Keywords: dynamic analysis, chains, slack cables, Karhunen-Loéve decomposition, reduced order model.

Chains and cables are structural elements of wide applications in mechanical and civil engineering. They are employed as mooring devices as well as in other structural applications. The dynamic response of structural systems (e.g. floating platforms) that include chains or cables are influenced by the strong nonlinearity which is of geometric rather than material type. In this paper, the dynamic response of chains and slack cables subjected to self-weight and prescribed motion at their ends is particularly addressed. First, the inextensible chain plane dynamics is studied using a rigid link model and the spatial coordinates as unknowns. A temporal power series approach provides a simple solution method to this problem governed by differential algebraic equations (DAE). The results are contrasted with the same system solved with the modified extended backward differentiation formula implicit (MEBDFI) routine implemented in Maple. The advantages and disadvantages of both solvers are discussed. Additionally, other existing models of concentrated masses connected with springs and relative angles as unknowns, are used for comparison. Afterwards, the extensible cable dynamics in the plane are addressed. The strongly nonlinear, governing partial differential equations are derived including various constitutive equations. In order to find the solution to this problem, a Galerkin method is employed with optimal basis. The contribution of this work is to find the basis by means of the Karhunen Loéve Decomposition (KLD) from the knowledge of the dynamics of a simpler problem, i.e. the inextensible chain dynamics with rigid links. This method essentially provides an orthonormal basis, proper orthogonal modes, (POMs) that represent the given data in a certain least squares optimal sense, and has lately gained popularity in several fields. The problem under study need not be linear although the approach constitutes a linear process and, when used to analyze experimental or computational data, allows the extraction of the dominant patterns of the dynamic response. The POMs and the proper orthogonal values (POVs) are found as the eigenmodes and eigenvalues, respectively, of a certain matrix constructed with the chain dynamics data. The KLD combined with a Galerkin approximation is useful for the construction of a reduced order model, i.e. very few modes are usually necessary to appropriately describe the dynamics of the cable. Some numerical examples are presented to demonstrate the capabilities and potential of the proposed method as well as to show low-rank approximations for the dynamic behavior of slack cables and the performance of the algorithms. Results are compared with others obtained for the extensible cable using a trigonometric basis. It can be concluded that the methodology is very efficient and reliable.

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