Active Vibration Control of a Flexible Beam Using a Buckling-Type End Force

[+] Author and Article Information
Shahin Nudehi, Steven W. Shaw

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226

Ranjan Mukherjee

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226mukherji@egr.msu.edu

In fact, this buckling load can easily be derived by noting that the line of action of the end load always passes through the two end points of the beam, resulting in a situation that is equivalent to the buckling problem of a beam with pinned ends.

If the force is applied for relatively short durations, it may be possible to utilize loads larger than the buckling load. However, in this preliminary study, we limit ourselves to the conservative assumption.

C1 should not be confused with the positive definite square matrix C in Eq. 25.

Digital signal processor.

The settling time was defined as Ts=4ζω, and both ζ and ω were computed numerically from the vibration plots.

J. Dyn. Sys., Meas., Control 128(2), 278-286 (Mar 25, 2005) (9 pages) doi:10.1115/1.2192836 History: Received May 12, 2004; Revised March 25, 2005

In this paper, we explore the use of end forces for vibration control in structural elements. The process involves vibration measurement and observer-based estimation of modal amplitudes, which are used to determine when to apply an end load such that it will remove vibration energy from the structure. For this study, we consider transverse vibration of a cantilever beam with a buckling-type end load that can be switched between two values, both of which are below the buckling load. The stability of the control system is proven using Lyapunov stability theory and its effectiveness is demonstrated using simulations and physical experiments. It is shown that the effectiveness of the approach is affected by the bandwidth of the actuator and the attendant characteristics of the filter, the level of the control force, and the level of bias in the end force. The experiments employ a beam fitted with a cable mechanism and motor for applying the end force, and a piezoelectric patch for taking vibration measurements. It is shown that the first two modes of the beam, whose natural frequencies are less than the bandwidth of the motor, are very effectively controlled by the proposed scheme.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

A flexible cantilever beam with an end force

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Figure 2

Simulation of (a) decay in modal amplitude a1 due to damping, (b), (c) decay in modal amplitudes a1 and a2 due to control in the presence of damping, and (d) plot of the control action.

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Figure 3

Control design based on output filtering

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Figure 4

Plot of modal amplitudes a1 and a2, and the control action u for the modified control design in Sec. 4.

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Figure 5

Control design based on bias tension and output filtering

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Figure 6

A comparion of the “smf” function of MATLAB and the memory-less nonlinearity in Figs.  35

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Figure 7

Experimental setup

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Figure 8

Free vibration of the beam in the (a) absence of bias tension, and (b) presence of 20N bias tension

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Figure 9

Vibration suppression using active control

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Figure 10

Vibration suppression using a one-mode dynamic model results in spillover

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Figure 11

The role of the low-pass filter in reducing the effect of spillover




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