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A novel hybrid sub-step explicit time integration method with cubic B-spline interpolation and momentum corrector technique for linear and nonlinear dynamics

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Abstract

In this study, a novel hybrid sub-step explicit time integration method based on cubic B-spline interpolation and integration procedure is proposed. In the procedure, a “momentum corrector” technique of load integration is employed to achieve improved solution accuracy for both linear and nonlinear dynamic problems. New method has three free parameters, and can flexibly control basic algorithm properties including accuracy, stability and numerical dissipation. With optimized algorithmic parameters, new method can achieve at least second-order accuracy with or without physical damping, and achieve third-order accuracy without physical damping. Numerical examples demonstrate, compared with other competitive explicit methods, new method shows far higher solution accuracy for general linear dynamic problems, discontinuous applied load, wave propagation problem and nonlinear dynamic problems.

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References

  1. Bathe, K.J.: Finite Element Procedures (2nd Edition) Prentice-Hall, (2014)

  2. Humar, J.: Dynamics of Structures: Second Edition Crc Press, (2002)

  3. Giorgio, I., Del Vescovo, D.: Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators. Math. Mech. Compl. Syst. 7, 159–174 (2019). https://doi.org/10.2140/memocs.2019.7.159

    Article  MathSciNet  MATH  Google Scholar 

  4. Turco, E., Barchiesi, E., dell’Isola, F.: A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams. Math. Mech. Solids 27, 22–48 (2022). https://doi.org/10.1177/10812865211010877

    Article  MathSciNet  MATH  Google Scholar 

  5. Subbaraj, K., Dokainish, M.A.: A survey of direct time-integration methods in computational structural dynamics—II. Implicit methods. Comput. Struct. 32, 1387–1401 (1989). https://doi.org/10.1016/0045-7949(89)90315-5

    Article  MathSciNet  MATH  Google Scholar 

  6. Wen, W.B., Wei, K., Lei, H.S., Duan, S.Y., Fang, D.N.: A novel sub-step composite implicit time integration scheme for structural dynamics. Comput Struct 182, 176–186 (2017). https://doi.org/10.1016/j.compstruc.2016.11.018

    Article  Google Scholar 

  7. Zhang, H., Zhang, R., Masarati, P.: Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods. Comput. Mech. 67, 289–313 (2021). https://doi.org/10.1007/s00466-020-01933-y

    Article  MathSciNet  MATH  Google Scholar 

  8. Dokainish, M.A., Subbaraj, K.: A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods. Comput. Struct. 32, 1371–1386 (1989). https://doi.org/10.1016/0045-7949(89)90314-3

    Article  MathSciNet  MATH  Google Scholar 

  9. Har, J., Tamma, K.K.: Thirteen. Time Discretization of Equations of Motion: Overview and Conventional Practices, Wiley, Ltd, (2012)

  10. Tseng, J.C., Hwu, J.G.: Fourteen. Time Discretization of Equations of Motion: Recent Advances, Wiley (2012)

  11. Gebhardt, C.G., Romero, I., Rolfes, R.: A new conservative/dissipative time integration scheme for nonlinear mechanical systems. Comput. Mech. 65, 405–427 (2020). https://doi.org/10.1007/s00466-019-01775-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Chung, J., Lee, J.M.: A new family of explicit time integration methods for linear and non-linear structural dynamics. Int. J. Numer. Meth. Eng. 37, 3961–3976 (1994). https://doi.org/10.1002/nme.1620372303

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhai, W.M.: Two simple fast integration methods for large-scale dynamic problems in engineering. Int. J. Numer. Meth. Eng. 39, 4199–4214 (1996). https://doi.org/10.1002/(SICI)1097-0207(19961230)39:24%3c4199::AID-NME39%3e3.0.CO;2-Y

