Ship Hydrostatics and Stability 2010 Part 9 pdf

Ships can capsize in head seas - that is waves travelling against the ship - and especially in following seas - that is waves travelling in the same direction with the ship.. Kerwin 1955

Additional regulations mentioned in this chapter are a code for small boats issued in the UK, and codes for internal-navigation vessels issued by theEuropean Parliament and by the Swiss Parliament.

gen-in Section 3.4 The analysis of the results leads to the followgen-ing conclusions:

1 The area under the GZ& curve, up to 30°, is 0.043 mrad, less than the

required 0.055 The area up to 40° equals 0.084mrad, less than the required0.09 mrad The area between 30° and 40° equals 0.04mrad, more than therequired 0.03 mrad

Table 8.1 Small cargo ship - the IMO general requirements

b GZefi Area under righting arm

0.000 0.019 0.043 0.072 0.109 0.153 0.197 0.233 0.262 0.283 0.298 0.302 0.288 0.253 0.201 0.139 0.073 0.009 -0.046

0.000 0.001 0.004 0.009 0.017 0.028 0.043 0.062 0.084 0.107 0.133 0.159 0.185 0.208 0.228 0.243 0.252 0.256 0.254

Intact stability regulations I 199

2 The righting arm lever equals 0.2 m at 30°; it meets the requirement at limit

3 The maximum righting arm occurs at an angle exceeding the required 30°

4 The initial effective metacentric height is 0 1 2 m, less than the required 0.15m

Example 8.2 - Application of the IMO weather criterion for cargo and senger ships

pas-We continue the preceding example and illustrate the application of the weathercriterion to the same ship, in the same loading condition The main dimensions

are L = 75.4, B = 11.9, Tm = 4.32, and the height of the centre of gravity

is KG = 5, all measured in metres The sail area is A = 175m2, the height

of its centroid above half-draught Z = 4.19m, and the wind pressure P =

504 N m~~2 The calculations presented here are performed in MATLAB keepingthe full precision of the software, but we display the results rounded off to thefirst two or three digits To keep the precision we define at the beginning the

constants, for example L = 75.4, and then call them by name, for example L.

The wind heeling arm is calculated as

To find the roll period we first calculate the coefficient

C = 0.373 + 0.023 x ( - - 0.043 x ( = 0.404

With GMeff = 0.12 m the formula prescribed by the code yields the roll period

T = <2 ° B = 27.752s

With this value we enter Table 3.2.2.3-4 and retrieve s = 0.035 Then, the angle

of roll windwards from the angle of statical stability, under the wind arm /wi, is

0i = W9kXiX2Vrs = 16.34°

Visual inspection of Figure 8.2 shows that the weather criterion is met This fact

is explained by the low sail area of the ship

Example 8.3 - The 1MO turning criterion

To illustrate the IMO criterion for stability in turning we use the data of the samesmall cargo ship that appeared above Cargo ships are not required to meet thiscriterion, but we can assume, for our purposes, that the ship carries more than

12 passengers

The ship length is L = 75.4 m, the mean draught Tm — 4.32 m, the ship speed

VQ = 16 knots, and the vertical centre of gravity KG = 5.0m The speed in

ms"1 is

V 0 = 16 x 0.5144 = 8.23 ms"1

and the heel arm due to the centrifugal force is

1 T = 0.02^- (~KG - ^p j - 0.051 m

Figure 8.1 shows the resulting statical stability curve We see that the heel angle

is slightly larger than 11°

Example 8.4 - The weather criterion of the US Navy

To allow comparisons between various codes of stability we use again the data

of the small cargo ship that appeared in the previous examples We initiatethe calculations by defining the wind speed, Vw = 80 knots, the sail area,

A = 175m2, the height of its centroid above half-draught, £ = 4.19m, and

the displacement, A = 26251 The corresponding stability curve is shown inFigure 8.4 The wind heeling arm is given by

cos 2 6

1QQQA

At the intersection of the righting-arm and the wind-arm curves we find thefirst static angle, </>i ~ 7.5°, and the righting arm at that angle equals 0.03 m

Intact stability regulations I 201

Rolling 25° windwards from the first static angle the ship reaches —17.5° Thesecond static angle is <ftst2 — 85.7° The ratio of the GZ value at the first static angle to the maximum GZ is 0.03/0.302, that is close to 0.1 and smaller than the maximum admissible 0.6 The area b equals 0.235 mrad, and the area a equals 0.024 mrad The ratio of the area b to the area a is nearly 10, much larger than

the minimum admissible 1 4 We conclude that the vessel meets the criteria ofthe US Navy

