Research Papers

Impulsive Steering Between Coexisting Stable Periodic Solutions With an Application to Vibrating Plates

[+] Author and Article Information
Daniël W. M. Veldman

Department of Mechanical Engineering,
Eindhoven University of Technology,
PO Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: d.w.m.veldman@tue.nl

Rob H. B. Fey

Department of Mechanical Engineering,
Eindhoven University of Technology,
PO Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: r.h.b.fey@tue.nl

Hans Zwart

Department of Applied Mathematics,
University of Twente,
PO Box 217,
Enschede 7500 AE , The Netherlands;
Department of Mechanical Engineering,
Eindhoven University of Technology,
PO Box 513,
Eindhoven 5600 MB, The Netherlands
e-mail: h.j.zwart@tue.nl

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 16, 2016; final manuscript received July 5, 2016; published online September 6, 2016. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 12(1), 011013 (Sep 06, 2016) (10 pages) Paper No: CND-16-1140; doi: 10.1115/1.4034273 History: Received March 16, 2016; Revised July 05, 2016

Single-degree-of-freedom (single-DOF) nonlinear mechanical systems under periodic excitation may possess multiple coexisting stable periodic solutions. Depending on the application, one of these stable periodic solutions is desired. In energy-harvesting applications, the large-amplitude periodic solutions are preferred, and in vibration reduction problems, the small-amplitude periodic solutions are desired. We propose a method to design an impulsive force that will bring the system from an undesired to a desired stable periodic solution, which requires only limited information about the applied force. We illustrate our method for a single-degree-of-freedom model of a rectangular plate with geometric nonlinearity, which takes the form of a monostable forced Duffing equation with hardening nonlinearity.

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Fig. 1

Domains of attraction in the Poincaré section

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Fig. 2

Illustration of Algorithm 1

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Fig. 3

Illustration of Algorithm 2

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Fig. 4

Dimensions and loads of the considered rectangular plate

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Fig. 5

Corresponding in-plane shape functions U(x, y) and V(x, y)

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Fig. 6

Frequency–amplitude plot of Eq. (71) for the parameter values in Table 1, computed in MATCONT [21]

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Fig. 7

Time history of an application of Algorithm 1 (Δt=0.050T)

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Fig. 8

Determining the maximal duration Δt for Algorithm 1 (Δt=0.050T)

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Fig. 9

Determining the maximal duration for Algorithm 2 (Δt=0.13T)




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