Dynamic modeling of cable deployment/retrieval based on ALE-ANCF and adaptive step-size integrator

The towing vehicle must maintain a stable distance from the seafloor to ensure the accuracy and safety of deeptowed seismic systems. However, due to equipment limitations, controlling the towing depth is achievable solely by adjusting the length of the towing cable. In this study, an Absolute Nodal Coordinate Formulation (ANCF) with Arbitrary Lagrange-Euler (ALE)description is employed to model variable-length flexible towing cables. The inertial and elastic force formulas for ALE-ANCF, along with their corresponding Jacobian matrices, are derived in detail, and their coefficient matrices are precomputed to avoid Gaussian integration. The proposed adaptive step-size strategy enhances computational efficiency. Detailed investigation into the impact of cable deployment/retrieval on towing depth is conducted, revealing an almost linear relationship between length change and towing depth. In Addition, a proportional, integral, derivative (PID) controller with compensation is proposed for efficiently tracking the target depth and alleviating the perturbations from the mothership, illustrating the feasibility of PID control. These findings hold practical significance for enhancing the operational safety and efficiency of deep-towed seismic systems.