    Article  MathSciNet  MATH  Google Scholar 

  14. Hulbert, G.M., Chung, J.: Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Comput. Methods Appl. Mech. Eng. 137, 175–188 (1996). https://doi.org/10.1016/S0045-7825(96)01036-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Tchamwa, B., Conway, T. and Wielgosz, C.: Accurate explicit direct time integration method for computational structural dynamics Recent Advances in Solids and Structures - 1999 (The ASME International Mechanical Engineering Congress and Exposition) ASME, Fairfield, NJ, United States, pp 77–84, (1999)

  16. Wen, W.B., Duan, S.Y., Yan, J., Ma, Y.B., Wei, K., Fang, D.N.: A quartic B-spline based explicit time integration scheme for structural dynamics with controllable numerical dissipation. Comput. Mech. 59, 403–418 (2017). https://doi.org/10.1007/s00466-016-1352-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Soares, D., Jr.: A novel time-marching formulation for wave propagation analysis: The adaptive hybrid explicit method. Comput. Methods Appl. Mech. Eng. (2020). https://doi.org/10.1016/j.cma.2020.113095

    Article  MathSciNet  MATH  Google Scholar 

  18. Noh, G., Bathe, K.-J.: An explicit time integration scheme for the analysis of wave propagations. Comput. Struct. 129, 178–193 (2013). https://doi.org/10.1016/j.compstruc.2013.06.007

    Article  Google Scholar 

  19. Kim, W., Reddy, J.N.: Novel explicit time integration schemes for efficient transient analyses of structural problems. Int. J. Mech. Sci. 172, 105429 (2020). https://doi.org/10.1016/j.ijmecsci.2020.105429

    Article  Google Scholar 

  20. Li, J., Yu, K.: Development of composite sub-step explicit dissipative algorithms with truly self-starting property. Nonlinear Dyn 103, 1911–1936 (2021). https://doi.org/10.1007/s11071-021-06202-y

    Article  Google Scholar 

  21. Zhang, H., Zhang, R., Zanoni, A., Xing, Y., Masarati, P.: A novel explicit three-sub-step time integration method for wave propagation problems. Arch. Appl. Mech. (2022). https://doi.org/10.1007/s00419-021-02075-0

    Article  Google Scholar 

  22. Wen, W., Deng, S., Duan, S., Fang, D.: A high-order accurate explicit time integration method based on cubic b-spline interpolation and weighted residual technique for structural dynamics. Int. J. Numer. Meth. Eng. 122, 431–454 (2021). https://doi.org/10.1002/nme.6543

    Article  MathSciNet  Google Scholar 

  23. Zhang, J., Liu, Y., Liu, D.: Accuracy of a composite implicit time integration scheme for structural dynamics. Int. J. Numer. Meth. Eng. 109, 368–406 (2017). https://doi.org/10.1002/nme.5291

    Article  MathSciNet  Google Scholar 

  24. Zhang, J.: A-stable two-step time integration methods with controllable numerical dissipation for structural dynamics. Int. J. Numer. Meth. Eng. 121, 54–92 (2020). https://doi.org/10.1002/nme.6188

    Article  MathSciNet  Google Scholar 

  25. Hilber, H.M., Hughes, T.J.R.: Collocation, dissipation and [overshoot] for time integration schemes in structural dynamics. Earthq. Eng. Struct. Dynam. 6, 99–117 (1978). https://doi.org/10.1002/eqe.4290060111

    Article  Google Scholar 

  26. Xie, Y.M.: An assessment of time integration schemes for non-linear dynamic equations. J. Sound Vib. 192, 321–331 (1996). https://doi.org/10.1006/jsvi.1996.0190

    Article  Google Scholar 

  27. Kim, W.: Higher-order explicit time integration methods for numerical analyses of structural dynamics. Lat. Am. J. Solids Struct. 16, 29 (2019). https://doi.org/10.1590/1679-78255609

    Article  Google Scholar 

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Acknowledgements

This research is substantially supported by the National Natural Science Foundation of China (No. 12072375).