Example 8.5 - The turning criterion of the US Navy

We continue the calculations using the data of the same ship as above We

assume the speed of 16 knots, and the vertical centre of gravity, KG = 5m,

as in Example 8.3 In the absence of other recommendations we consider, as

in NES 109, that the speed in turning is 0.65 times the speed on a straight-linecourse, that is

8.10 Exercises

Exercise 8.1 - IMO general requirements

Let us refer to Example 8.1 Find the KG value for which the general requirement

4 is fulfilled Check if with this value the first general requirement is also met

Exercise 8.2 - The IMO turning criterion

Return to the example in Section 8.9 and find the limit speed for which theturning criterion is met

Exercise 8.3 - The IMO turning criterion

Return to the example in Exercise 8.2 and check if with the vertical centre of

gravity, KG, found in Section 8.10 the turning criterion is met.

Exercise 8.4 - The US-Navy turning criterion

Return to Example 8.4 and redo the calculations assuming a wind speed of

100 knots

Exercise 8.5 - The code for small vessels

Check that for ^czmax — 15° and 30°, Eq (8.16) yields the values specified incriterion 1 for multihull vessels (Section 8.6)

if yes, how? Arndt and Roden (1958) and Wendel (1965) cite French engineersthat discussed this question at the end of the nineteenth century (J Pollard and

A Dudebout, 1892, Theorie du Navire, Vol Ill, Paris) In the 1920s, Doyere

explained how waves influence stability and proposed a method to calculate thatinfluence After 1950 the study of this subject was prompted by the sinking of afew ships that were considered stable

At a first glance beam seas - that is waves whose crests are parallel to the

ship - seem to be the most dangerous In fact, parallel waves cause large angles

of heel; loads can get loose and endanger stability However, it can be shownthat the resultant of the weight force and of the centrifugal force developed inwaves is perpendicular to the wave surface Therefore, a correctly loaded vesselwill never capsize in parallel waves, unless hit by large breaking waves Ships

can capsize in head seas - that is waves travelling against the ship - and

especially in following seas - that is waves travelling in the same direction with the ship This is the lesson learnt after the sinking of the ship Irene Olden- dorff in the night between 30 and 31 December 1951 Kurt Wendel analyzed the

case and reached the conclusion that the disaster was due to the variation of therighting arm in waves Divers that checked the wreck found it intact, an obser-

vation that confirmed Wendel's hypothesis Another disaster was that of Pamir.

Again, the calculation of the righting arm in waves surprised the researchers(Arndt, 1962)

Kerwin (1955) analyzed a simple model of the variation of GM in waves and

its influence on ship stability His investigations included experiments carriedout at Delft and he reports difficulties that we attribute to the equipment available

at that time

To confirm the results of their calculations, researchers from Hamburg carriedout model tests in a towing tank (Arndt and Roden, 1958) and with self-propelled

models on a nearby lake (Wendel, 1965) Post-mortem analysis of other marine

disasters showed that the righting arm was severely reduced when the ship was

on the wave crest Sometimes it was even negative

Paulling (1961) discussed The transverse stability of a ship in a longitudinalseaway'.

Storch (1978) analyzed the sinking of thirteen king-crab boats In one case hediscovered that the righting arm on wave crest must have been negative, and intwo others, greatly reduced

Lindemann and Skomedal (1983) report a ship disaster they ^attribute to the

reduction of the stability in waves On 1 October 1980 the RO/RO (roll-on/roll' off) ship Finneagle was close to the Orkney Islands and sailing mfollowing seas,

that is with waves travelling in the same direction as the ship All of a suddenthree large roll cycles caused the ship to heel up to 40° It is assumed that thislarge angle caused a container to break loose Trimethylphosphate leaked fromthe container and reacted with the acid of a car battery Because of the resultingfire the ship had to be abandoned

Chantrel (1984) studied the large-amplitude motions of an offshore supplybuoy and attributed them to the variation of properties in waves leading to thephenomenon of parametric resonance explained in this chapter Interesting exper-imental and theoretical studies into the phenomenon of parametric resonance oftrimaran models were performed at the University College of London, withinthe framework of Master's courses supervised by D.C Fellows (Zucker, 2000).The influence of waves on ship stability can be modelled by a linear differential

equation with periodic coefficients known as the Mathieu equation Under

certain conditions, known as parametric resonance, the response of a systemgoverned by a Mathieu equation can be unstable, that is, grow beyond any limits.For a ship, unstable response means capsizing This is a new mode of ship

capsizing; the first we learnt are due to insufficient metacentric height and

to insufficient area under the righting-arm curve This chapter contains a

practical introduction to the subjects of parametric excitation and resonance

known also as Mathieu effect.

