### THE ENHANCED HOMOTOPY PERTURBATION METHOD FOR AXIAL VIBRATION OF STRINGS

Ji-Huan He, Yusry O. El-Dib

DOI Number
https://doi.org/10.22190/FUME210125033H
First page
735
Last page
750

#### Abstract

A governing equation is established for string axial vibrations with temporal and spatial damping forces by the Hamilton principle. It is an extension of the well-known Klein-Gordon equation. The classical homotopy perturbation method (HPM) fails to analyze this equation, and a modification with an exponential decay parameter is suggested. The analysis shows that the amplitude behaves as an exponential decay by the damping parameter. Furthermore, the frequency equation is established and the stability condition is performed. The modified homotopy perturbation method yields a more effective result for the nonlinear oscillators and helps to overcome the shortcoming of the classical approach. The comparison between the analytical solution and the numerical solution shows an excellent agreement.

#### Keywords

Homotopy Perturbations Method, Exponential Decay Parameter, Damping Duffing Equation, Damping Nonlinear Klein-Gordon Equation

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DOI: https://doi.org/10.22190/FUME210125033H

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