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Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410083, People’s Republic of China

    Weibin Wen, He Li, Tianhao Liu & Shanyao Deng

  2. Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Shengyu Duan

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  1. Weibin Wen
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Correspondence to Shengyu Duan.

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The raw/processed data can be obtained by contacting the first author or corresponding author by email. (Weibin Wen: wenwbin@126.com; Shengyu Duan: shengyu_duan@126.com).

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Appendices

Appendix A

The expression of amplification matrix \({\varvec{A}}\) is obtained as follows,

$$ {\varvec{A}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {A_{11} } & {A_{21} } & {A_{31} } \\ \end{array} } \\ {\begin{array}{*{20}c} {A_{21} } & {A_{22} } & {A_{23} } \\ \end{array} } \\ {\begin{array}{*{20}c} {A_{31} } & {A_{32} } & {A_{33} } \\ \end{array} } \\ \end{array} } \right] $$
(40)

where

$$ A_{11} = \frac{{\left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{12}{\Omega }^{4} + \frac{{\left( {p + 1} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{12}\xi {\Omega }^{3} + \frac{{\left( {p - 1} \right)}}{12}{\Omega }^{2} + 1 $$
(41)
$$ \begin{aligned} A_{12} = & \frac{{\left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{12}\Omega^{4} + \frac{{\left( {6p^{2} + 9p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{6}\xi \Omega^{3} \\ & + \frac{{\left( {p - 1} \right)\left( {p^{2} - p + 1 - \alpha \left( {p - 1} \right)^{2} } \right)}}{2}\Omega^{2} + 2\left( {p + 1} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)\left( {\xi \Omega } \right)^{2} \\ \end{aligned} $$
(42)
$$ \begin{aligned} A_{13} = & \frac{{p^{2} \left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{24}{\Omega }^{4} + \frac{{p\left( {3p^{2} + 6p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{6}\xi {\Omega }^{3} + \frac{{p\left( {p - 1} \right)\left( {p^{2} - p + 1 - \alpha \left( {p - 1} \right)^{2} } \right)}}{4}{\Omega }^{2} \\ & + 2p\left( {p + 1} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)\left( {\xi {\Omega }} \right)^{2} + p\left( {p - 1} \right)\left( {\alpha \left( {p - 1} \right)^{2} - p^{2} + 2p} \right)\xi {\Omega } + \frac{p}{2} \\ \end{aligned} $$
(43)
