A Reduced Order Model of a 3D Cable Using Proper Orthogonal Decomposition

Cables and flexible pipes are found in many applications of civil, aerospace, and mechanical engineering. Many offshore engineering problems involve slender rods or cables (pipelines, risers and mooring lines) and also are commonly used in robotics to bring energy to the actuators of a robotic arm. Risers are used, for example, to convey fluids from the bottom of the sea up to the surface. Engineers need numerical models of nonlinear thin deformable structures in order to predict their dynamics and the internal stresses. To accurately simulate the motion of slack marine cables, it is necessary to capture the effects of the cables bending and torsional stiffness. Hence, the need for numerical nonlinear models is still an intense subject of research. In this work, a finite element simulation of cables is used to get a reduced order model by means of the Proper Orthogonal Decompositions (POD). The finite element cable model adopted makes use of a twisted cubic spline element with a lumped mass approximation; it provides a representation of both the bending and torsional effects. Cables are modeled as slender flexible rods that can support environmental, gravitational, and buoyancy forces. The nonlinear equations of motion for the continuous cable are developed in terms of an inertial frame of reference using the Frenet equations. The model permits large deflections and finite rotations and accounts for tension variation along its length as well as nonlinearities arising from applied loads or constraints. The proper orthogonal modes, in conjunction with the Galerkin approach, permit the construction of a lower-dimensional model that is very efficient with respect to the ones constructed with the typical finite elements basis. Different proper orthogonal modes computed from time-series at different excitation frequencies are used and different approximations are compared.