Weak Instability in Direct Integration Methods for Structural Dynamics

Authors: Shuenn-Yih Chang, Chiu-Li Huang

Abstract:

Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.

Keywords: Dynamic analysis, high frequency, integration method, overshoot, weak instability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3299759

References:

[1] S. Y. Chang, “Explicit pseudodynamic algorithm with unconditional stability,” Journal of Engineering Mechanics, ASCE, vol. 128, no. 9, pp. 935-947, 2002.
[2] S. Y. Chang, “Improved explicit method for structural dynamics,” Journal of Engineering Mechanics, ASCE, vol. 133, no. 7, pp. 748-760, 2007.
[3] S. Y. Chang, “An explicit method with improved stability property,” International Journal for Numerical Method in Engineering, vol. 77, no. 8, pp. 1100-1120, 2009.
[4] S. Y. Chang, “A family of non-iterative integration methods with desired numerical dissipation,” International Journal for Numerical Methods in Engineering, vol. 100, no. 1, pp. 62-86, 2014.
[5] S. Y. Chang, “Dissipative, non-iterative integration algorithms with unconditional stability for mildly nonlinear structural dynamics,” Nonlinear Dynamics, vol. 79, no. 2, pp. 1625-1649, 2015.
[6] S. Y. Chang, “Elimination of overshoot in forced vibration response for Chang explicit family methods,” Journal of Engineering Mechanics, ASCE, vol. 144, no. 2, pp. 04017177, 2017.
[7] S. Y. Chang, “An amplitude growth property and its remedy for structure-dependent integration methods,” Computer Methods in Applied Mechanics and Engineering, vol. 330, pp. 498-521, 2018.
[8] Y. Gui, J. T. Wang, F. Jin, C. Chen, and M. X. Zhou, “Development of a family of explicit algorithms for structural dynamics with unconditional stability,” Nonlinear Dynamics, vol. 77, no. 4, pp. 1157-1170, 2014.
[9] C. Chen, and J. M. Ricles, “Development of direct integration algorithms for structural dynamics using discrete control theory,” Journal of Engineering Mechanics, ASCE, vol. 134, no. 8, pp. 676-683, 2008.
[10] Y. Tang, and M. L. Lou, “New unconditionally stable explicit integration algorithm for real-time hybrid testing,” Journal of Engineering Mechanics, ASCE, vol. 143, no. 7, pp. 04017029, 2017.
[11] S. Y. Chang, “A new family of explicit method for linear structural dynamics,” Computers & Structures, vol. 88, no. 11-12, pp. 755-772, 2010.
[12] S. Y. Chang, “Family of structure-dependent explicit methods for structural dynamics,” Journal of Engineering Mechanics, ASCE, vol. 140, no. 6, 06014005, 2014.
[13] G. L. Goudreau, and R. L. Taylor, “Evaluation of numerical integration methods in elasto-dynamics,” Computer Methods in Applied Mechanics and Engineering, vol. 2, pp. 69-97, 1972.
[14] E. L. Wilson, A Computer Program for the Dynamic Stress Analysis of Underground Structures, SESM Report No.68-1, Division of Structural Engineering Structural Mechanics, University of California, Berkeley, USA, 1968.
[15] H. M. Hilber, and T. J. R. Hughes, “Collocation, dissipation, and ‘overshoot’ for time integration schemes in structural dynamics,” Earthquake Engineering and Structural Dynamics, vol. 6, pp. 99-118, 1978.