Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

The numerical solution of dynamic response of SDOF systems using cubic B-spline polynomial functions

  • Shojaee, S. (Civil Engineering Department, Shahid Bahonar University of Kerman) ;
  • Rostami, S. (Civil Engineering Department, Islamic Azad University) ;
  • Moeinadini, A. (Civil Engineering Department, Islamic Azad University)
  • Received : 2010.08.17
  • Accepted : 2010.12.09
  • Published : 2011.04.25
⟨ Previous Next ⟩

Abstract

In this paper, we present a new explicit procedure using periodic cubic B-spline interpolation polynomials to solve linear and nonlinear dynamic equation of motion governing single degree of freedom (SDOF) systems. In the proposed approach, a straightforward formulation was derived from the approximation of displacement with B-spline basis in a fluent manner. In this way, there is no need to use a special pre-starting procedure to commence solving the problem. Actually, this method lies in the case of conditionally stable methods. A simple step-by-step algorithm is implemented and presented to calculate dynamic response of SDOF systems. The validity and effectiveness of the proposed method is demonstrated with four examples. The results were compared with those from the numerical methods such as Duhamel integration, Linear Acceleration and also Exact method. The comparison shows that the proposed method is a fast and simple procedure with trivial computational effort and acceptable accuracy exactly like the Linear Acceleration method. But its power point is that its time consumption is notably less than the Linear Acceleration method especially in the nonlinear analysis.

Keywords

References

  1. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall Inc.
  2. Belytschko, T. and Hughes, T.J.R. (1983), "Computational methods for transient analysis", Comput. Method. Mech., 1, Northhooland Elsevier, Amsterdam.
  3. Caglar, N. and Caglar, H. (2006a), "B-spline solution of singular boundary value problems", Appl. Math. Comput., 182, 1509-1513. https://doi.org/10.1016/j.amc.2006.05.035
  4. Caglar, H., Caglar, N. and Elfaituri, K. (2006b), "B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems", Appl. Math. Comput., 175, 72-79. https://doi.org/10.1016/j.amc.2005.07.019
  5. Caglar, N. and Caglar, H. (2009), "B-spline method for solving linear system of second-order boundary value problems", Comput. Math. Appl., 57, 757-762. https://doi.org/10.1016/j.camwa.2008.09.033
  6. Chang, S.Y. (1997), "Improved numerical dissipation for explicit methods in pseudo-dynamic tests", Earthq. Eng. Struct. D., 26, 917-929. https://doi.org/10.1002/(SICI)1096-9845(199709)26:9<917::AID-EQE685>3.0.CO;2-9
  7. Chang, S.Y. (2002), "Explicit pseudo-dynamic algorithm with unconditional stability", J. Eng. Mech.-ASCE, 9, 128, 935-947.
  8. Chang, S.Y. (2010), "A new family of explicit methods for linear structural dynamics", Comput. Struct., 88, 755-772. https://doi.org/10.1016/j.compstruc.2010.03.002
  9. Chopra, A.K. (1995), Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice Hall Inc., Englewood Cliffs, NJ.
  10. Chung, J. and Lee, J.M. (1994), "A new family of explicit time integration methods for linear and non-linear structural dynamics", Int. J. Numer. Meth. Eng., 37, 3961-3976. https://doi.org/10.1002/nme.1620372303
  11. De Boor, C. (1978), A Practical Guide to Splines, Springer-Verlag, New York.
  12. Dokainish, M.A. and Subbaraj, K. (1989), "A survey of direct time integration methods in computational structural dynamics. I. Explicit methods", Comput. Struct., 32(6), 1371-1386. https://doi.org/10.1016/0045-7949(89)90314-3
  13. Ebeling, R.M., Green, R.A. and French, S.E. (1997), "Accuracy of response of single-degree-of-freedom systems to ground motion", Technical Report ITL-97-7, U.S. Army Corps of Engineers.
  14. Hilber, H.M., Hughes, T.J.R. and Taylor, R.L. (1977), "Improved numerical dissipation for time integration algorithms in structural dynamics", Int. J. Earthq. Eng. Struct. D., 5, 283-292. https://doi.org/10.1002/eqe.4290050306
  15. Houbolt, J.C. (1950), "A recurrence matrix solution for the dynamic response of elastic aircraft", J. Aeronaut Sciences, 17, 540-550. https://doi.org/10.2514/8.1722
  16. Hughes, T.J.R. (1987), The Finite Element Method-linear Static and Dynamic Finite Element Analysis, Prentice Hall Inc.
  17. Inoue, T. and Sueoka, A. (2002), "A step-by-step integration scheme utilizing the cardinal B-splines", JSME Int. J. Series C, 45(2), 433-441. https://doi.org/10.1299/jsmec.45.433
  18. Xiang, J., He, Z., He, Y. and Chen, X. (2007), "Static and vibration analysis of thin plates by using finite element method of B-spline wavelet on the interval", Struct. Eng. Mech., 25(5), 613-629. https://doi.org/10.12989/sem.2007.25.5.613
  19. Liu, J.L. (2002), "Solution of dynamic response of framed structure using pricewise Birkhoff polynomial", J. Sound Vib., 251(5), 847-857. https://doi.org/10.1006/jsvi.2001.4014
  20. Liu, J.L. (2001), "Solution of dynamic response of SDOF system using pricewise Lagrange polynomial", Int. J. Earthq. Eng. Struct. D, 30, 613-619. https://doi.org/10.1002/eqe.24
  21. Mullen, R. and Belytschko, T. (1983), "An analysis of an unconditionally stable explicit method", Comput. Struct., 16(6), 691-696. https://doi.org/10.1016/0045-7949(83)90060-3
  22. Newmark, N.M. (1959), "A method of computational for structural dynamics", J. Eng. Mech. Div., 8, 67-94.
  23. Park, K.C. (1975), "An improved stiffly stable method for direct integration of nonlinear structural dynamic equations", J. Appl. Mech., 42, 464-470. https://doi.org/10.1115/1.3423600
  24. Rio, G., Soive, A. and Grolleau, V. (2005), "Comparative study of numerical explicit time integration algorithms", Adv. Eng. Softw., 36, 252-265. https://doi.org/10.1016/j.advengsoft.2004.10.011
  25. Rogers, D.F. (2001), An Introduction to NURBS, with Historical Perspective, Morgan Kaufmann Publishers, San Fransisco.
  26. Wilson, E.L., Farhoomand, I. and Bathe, K.J. (1973), "Nonlinear dynamics analysis of complex structures", Int. J. Earthq. Eng. Struct. D., 1, 241-252.
  27. Yingkang, H. and Xiang, M.Y. (1995), "Discrete modulus of smoothness of spline with equally spaced knots", SIAM Rev., 32(5), 1428-1435.

