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Research Papers

Accounting for Nonlinearities in Open-Loop Protocols for Symmetry Fault Compensation

[+] Author and Article Information
Louis A. DiBerardino, III

e-mail: diberard@illinois.edu

Harry Dankowicz

e-mail: danko@illinois.edu
Department of Mechanical Science
and Engineering,
University of Illinois at Urbana–Champaign,
Urbana, IL 61801

1Corresponding author.

ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC2012-70387)

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 3, 2012; final manuscript received July 28, 2013; published online September 12, 2013. Assoc. Editor: Henryk Flashner.

J. Comput. Nonlinear Dynam 9(2), 021002 (Sep 12, 2013) (10 pages) Paper No: CND-12-1121; doi: 10.1115/1.4025193 History: Received August 03, 2012; Revised July 28, 2013

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom, and with desirable symmetry properties, and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may coexist. We further formulate a candidate optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol may exhibit discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.

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References

Figures

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Fig. 1

A two-degree-of-freedom coupled mechanical oscillator used to illustrate the fault-correcting strategies and the influence of nonlinearity on the robustness of such a compensatory scheme.

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Fig. 2

The response amplitude ‖x1‖∞ versus ω for the symmetric system (α0 = 0) for a1 = {0.1,0.25,0.75,1.5,3}.

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Fig. 3

The response amplitude ‖x1‖∞ versus ω for a system near symmetry (α0 = 0.1) for a1 = {0.1,0.25,0.75,1.5,3} (a) and corresponding Csym values from Eq. (18) (b).

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Fig. 4

The response amplitude ‖x1‖∞ versus ω for an asymmetric system (α0 = 2) for a1 = {0.1,0.25,0.75,1.5,3} (a) and corresponding Csym values from Eq. (18) (b). Rectangle denotes plot range of Fig. 5.

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Fig. 5

The response amplitude ‖x1‖∞ versus ω for α0 = 2 near a branch-point bifurcation, inside the rectangle shown in Fig. 4. Arrows indicate increasing a1 (a1∈[0.1,0.98]).

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Fig. 6

The graph of f(xb(Δω,a1),Δω,a1) versus Δω of the symmetric system (Δα = 0) for a1 = {0.2,0.3,0.4}.

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Fig. 7

The graph of f(xb(Δω,a1),Δω,a1) versus Δω of an asymmetric system (Δα = 2) for a1 = {0.1,0.175,0.25}.

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Fig. 8

The response amplitude, x (similar to ‖x1‖∞ in previous section), versus Δω for the symmetric system, for a1 = {0.1,0.25,0.75,1.5,3}.

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Fig. 9

The response amplitude, x (similar to ‖x1‖∞ in previous section), versus Δω for an asymmetric system (Δα = 2), for a1 = {0.1,0.25,0.75,1.5,3}. Rectangle denotes plot range of Fig. 10.

Grahic Jump Location
Fig. 10

The response amplitude, x (similar to ‖x1‖∞ in previous section), versus Δω for an asymmetric system (Δα = 2) near a branch-point bifurcation, inside the rectangle shown in Fig. 9. Arrows indicate increasing a1 (a1∈[0.1,1]).

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Fig. 13

The response amplitude, x (similar to ‖x1‖∞ in previous section), versus Δω for an asymmetric system (Δα = 2) near a branch-point bifurcation, inside the rectangle shown in Fig. 12. Arrows indicate increasing a1 (a1∈[0.1,2]).

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Fig. 12

The response amplitude, x (similar to ‖x1‖∞ in previous section), versus Δω for an asymmetric system (Δα = 2), for a1 = {0.1,0.25,0.75,1.5,3}. Rectangle denotes plot range of Fig. 13.

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Fig. 11

The graph of f(xb(Δω,a1),Δω,a1) versus Δω of an asymmetric system (Δα = 2) for a1 = {0.2, 0.245, 0.275}

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Fig. 15

Forcing amplitude a1 and frequency ω versus asymmetry α0 of the periodic solution based on the numerical method to maintain at the local frequency-response amplitude maximum. Arrows indicate direction of continuation of the solution.

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Fig. 14

Response amplitude ‖x11‖∞ versus forcing frequency ω of the periodic solution at α0 = 2 prior to applying control strategy.

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