NON-LINEAR FREE VIBRATION ANALYSIS OF A STRING UNDER BENDING MOMENT EFFECTS USING THE PERTURBATION METHOD

In this paper, the non-linear free vibration of a string with large amplitude is considered. The initial tension, lateral vibration amplitude, cross-section diameter and the modulus of elasticity of the string have main effects on its natural frequencies. Increasing the lateral vibration amplitude makes the assumption of constant initial tension invalid. Therefore, it is impossible to use the classical equation of transverse motion assuming a small amplitude. On the other hand, by increasing the string cross-sectional diameter, the bending moment effect will increase dramatically, and it will act as an impressive restoring moment. Considering the effects of the bending moments, the non-linear equation governing the large amplitude transverse vibration of a string is derived. The time-dependent portion of the governing equation has the form of the Duffing equation. Due to the complexity and non-linearity of the derived equation and the fact that there is no established exact solution method, the equation is solved using the perturbation method. The results of the analysis are shown in appropriate graphs, and the natural frequencies of the string due to the non-linear factors are compared with the natural frequencies of the linear vibration of a string without bending moment effects.