The Finite Difference Scheme for the Suspended String Equation with the Nonlinear Damping Term

Authors: Jaipong Kasemsuwan

Abstract:

A numerical solution of the initial boundary value problem of the suspended string vibrating equation with the particular nonlinear damping term based on the finite difference scheme is presented in this paper. The investigation of how the second and third power terms of the nonlinear term affect the vibration characteristic. We compare the vibration amplitude as a result of the third power nonlinear damping with the second power obtained from previous report provided that the same initial shape and initial velocities are assumed. The comparison results show that the vibration amplitude is inversely proportional to the coefficient of the damping term for the third power nonlinear damping case, while the vibration amplitude is proportional to the coefficient of the damping term in the second power nonlinear damping case.

Keywords: Finite-difference method, the nonlinear damped equation, the numerical simulation, the suspended string equation

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1075759

References:

[1] N. S. Koshlyakov, E. V. Gliner and M. M. Smirnov, Differential Equations of Mathematical Physics, Moscow, 1962 (in Russian). English Translation : North-Holland Publ. Co, 1964.
[2] M. Yamaguchi, T. Nagai and K. Matsukane, Forced oscillations of nonlinear damped equation of suspended string, Journal of Mathematical Analysis and Applications. 342, No.1 (2008), 89-107.
[3] J. Kasemsuwan, Exponential decay for nonlinear damped equation of suspended string, Proceedings of 2009 International Symposium on Computing, Communication, and Control, 2009, 308-312.
[4] J. Kasemsuwan, P. Chitsakul and P. Chaisanit, Simulation of suspended string equation, The 3rd Thai-Japan International Academic Conference, 2010, 60-61.
[5] K. Subklay and J. Kasemsuwan, Numerical simulation of suspended string Vibration, The 8th Kasetsart University Kamphaeng Saen Campus Conference, 2011, 1486-1491.
[6] J. Kasemsuwan, Numerical Solution of the Damped Vibration of Suspended String, University of the Thai Chamber of Commerce Journal, 2012, peer-review.