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Technical Briefs

Active Wave Control of String Near Fixed Boundary

[+] Author and Article Information
Muneharu Saigo

 National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1, Namiki, Tsukuba, Ibaraki 305-8564, Japanm.saigo@aist.go.jp

Kiyoshi Takagi

 National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1, Namiki, Tsukuba, Ibaraki 305-8564, Japantakagi.k@aist.go.jp

Nobuo Tanaka

 Tokyo Metropolitan University, 6-6, Asahigaoka, Hino, Tokyo 191-0065, Japanntanaka@ccdms.cc.tmit.ac.jp

J. Vib. Acoust 130(1), 014502 (Nov 12, 2007) (4 pages) doi:10.1115/1.2731418 History: Received December 25, 2005; Revised February 13, 2007; Published November 12, 2007

We studied a wave control of a string near a fixed end. The equation of motion of a string is approximated as a lumped-parameter spring-and-mass system using the finite difference method. Finite difference equations (FDEs) for interior node points have the same set of coefficients, whereas the FDEs for boundary node points have a different set of coefficients. By using the control, the latter equation is modified to be as the same equation as the former one. The propagating wave is absorbed through a control actuator as if no boundary existed, and virtually, an infinite system is thus realized. Control is at the string position of the finite difference mesh spacing inside the fixed boundary that is free from supporting loads. We confirmed the effectiveness of the controller by numerical simulation for the traveling string and also did that by numerical simulation and by an experiment for the nontraveling string.

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Figures

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Figure 1

Stretched traveling string

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Figure 2

Finite difference model for traveling string

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Figure 3

Phase of characteristic root (Eq. 4); B+(κ): β+; B−(κ): β−

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Figure 4

Controlled response of a 100-DOF finite difference system for disturbance at node 30; G30 (i): gain of node 30 controlled at node i(=100,1); G30 (NC): gain of node 30 without control

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Figure 5

Controlled mode shape of a 100-DOF finite difference system for disturbance at node 30; Ωi(j): controlled at node j at ith natural frequency

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Figure 6

Controlled response of a 100-DOF finite difference system for disturbance at node 30; G30 (1+100): gain of node 30 controlled at both nodes 1 and 100, G30 (i): gain of node 30 controlled at node i(=100,1)

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Figure 7

Controlled mode shape of a 100-DOF finite difference system for disturbance at node 30: (a) left end controlled and (b) both ends controlled

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Figure 8

Simulation waveforms at first resonance frequency excitation: (a, b) wave control and (c) normal damping with optimal damping coefficient

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Figure 9

Experimental apparatus. Gap sensor outputs: Ch1 and Ch2 for arms of a control actuator, Ch4–Ch8 for string deflection, Ch3=(Ch1+Ch2)∕2.

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Figure 10

Experimental results at first resonance disturbance frequency: (a) timing charts and (b) mode shape

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