International Journal of Nonlinear Analysis and Applications

The use of the trapezoidal method for solving the Tacoma Narrows Bridge model

Document Type : Research Paper

Authors

1 Department of Mathematics, Tafresh University, Tafresh 39518 79611, Iran

2 Department of Mathematics and Statistics, Faculty of Basic Sciences, Gonbad Kavous University, Gonbad Kavous, Iran

Abstract

In this paper, an efficient method is developed for the approximate solution of a benchmark non-smooth dynamical system. In the proposed method, the trapezoidal method is utilized for solving the Tacoma Narrows Bridge equation. For this purpose, at first, the integral form of the dynamical equation is considered. Afterwards, the obtained integral equation is discretized by the trapezoidal method. The accuracy and performance of the proposed method are examined by means of some numerical experiments.

Keywords

[1] G. Arioli and F. Gazzola, A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge, Appl. Math. Model. 39 (2015), no. 2, 901–912.
[2] R. Burden and J. Faires, Numerical analysis, Available Titles CengageNOW Series, Cengage Learning, 2004.
[3] M. Ghaznavi and M.H. Noori Skandari, An efficient pseudo-spectral method for nonsmooth dynamical systems, Iran. J. Sci. Technol. Trans. A: Sci. 42 (2018), no. 2, 635–646.
[4] A. C Lazer and P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), no. 4, 537–578.
[5] S. Mahmoud and X. Chen, A verified inexact implicit Runge-Kutta method for nonsmooth ODEs, Numer. Algorithms 47 (2008), no. 3, 275–290.
[6] P.J. McKenna, Large torsional oscillations in suspension bridges revisited: fixing an old approximation, Amer. Math. Month. 106 (1999), no. 1, 1–18.
[7] M.A. Mehrpouya and S. Fallahi, A modified control parametrization method for the numerical solution of bang-bang optimal control problems, J. Vibr. Control 21 (2015), no. 12, 2407–2415.
[8] Mohammad A Mehrpouya, A modified pseudospectral method for indirect solving a class of switching optimal control problems, Proc. Inst. Mech. Engin. Part G: J. Aerospace Engin. 234 (2020), no. 9, 1531–1542.
[9] M.A. Mehrpouya and M. Khaksar-e Oshagh, An efficient numerical solution for time switching optimal control problems, Comput. Meth. Differ. Equ.9 (2021), no. 1, 225–243.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 8
August 2023
Pages 67-72
  • Receive Date: 06 March 2022
  • Revise Date: 15 August 2022
  • Accept Date: 08 September 2022