Free vibrations of shallow inextensible cables: a perturbation approach

The statics and dynamics of catenary cables are classic matters of theoretical and applied mechanics. The paper systematizes a methodological strategy to achieve analytical solutions for the nonlinear static problem and linearized free dynamic problem of inextensible shallow cables. First, the one-dimensional continuum model of elastically extensible, perfectly flexible cables is revisited to state the nonlinear differential equations governing the static and dynamic equilibria as parametric expressions of the cable shallowness and extensibility. Second, an original hierarchical generalization of the Force Method is presented as methodological solution strategy. The key is the systematic application of perturbation schemes to equilibrium equations, indeformability constraints and geometric compatibility conditions. As a principal point of strength, the proposed strategy allows the unified and consistent treatment of the static and dynamic problems, while requiring the sole assumption of cable shallowness as postulate a priori. As major achievements, fully analytical high-order solutions are obtained for the asymptotic approximation of (i) the catenary configuration in the static field, and (ii) the natural frequencies and classical modal forms in the linearized dynamic field. Parametric analyses of the results highlight that high-order terms determine significant qualitative and quantitative effects on the modal solutions, including competing softening or hardening effects in the natural frequencies.