+oIIIcJ> + 02cJ>3 +. .. +co 02q:> + a3q:>3 + asq:>S + ... = 2 1tsw[(aoco 0 2 - alco )COS(cot)- 2/la 2 COSin(cot)] (4)
The RHS is sometimes furtherly simplified since the last contribution is of order O(1/co) with respect to the former. Therefore the RHS becomes
coefficients' Xq:>j put in front of each contribution of the exciting cause. In this way an important feature of wave load is lost when the wave lenght becomes comparable with the transversal dimension of the body, i.e. wave diffraction. Newman (ISJ has shown that for 2D bodies the amplitude X of wave loads follows an asymptotic behaviour X = O(co -1/2) when co ~ 00 . Evidently this is not found in eqn (3). The asymptotic behaviour of the global wave load has been experimentally evidenced by Vugts (16] for 2D fixed bodies of simple shapes. Thus some inconsistencies must be expected when using eqn (3) in a wide frequency range. This is generally true but it is equally true that restricting the analysis to the roll resonance range, that for the experimental test here presented, corresponds to the steepest part of the wave load amplitude function, eqn (3) seems to be fully consistent with the expected actual behaviour of the wave load. For sake of completeness, the original meaning of the 'effective wave coefficients' is the following:
(S)
Despite its simplicity an even more simplified RHS is usually found in literature where no explicit dependence of the amplitude of the excitation on the wave frequency appears. In this case the RHS reads as follows: (6)
If 0.0 * is assumed to be a quantity not dependent on the excitation parameters, this modelling works resonably well provided it is applied in a narrow frequency band including the resonant peak, whereas a quite strong difference is found between theory and experiments in the peak tails [17J.
A full consistency with the hydrodynamic approach to the wave-hull interaction. would require an explicit or implicit frequency dependency of the added inertia and damping terms too. On the other hand, it is well known that the damping term begins to play its role only in a narrow frequency range centered in the roll peak frequency, therefore outside this range it can be practically neglected. Moreover the added inertia term for the roll motion is usually within some units percent of the mass inertia (let say < 20%). Again Vugts [16] has given experimental evidence that the variation of the added inertia term 51 for 2D bodies of different shapes is generally very limited, excluding the lowest frequency range tested
0)
~ < 0.4.
If these conditions are
unknown amplitude C and phase \jJ is obtained. The result can then be expressed in the compact form:
[ O)Oeq 2 _0)2 {
-lJ.eqO)C
Jc =
where the nonlinear restoring characteristics and the consequent deviation of the natural roll frequency from 0) 0, are summarized in the equivalent roll frequency 0) 0 2, given by eq
(10)
(7)
1tSw ((J.oO) 0 2 - (J.IO) 2 ) cos(O)t)
PERTURBATIVE SOLUTIONS DOMAINS OF ATTRACTION
(ll)
AND
The numerical solution of eqn (8) is straightforward. Nevertheless it can be very attractive to obtain approximate solutions in closed form. The reason for this interest is to be searched in the stability analysis of the steady state solutions (domains of attraction DOA). Moreover simple expression for the roll amplitude (steady state solutions - SSS) can be used to develop semi-empirical approaches for the statement of upper bounds in the frame of safety criteria or in the improvement of seakeeping codes. In the following, we will first analyse the possibility to use some simple analytic expressions for the SSS problem. The stability analysis of SSS will be then carried out through the DOA. Steady state solutions Approximate solutions of eqn (7) are quite difficult to be obtained if resonance zones different from the synchronism are involved (ultra and sub harmonic oscillations) [8]. On the other hand, a good quality steady state solution in the region of synchronism can be obtained by means of several methods, assuming a solution of the form: q>(t) =Ccos(O)t)
(9)
and the linear and non-linear damping features are included in the equivalent damping lJ.eq
11.
~+21J.J = j=1
cos(\jJ)
= 1tS w(J.O *0) 02sin(\I')
fulfilled, a good silmulation capability can be obtained also with constant damping and added inertia coefficients. In the application case here presented, the roll motion equation used is the following: 3
1tSw (J.O *0)0 2
(8)
After substituting this function and its derivatives in eqn (7), neglecting the rapidly varying terms of frequency 30), 5(0, 7(0, ... and applying the harmonic balance method, a system of two equations in the
The main advantage to follow the analytical way through eqn (9) stands in the capability of the method to detect the possibility of bifurcation, i.e. the existance of multiple solutions at the same excitation frequency. If this happens, two strictly correlated questions arise. The problems to be solved can be shortly stated as: - identification of the danger for the survival or for the operability of the system arising from the possibility of bifurcations; - identification of the probability of falling in each of the stable steady states or of switching back and forth between them. Transient solutions Both problems have no simple answer and need an analysis of the transient phase of the oscillations. In the following we will concentrate on the second problem, and particularly on the deterministic, non chaotic case. The problem of which steady state is reachable can be solved in terms of strict dependence on initial conditions of the motion, that in the case of eqn (8) means q>0 =q>(tO) and 0 =(t) =C cos(O)t + \jJ)
(12)
with C and II' slowly varying functions of time. This assumption allows to obtain the evolutionary equations for C and \jI in the following form:
experimental roll amplitudes measured at Ndata wave frequencies co i, num i are the Ndata stationary numerical solutions of the equation of motion adopted. The minimization process is evidently a strongly nonlinear one. Here we are interested in non-linear restoring characteristics of the ship. In this case even a small change in the set of parameters p j can lead to completely different numi solutions in the peak zone.