9.2 The influence of waves on ship stability

In this section we explain why the metacentric height varies when a wave travelsalong the ship We illustrate the discussion with data calculated for a 29-m fastpatrol boat (further denoted as FPB) whose offsets are described by Talib andPoddar (1980) For hulls like the one chosen here the influence of waves isparticularly visible Figure 9.1 shows an outline of the boat and the location

of three stations numbered 36, 9, and 18 This is the original numbering inthe cited reference The shapes of those sections are shown in Figure 9.2 Wecalculated the hydrostatic data of the vessel for the draught 2.5 m, by means ofthe same ARCHIMEDES programme that Talib and Fodder used The waterlinecorresponding to the above draught appears as a solid line in Figures 9.1 and 9.2.Let us see what happens in waves Calculations and experiments show that themaximum influence of longitudinal waves on ship stability occurs when the

Parametric resonance 205

T~ _

Figure 9.1 Wave profiles on a fast patrol boat outline - S = still water,

T = wave trough, C = wave crest

wave length is approximately equal to that of the ship waterline Consequently,

we choose a wave length

The dot-dot lines in Figures 9.1 and 9.2 represent the waterline corresponding

to the situation in which the wave crest is in the midship-section plane We say that the ship is on wave crest In Figure 9.2 we see that in the midship section

the waterline lies above the still-water line The breadth of the waterline almostdoes not change in that section In sections 36 and 18 the waterline descendsbelow the still-water position In section 18 the breadth decreases This effectoccurs in a large part of the forebody In the calculation of the metacentric radius,jBM, breadths enter at the third power (at constant displacement!) Therefore,the overall result is a decrease of the metacentric radius

The dash-dash lines in Figures 9.1 and 9.2 represent the situation in whichthe position of the wave relative to the ship changed by half a wave length The

trough of the wave reached now the midship section and we say that the ship is

in a wave trough In Figure 9.2 we see that the breadth of the waterline increased

significantly in the plane of station 18, decreased insignificantly in the midshipsection, and increased slightly in the plane of station 36 The overall effect is anincrease of the metacentric radius

A quantitative illustration of the effect of waves on stability appears in ure 9.3 For some time the common belief was that the minimum metacentricradius occurs when the ship is on a wave crest It appeared, however, that for

Figure 9.2 Wave profiles on FPB transverse sections - S = still water,

T = wave trough, C = wave crest

Figure 9.3 The influence of waves on KM

forms like those of the FPB the minimum occurs when the wave crest is imately 0.31/pp astern of the midship section Calculations carried out by usfor various ship forms showed that the relationships can change Figure 9.3

approx-shows, indeed, that for draughts under 1.6 m, KM is larger on wave crest than

in wave trough Similar conclusions can be reached for the righting-arm curves

in waves For example, the righting arm in wave trough can be the largest in acertain heeling-angle range, and ceases to be so outside that range The reader isinvited to use the data in Exercise 1 and check the effect of waves on the righting

arm of another vessel, named Ship No 83074 by Poulsen (Poulsen, 1980).

More explanations of the effect of waves on righting arms can be found inWendel (1958), Arndt (1962) and Abicht (1971) Detailed stability calculations

in waves, for a training ship, are described by Arndt, Kastner and Roden (1960),

and results for a cargo vessel with CB — 0.63, are presented by Arndt (1964).

A few results of calculations and model tests for ro-ro ships can be found inSjoholm and Kjellberg (1985)

To develop a simple model of the influence of waves we assume that the wave

is a periodic function of time with period T Then, also GM is a periodic function with period T We write

GM(t) = GM0 + 5GM(t)

Parametric resonance 207where

SGM(t) = 5GM(t + T)

for any t In Section 6.7 we developed a simple model of the free rolling motion.

