An Unusual Damped Stability Property and its Remedy for an Integration Method
DOI:
https://doi.org/10.13052/ejcm2642-2085.3112Keywords:
dynamic analysis, time integration, responses of building, viscous damping, structure vibrationsAbstract
An unusual stability property is found for a structure-dependent integration method since it exhibits a different nonlinearity interval of unconditional stability for zero and nonzero damping. Although it is unconditionally stable for the systems of stiffness softening and invariant as well as most systems of stiffness hardening, an unstable solution that is unexpected is obtained as it is applied to solve damped stiffness hardening systems. It is found herein that a nonlinearity interval of unconditional stability for a structure-dependent method may be drastically shrunk for nonzero damping when compared to zero damping. In fact, it will become conditionally stable for any damped stiffness hardening systems. This might significantly restrict its applications. An effective scheme is proposed to surmount this difficulty by introducing a stability factor into the structure-dependent coefficients of the integration method. This factor can effectively amplify the nonlinearity intervals of unconditional stability for structure-dependent methods. A large stability factor will result in a large nonlinearity interval of unconditional stability. However, it also introduces more period distortion. Consequently, a stability factor must be appropriately selected for accurate integration. After choosing a proper stability factor, a structure-dependent method can be widely and easily applied to solve general structural dynamic problems.
Downloads
References
Houbolt, J.C. A recurrence matrix solution for the dynamic response of elastic aircraft. Journal of the Aeronautical Sciences 17 (1950) 540–550. https://doi.org/10.2514/8.1722
Newmark, N.M. A method of computation for structural dynamics. Journal of Engineering Mechanics Division, 85(3) (1959) 67–94. https://doi.org/10.1061/JMCEA3.0000098
Hilber, H.M., Hughes, T.J.R., Taylor, R.L. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 5 (1977) 283–292. https://doi.org/10.1002/eqe.4290050306
Wood, W.L., Bossak, M., Zienkiewicz, O.C. An alpha modification of Newmark’s method. International Journal for Numerical Methods in Engineering 15 (1981) 1562–1566. https://doi.org/10.1002/nme.1620151011
Chung, J., Hulbert, G.M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. Journal of Applied Mechanics 60(6) (1993) 371–375. https://doi.org/10.1115/1.2900803
Noels, L., Stainier, L., Ponthot, J.P., Bonini, J. Combined implicit-explicit algorithms for non-linear structural dynamics, European Journal of Computational Mechanics 11(5) (2002) 565–591. https://journals.riverpublishers.com/index.php/EJCM/article/view/2551
Papathanasiou, T.K. A Linearised θ Numerical Scheme for the Vibrations of Inextensible Beams. European Journal of Computational Mechanics 30(1) (2021) 121–144. https://doi.org/10.13052/ejcm1779-7179.3015
Belytschko, T. Hughes, T.J.R., Computational Methods for Transient Analysis, Elsevier Science Publishers B.V., North-Holland, (1983).
Hughes, T.J.R., The Finite Element Method, Prentice-Hall, Inc., Englewood Cliffs, N.J., U.S.A. (1987).
Chang, S.Y. Explicit pseudo-dynamic algorithm with unconditional stability, Journal of Engineering Mechanics, ASCE 128(9) (2002) 935–947. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:9(935)
Chang, S.Y. Improved explicit method for structural dynamics, Journal of Engineering Mechanics, ASCE 133(7) (2007) 748–760. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(748)
Chang, S.Y. An explicit method with improved stability property, International Journal for Numerical Method in Engineering 77(8) (2009) 1100–1120. https://doi.org/10.1002/nme.2452
Chang, S.Y., Yang, Y.S., Hsu, C.W. A family of explicit algorithm for general pseudo-dynamic testing, Earthquake Engineering and Engineering Vibration, 10(1) (2011) 51–64. http://www.springerlink.com/content/1647t81504715251/
Chang, S.Y. An explicit structure-dependent algorithm for pseudo-dynamic testing, Engineering Structures, 46 (2013) 511–525. http://dx.doi.org/10.1016%2Fj.engstruct.2012.08.009
Chang, S.Y. A dual family of dissipative structure-dependent integration methods, Nonlinear Dynamics, 98(1) (2019) 703–734.
Chang, S.Y. Non-iterative methods for dynamic analysis of nonlinear velocity-dependent problems.” Nonlinear Dynamics, 101 (2020) 1473–1500.
Dahlquist, G. Convergence and stability in the numerical integration of ordinary differential equations, Mathematica Scandinavica 4 (1956) 33–53. https://doi.org/10.7146/math.scand.a-10454
Chang, S.Y. An eigen-based theory for structure-dependent integration methods for nonlinear dynamic analysis, International Journal of Structural Stability and Dynamics 20(12) (2020) 2050130. https://doi.org/10.1142/S0219455420501308
Chang, S.Y. An unusual amplitude growth property and its remedy for structure-dependent integration methods, Computer Methods in Applied Mechanics and Engineering, 330 (2018) 498–521. https://doi.org/10.1016/j.cma.2017.11.012
Clough, R.W., Penzien, J., Dynamics of Structures, 2nd Edition, McGraw-Hill, 1993. https://doi.org/10.12989/sem.2012.43.1.001
Chang, S.Y. A general technique to improve stability property for a structure-dependent integration method, International Journal for Numerical Methods in Engineering, 101(9) (2015) 653–669. https://doi.org/10.1002/nme.4806
