showing period-doubling, saddle-node, and grazing bifurcation curves corresponding to periodic solutions with one or several segments. Here, PD-n, SN-n, and G-n represent the period-doubling, saddle-node, and grazing bifurcation curves corresponding to periodic solutions with n impacts per period. Dotted, dashed, and solid curves represent period-doubling, saddle-node, and grazing bifurcation curves, respectively.

Grahic Jump Location
Figure 9

Grazing bifurcation curves obtained through the two-parameter continuation

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