| A theoretical model is presented which describes the three-dimensional nonlinear motion of a sagged cable that supports an array of discrete masses. An asymptotic form of this general model is derived for the linear response of a cable/mass suspension having small equilibrium curvature and horizontal supports. While the asymptotic model remains rich enough to capture dominant sagged cable effects, it is simple enough to permit closed-form analysis. A free vibration analysis is pursued that leads to closed-form solutions for problems which, heretofore, were analyzed using purely numerical methods. A second asymptotic model is developed which partially relaxes the quasi-static stretching assumption used in the first asymptotic model. Solutions obtained from the second model are used to establish the theoretical limit of the first model. This comparison reveals that the first (simpler) model remains an excellent approximation for all cable/mass system modes having natural frequencies below a critical frequency for elastic mode response. Among the advantages of the asymptotic model is its ability to provide results for: (1) complex mass arrays, (2) high-order modes, and (3) dynamic cable tension. Furthermore, this model is readily extended to predict (1) forced response and (2) the response of cables with arrays of offset bodies (pendula). Examples highlight the key role played by mass array symmetry and lead to new conclusions regarding free and forced response. Experimental measurements of forced response provide measured frequency response functions and natural frequencies that are in agreement with results from the asymptotic model. |
