Calculating Arc Length and Curvature of Helical Elements in Bent Cables and Umbilicals Using Fourier Series

Mechanical cable analyses range from simple hand calculations to complex, three-dimensional finite element analyses. This paper considers numerical calculations of analytical models as an attractive compromise between these extremities. The paper derives numerical calculations of the arc lengths and the local bending curvatures of helical elements in bent cables and umbilicals, and compares the results with analytical approximations found in the scientific literature. The numerical calculations are highly efficient, requiring only a very few function evaluations to give excellent accuracy. On today’s PCs and laptops the calculations require only a few milliseconds CPU time. The excellent convergence properties of Fourier series for smooth, periodic functions are used to achieve these favorable results. In addition to high accuracy, the numerical calculations disclose behaviors that are not captured by the analytical approximations.