An analysis of ship stability based on transient motions

Numerical analysis ofthe steady_{state and transient motions of the semi-empirical}

nonlineardifferential equations, which have been used to model the resonant rolling motions of two ships, are presented in this paper. Examination of the safe basin in the space of the starting conditions shows that transient capsizes can occur ai awave height that is a small fraction of that_{at which the final steady state motions}

lose their stability. It is seen7tt

_{basirtis eroded quite suddenly througut}

its cen tral region by gross striations, implying that transient capsize might be areasonablyrepeatabiephenomenon,offtring a new approach tothquantification ofship stability_{in waves. Such an approach has the}

twin advantages of being both conceptually simpler, and at the same time more relevant,_{than one based on the steady} state rolling motions. The latter analysis can be dangerously non conservative.

i INTRODUCTION

Although sea states are essentially random processes (but not necessarily stationary ones),

_{a short train or}

pu/se of regular waves that can excite resonant motions, can usefully be viewed as a worst-case scenario when considering capsize. For practical purposes a long train of regular waves can be considered to have a probability of zero. Despite this, most researchers in the extensive literature on ship capsize under regular forcing, focus on just the single predominant

steady state motion, he it harmonic, subharnionic or even chaotic. In this, they follow the tradition of classical analysis, despite the fact that for a boat, with its relatively light damping, regular

waves will manifestly never persist long enough for

transients to decay substantially. Not only is steadystate analysis inapplicable, for this reason, but we show that it is also grossly non-conservative.

In this paper we focus attention on the transient

motions of a ship which we investigate against the

background of the steady-state behaviour Ill. Firstly,we

present a steady state bifurcation diagram, in the control

space of a wave height

_{parameter, I-I, against wave}

frequency, o, at which

_{distinct local bifurcational}

phenomena take place. These typically include a jump to resonance at a cyclic fold bifurcation (saddle-node);

a build-up of subhai

_{oscillations at a flip}

bifurcation, as well

_{as a stability boundary of capsize}

conditions.

Deparrment of Civil Engineering, University College London, University of London, England.

MOHAMED S. SOLIMAN

ABSTRACT 183

-TE

_{Laboratorium 'icor}

Scheepshydromethcna

rchlef

Meke!weg 2,2623 CD Do!t

iL U

-Tdd87 -Fa- 015 .1SI

Secondly, we consider the transient motions of a ship

subjected to a short pulse of regular

_{waves: and since} starting conditions of a ship at the beginning of a pulse may vary widely, and in any event are unknown, we look at all possible transient motions. The simplest and most

direct way to do this is to take a grid in the starting

conditions of roll angle, 8, and angular roll velocity, Running simulations from eachgrid point, we can easily map out the safe basin from which transient motions do not lead to capsize within the specified duration of the

pulse. Now as Soliman and

_{Thompson [2]}

_have

identified and quantified for

_{an archetypal driven}

oscillator there can arise a loss of engineering integrity accompanying the rapid erosion and stratification of the safe basin as a control_{parameter is varied. We show}

here

that this behaviour does indeed take place in

the analytical models oftwo real ships givingacritical_wave height, Ht, at which the ship loses the bulk of its _calm water stability. We use engineering integrity curves and transient capsize diagrams to quantify this behaviour.

2 GENERAL ROLL EQUATION

We consider, in common with many authors _[3,4], that the roll motion of a ship, when subjected to wave and wind moments, can be modelled by the non-linear differential equation

I +B(Ó)+ C(B)

_{= M(r)+ w(e,r)}

₍₁₎ where lis the roll inertia (included added hydrodynamic

r

AN ANALYSIS OF SI

_{lIP STABILITY BASED ON}

TRANSIENT MOTIONS.

inertia). O is the roll angle (3.0 being the angularroll

velocity and acceleration respectively). B (13) is the non-linear clamping moment and C(0) is thenoii-linear restoring moment: these represent the stabilizing moments. The wind moment. lV,(0. t). and the way. moment, v1(t). represent the dc-stabilizing moments. Both these moments occur randonìlv in real seas hut for the sake of simplicity, and for a worst case scenario.they can he considered to he deterministic quantities.

