Solving Mathematical Models of String Vibrations with Radial Basis Function Networks

Alifia Nisa Maghfiroh; Ari Kusumastuti

Alifia Nisa Maghfiroh Department of Mathematics, Faculty of Science and Technology, Maulana Malik Ibrahim State Islamic University of Malang
Ari Kusumastuti Department of Mathematics, Faculty of Science and Technology, Maulana Malik Ibrahim State Islamic University of Malang

Keywords: Vibration String Model; Second-Order Differential Equation; Radial Basis Function (RBF); Numerical Solution; Error Analysis

Abstract

String vibration phenomena on suspension bridges describe oscillations due to external forces and gravity, which can affect the stability and safety of the structure. The mathematical model for this vibration problem is stated by Mckenna (1999) in the form of a second-order ordinary differential equation for deflection . This study aims to analyze the behavior of the model and solve the model solution with Radial Basis Function (RBF) and compare it with the analytical solution. In RBF, the basis function is selected in the form of a 1st order multiquadratic function. Error analysis is carried out by calculating the difference between RBF and the analytical solution at a time interval with parameters . The results show that the numerical solution of RBF to the exact solution produces a difference of at time for , and a difference of at time for . With this level of accuracy, the RBF method can be said to be quite effective in solving string vibration models, and has the potential to be applied to the stability analysis of engineering structures such as suspension bridges.

References

[1] K. Aliy, A. Shiferaw, and H. Muleta, "Radial basis functions based differential quadrature method for one dimensional heat equation," American Journal of Mathematical and Computer Modelling, vol. 6, no. 2, pp. 35–42, 2021. doi: 10.11648/j.ajmcm.20210602.12.
[2] R. J. Howlett and P. L. C. Jain, Radial Basis Function Networks 2, vol. 67. 2001.
[3] M. Jamhuri, "Penyelesaian numerik persamaan differensial biasa menggunakan jaringan fungsi radial basis," no. 2, 2011.
[4] P. J. McKenna, "Large torsional oscillations in suspension bridges revisited: Fixing an old approximation," The American Mathematical Monthly, vol. 106, no. 1, pp. 1–18, 1999.
[5] W. Pandia and I. Sitepu, "Penentuan galat persamaan diferensial biasa orde 1 dengan metode numerik," Jurnal Mutiara Pendidikan Indonesia, vol. 6, no. 1, pp. 1–9, 2021. doi: 10.51544/mutiara%20pendidik.v6i1.1907.
[6] M. Thahiruddin and M. Jamhuri, "Penyelesaian numerik model pemangsa-mangsa dengan metode jaringan fungsi radial basis menggunakan trigonometric shape parameter," Jurnal Arjuna: Publikasi Ilmu Pendidikan, Bahasa dan Matematika, vol. 1, no. 4, pp. 248–263, 2023. doi: 10.61132/arjuna.v1i4.135.
[7] D. Varberg, E. J. Purcell, and S. E. Rigdon, Calculus, 9th ed., vol. I, translated by I. N. Susila. Jakarta: Erlangga, 2007.
[8] H. W. Permata, A. Kusumastuti, and Juhari, “Solusi numerik model gerak osilasi vertikal dan torsional pada jembatan gantung,” Jurnal Riset Mahasiswa Matematika, vol. 1, no. 1, pp. 1–13, 2021, doi: 10.18860/jrmm.v1i1.13409.