The Influence of Practical Time Integration Schemes on Dynamic Mooring Line Analysis
Authors: Hearn G.E. , Thomas D.O. 
Published: October 2008

ABSTRACT
This paper examines the influence of using different time integration schemes to solve the dynamic equations of motion applicable to a mooring line. The four time integration schemes investigated are the Central Difference, Houbolt, Wilson-? and Newmark schemes. The derivation of each scheme is outlined and the expected stability characteristics of each scheme is indicated when applied to linear analysis. For non-linear formulations, the stability and accuracy of the schemes can only be assessed from the solution of the governing equations. The general solution of the equations of motion for a cable structure is presented using all four schemes, and an assessment of the stability, accuracy and the influence of time step size for each scheme is discussed.

INTRODUCTION
For many years the problem of calculating the response of floating structures subjected to regular and irregular wave excitation has been addressed and significant advances have been made in the prediction of these responses. When considering the case of moored offshore structures, there is the additional problem of accounting for the influence of the moorings upon the floater motions. Previously, mooring line influences have been included by modelling the tension of the mooring cable at its point of attachment, as an equivalent contribution to the stiffness (hydrostatic restoration) matrix of the floater. The tension is usually derived from the solution of the cable catenary equations, which yield the static tension at any point in the cable(l)*.
To satisfy the requirements of the certifying authorities, a factor of safety is introduced which weights the static tension prior to determining the motions of the floater. The inclusion of a safety factor is clearly a simplified procedure for accounting for the dynamic effects inherently present, and for any other uncertainties associated with the use of a static tension. Bergdahl and Rask(2) indicate that a safety factor of 3 is to be used for intact moorings, whilst a value of 1.4 would be acceptable during transient floater motions after one cable failure. Boom et al(3) suggest a safety factor of 3 for operational conditions and a value of 2 for survival. Lloyds(4) have a series of safety factors ranging from 1.4 for survival with one line failure under maximum environmental loading, to 3 for intact operating conditions. The application of safety factors to a static tension is unsatisfactory in a number of respects, namely :The static influence of mooring lines upon the floater motions is not significant, but floater motions do produce a significant dynamic response in the mooring lines. Hence the dynamic analysis of mooring lines is usually separated from the dynamic analysis of moored floaters.The dynamics of cables has been studied for many years with the aim of ensuring the structural integrity of the cable. Results show that there is significant variation in the dynamic tension and it is evident that this may have an influence upon the floater motions, particularly by affecting the low frequency damping values, see Huse(5).