Linearized Dynamic Analysis of Weightless Sagging Planar Elastic Cables

Abstract

Nonlinear dynamic analysis of elastic structures is known to be much more complex than their linear analysis. There are many sources of nonlinearity of the structural response of elastic cables, viz., physical nonlinearity due to nonlinear tension–extension relations, geometric nonlinearity associated with finite elastic displacements and nonlinearity of nodal load–displacement relations due to the presence of self-weight. Incremental second-order differential equations of motion are used to predict the vibration amplitudes relative to the equilibrium state caused by additional dynamic forces. Generally, the tangent stiffness matrices are determined by adding the tangent elastic and geometric stiffness matrices. Many a time, an approximate linearized dynamic analysis is attempted. In this paper, the initial tangent stiffness matrix corresponding to the equilibrium state is used in the second-order linear differential equation of motion. The dynamic response relative to the equilibrium state of the structure subjected to additional dynamic loads is predicted. The predictions of linearized dynamic analysis are generally considered acceptable for small elastic displacements from the equilibrium state. The validity of such linearized dynamic analysis for elasto-flexible cables obeying third-order differential equation of motion is explored.