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Research Papers

Vibration Control and Trajectory Tracking for General In-Plane Motion of an Euler–Bernoulli Beam Via Two-Time Scale and Boundary Control Methods

[+] Author and Article Information
Amir Lotfazar

Department of Mechanical Engineering, Shiraz University, Mollasadra Avenue, Shiraz, Fars 71348-51154, Iranlotfavar@shirazu.ac.ir

Mohammad Eghtesad

Department of Mechanical Engineering, Shiraz University, Mollasadra Avenue, Shiraz, Fars 71348-51154, Iraneghtesad@shirazu.ac.ir

Ali Najafi

Department of Mechanical Engineering, Shiraz University, Mollasadra Avenue, Shiraz, Fars 71348-51154, Iranalinajafi62@gmail.com

J. Vib. Acoust 130(5), 051009 (Aug 14, 2008) (11 pages) doi:10.1115/1.2948406 History: Received July 01, 2007; Revised March 30, 2008; Published August 14, 2008

In this paper, general in-plane trajectory tracking problem of a flexible beam is studied. To obtain the dynamic equations of motion of the beam, Hamiltonian dynamics is used and then Lagrange’s equations of beam dynamics and corresponding expressions for boundary conditions are derived. Resulting equations show that the coupled beam dynamics including beam vibration and its rigid in-plane motion take place in two different time domains. By using two-time scale (TTS) control theory, a control scheme is elaborated that makes the orientation and position of the mass center of the beam track a desired trajectory while suppressing its vibration. TTS composite controller has two parts: one is a tracking controller designed for the slow (rigid) subsystem, and the other one is a stabilizing controller for the fast (flexible) subsystem. For the fast subsystem, the proposed boundary control (BC) method does not require any information about vibration along the beam except at the end points, nor requires discretizing the partial differential equation of beam vibration to a set of ordinary differential equations. So, the method avoids the need for instruments to measure data from vibration of any point along the beam or designing an observer for estimating this information. Also, the proposed method prevents control spillover due to discretization. Simulation results show that fast BC is able to remove undesirable vibration of the flexible beam and the slow controller provides very good trajectory tracking with acceptable actuating forces/moments.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

A flexible beam with boundary forces/moments

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Figure 2

Rigid position tracking without fast BC

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Figure 3

Rigid velocity tracking without fast BC

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Figure 4

Rigid position tracking error without fast BC

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Figure 5

Lateral vibration at the center point and the second end points without fast BC

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Figure 6

Control commands at the first end without fast BC

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Figure 7

Control commands at the second end without fast BC

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Figure 8

Rigid position tracking with fast BC

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Figure 9

Rigid velocity tracking with fast BC

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Figure 10

Rigid position tracking error with fast BC

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Figure 11

Lateral vibration at the center point and the second end with fast BC

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Figure 12

Control commands at the first end with fast BC

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Figure 13

Control commands at the second end with fast BC

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