Vessel Motion Control Through Anchor-Line Design
Authors: Klotz J.A. 
Published: December 2007

In contending with the surge, pitch, and heave of a vessel we cannot control the size and frequency of the waves, nor the mass of the vessel; but we can control the anchoring system - by changing the pre-tension, by changing the weight of the anchor lines, and by using a compound anchor line consisting of an upper, wire-line section and a lower, chain section.

Introduction
This paper concerns an attempt to gain insight into the use of chain anchor lines and wire-line anchor lines to keep a floating work vessel on station in a sea-way. Interviews with barge captains seem to reveal a wide diversity of opinion. Some barge captains advocate slackening lines before a storm; others advocate tightening the lines. Some operators maintain that chain anchor lines are superior to wire anchor lines; others prefer wire. Some aspects of the problems of vessel motion in a sea-way have been described by J. J. Leendertse. The material presented here is based in part upon his analysis; it includes, in addition, a consideration of varying forms of anchor line (chain, wire line, or a combination of wire line and chain), and it attempts to show the influence of varying anchor-line tension on ship motion. As usual, when we attempt to apply formal mathematics to a complicated problem, we learn that many assumptions and simplifications are necessary to reduce the complexity of the mathematics to a point that some practical calculations can be made. Thus this paper does not pretend to solve all of the problems of anchoring systems, but it should throw considerable light upon the relationships among vessel motions, anchor-line tension variation, and composition of the anchor line. Details of these relationships are presented in Appendices A and B.

A Mass Between Springs
A vessel anchored in a sea-way acts somewhat like a mass suspended between springs. If such a mass is excited by an alternating force field, the mass will vibrate. The amplitude of the vibration depends upon several things: size of the mass, spring constant of the springs, and frequency and amplitude of the alternating force field. The mathematics of such motion is described in most books dealing with the subject of mechanics. The book by W. T. Thomson is an example.

Anchor-Line Spring Coefficients
The array of anchor lines holding a vessel in position can be considered to act like a spring. One critical assumption is necessary. For the work described here, we assume that the anchor lines behave like a linear spring; that is, that the spring coefficient does not change over the range of motion of the vessel. This is not a very good assumption, but it makes the mathematics simpler; the use of it does not alter the conclusions of this paper. But if we assume that anchor lines behave as linear springs, paper. But if we assume that anchor lines behave as linear springs, the specific values that we calculate for vessel motion will not be correct. The spring coefficient of an anchor line depends upon several things: length of the line, weight of the line (1 ft of anchor line measured in water), and tension in the line measured at the ship while it rests in calm water. We assume that anchor lines always are sufficiently long so that a portion of the line lies along the bottom of the sea; the line is slack, and no vertical forces are imposed upon the anchor. For illustration we will consider a vessel held by two anchor lines as shown in Fig. 1.
JPT
P. 685