Cited by

  1. A Family of Cubic B-Spline Direct Integration Algorithms with Controllable Numerical Dissipation and Dispersion for Structural Dynamics 2018, https://doi.org/10.1007/s40996-017-0083-y
  2. An unconditionally stable implicit time integration algorithm: Modified quartic B-spline method vol.153, 2015, https://doi.org/10.1016/j.compstruc.2015.02.030
  3. Alpha-Modification of Cubic B-Spline Direct Time Integration Method vol.17, pp.10, 2017, https://doi.org/10.1142/S0219455417501188
  4. Development of a direct time integration method based on Bezier curve and 5th-order Bernstein basis function vol.194, 2018, https://doi.org/10.1016/j.compstruc.2017.08.015
  5. Insight into an implicit time integration method based on Bezier curve and third-order Bernstein basis function for structural dynamics 2017, https://doi.org/10.1007/s42107-017-0001-4
  6. An explicit time integration method for structural dynamics using cubic -spline polynomial functions vol.20, pp.1, 2011, https://doi.org/10.1016/j.scient.2012.12.003
  7. Development of a Direct Time Integration Method Based on Quartic B-spline Collocation Method vol.43, pp.suppl1, 2019, https://doi.org/10.1007/s40996-018-0193-1
  8. Further Insights Into Time-Integration Method Based on Bernstein Polynomials and Bezier Curve for Structural Dynamics vol.19, pp.10, 2019, https://doi.org/10.1142/s021945541950113x
  9. An effective locally-defined time marching procedure for structural dynamics vol.73, pp.1, 2011, https://doi.org/10.12989/sem.2020.73.1.065
  10. Survey of cubic B-spline implicit time integration method in computational wave propagation vol.79, pp.4, 2011, https://doi.org/10.12989/sem.2021.79.4.473