Karhunen-Loéve Decomposition and Model Order Reduction applied to the Non-Linear Dynamics of an Extensible Cable

摘要

Chains and cables are employed as mooring devices as well as in other structural applications. The dynamic response of structural elements joined to the chains/cables are influenced by the strong nonlinearity which is of geometric rather than material type. In this paper, the dynamic response of slack elastic cables subjected to self-weight and prescribed motion at their ends, is addressed. The dynamic response of the slack cable governed by a strongly nonlinear system of partial differential equations is solved by means of Galerkin method. The contribution of this work consists on using a set of trial orthogonal functions that is obtained by using a Karhunen Loeve (KL) Decomposition. This method essentially provides an orthonormal basis to represent the given data in a least squares optimal sense. Experimental or computational data allows to extract dominant patterns of the dynamic responses. Herein, the study is tackled by means of this advantageous technique and starting from the knowledge of the dynamics of a simpler problem with similar characteristics, i.e. the dynamical study of an inextensible chain with rigid links. First, the chain problem is solved with different approaches considering inextensibility (Differential-Algebraic Equations, DAE) and extensibility (Differential Equations problem). Then, the response of the chain problem is employed to find the KL optimum basis. The extensible cable model and the governing equations are then stated and, using a Galerkin approximation with that basis, a low dimensional model is obtained. Some numerical examples are presented to demonstrate the capabilities and potentiality of the proposed method. Results are compared with others obtained for the extensible cable by using a basis of trigonometric orthogonal functions in the Galerkin Method.