Cable dynamics
During search operations, a tow platform (rotorcraft) pulls the submerged load with a partially immersed cable. The tow cable is a long (hundreds of feet) flexible structure that exhibits varying levels of curvature along its length in response to hydrodynamic drag, gravity, buoyancy and tension. Transverse deflections are not small in comparison to the length and "bending" slopes can exceed 60 degrees. Hydrodynamic drag is significantly influenced by cable orientation, necessitating the use of a "large-deflection" analysis.
The length-to-diameter ratio of the structure is very large (over 100), and is modeled as an Euler-Bernoulli beam. I implemented two different formulations for the cable dynamics. The first model uses small deformations within each beam finite element together with floating reference frames to model locally small (but globally large) angles. Two famous static test cases are shown below. The first test case is a uniform cantilever beam under tip bending moment, which bends into a perfect circle as shown in the first figure.
The second test case involves bending under the action of a vertical tip load. While the material of the beam influences small strain levels and operates in the linear elastic region, the geometry of the beam induced by bending introduces nonlinearities in the load-displacement curve due to axial fore-shortening.
This large deflection model is applied to a long cable that is hinged to the helicopter and the towed body. The root angle at the tow point is modeled as a separate state to circumvent the "small angle" formulation inherent to traditional beam dynamics implementations. When the tow point moves at increasing speeds (from left to right), cable drag causes varying tension along the span. Shapes of the cable at various speeds are represented as dashed lines, and the numbers in the adjacent figure represent tow speed in knots corresponding to a particular geometry.
