THE ENHANCED HOMOTOPY PERTURBATION METHOD FOR AXIAL VIBRATION OF STRINGS

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.

Automated summary

In this study, a governing equation for string axial vibrations with temporal and spatial damping forces is established using the Hamilton principle, which is an extension of the well-known Klein-Gordon equation. The classical homotopy perturbation method is found to be ineffective in analyzing this equation, prompting a modification with an exponential decay parameter. The analysis reveals that the amplitude decays exponentially with the damping parameter, and a frequency equation is established along with a stability condition analysis. The modified homotopy perturbation method proves to be more effective for nonlinear oscillators and successfully overcomes the shortcomings of the classical approach, as evidenced by the excellent agreement between analytical and numerical solutions.