Explicit High-Accuracy Approximations for the Period of Large-Amplitude String Vibrations

Strings (cables) are basic structural elements in many engineering systems, including cranes, maritime vessels, and bridge structures, where dynamic loads can induce large-amplitude transverse vibrations. Such nonlinear vibrations may cause fatigue damage and eventually lead to cable or structural failure. Therefore, the nonlinear large-amplitude vibration of taut strings is crucial for engineering design and safety assessment. This technical note derives explicit and highly accurate analytical expressions for the period of transverse vibration of flexible strings. The proposed method consists of two main steps: (i) expressing the exact period in integral form, and (ii) applying a variable transformation along with the two-node and three-node Gaussian–Legendre quadrature rules. The obtained period expression is explicit and maintains high accuracy over the entire amplitude range from small to large. As the amplitude tends to infinity, the approximate period shows an error of only 0.00081% relative to the exact period. The results presented in this study will provide theoretical support and useful guidance for string design.