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Research Papers

A High Precision Direct Integration Scheme for Nonlinear Dynamic Systems

[+] Author and Article Information
Kuinian Li

School of Civil and Environmental Engineering, University of the Witwatersrand, Johannesburg, 2050 Johannesburg, South Africakuinian.li@wits.ac.za

Antony P. Darby1

Department of Architecture and Civil Engineering, University of Bath, Bath, BA2 7AY, UKabsapd@bath.ac.uk

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041008 (Aug 25, 2009) (10 pages) doi:10.1115/1.3192129 History: Received May 29, 2008; Revised February 03, 2009; Published August 25, 2009

Based on the high precision direct (HPD) integration scheme for linear systems, a high precision direct integration scheme for nonlinear (HPD-NL) dynamic systems is developed. The method retains all the advantages of the standard HPD scheme (high precision with large time-steps and computational efficiency) while allowing nonlinearities to be introduced with little additional computational effort. In addition, limitations on minimum time step resulting from the approximation that load varies linearly between time-steps are reduced by introducing a polynomial approximation of the load. This means that, in situations where a rapidly varying or transient dynamic load occurs, a larger time-step can still be used while maintaining a good approximation of the forcing function and, hence, the accuracy of the solution. Numerical examples of the HPD-NL scheme compared with Newmark’s method and the fourth-order Runge–Kutta (Kutta 4) method are presented. The examples demonstrate the high accuracy and numerical efficiency of the proposed method.

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Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Response of a linear SDOF system subject to nonlinear loading

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Figure 3

Time history of angular rotation response calculated with different time-steps: HPD-NL versus Kutta 4

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Figure 4

Time history responses obtained with HPD-NL, Kutta 4, and Newmark’s method (time-step τ=0.00001 s)

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Figure 5

Phase portraits (time-step τ=0.00001 s)

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Figure 6

Time history responses obtained with HPD-NL, Kutta 4, and Newmark’s method, comparison between τ=0.00001 s and 0.015 s

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Figure 7

Phase portraits time-step comparison between τ=0.00001 s and 0.015 s

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Figure 8

Time history responses obtained with HPD-NL, Kutta 4, and Newmark’s method (time-step τ=0.05 s)

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Figure 9

Phase portraits (time-step τ=0.05 s)

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Figure 10

Spherical pendulum damper system

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Figure 11

Displacement response calculated with a time-step of 0.001 s: HPD-NL versus Kutta 4

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Figure 12

Displacement response calculated with a time-step of 0.006 s: HPD-NL versus Kutta 4

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