A Comparison of Numerical Integration Algorithms for Flexible Variable-Length Cable Multibody Dynamics

Flexible cables exhibit advantages such as high flexibility, superior strength, and ease of altering force transmission directions, making them widely applicable in aerospace and civil fields, including large deployable antennas, cable-driven parallel robots for in-orbit operations, and cranes. Motor-driven winches are commonly combined with flexible cables, resulting in variable-length characteristics. For cable systems featuring large deformations and time- varying lengths, the arbitrary Lagrangian-Eulerian (ALE) description method under the Absolute Nodal Coordinate Formulation (ANCF) framework is particularly suitable for modeling them as differential-algebraic equations (DAEs) with constraints. This study aims to compare the capabilities of several numerical methods in representing the dynamic characteristics of such systems, including accuracy, stability, computational efficiency, and constraint violation correction. The evaluated methods include the Runge-Kutta method, generalized-alpha method, and backward differentiation formula (BDF) method. Numerical simulations under multiple working conditions yield the following conclusions: All three methods stably generate accurate dynamic results for flexible variable-length cables. Regarding constraint violation correction, the generalized-alpha and BDF methods demonstrate higher accuracy without requiring the conversion of DAEs into ordinary differential equations. In terms of step size, the generalized-alpha method allows larger time steps. However, the Runge-Kutta method exhibits superior computational efficiency. The generalized-alpha method strikes a balance for scenarios requiring strict constraint enforcement and moderate computational resources, whereas the Runge-Kutta method is preferable for efficiency-critical applications with minor constraint violations.