ROV on a Tether

What is modelled

The tether is cut into 20 stiff segments, each with its share of weight in water and the drag of the current on the part of the cable that is square to the flow. The ship end and the vehicle end are fixed, and the model is run forward in time with strong damping until the tether stops swaying: the shape and the tensions at rest are the answer. The thrusters must supply whatever is left after the pull of the tether, the drag on the vehicle itself and its small net weight, so the thrust needed rises with the square of the current and with the length of cable out.

Equations

• Cable drag per length f = ½ ρ Cd d |un| un with the normal relative velocity u_n, C_d = 1.2.
• Vehicle drag D = ½ ρ (CdA) U2.
• Segment tension T = (EA / Δs) (l − Δs), never negative.
• Still-water cable between the ends: catenary with √(L2 − H2) = 2a sinh(X / 2a), horizontal tension w a.
• Thrust = −(Tcable + D + W), limit current where it equals the thruster force.

Checks run in the page

• The relaxed tether in still water against the exact catenary.
• The force balance of all tether segments.
• The change of thrust when the current doubles against the square law.
• The limit current found by bisection and verified by a fresh solution.

Honest simplifications

The current is uniform and steady, the ship does not move, the tether is a single uniform cable that never goes slack on its own, only the drag normal to the cable is counted, and the vehicle is held exactly at the chosen point with thrust in any direction. Real tethers have floats, changing current with depth and ship motion.

• Press Strong current and watch the thrust climb past the limit.
• Press Deep dive to see the cost of a long tether.
• Press Thin light tether and compare the thrust needed.