In order to illustrate the ideas presented in this paper we hase considered two different ships which have been

well documented and

researched following their capsizes. Both have the following specific equation of niotion:

I I

IO2 I o i

iei

o-1()

= + T

The first is the Gaul(5,with a GZ curve and damping characteristics taken from 16(. Here / =0.0555, h2 = 0.1659,c = 0.2227,t. = 0.0, e1 =0.0694,(4 = RO.

e5 = 0.0131 and 1=64489. The second is the Edith

Terkol (7(. Here h =

0.0043, h2 0.0225, e1 =(1.385.

= 0.1300, c = 1.0395, e4 =4.070,

c =

2.4117 and = 1174 (taken from (X(). We have approximated from these that their linear natural frequencies, con, are0.47

and 0.62

radians per second respectively and their

equivalent linear damping ratios. Ç, are0.075and 0.01. The latter value of damping is unrealistically low. hut is adopted uncritically in this study to illustrate the effect

of damping on our analses. We have assumed in

common with other authors that M(t) = A sin o)1-t where

O), is the wave frequency and A is the amplitude of the

wave moment which in general will be a function both of the wave frequency and height(91. We have also assumed for the sake of simplicity that the wind moment is a constant value independent of roll angle and time: it is zero unless otherwise stated. We refer to the ratio of the forcing frequency to the linear natural frequency as co. such thatco = co/w, and refer to a wave height

parameter H, such that H=AlIw.

3 BAcK(;RouNt) THEORY

Before summarizing the results, a brief review of the

dynamics theory, mapping techniques

and terms employed is appropriate (lO(. Considering the

single-degree-of-freedom system (I) it is well known that to completely define the motion of a ship tinder given environmental conditions (such as wave height,

period,etc) and from certain initial conditions(roll angle and angular velocity), the three dimensional trajectory

in (O, , r)

phasepce must he determined. lraries

who not lead to capsize. will eventually settle down to a bounded stable motion (for example periodic or

(2)

- tR4

-subhanvionic oscillations) Suc h stable steady state

lflOtiOO is called zm attractor. All starting conditious

which generate traectorthat tend

towards an

attractor. thus define its ha.ci,i ordomain (/ attraction

There may he

alternative co-existing _attractors, depending on the starting conditions. but we shall define the union of the hasinsof all the non-capsizing attractors as the saje hasin.

In the case of a ship rolling in regtilar waves, the concept of phase space is extended by the Poincaré map for which the continuous trajectory is replaced by

succession of points obtained by stroboscopically

sampling the motion of the ship at the wave period. Th5

sampling technique produces a sequence of points

(Poinctiré points) in the (O. ) plane which may converge to a fixed point corresponding to a stable periodic state, converge to alternating points for .suhharnionic motion (a period N attractor with N points visited in sequence

or possibly to a chaotic

attractor. Such sampling techniques have been employed in the field of non-linear dynamics for their obvious advantages of summarizing the motion in a relatively simple fashion.

90 -80 - roll argle

70-60 30 20

-03 0 4 0 5 0.5 0.7 0.8 0.9

n0rr'alsed wave heçuercy

Figure 1. Typical resonance response curve for the Gaul(H=0.24).Here the solid lines represent stable steady state response: arrows indicate a

jump to and from resonance. Dashed lines

represent the unstable steady state response.

4 STEADY STATE BEHAVIOUR. 4.1 Bifurcation diagram

Excitation, corresponding to a slowly evolving sea sUItC.

can lead to resonance or large amplitude rolling :iS shown, in figure 1. As we slowly vary the frequency, s that transients have always effectively decayed, we see that the roll response is a smooth function ofcoat all but

two values. Atw=0.77there exists a dangerous hut not

fatal jump ro resonance, in the sense that the ship

restabilizes at a greater amplitude of oscillation. At

i ',1

¡A

50

-$

--s. le rs le p. lv is ts n

0.71) there o ¿LttiInp/ro)n resontncc. I Iied:ihed line ShO\VS the unata/,Ic steady state \Olutions. and although they are no ph y si call y real i zahl e. and indeed will not

appear in the direct time domain simulations. they

provide useful information ahoit the global behaviour and for example play a key role in determining domains of attraction 11). lt is observed that due lo the accra!! softening nature of the restoring moment curve the peak amplitude of oscillation occurs below the mear natural frequency. Such observations clearly illustrate that resonant frequencies should he avoided, S uch resonant behaviour can also be observed h gradually increasing the wave height from the fundamental ii O = ú = o

state until the ship capsizes for a fixed value of wave frequency. Complex bifurcations of the steady states wereobserved as, forexample at o=O.80, there is ajunip

to resonance at H'=O.2l and then a flip to an n=2

subharmonic at H'=O.44. A further increase in H results in a period-doubling cascade and chaotic motion leading tocapsize at Hs=0.45. lt was also found that the optimal capsize condition, corresponding to capsize under a

flliniflltIlTl H. occurs at about w=().70.