$$ \begin{aligned} A_{21} = & - \frac{{\left( {3p + 2} \right)\left( {p - 1} \right)^{4} \left( {\alpha - 1} \right)\left( {2\beta - 1} \right)}}{24}{\Omega }^{6} - \frac{{\left( {p - 1} \right)^{3} \left( {2\beta - 1} \right)\left( {6\alpha \left( {p^{2} - 1} \right) - 6p^{2} + 3p + 8} \right)}}{12}\xi {\Omega }^{5} \\ & - \frac{{\left( {p - 1} \right)^{3} \left( {p + 1} \right)\left( {2\beta - 1} \right)}}{12}\xi^{2} {\Omega }^{4} - \frac{{\left( {p - 1} \right)^{2} \left( {2\beta - 6\beta p - 3} \right)}}{12}{\Omega }^{4} + \frac{{\left( {p - 1} \right)\left( {p - 2\beta p + 4\beta p^{2} - 2} \right)}}{2}\xi {\Omega }^{3} + \frac{{\left( {p - 2} \right)}}{2}{\Omega }^{2} \\ \end{aligned} $$
(44)
$$ \begin{aligned} A_{22} = & - \frac{{p\left( {3p + 2} \right)\left( {p - 1} \right)^{4} \left( {\alpha - 1} \right)\left( {2\beta - 1} \right)}}{24}\Omega^{6} - \frac{{\left( {p - 1} \right)^{3} \left( {2\beta - 1} \right)\left( {\alpha \left( {p - 1} \right)\left( {6p^{2} + 9p + 2} \right) - 6p^{2} + 9p + 2} \right)}}{12}\xi \Omega^{5} \\ & - \frac{{\left( {p - 1} \right)^{3} \left( {2\beta - 1} \right)\left( {6\alpha \left( {p^{2} - 1} \right) + 9p + 8} \right)}}{6}\xi^{2} \Omega^{4} - \frac{{\left( {p - 1} \right)^{2} \left( { - 3\left( {p - 1} \right)^{2} \left( {2\beta - 1} \right)\alpha + 6\beta + 3p - 10\beta p - 3p^{2} - 3} \right)}}{12}\Omega^{4} \\ & - 2\left( {p - 1} \right)^{3} \left( {p + 1} \right)\left( {2\beta - 1} \right)\left( {\xi \Omega } \right)^{3} - \frac{{\left( {p - 1} \right)\left( {20\beta p - 3p - 8\beta - 6\beta p^{2} - 12\beta p^{3} + 6} \right)}}{6}\xi \Omega^{3} \\ & + \left( {p - 1} \right)\left( {p - 2\beta p + 4\beta p^{2} - 2} \right)\left( {\xi \Omega } \right)^{2} - \frac{1}{2}\Omega^{2} \\ \end{aligned} $$
(45)
$$ \begin{aligned} A_{23} = & - \frac{{p^{2} \left( {3p + 2} \right)\left( {p - 1} \right)^{4} \left( {\alpha - 1} \right)\left( {2\beta - 1} \right)}}{48}{\Omega }^{6} - \frac{{p\left( {p - 1} \right)^{3} \left( {2\beta - 1} \right)\left( {2\alpha \left( {p - 1} \right)\left( {3p^{2} + 6p + 2} \right) - 6p^{3} - 3p^{2} + 10p + 4} \right)}}{24}\xi {\Omega }^{5} \\ & - \frac{{p\left( {p - 1} \right)^{3} \left( {2\beta - 1} \right)\left( {6\alpha \left( {p^{2} - 1} \right) - 3p^{2} + 6p + 8} \right)}}{6}\xi^{2} {\Omega }^{4} + \frac{{p\left( {p - 1} \right)^{2} \left( { - 3\left( {p - 1} \right)^{2} \left( {2\beta - 1} \right)\alpha + 6\beta + 3p - 10\beta p - 3p^{2} - 3} \right)}}{24}{\Omega }^{4} \\ & - \frac{{p\left( {p - 1} \right)\left( {6\alpha \left( {p - 1} \right)^{3} \left( {2\beta - 1} \right) + 2\beta + 12p - 8\beta p + 24\beta p^{2} - 24\beta p^{3} - 18p^{2} + 6p^{3} + 3} \right)}}{6}\xi {\Omega }^{3} \\ & - 