A Reduced Order Model of Taut Cable Dynamics Using Karhunen-Loeve Decomposition

Abstract

In a previous work carried out by the research group, the dynamics of slack extensible cables subjected to end prescribed motion was tackled through a reduced order model with a Karhunen-Loeve (KL) basis (also known as POMs, Proper Orthogonal modes). The latter was found from data of the dynamic response of an inextensible chain under similar conditions. The use of the chain problem was a natural selection due to the similarity between the slack cable and the chain. Now, the same approach is applied to a taut cable which exhibits small sag. Here, the pretension level is higher, the extensibility is more noticeable and the range of validity of the chain basis should be verified. In the present work, both the chain and the cable will be exposed to horizontal forces and prescribed dynamic motion at the ends. This situation resembles the case of the guys in a guyed structure as in the case of communications towers. The chain dynamics are solved with two DAE approaches and then, the KL basis extracted from the obtained data. It should be mentioned that this basis is optimal for the problem under study, in the sense of the least squares criterion. Once the basis is found along with the eigenvalues (POVs, Proper Orthogonal values, sometimes named "energy"), the POMs are introduced in a Galerkin approach to represent the dynamics of the extensible cables. Given its hierarchical grouping, the KL decomposition allows to choose only a few modes to be used in a Galerkin approximation, keeping the most important component of the dynamic response information. An illustrative example is worked out using the data from a real communication tower guy. The availability of a reduced dimensional model is very useful for structures that have large or infinite degrees of freedom. The advantages are apparent: considerable saving of computing times makes simulations more economical and parametric and bifurcation studies feasible.