Having outlined that steady states can undergo various complicated bifurcations including folds, flips and ultimate capsize, we show how the ship motion in a slowly evolving sea state may he summarised using a sread' state hifi i'rcatioli diagram. Such an analysis may help in predicting instabilities ¿md capsize. Regions

showing when and how the ship capsizes may be

determined. Dangerous and fatal jumps to resonance, subhannonic oscillations and chaotic behaviour may he determined,

all of which can add to the

overall understanding

of

ship behaviour and capsize phenomena. Before summarising the results, we give a brief review of the analysis.

4.2 Analysis

The relationship between two consecutive Poincaré

points in the Poincaré map will he governed by

a

complex non-linear relationship, hut close to a fixed point (whether it he periodic or subharmonic)we may approximate the Poincaré map by a 2-dimensional linear map in the forro

O, = aB, + h ₍₃₎

+ = CO, + dO

i n which O _{, * can} be evaluated numerically t'or ¿mv (O,, O) by making a numerical time integration through one forcing period. In this variational equation it is

understood that O and O are measured from the fixed point. The nature of the stability of the system may he determined by calculating the eigenvalues, X1, X-,. of the Jacobian matrix

H "1.

For stability both ofthesemust

LC

dj

-he less than one in niodulus. T-he stability can -he

charactet'ized h' the position of the eigenvalues in the coniplex pIalle I (t I.

In ¿t changing sea-state both the fixed point and the

coefticients (1f the linear map will var' so that the

eigenvalues will describe a path in the coniplex plane. Il' the eigenvalues are real one of them can cross the stability boundary at + I. a cyclic fold (a saddle-node l'ifurcariopi), or at - i producing a flip hifurcazion (a transition to resonance oforder n=2). These events are clearly of interest to the naval architect 121. The fold

Piit (points A and B in figure

1 ) corresponds to a resonant hysteresis jump which may cause the ship to capsize if the resulting transients are large enough to carry the ship beyond its righting moment limit. or mas' cause the ship to oscillate ¿it a different (and often a considerably larger) amplitude. The crossing at- I results

i!) the ship oscillating in a n=2 subharnionic manner. This,ashas been shown, isoften the precursor ofehaojic oscillations and hence capsize ) 13).

Using such stability properties we have drawn a steady state

bifurcation diagram which

summaries the hifurcational behaviour of the ship over a whole range

of frequencies and wave heights. \Ve focus

most attention just below w=1 as resonant phenomena will

normally govern ship safety. Figure 2a shows the

bifurcation diagram for the Gaul such that W=O.

Bifurcations can easily he seen from this diagram:e.g.

at (t) = 0.75 the ship initially oscillates in a periodic manner, bitt as H is increased the ship makes adangerous but not fatal jump to resonance at A1 in which the ship starts to oscillate at an increased amplitude. A reduction of H ¿ut this stage would cause ajump from resonance at

B1, giving rise to ¿t region of resonant hysteresis as

typically shown in figure 1. A further increase of H beyond A1 results in a flip bifurcation at C and as can he seen the ship capsizes shortly afterwards at H5=O.38. By keeping H constant and varying tile frequency also allows us to determine the regions of resonant hysteresis in the frequency plane. Indeed in real situations hot/i the wave frequency and height change simultaneously and such behaviour can be interpreted by this diagram. The steady state stability boundary, H5(o) indicates the

region of inevitable capsize. Figure 2h

_{shows the} bifurcation diagram for the same ship subjected to a steady wind nionient by incorparating a static bias term in eqn (I). lt can be seen tht this asymmetr increases the likelihood of capsize by lowering the steady state stability boundary. At o = 0.85, H5=0.26 for the biased svstenl. whereas Hs=O.54 for the unbiased case.