2p\left( {p - 1} \right)^{3} \left( {p + 1} \right)\left( {2\beta - 1} \right)\left( {\xi {\Omega }} \right)^{3} + p\left( {p - 1} \right)\left( {2\beta p - p - 2\beta + 2\beta p^{2} + p^{2} - 1} \right)\left( {\xi {\Omega }} \right)^{2} \\ & - \frac{p}{4}{\Omega }^{2} + \frac{{p\left( {2\beta + p - 6\beta p + 4\beta p^{2} - 3} \right)}}{2}\xi {\Omega } + \frac{p}{2} \\ \end{aligned} $$
(46)
$$ \begin{aligned} A_{31} = & - \frac{{\left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{12}{\Omega }^{6} - \frac{{\left( {p - 1} \right)^{2} \left( {2\beta - 1} \right)\left( {6\alpha \left( {p^{2} - 1} \right) + 3p - 6p^{2} + 8} \right)}}{6}\xi {\Omega }^{5} \\ & - 2\left( {p - 1} \right)^{2} \left( {p + 1} \right)\xi^{2} {\Omega }^{4} - \frac{{\left( {p - 1} \right)}}{2}{\Omega }^{4} - \left( {p - 2} \right)\xi {\Omega }^{3} - {\Omega }^{2} \\ \end{aligned} $$
(47)
$$ \begin{aligned} A_{32} = & - \frac{{p\left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{12}{\Omega }^{6} - \frac{{\left( {p - 1} \right)^{2} \left( {\left( {p - 1} \right)\left( {6p^{2} + 9p + 2} \right)\alpha - 6p^{3} + 9p + 2} \right)}}{6}\xi {\Omega }^{5} \\ & - \frac{{\left( {p - 1} \right)^{2} \left( {6\alpha \left( {p^{2} - 1} \right) + 9p + 8} \right)}}{3}\xi^{2} {\Omega }^{4} - \frac{{\left( {p - 1} \right)\left( { - \left( {p - 1} \right)^{2} \alpha + p^{2} - p + 1} \right)}}{2}{\Omega }^{4} - 4\left( {p - 1} \right)^{2} \left( {p + 1} \right)\left( {\xi {\Omega }} \right)^{3} \\ & + \left( {2 - p} \right)\xi {\Omega }^{3} + 2\left( {2 - p} \right)\left( {\xi {\Omega }} \right)^{2} - {\Omega }^{2} - 2\xi {\Omega } \\ \end{aligned} $$
(48)
$$ \begin{aligned} A_{33} = & - \frac{{p^{2} \left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)}}{24}{\Omega }^{6} - \frac{{p\left( {p - 1} \right)^{2} \left( {2\left( {p - 1} \right)\left( {3p^{2} + 6p + 2} \right)\alpha - 6p^{3} - 3p^{2} + 10p + 4} \right)}}{12}\xi {\Omega }^{5} \\ & - \frac{{p\left( {p - 1} \right)^{2} \left( {6\alpha \left( {p^{2} - 1} \right) - 3p^{2} + 6p + 8} \right)}}{3}\xi^{2} {\Omega }^{4} + \frac{{p\left( {p - 1} \right)\left( {\left( {p - 1} \right)^{2} \alpha - p^{2} + p - 1} \right)}}{4}{\Omega }^{4} \\ & - \frac{{p\left( {2\alpha \left( {p - 1} \right)^{3} - 2p^{3} + 6p^{2} - 4p - 1} \right)}}{2}\xi {\Omega }^{3} - 4p\left( {p - 1} \right)^{2} \left( {p + 1} \right)\left( {2\beta - 1} \right)\left( {\xi {\Omega }} \right)^{3} \\ & + 2\left( {1 + p - p^{2} } \right)\left( {\xi {\Omega }} \right)^{2} - \frac{1}{2}{\Omega }^{2} - p\xi {\Omega } \\ \end{aligned} $$
(49)

