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TECHNICAL PAPERS

Residual Vibration Suppression for Duffing Nonlinear Systems With Electromagnetical Actuation Using Nonlinear Command Shaping Techniques

[+] Author and Article Information
Kuo-Shen Chen1

Department of Mechanical Engineering, National Cheng-Kung University, Tainan, Taiwan, R.O.C. 70101kschen@mail.ncku.edu.tw

Tian-Shiang Yang, Jui-Feng Yin

Department of Mechanical Engineering, National Cheng-Kung University, Tainan, Taiwan, R.O.C. 70101

1

Corresponding author.

J. Vib. Acoust 128(6), 778-789 (Mar 03, 2006) (12 pages) doi:10.1115/1.2203340 History: Received November 01, 2005; Revised March 03, 2006

Residual vibration control is crucial for numerous applications in precision machinery with negligible damping such as magnetically actuated systems. In certain magnetically actuated applications, the systems could also be highly nonlinear and conditionally stable. Although traditional command shaping techniques work well for linear and weakly nonlinear systems, they show little effects for dealing with systems with both strong structural and actuation nonlinearities. In this paper, a general input shaper design methodology for single degree of freedom systems with both Duffing spring and electromagnetic forcing nonlinearities is successfully devised using an energy approach. Following this method, two-step and three-step shapers are developed, which in the linear limit reduce to the traditional zero-vibration (ZV) and zero-vibration-and-derivative (ZVD) shapers, respectively. The robustness of these nonlinear shapers is investigated numerically through several case studies and the results show that the three-step shaper is sufficiently robust to resist significant amounts of parameter variations without exciting significant residual vibration. The two-step shaper, however, is somewhat less robust with respect to parameter variations. Meanwhile, an electromagnetically driven Duffing mechanical system is also constructed so that the performances and robustness of the nonlinear shapers in vibration suppression can be examined. It is shown that the nonlinear shapers result in a significant improvement in residual vibration suppression and settling time reduction in comparison with the traditional linearized ZV and ZVD shapers.

Copyright © 2006 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic plot of the system dynamics of an electromagnetically actuated system

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Figure 2

Schematic plot to define the design parameters of input shapers

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Figure 3

Input shaper design flow

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Figure 4

Energetics of two-step input shaping

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Figure 5

Energetics of three-step input shaping

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Figure 6

Optimal choices of the parameter α in three-step input shaping as functions of the nonlinearity parameter kr, for various values of G0

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Figure 7

The responses of a magnetically actuated Duffing (kr=1, G0=4) system subjected to different shaped step commands (amplitude=1)

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Figure 8

The residual vibration of linearized ZV and ZVD shapers under different kr and G0

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Figure 9

Effect of mass parameter uncertainty for a linear structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 10

Effect of damping uncertainty for a linear structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 11

Effect of force coefficient uncertainty for a linear structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 12

Effect of mass parameter uncertainty for a pure cubic structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 13

Effect of damping uncertainty for a cubic structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 14

Effect of force coefficient uncertainty for a cubic structure under different initial air gaps. (a) Two-step shaped and (b) three-step shaped responses.

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Figure 15

The electromagnetic actuated fixed-fixed beam (a) schematic plot and (b) experimental apparatus

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Figure 16

Two-step and linearized ZV shaped step response of the experimental apparatus at different amplitudes (a) 1mm and (b) 2mm

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Figure 17

Three-step and linearized ZVD shaped step response of the experimental apparatus at different amplitudes (a) 1mm and (b) 2mm

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Figure 18

The experimental results of the robustness of the nonlinear shapers against the mass uncertainty. (a) Two-step, and (b) three-step shaper.

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