5 TRANSIENT BEhAVIOUR

5.1 Safe basins

o'

pVflAí C*2S

B,

(a) Unbiased system (W1=O)

fio

a multitude of modes of capsize[14] as well as having various types of stable steady states. Indeed these steady states can undergo intricate bifurcating patterns until they reach the point of capsize [15].

In this section we consider the transient motions of a ship subjected to a short pulse of waves, by making safe basin studies, as they are both easier to perform, and at the same time more relevant to ship capsize than the steady state analyses. There are several reasons for this. Firstly, for a ship with its relatively light damping, regular waves will evidently never persist long enough for transients to have decayed substantially for steady state behaviour to take place. A short pulse of regular waves can thus be viewed not only as a worst-case scenario, but as a more realistic representation of a sea state than a long train of regular waves. Secondly since the starting conditions of a ship at the beginning of a pulse may vary widely, and are indeed unknown, we must look at all motions rather than focus obsessively on one predominant steady state.

Finally we show that by making such a transient analysis the area of the safe basin, A(H,oj), can faIl dramatically at a steep cliff at H', which can often be at a small fraction of Hs, the wave height at which the final attracting steady state loses its stability.

By acknowledging that a ship from can experience various combinations of wave height, wave frequency and wind moments, we can say that the five dimensional phase-control space spanned by

(On, ,H, co, WM)defines the ensuing motion. (Poincaré

-

186

05

al

-05 055 06 055 07 075 00 Iwnabsal *afl frequency

Figure 2. Steady state Bifurcation diagrams for the Gaul depicting: Dangerous folds (jumps to resonance (A1); from resonance (B1)): Fatal folds (instantaneous capsize (A2)); Subharmonic instabilities (period doubling (C)); Capsize from a chaotic attractor; Regions of inevitable capsize. Contours of transient stability, (H),,o), arc also shown in figure 2a.

phase space has already been defined as O, Ó space while control space refers to the external parameters such as H,coor WM). To determine a safe basin, we use fourth

order Runge Kutta numerical time-integrations from a

simple grid of starts, typically 100 by 100. Eah

integration is continued until either the roll angle exceeds

a capsizing criterion, 0c' at which point the ship is

deemed to have capsized, or the maximum allowable number of cycles, m(=16), is reached, in which case it is assumed that the ship will remain upright under these conditions [161. In this way we can define a set of points in the five dimensional space that do not capsize in m

cycles and hence define a transient safe basin[ 17]. In this

study we have chosen 8 = it for the Gaul and 8 = 0.88 for the Edith Terkol.

Specifying the controls and taking a grid in the (Os, )

plane allows us to draw the conventional cross-sections in the phase-space of the starting conditions: while specifying 8, l (say (0,0) in the case of the ship starting originally in its equilibrium position in calm water) and takin

a grid in (H,co) plane allows us to draw the

cross-section of the transient basins in the

two-dimensional control space[l].

In contrast to the steady state analysis, we are making no note of the final steady state motion (attractor) of the ship to which the non-capsizing motions might settle, whether it is harmonic, subharmonic, small amplitude, large amplitude or chaotic oscillations. At many control

settings there can be of course many competing

attractors, as in a region of resonant hysteresis, some with exceedingly small regions ot attraction.

IY% Ql pQ(C0flrÇ Sud caOS (b) Baised system (WM=2000)

i

o-9 055 07 075 08 085 rtTflaJ]sO flt fr.qancy 05 050 08

r

O.00COO -Q.O7366 O.I7J3 O.??O99 hO.79'66

3 -O.565

Figure 3. Safe basins in phase pacc for n=t).85. Itere white represents initial conditions whose trajectories do Flot capsize within I 6 forcing cvcics and black represents initial conditions whose trajectoriesdocapsize within I 6 forcingcycics. Foicuclt hasin the norinalised wave height parameter h=H/Hs is shown W1=0.() with:

1.74< e <1.74 1.05 <

< 1.05:

