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Research Papers

Effective Higher-Order Time Integration Algorithms for the Analysis of Linear Structural Dynamics

[+] Author and Article Information
Wooram Kim

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123
e-mail: kim.wooram@yahoo.com

J. N. Reddy

Oscar S. Wyatt Jr. Chair
Distinguished Professor
Life Fellow ASME
Department of Mechanical Engineering,
Texas A&M University,
MS 3123,
College Station, TX 77843-3123
e-mail: jnreddy@tamu.edu

1Present address: Department of Mechanical Engineering, Korea Army Academy at Yeongcheon, Yeongcheon 38900, South Korea.

2Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 17, 2017; final manuscript received May 21, 2017; published online June 7, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(7), 071009 (Jun 07, 2017) (13 pages) Paper No: JAM-17-1260; doi: 10.1115/1.4036822 History: Received May 17, 2017; Revised May 21, 2017

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

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References

Figures

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Fig. 1

Schematic presentation of time element obtained from pth-order Hermite interpolation functions

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Fig. 2

Spectral radii versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

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Fig. 3

Relative period errors versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

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Fig. 4

Damping ratios versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

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Fig. 5

Description of three-degrees-of-freedom spring system used by Bathe and Nho

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Fig. 6

Displacement of second node for various methods

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Fig. 7

Velocity of second node for various methods

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Fig. 8

Acceleration of second node for various methods

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Fig. 9

Acceleration of second node for various methods (enlarged image of Fig. 8 without the trapezoidal rule)

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Fig. 10

Computational domain and mesh used to analyze 2D wave equation

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Fig. 11

Comparison of center displacements at 0≤t≤5 for various methods

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Fig. 12

Comparison of center displacements at 160≤t≤165 for various methods

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