in which \(\Omega = \omega \Delta t\).

The expressions of \({A}_{1}\), \({A}_{2}\) and \({A}_{3}\) in Eq. (23) are obtained as

$$ \begin{aligned} A_{1} = & - \frac{{p\left( {3p + 2} \right)\left( {p - 1} \right)^{3} \left( {\alpha - 1} \right)\left( {2\beta p - 2\beta + 1} \right)}}{48}{\Omega }^{6} \\ & - \frac{{\left( {p - 1} \right)^{2} }}{24}\left( {\begin{array}{*{20}c} {\alpha \left( {p - 1} \right)\left( {11p - 4\beta - 14\beta p + 6\beta p^{2} + 12\beta p^{3} + 9p^{2} + 2} \right)} \\ { + 11p - 4\beta - 14\beta p + 18\beta p^{2} + 12\beta p^{3} - 12\beta p^{4} + p^{2} - 9p^{3} + 2} \\ \end{array} } \right)\xi {\Omega }^{5} \\ & - \frac{{\left( {p - 1} \right)^{2} }}{12}\left( {\begin{array}{*{20}c} {6\alpha \left( {p - 1} \right)\left( {p + 1} \right)\left( {p - 2\beta + 2\beta p + 1} \right)} \\ { + 2\left( {9p + 8} \right)\left( {p - 1} \right)\beta - 6p^{3} + 3p^{2} + 17p + 8} \\ \end{array} } \right)\xi^{2} {\Omega }^{4} \\ & + \frac{{\left( {p - 1} \right)}}{24}\left( {\begin{array}{*{20}c} {\left( {p - 1} \right)^{2} \left( {3p - 6\beta + 6\beta p + 5} \right)\alpha } \\ { + 2\left( {5p - 3} \right)\left( {p - 1} \right)\beta - 3p^{3} + p^{2} + 4p - 5} \\ \end{array} } \right){\Omega }^{4} \\ & - \left( {p + 1} \right)\left( {p - 1} \right)^{2} \left( {p - 2\beta + 2\beta p + 1} \right)\left( {\xi {\Omega }} \right)^{3} \\ & + \frac{{\left( {6\alpha \left( {p - 1} \right)^{3} \alpha + 2\left( {p - 1} \right)\left( {6p^{3} + 3p^{2} - 10p + 4} \right)\beta - 6p^{3} + 15p^{2} - 18p + 12} \right)}}{12}\xi {\Omega }^{3} \\ & + \frac{{\left( {2p\left( {p - 1} \right)\left( {2p - 1} \right)\beta - 2p^{3} + 3p^{2} - p + 2} \right)}}{2}\left( {\xi {\Omega }} \right)^{2} - \frac{1}{2}{\Omega }^{2} - \xi {\Omega } + 1 \\ \end{aligned} $$
(50)
$$ \begin{aligned} A_{2} = & - \frac{{\left( {p - 1} \right)^{2} }}{12}\left( {\begin{array}{*{20}c} {\left( {p - 1} \right)\left( {8\beta + 5p - 26\beta p + 6\beta p^{2} + 12\beta p^{3} - 3p^{2} - 4} \right)\alpha } \\ { - \left( {12p^{4} - 18p^{2} + 22p - 6} \right)\beta + 3p^{3} - 5p^{2} + 5p - 3} \\ \end{array} } \right)\xi {\Omega }^{5} \\ & - \frac{{\left( {p - 1} \right)^{2} }}{3}\left( {\begin{array}{*{20}c} {3\left( {p - 1} \right)\left( {3p - 2\beta - 2\beta p + 4\beta p^{2} + 1} \right)\alpha } \\ { - \left( {2\beta - 5p + 6\beta p - 6\beta p^{2} + 6\beta p^{3} + 3p^{2} - 1} \right)} \\ \end{array} } \right)\xi^{2} {\Omega }^{4} \\ & + \frac{{\left( {p - 1} \right)^{3} }}{12}\left( {\begin{array}{*{20}c} {\left( {3p - 6\beta + 6\beta p - 1} \right)\alpha } \\ { + 4\beta - 3p + 1} \\ \end{array} } \right){\Omega }^{4} + 2\left( {p - 1} \right)^{2} \left( { - 2p^{2} - 2p + 2\beta - 1} \right)\left( {\xi {\Omega }} \right)^{3} \\ & + \frac{{\left( {6\alpha \left( {p - 1} \right)^{3} \alpha + 2\left( {p - 1} \right)\left( {6p^{3} + 3p^{2} - 10p + 4} \right)\beta - 6p^{3} + 15p^{2} - 18p + 6} \right)}}{6}\xi {\Omega }^{3} \\ & + 2\left( {2p\left( {2p - 1} \right)\left( {p - 1} \right)\beta - 2p^{3} + 3p^{2} - p + 1} \right)\left( {\xi {\Omega }} \right)^{2} - 2\xi {\Omega } + 1 \\ \end{aligned} $$
(51)
$$ \begin{aligned} A_{3} = & \frac{{p\left( {p - 1} \right)^{2} \left( {2\beta - 1} \right)\left( {6\alpha \left( {p - 1} \right)^{2} - 6p^{2} + 9p - 5} \right)}}{6}\xi^{2} {\Omega }^{4} \\ & - 2p^{2} \left( {p - 1} \right)^{2} \left( {2\beta - 1} \right)\left( {\xi {\Omega }} \right)^{3} - p\left( {p - 1} \right)\left( {2p - 1} \right)\left( {2\beta - 1} \right)\left( {\xi {\Omega }} \right)^{2} \\ \end{aligned} $$
(52)

