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Research Papers

Estimating Impulsive Loads in Duffing's Equation Using Two Methods

[+] Author and Article Information
Dong-Cherng Lin1

Department of Product Development and Design,  Taiwan Shoufu University, Matou, Tainan 72153, Taiwan, R. O. C.dclin@tsu.edu.tw

1

Mailing address: Department of Product Development and Design, Taiwan Shoufu University, Matou, Tainan 72153, Taiwan, Republic of China.

J. Vib. Acoust 134(3), 031001 (Apr 24, 2012) (8 pages) doi:10.1115/1.4005653 History: Received November 17, 2009; Revised October 12, 2011; Published April 23, 2012; Online April 24, 2012

This work determines the time-varying impulsive loads, called inputs, in a nonlinear system using two novel input estimation inverse algorithms. Both algorithms use the extended Kalman filter with two different recursive estimators to determine impulsive loads. The extended Kalman filter generates the residual innovation sequences. The estimators use the residual innovation sequences to evaluate the magnitudes and, therefore, the onset time histories of the impulsive loads. Based on the two regression equations, a recursive least-squares estimator with a tunable fading factor is called a conventional input estimation with an adaptive weighting fading factor called an adaptive weighting input estimation. Both are used to estimate on-line inputs involving measurement noise and modeling errors. Numerical simulations of a nonlinear system, Duffing’s equation, demonstrate the accuracy of the proposed methods. Simulation results show that the proposed methods accurately estimate impulsive loads, and the AWIE approach has superior robust estimation capability than the CIE method in the nonlinear system.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The normalized estimated errors of the estimated half-sine impulsive loads versus fading factors at R = 1 × 10−6 to 1 × 10−14 with Q = 1 × 10−6 for the Duffing equation

Grahic Jump Location
Figure 2

The normalized estimated errors of the estimated rectangular impulsive loads versus fading factors at R = 1 × 10−6 to 1 × 10−14 with Q = 1 × 10−6 for the Duffing equation

Grahic Jump Location
Figure 3

The normalized estimated errors of the estimated triangular impulsive loads versus fading factors at R = 1 × 10−6 to 1 × 10−14 with Q = 1 × 10−6 for the Duffing equation

Grahic Jump Location
Figure 4

Input estimation for Duffing equation subjected to half-sine impulsive load: (a) AWIE method, (b) CIE method (γ = 0.1), and (c) CIE method (γ = 0.9) (force unit, kg-m-ms−2 )

Grahic Jump Location
Figure 5

Input estimation for Duffing equation subjected to rectangular impulsive load: (a) AWIE method, (b) CIE method (γ = 0.1), and (c) CIE method (γ = 0.9) (force unit, kg-m-ms−2 )

Grahic Jump Location
Figure 6

Input estimation for Duffing equation subjected to triangular impulsive load: (a) AWIE method, (b) CIE method (γ = 0.1), and (c) CIE method (γ = 0.9) (force unit, kg-m-ms−2 )

Grahic Jump Location
Figure 7

Input estimation for Duffing equation subjected to consecutive impulsive loads: (a) AWIE method, (b) CIE method (γ = 0.1), and (c) CIE method (γ = 0.9) (force unit, kg-m-ms−2 )

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