Quantitative comparison of the accuracy of the 3rd order numerical acceleration method with other numerical methods

Document Type : Research Note

Authors

masoud mahmoudabadi

fateme hatefi ardakani

fateme ansari

Department of Civil Engineering, University of Qom, Qom, Iran.

10.22067/jfcei.2024.85785.1276

Abstract

In this article, the 3rd order numerical acceleration method is presented. The basic assumption of this method is that the acceleration changes in the time interval Δt is in the form of a third degree polynomial, and as a result, its displacement equation, which is obtained by integrating the acceleration equation twice, is a fifth degree curve that has six The coefficient is unknown. To quantitatively compare the error rate of this method with other numerical methods, a linear one-degree-of-freedom system with a frequency of one hertz and specific dampings of 1, 2, 5, 10 and 20% was considered. Then, sinusoidal harmonic loading with a specific frequency was applied to this system. The exact answer of this system is available in the books of dynamics of structures. The ratio of the loading frequency to the natural vibration frequency of the system varied from 0.01 to 3 with an increase of 0.01. Using numerical methods of second-order acceleration and third-order acceleration and other popular methods, and using the root mean square error criterion, they were quantitatively compared with each other. The comparison was made for ten Δt, the smallest of which was equal to 0.02 seconds and the largest was equal to 0.20 seconds. The amount of error reduction for the third-order acceleration method was insignificant compared to the second-order acceleration method. The lowest error was related to the upgraded Jennings method, in which the error rate was much lower than other known methods.

Keywords

3rd order acceleration method

weighted function

weighted residual integral

root mean square error

curve fitting

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References

[1] A. A. Gholampour, M. Ghasemiye, H. Fallahnejad, “Direct Time Integration Method for Nonlinear Structural Problems,” Proceedings of the 6th National Congress on Civil Engineering, Iran, Semnan, Semnan University, [Spring 2011].

[2]   S. Y.Chang, “Accurate integration of nonlinear systems using Newmark explicit method,” Journal of Mechanics, vol. 25, no. 3, pp 289-297, 2009.

[3] M. Mahmoudabadi, B. Taghavi, S. M. R. Hassani, S. M. J. Hosseini, “Improvement of the Linear Interpolation Method of Excitation Using the Spline Interpolation Function to Numerically Calculate the Response Coefficient of Linear One-Degree-Of-Freedom Systems,” ‌Bulletin of Earthquake Science and Engineering,Volume 10, Issue 4, Pages 121-133, [ March 2024]: (in Persian).

[4]   M. Mahmoodabadi, S. M. J. Hosseini, “Comparison nonlinear response spectra of elastoplastic systems resulting from two excitation interpolation methods,” Ferdowsi Civil Engineering Journal, vol. 36, no. 4 , pp. 21-34, [winter 2024].

[5] A. K. Chopra, “Dynamics of Structures: Theory and Applications to Earthquake Engineering, 3^rded,” Prentice-Hall, Upper Saddle River, New Jersey,2007.