Journal of Central South University

, Volume 24, Issue 5, pp 1090–1097 | Cite as

Comparison and assessment of time integration algorithms for nonlinear vibration systems

  • Chao Yang (杨超)
  • Bao-zhu Yang (杨宝柱)
  • Tao Zhu (朱涛)
  • Shou-ne Xiao (肖守讷)


A corrected explicit method of double time steps (CEMDTS) was introduced to accurately simulate nonlinear vibration problems in engineering. The CEMDTS, the leapfrog central difference method, the Newmark method, the generalized-α method and the precise integration method were implemented in typical nonlinear examples for the purpose of comparison. Both conservative and non-conservative systems were examined. The results show that it isn’t unconditionally stable for the precise integration method, the Newmark method and the generalized-α method in nonlinear systems. The stable interval of the three methods is less than that of the CEMDTS. When a small time step (Δt≤T min/20) is employed, the precise integration method is endowed with the highest accuracy while the CEMDTS possesses the smallest computation effort. However, the CEMDTS becomes the most accurate one when the time step is large (Δt≥0.3T min) or the system is strongly nonlinear. Therefore, the CEMDTS is more applicable to the nonlinear vibration systems.

Key words

nonlinear vibration conservative system explicit algorithm accuracy 


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  1. [1]
    Zhai W M, Gao J, Liu P, Wang K. Reducing rail side wear on heavy-haul railway curves based on wheel-rail dynamic interaction [J]. Vehicle System Dynamics, 2014, 52(s): 440–454.CrossRefGoogle Scholar
  2. [2]
    Zhai W M, Wang K Y, Cai C B. Fundamentals of vehicle-track coupled dynamics [J]. Vehicle System Dynamics, 2009, 47(11): 1349–1376.CrossRefGoogle Scholar
  3. [3]
    Zhai Wan-ming, Xia He. Train-track-bridge dynamic interaction: Theory and engineering application [M]. Beijing: Science Press, 2011. (in Chinese)Google Scholar
  4. [4]
    Hilber H M, Hughes T J R, Taylor R L. Improved numerical dissipation for time integration algorithms in structural dynamics [J]. Earthquake Engineering and Structural Dynamics, 1977, 5: 283–292.CrossRefGoogle Scholar
  5. [5]
    Chung J, Hulbert G M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-? method [J]. Journal of Applied Mechanics, 1993, 60(2): 371–375.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Wen W, Jian K, Luo S. An explicit time integration method for structural dynamics using septuple B-spline functions [J]. International Journal for Numerical Methods in Engineering, 2014, 97(9): 629–657.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Xie Y M. An assessment of time integration schemes for non-linear dynamic equations [J]. Journal of Sound and Vibration, 1996, 192(1): 321–331.CrossRefGoogle Scholar
  8. [8]
    Zhong W X, Williams F W. A precise time step integration method [J]. Proc IMechE, Part C: Journal of Mechanical Engineering Science, 1994, 208: 427–430.CrossRefGoogle Scholar
  9. [9]
    Zhong Wan-xie. On precise time integration method for structural dynamics [J]. Journal of Dalian University of Technology, 1994, 34(2): 131–135. (in Chinese)MathSciNetMATHGoogle Scholar
  10. [10]
    Chen R L, Zeng Q Y, Zhang J Y. New algorithm applied to vibration equations of time-varying systems [J]. Journal of Central South University of Technology, 2008, 15(s1): 57–60.CrossRefGoogle Scholar
  11. [11]
    Chang S Y. An explicit method with improved stability property [J]. International Journal for Numerical Methods in Engineering, 2009, 77(8): 1100–1120.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Chang S Y. A family of noniterative integration methods with desired numerical dissipation [J]. International Journal for Numerical Methods in Engineering, 2014, 100(1): 62–86.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Masuri S U, Hoitink A, Zhou X, Tamma K K. Algorithms by design: A new normalized time-weighted residual methodology and design of a family of energy-momentum conserving algorithms for non-linear structural dynamics [J]. International Journal for Numerical Methods in Engineering, 2009, 79: 1094–1146.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Yang C, Xiao S, Lu L, Zhu T. Two dynamic explicit methods based on double time steps [J]. Proc IMechE, Part K: Journal of Multi-body Dynamics, 2014, 228(3): 330–337.Google Scholar
  15. [15]
    Zhang Xiong, Wang Tian-shu. Computational dynamics [M]. Beijing: Tsinghua University Press, 2007: 147–184. (in Chinese)Google Scholar
  16. [16]
    Leontiev V A. Extension of LMS formulations for L-stable optimal integration methods with U0-V0 overshoot properties in structural dynamics: The level-symmetric (LS) integration methods [J]. International Journal for Numerical Methods in Engineering, 2007, 71: 1598–1632.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Shao Hui-ping, Cai Cheng-wen. A three parameters algorithm for numerical integration of structural dynamic equations [J]. Chinese Journal of Applied Mechanics, 1988, 5(4): 76–82. (in Chinese)Google Scholar
  18. [18]
    Lu He-xiang, Yu Hong-jie, Qiu Chun-hang. An integral equation of nonlinear dynamics and its solution method [J]. Acta Mechanica Solida Sinica, 2001, 22(3): 303–308. (in Chinese)Google Scholar
  19. [19]
    Chang S Y. A new family of explicit methods for linear structural dynamics [J]. Computers & Structures, 2010, 88(11, 12): 755–772.CrossRefGoogle Scholar
  20. [20]
    Yang Chao, Xiao Shou-ne, Yang Guang-wu, Zhu Tao, Yang Bing. Non-dissipative explicit time integration methods of the same class [J]. Journal of Vibration Engineering, 2015, 28(3): 441–448. (in Chinese)Google Scholar

Copyright information

© Central South University Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Chao Yang (杨超)
    • 1
    • 2
  • Bao-zhu Yang (杨宝柱)
    • 1
  • Tao Zhu (朱涛)
    • 1
  • Shou-ne Xiao (肖守讷)
    • 1
  1. 1.State Key Laboratory of Traction PowerSouthwest Jiaotong UniversityChengduChina
  2. 2.School of Mechanical, Electronic and Control EngineeringBeijing Jiaotong UniversityBeijingChina

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