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Research Papers

Free Response and Absolute Vibration Suppression of Second-Order Flexible Structures—The Traveling Wave Approach

[+] Author and Article Information
Lea Sirota

Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israellsirota@technion.ac.il

Yoram Halevi1

Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israelyoramh@technion.ac.il

1

Corresponding author.

J. Vib. Acoust 132(3), 031008 (Apr 23, 2010) (10 pages) doi:10.1115/1.4000771 History: Received November 30, 2008; Revised September 13, 2009; Published April 23, 2010; Online April 23, 2010

The paper considers the problem of suppressing the free vibration, induced by nonzero initial conditions, in a flexible system governed by the wave equation. First an exact response of the system, with general linear boundary conditions, is derived in terms of propagating waves that are reflected from the boundaries. The solution is explicit and with clear physical interpretation. The general expressions for the response are then used to investigate the behavior of the system under control with the absolute vibration suppression controller, which was originally designed for tracking control. It is shown that the vibration suppression properties of this controller apply also to nonzero initial conditions. In cases where the load end is free or contains only damping, the vibration stops completely in finite time and if it contains only inertia and damping it decays exponentially without vibration.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

A schematic diagram of the loaded rod

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Figure 2

Snapshots of the displacement distribution along the rod of the system in the example: total displacement—solid, progressing wave—dashed, regressing wave—dotted

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Figure 3

Response at x=L/4 of the system in the example 1

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Figure 4

The closed loop control scheme in the case of nonzero initial conditions

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Figure 5

System with free actuating end J2=0 and D2=ϕ/3: (a) open loop response at x=L, (b) closed loop response at x=L, and (c) control signal

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Figure 6

System with free actuating end J2=1 and D2=0: (a) open loop response at x=L, (b) closed loop response at x=L, and (c) control signal

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Figure 7

Animation of the displacement response of a free-fixed string with AVS controller

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