ROLL MOTION OF MANOEUVRING SHIPS
Peter Trägårdh (SSPA Sweden AB)
Abstract: An analysis of a number of manoeuvring tests with freesailing models carried out in SSPA Maritime Dynamics Laboratory reveals interesting relations regarding roll motion in turning and zigzag tests. A large overshoot at the zigzag test in combination with the increasing yaw rate at the pullout from the turning circle test should mean a considerable degree of course instability. A large roll angle at these situations contributes to the poor course stability and vice versa. Further, a large roll angle, say in excess of 10°, should be a problem with respect to the comfort and safety of crew, passengers and cargo. Shifting of cargo due to excessive heel may in the worst case lead to a disaster with loss of lives and ship and environmental damage. This paper gives a simple method to make a preliminary assessment of the course stability and combined roll behaviour of a ship.
1. INTRODUCTION
Sometimes an extreme rolling behaviour of ships at manoeuvring has been found at freesailing model tests. For the most extreme case the maximum roll angle was 26°and the corresponding overshoot 60° at the 20/20°zigzag test. A quite strange behaviour was observed at the standard turning circle test ending with a pullout. The maximum roll angle was abt 28°and when the rudder was put amidships for the pullout, the yaw rate and roll angle were first decreased as expected, but then increased as speed was picked up. The test had to be stopped before steady state was reached due to restrictions of test area, and by then the roll angle was abt 20°and both roll angle and yaw rate had an increasing trend.
The reason for this unstable behaviour was not obvious. The above example was a single skeg RoRo vessel with a normal block coefficient and rudder area. Thus it was decided to study this phenomenon in view of results from freesailing manoeuvring tests with 24 selected ships that had been tested at SSPA. The selected ships were of different types such as RoRo, LNG, Cruising and Container vessels.
It should be noted that only results from model tests at even keel condition have been used in this study. For reasons of confidentiality no details of the ships or test results others than given here can be published.
2. MODEL TESTS
The 20/20°zigzag tests only were chosen for this study. Results for the 10/10°zigzag tests were not available for all ships and the turning tests did not really add anything.
2.1 Analysis of model test results
After compiling the data, relations between different variables or combinations of variables were studied in order to find explanations to the unstable behaviour.
By plotting the max roll angle versus the 1^{st} overshoot some trend could be identified although the scatter is large (fig 1). A large overshoot is an indication of poor course stability and it seems reasonable that the maximum roll angle should increase with increasing overshoot.
Fig.1 
Max angle of roll plotted versus 1^{st} overshoot (IMO limit 25°). 
A more logic relation would be expected if the rollmax is plotted versus GM.
Fig.2 Max angle of roll plotted versus metacentric height GM.
By dividing the rollmax value by the square of the approach speed the scatter is reduced.
Fig.3 
Max angle of roll divided by speed squared plotted versus metacentric height. 
In the search for more relations also including hull characteristics the following interesting trend was found.
Fig.4 
1^{st} overshoot versus length to beam ratio with linear trend line. 
After further exercises in order to minimise the scatter, the derived relation and trend is given in fig 5. Here the Froude number and the slenderness ratio have been used instead of speed and length to beam ratio respectively.
Fig.5 Max angle of roll and curve fit.
An empirical curve fit is given by
y = 50 + 500・x + (x + 0.034)^{5}/20000 (1)
where y = max roll angle/Fnl/Fnl
x = GM/KG/Slr
Fnl = Froude number = V・0.51444/sqrt(g・L)
V = approach speed of ship[knots]
L = waterline length of ship[m]
GM = metacentric height[m]
KG = vertical centre of gravity[m]
Slr = slenderness ratio = L/Displacement^{0.333}
Thus a preliminary maximum roll angle may be calculated by just using the above parameters.
The 1^{st} overshoot angle was analysed in a similar way and the final result given in fig 6. To get the least scatter the speed was here used instead of Froude number. It should also be noted that the GM/KG ratio is here multiplied by the slenderness ratio instead of divided as for the roll.
Fig.6 1^{st} overshoot and curve fit.
The empirical curve fit is given by
y = 0.02 + 0.005・x + (x + 0.3)^{2}/20 (2)
where y = 1^{st} overshoot/V/V
x = GM/KG・Slr
Thus a preliminary 1st overshoot for the 20/20°zig  zag test could be calculated.