_{0( =3.142.1/0 04}

striations penetrating into the very heart of the central zone resulting in a dramatic erosion of the safe basin. Here we can quantify the size of the basin. A, within the

window represented by our grid, by recording the

proportion of starts that do not capsize within niwave periods. As can be seen from the integrity curves (figure 4) the ship retains practically all of its still water stability up to a critical value H1 (=0.32), and thereafter loses almost all of it. Indeed, the value of FI' in which the ship toses most of its calm water stability is sometimes not so well defined. It is thus more convenient to plot in the

(H,co)plane, contours of H., where we defineH as the value of wave height in which the ship has lost P% of its calm water stability. Indeed, the three-dimensional (A, H, co)surface completely defines the area of the safe basin for any given H andco. By fixing co, and taking a

cross-section in the (A.H) platie gives us the typical integrity curves, whereas fixing A we may obtain a

contour of transient stability

H.

H1 then represents the contour in which the ship has lost all of its calm water stahility i.e. H just above H5, 'liereas H represents the contour in which the ship retains all of its calm water sta hilitv. These two contours represent upper and lower hounds on the transient stability; H,, can then be chosen

for the required margin of safety. Figure 2a shows

5.2 Erosion ofthe

basins and ftc transient capsize

dia gram

Figure 3 shows a sequence of safe basins, for fifteen equally spaced values of the normalised wave height parameter,h=H/Hs,for the Gaul. Here the frequency is

kept fixed at ci)=0.S5; a value of interest as it is close to the optimal capsize condition. lt can he seen that there little change in size or position of the transient basinsup

to h=0.60. However after that the basin boundary

becomes fractal (i nfl n itety tex tu red) due to a /zo,noclinic

tangling at H1 in the underlying dynamics!] 81. Starts within this fractal zone lead to chaotic transients which oscillate hesitatingly for an arbitrary length of time before the ship either capsizes or settles to a safe steady state harmonic rolling. Moreover, fractal zones are particularly sensitive to initial conditions: external forces such as an impact load or random noise can often push trajectories across basin boundaries and thus cause a ship oscillating originally in a safe basin to oscillate in an unsafe basin and hence cause capsize 119,201. This phenomena is not serious in itself, provided that the

fractal zone to which it is confined remains as a thin laver around the edge of the boundary, as it does for 1-I just above HM. However as H is further increased the fractal boundary soon becomes incursive with thick finger-like

0 45 04 0.35 03 025 -02 0,15 01 005 -o nteçrty 005 01 015 02 025 m-8.16

several transient contours. It can he seen that steepest cliff occurs at about =0. 70.

This behaviour clearly illustrates how the steady state analysis that predicts the final capsize, say Hs=O.54 at w=O.85, dangerously over-estimates the overall stability of the ship. Such results clearly display, for a designer, that H should be adopted in preference to Hs in defining the operational locus in the (H,w) domain.

Similar studies were made for the biased system. The effect, as expected is to reduce the value of H [I ¡. We have also made a safe basin study for the Edith Terkol (figures 5, 6). Here we obsei'e that _{unlike the previous} case considered, the erosion of the safe_{basin is both less} dramatic and starts to take _{place at i relatively small} value of H. The reasons for such behaviour are discussed in the next section.

cc

a

tuo

-06 0.7

Figure 5. _{Similar figure o fig. 3 lou for fie Filiifi 'le,'kol. \V1=(f.() and}

0,

<U <1).X 0.52<1) <1)52:

1) . U . l/ (1.22

mt5

--a

t U-4

08 0.9 1 1,1

Figure 4. Integrity curves representing the erosion ut the safe basin described in figure 3.

6 TIlE EFFECT OF ROLL

_DAMPING.

As is to be expected, both the steady state and transient

behaviour of the Gaul and the

Edit/i Ter/-o! \vere

different. However by examining the the normalised integrity curves of each ship we may make a relative, if not completelyjustified, comparison._{As can be seen the} Edith Terkol loses its integrity at a much lower value of H/IIS than _{the Gaul. We believe, that this}

feature is mainly due to the fact that the Edit/i Terkol was much more lightly damped (using the model considered here). than the Gaul.

