Free Vibration of Flexible Cables

The free undamped vibrations of cables undergoing stretching, bending and twisting are investigated. To this end, a geometrically exact model of elastic cables accounting for bending and torsional stiffness is employed. The cable kinematics retain the full geometric nonlinearities. Starting from a prestressed catenary configuration, the nonlinear equations of motion are linearized about the initial equilibrium. In particular, two initial equilibrium states (shallow and taut) are considered while varying the cable elastic axial stiffness. The influence of the bending flexibility on the cable frequencies is assessed by direct comparisons with the frequencies predicted by classical cable theories of purely extensible cables.

Nonlinear Modeling of Cables with Flexural Stiffness

A geometrically exact formulation of cables suffering axis stretching and flexural curvature is presented. The dynamical formulation is based on nonlinearly viscoelastic constitutive laws for the tension and bending moment with the additional constitutive nonlinearity accounting for the no-compression condition. A continuation method, combined with a mixed finite-difference spatial discretization, is then employed to path-follow the static responses of cables subject to forces or support displacements. These computations, conducted in the quasistatic regime, are based on cables with linearly elastic material behaviors, whereas the nonlinearity is in the geometric stiffness terms and the no-compression behavior. The finite-difference results have been confirmed employing a weak formulation based on quadratic Lagrangian finite elements. The influence of the flexural stiffness on the nonlinear static responses is assessed comparing the results with those obtained for purely extensible cables. The properties of the frequencies of the linear normal modes of cables with flexural stiffness are also investigated and compared with those of purely extensible cables.