Appendix B

The theoretical displacement for the SDOF system shown in example 5.1.3 is obtained as

$$ u\left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{{F_{0} }}{{\omega^{2} }}\left( {1 - \cos \left( {\omega t} \right)} \right), } \hfill & {t \in \left[ {0,\frac{{t_{b} }}{2}} \right]} \hfill \\ {\frac{{F_{0} }}{{\omega^{2} }}\left( {2\cos \left( {\omega \left( {t - \frac{{t_{b} }}{2}} \right) - 1} \right) - \cos \left( {\omega t} \right) - 1} \right),} \hfill & {t \in \left[ {\frac{{t_{b} }}{2}, t_{b} } \right]} \hfill \\ {\frac{{F_{0} }}{{\omega^{2} }}\left( {\begin{array}{*{20}c} {1 - 2\cos \left( {\omega \left( {t - t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{t_{b} }}{2}} \right)} \right) - \cos \left( {\omega t} \right)} \\ \end{array} } \right), } \hfill & {t \in \left[ { t_{b} ,\frac{{3t_{b} }}{2}} \right]} \hfill \\ {\frac{{F_{0} }}{{\omega^{2} }}\left( {\begin{array}{*{20}c} {2\cos \left( {\omega \left( {t - \frac{{3t_{b} }}{2}} \right)} \right) - 2\cos \left( {\omega \left( {t - t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{t_{b} }}{2}} \right)} \right) - \cos \left( {\omega t} \right) - 1} \\ \end{array} } \right),} \hfill & {t \in \left[ {\frac{{3t_{b} }}{2},2t_{b} } \right]} \hfill \\ {\frac{{F_{0} }}{{\omega^{2} }}\left( {\begin{array}{*{20}c} {1 - 2\cos \left( {\omega \left( {t - 2t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{3t_{b} }}{2}} \right)} \right) - 2\cos \left( {\omega \left( {t - t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{t_{b} }}{2}} \right)} \right) - \cos \left( {\omega t} \right)} \\ \end{array} } \right), } \hfill & { t \in \left[ {2t_{b} ,\frac{{5t_{b} }}{2}} \right]} \hfill \\ {\frac{{F_{0} }}{{\omega^{2} }}\left( {\begin{array}{*{20}c} {2\cos \left( {\omega \left( {t - \frac{{5t_{b} }}{2}} \right)} \right) - 2\cos \left( {\omega \left( {t - 2t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{3t_{b} }}{2}} \right)} \right) - 2\cos \left( {\omega \left( {t - t_{b} } \right)} \right) + } \\ {2\cos \left( {\omega \left( {t - \frac{{t_{b} }}{2}} \right)} \right) - \cos \left( {\omega t} \right) - 1} \\ \end{array} } \right),} \hfill & {t \in \left[ {\frac{{5t_{b} }}{2},3t_{b} } \right]} \hfill \\ \end{array} } \right. $$
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Wen, W., Li, H., Liu, T. et al. A novel hybrid sub-step explicit time integration method with cubic B-spline interpolation and momentum corrector technique for linear and nonlinear dynamics. Nonlinear Dyn 110, 2685–2714 (2022). https://doi.org/10.1007/s11071-022-07740-9

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