Indeed making numericul simulations ofan archetypal capsize model with both linear and nonlinear stiffnesses and linear damping ratios of =0.005, _{0.025. 0.05 and}

0.15 reveals that the damping

_{level determines ¡he} process of the erosion of the safe basin_{(see figure 7) and} hence the transient capsize diagram. This studyuses the

09 -08 m-1 07 ,,ormahsed 06 _nterty 05 m-2 04 02 01 a il ₀ 0 63 4 4 4 4 01 0.2 03 04 05 03 0.35 04 0.45 05 055 06

wave heght parameter. H

0.9 0.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1

canonical escape equation of Thompson [15J both with and without the stiffness nonlinearity, the linearized

system being deemed to fail if the displacement x

exceeds unity: more details of this comparative analysis will be given in a forthcoming paper [211. This type of behaviour, although difficult to analyze practically due to the nonlinear damping effects, can help the naval architect in his design. Bilge keels can be designed such that a minimum damping level can be achieved for a given ship.

7 CONCLUSIONS

Using stead' state analysis, we have drawn an (H, co) bifurcation diagram. We have defined regions of inevitable capsize; jumps to resonance as characterised

by a fold point; and subharmonic instabilities as

characterised by a flip bifurcation.

The optimal

capsizing wave height occurs at about 70% of the linear natural frequency and obviously that frequency should be avoided. This behaviour is obviously very important

to the naval architect and such a diagram would

obviously be helpful to his dynamic analyses of ship

stability.

We have used a simple and direct method for

finding the critical wave height by analysing the

transient basins and engineering integrity curves. \Ve have shown that using this method several important deductions can be made.

Roll stability analysis using the classical methods,

such as harmonic balance, which locates the main

attractor, tests its stability (using a perturbation or

Liapunov analysis) and then follows the evolution of

daughter attractors, abandoning each in turn as

it becomes unstable, is both a daunting, if not impossible prospect (strictly impossible in

detail due to the

inevitable appearance of subharmonics of infinitely higher order). We have used numerical path following

0.2 4

NONLINEAR LINEAR

Normalized forcing magnitude

Figure 7. The effect of damping on the erosion

of the safe basins for linear and nonlinear

archetypal capsize models. Here the canonical escape equation and its linearization are considered with co = 0.85 and j3 = 2Ç.

routines to overcome this problem. However whichever technique is used the results can be misleading in terms of ship stability. We have shown that the ship up to H' retains all its caim water stability, and thereafter loses almost all of it, as exemplified by the integrity curves. Classical methods would find the ship stable up to Hs

which is obviously, for practical purposes, grossly

non-conservative. Using Liapunov functions, in estimating the domains of attraction, would also be quite impossible due to the homoclinic tangencies and hence the extremely complex shape of the safe basins.

By plotting H contours we can make a critical judgement for the safe operational locus (H', co) ofa ship

subjected to a short pulse of regular waves. The sudden loss of safe basin does moreover suggest that a transient capsize diagram can offer a useful and repeatable index of capsizability, that might have important implications for naval architects [16].

The roll damping plays a most critical part in the erosion of the safe basins. This was illustrated by making

a safe basin study on both linear and nonlinear

archetypal capsize models. Such behaviour demonstrates, that as well as designing a ship with a minimum criterion for certain characteristics of its righting lever arm, a minimum damping level should be included to ensure greater stability.

The stability analysis considered by making a safe basin study can equally be applied for non-capsizing but dangerous motions. Indeed a linear safe basin study can be made (section 6). For example we have considered that the Edith Terkol loses its stability at the angle of vanishing stability (O 46°): however for practical purposes motions can become dangerous to both the passengers and the structure of the ship when the ship

-

189 -CurvenA: ß-0.J Curves B: ß-0.05 Curves C: p-0.01 j .0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

rmaltsed wive heçhl parzme.r. h

Fitzure 6. Integrity curves representing the

LLfl _{IHUIC LCuiiUi'LC} anal sis tue ship can he deemed dangerousonce the motiofl exceeds a civen angle of roll (say 25°) and that ancle radier than the ancle of vanishing stabilitycan be

chosen as a limiting criterion. Indeed by comparinu realistic approach with the extreme limiting condition

i'es a feel for the Fnarç'in fsafiaty made in this transient

anaiysis analogues to elastic and plastic design inthe

field of structural engineering can he observed here. (i) Advances made in recent research in the field_of nonlineardynarnics can help to understand d namicship

plie n orne n a.

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Integrity measures quantifying the erosion of smooth and fractal basins of attraction. Journal

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