To include the variation of the metacentric height in waves we can rewrite theroll equation as

Going one step further we assume that the wave is harmonic (regular wave) so

that the free rolling motion can be modelled by

This is a Mathieu equation; those of its properties that interest us are described

in the following section

9.3 The Mathieu effect - parametric resonance

9.3.1 The Mathieu equation - stability

A general form of a differential equation with periodic coefficients is Hill's equation:

where h(t) = h(t + T) In the particular case in which the periodic function is

a cosine we have the Mathieu equation', it is frequently written as

<j> + (6 + e cos 2t)c/) = 0 (9.2)

This equation was studied by Mathieu (Emile-Leonard, French, 1835-1900)

in 1868 when he investigated the vibrational modes of a membrane with anelliptical boundary Floquet (Gaston, French, 1847-1920) developed in 1883

an interesting theory of linear differential equations with periodic coefficients.Since then many other researchers approached the subject; a historical summary

of their work can be found in McLachlan (1947)

A rigorous discussion of the Mathieu equation is beyond the scope of this book;for more details the reader is referred to specialized books, such as Arscott ( 1 964),Cartmell (1990), Grimshaw (1990) or McLachlan (1947) A comprehensive bib-liography on 'parametrically excited systems' and a good theoretic treatment aregiven by Nay f eh and Mook (1995) For our purposes it is sufficient to explainthe conditions under which the equation has stable solutions By 'stable' we

understand that the response, </>, is bounded Correspondingly, 'unstable' means

that the response grows beyond any boundaries For a ship whose rolling motion

is governed by the Mathieu equation, unstable response simply means that theship capsizes The reader may be familiar with the condition of stability of anordinary, linear differential equation with constant coefficients: A system is sta-ble if all the poles of the transfer function have negative real parts (Dorf andBishop, 2001) This is not the condition of stability of the Mathieu equation; the

behaviour of its solutions depends on the parameters e and 6 This behaviour

can be explained with the aid of Figure 9.4 In this figure, sometimes known

as Strutt diagram, but attributed by McLachlan (1947) to Ince, the horizontal axis represents the parameter 6, and the vertical axis, the parameter € The 5 - e plane is divided into two kinds of regions For 5, e combinations that fall in the

grey areas, the solutions of the Mathieu equation are stable The (5, e points inwhite regions and on the boundary curves correspond to unstable solutions The

diagram is symmetric about the 8 axis; for our purposes it is sufficient to show

only half of it

The theory reveals the following properties of the Strutt-Ince diagram

• The lines separating stable from unstable regions intercept the 6 axis in points

Parametric resonance 209

• As S grows larger, so do the stable regions.

• As e grows, the stable regions become smaller Remember, e is the bance'

'distur-Cesari (1971) considers the equation

The natural frequency of the 'undisturbed' equation - that is for e = 0 - is cr/27r,

while the frequency of the periodic disturbance is UJ/27T With the transformation

n — 1, 2, 3, The first dangerous situation is met when u = 2<j We reach

the important conclusion that the danger of parametric resonance is greatestwhen the frequency of the perturbation equals twice the natural frequency of theundisturbed system

This statement is rephrased in terms of ship-stability parameters in

Exam-ple 9.1 where a becomes the natural roll circular frequency, UJQ, of the ship, and

uj becomes o;E, the frequency of encounter, that is the frequency with which theship encounters the waves This theoretical conclusion was confirmed by basintests

Surprising as it may seem, the phenomenon of parametric excitation is well

known The main character in Moliere's Le bourgeois gentilhomme has been

Figure 9.5 Strutt-lnce diagram, Si - ei plane

writing prose for many years without being aware of it Similarly, readers arecertainly familiar with parametric excitation since their childhood Here are,indeed, three well-known examples

The motion of a pendulum is stable However, if the point from which thependulum hangs is moved up and down periodically, with a suitable amplitudeand frequency, the pendulum can be caused to overturn

Try to 'invert' a pendulum so that its mass is concentrated above the centre ofoscillation The pendulum will fall Still, at circus we see clowns that keep a long

rod clasped in their hand, as shown in Figure 9.6(a) The rod can be stabilized

by moving the hand up and down with a suitable amplitude and frequency

A third, familiar example of parametric excitation is that of a swing To increasethe amplitude of motion the person on the swing kneels close to the extremepositions and stands up in the middle position (Figure 9.6(b)) Thus, the dis-tance between the hanging point and the centre of gravity of the person varies

periodically The swing behaves like a pendulum with varying length.

More examples of parametrically excited systems can be found in Den tog (1956) That author also studies a case in which the periodic function is a

Har-rectangular ripple whose analytic treatment is relatively simple and allows the

derivation of an explicit condition of stability