In-plane free vibration analysis of an inclined taut cable with a point mass

Abstract

An analytical theory for predicting the free in-plane vibrations of an inclined taut cable carrying a point mass is presented in this paper. A linearized eigenvalue formulation of the problem is derived. It is based on the nonlinear equations of motion and the quasi-static stretching assumption, which assumes that the dynamic tension is piecewise constant to the left and right of the point mass. By neglecting the equation of motion for the longitudinal vibrations in favor of the constitutive cable equation, the equation governing the lateral vibrations can be solved using its boundary conditions. Two coupled linear equations, whose non-trivial solution gives the natural frequencies of the system, can be constructed from the jump and continuity conditions in the longitudinal direction. The comparison of the results with those of a known Galerkin procedure shows that the analytical theory is sufficiently accurate in the applicable parameter range. These results reproduce the curve-veering and the elastic mode transition phenomena known from previous studies. As such, the theory presented here extends the range of applicability of previous analytical cable theories to include inclined cables with point masses as well.