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Research Papers

A New Family of Higher-Order Time Integration Algorithms for the Analysis of 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,
MS 3123,
Texas A&M University,
College Station, TX 77843-3123
e-mail: jnreddy@tamu.edu

1Present address: Department of Mechanical Engineering, Korea Army Academy at Yeongcheon, Yeongcheon-si, Gyeongsangbuk-do 38900, Republic 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), 071008 (Jun 07, 2017) (17 pages) Paper No: JAM-17-1259; doi: 10.1115/1.4036821 History: Received May 17, 2017; Revised May 21, 2017

For the development of a new family of implicit higher-order time integration algorithms, mixed formulations that include three time-dependent variables (i.e., the displacement, velocity, and acceleration vectors) are developed. Equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables in the mixed formulations, and the time finite element method and the modified weighted-residual method are applied to the velocity–displacement and velocity–acceleration relations of the mixed formulations. Weight parameters are introduced and optimized to achieve preferable attributes of the time integration algorithms. Specific problems of structural dynamics are used in the numerical examples to discuss some fundamental limitations of the well-known second-order accurate algorithms as well as to demonstrate advantages of using the developed higher-order algorithms.

Copyright © 2017 by ASME
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References

Figures

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Fig. 1

Schematic presentation of time element obtained from fourth-degree Lagrange interpolations functions. Parameters τ1, τ2, and τ3 are used to adjust specific locations of the internal time nodes.

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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 bimaterial bar with continuous excitation on left edge

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Fig. 6

Comparison of center displacements at t = 100.0 (after hundred cycles of excitation)

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Fig. 7

Comparison of center velocities at t = 100.0 (after hundred cycles of excitation)

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Fig. 8

Comparison of center accelerations at t = 100.0 (after hundred cycles of excitation)

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Fig. 9

Comparison of center displacements at the beginning of excitation (0 ≤ t ≤ 1)

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Fig. 10

Comparison of center velocities at the beginning of excitation (0 ≤ t ≤ 1)

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Fig. 11

Comparison of center accelerations at the beginning of excitation (0 ≤ t ≤ 1)

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Fig. 12

Comparison of center displacements of the bimaterial bar with stiff and soft parts

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Fig. 13

Comparison of center velocities of the bimaterial bar with stiff and soft parts

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Fig. 14

Comparison of center accelerations of the bimaterial bar with stiff and soft parts

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Fig. 15

Enlarged picture of Fig. 14

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Fig. 16

Oscillation of moderately nonlinear simple pendulum with θmax = 90 deg

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Fig. 17

Oscillation of highly nonlinear simple pendulum with θmax = 179.9 deg

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Fig. 18

Comparison of angles for current (eighth- and tenth-order) and second-order algorithms (the Newmark method and the Baig and Bathe method) for mildly nonlinear case. Initial conditions are chosen to get the maximum angle π/2 = 90.0 deg at the quarter of period.

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Fig. 19

Comparison of angles for current (eighth- and tenth-order) and second-order algorithms (the Newmark method and the Baig and Bathe method) for highly nonlinear case. The second-order algorithms used Δt = T/10,000, and the current tenth-order algorithm used Δt = T/100, respectively, T being the period.

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Fig. 20

Comparison of angles for current (eighth- and tenth-order) and second-order algorithms (the Newmark method and the Baig and Bathe method) for highly nonlinear case. The second-order algorithms used Δt = T/20,000, and the current eighth- and tenth-order algorithms used Δt = T/100 and Δt = T/50, respectively, T being the period.

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