Summarizing, eqn (9) represents an approximate analytic solution of eqn (7). In particular the case of multiple solutions is easily handled. The analysis of their stability and their dependence from the initial conditions can be carried out through eqn (13). It results that the perturbative method is a powerful tool for obtaining the relevant parameters of the motions, provided the mathematical model adopted can be handled as above. An interesting question is that connected with the goodness of fit of the perturbative approximate solutions to the exact one, i.e. with the numerical integration of the equation of motion. The approximate steady state solution (eqn 9) has been obtained under the general condition C«I, while the evolutionary equations (13) require the additional assumption that C and \jI are slowly varying function of time. The comparison usually indicates that the steady state solution is an extremely good approximation while the evolutionary equations are strictly valid only in a neighbourhood of the steady state(s). As a consequence, the behaviour of the trajectories in the Van der Pol representation far from the steady state has to be regarded as a very approximate picture of the true behaviour. On the other hand, as will be discussed in the following, exact solutions including all the features of the nonlinear system are not easy to be computed due to the possible complex dynamics.
EXACT SOLUTION IDENTIFICATION
AND
PARAMETER
Haddara [19] developed a Parameter Identification Technique based on the least squares fitting of the numerical solution of the equation of motion to roll decrements of a ship in calm water. His method was specifically oriented to estimate damping coefficients. The extension of the method to the forced oscillations both for a single or even for a multi degree of freedom system [10] was developed by the authors. In the following we will refer to the single degree of freedom case applied to eqn (7). Following this approach, the best estimate of the unknown parameters of the mathematical model can be found by minimizing the function S given by:
where
pj
are
(/-1,01,02,··· ,0.0,···),
N param expi
unknown are
parameters stationary
Further strong complications derive from the initial conditions dependency of numi (DOA) within the same set of parameters. Since the minimization process requires the (numerical) computation of partial derivatives oS,j
Opj
=I, N param , it results that the whole
minimization process (identified parameters) depends on the capability of the time domain integration to capture the appropriate solutions.
PARAMETER IDENTIFICA TION IN EXPERIMENTAL DATA PRESENCE OF BIFURCATION
FROM THE
The tested model A I :50 scale model of a destroyer has been tested in a regular beam sea. The tests have been coducted at constant wave steepness in a range of frequencies 3.4:::; co :::; 6.0 rad / s including the resonance peak. The incident wave steepness has been set to sw=1I30 corresponding to the largest value available in the whole frequency range of the wavemaker. The model has been placed at middle tank in the beam sea condition. To avoid large drift, yaw and collision with the tank walls, the model has been so fly restrained in surge, yaw and drift by the use of elastic ropes. They were not pretensioned, their length being chosen in order to leave the model free from sway and heave according to the orbital motion of a fluid particle belonging to the free surface. The situation was particularly favourable since the model is a slender one and it doesn't show strong differences in the fore and aft part of the hull that can be responsible oflarge yaw. Ship rolling was measured by a Single AxisVibrating Structure-Gyroscope by British AerospaceSystems & Equipments. The output of this device is the istantaneous angular velocity within the range 11 :::; 200 deg/ s. Numerical time integration was then performed accounting for some unavoidable bias. Startup conditions were measured by a high quality clinometer by Lucas Schaevitz. An analogous device was used for the pitch motion. Sway and heave accelerations were measured too in a body fixed frame of reference by means of two accelerometers. Such measures were processed by numerical integration and coordinate transformation to give absolute values. Though not directly interested in sway and heave amplitudes, they
were checked anyhow to veritY the influence of the model restraining. Following the indications of 19th ITTC Panel on Validation Procedures, great attention has been paid to the evaluation of experimental uncertainty. Fig. I shows the measured steady roll amplitudes as a function of wave frequency. The frequency range between 3.50 and 4.00 radls (including the peak zone only) is expanded in Figure 2. Two distinct frequencies, (0=3.6424 radls and (0=3.6720 radls respectively, feature two different measured steady roll amplitudes. Figure 3 and Figure 4 respectively report the corresponding time hystories. Though so close each other in the frequency domain, the two experimental situations were thoroughly different. At the lower frequency (0=3.6424 rad/s, the amplitude jump from 17.9 to 36.6 deg was simply obtained by knocking the deck of the model at a side by means of a rod (external shock excitation). The antiresonant (low) and resonant (high) conditions are well evidenced. Any further attempt to bring the roll amplitude back to the antiresonant state was unsuccessfull suggesting that the resonant state "resistant to disturbances" was the condition with the highest probability possible according to the existing wave parameters. Much more difficult was to obtain two different steady rolling amplitudes at (0=3.6720 rad/s. At this frequency, the still model/water initial condition made the roll amplitude converge unavoidably after a long transient to the resonant state. To obtain the antiresonant condition the initial conditions of the motion were artificially modified. This was accomplished by keeping a low phase angle between the response and the exciting cause, i.e. keeping the model parallel to the istantaneous free surface (antiresonant condition). Fig. 5 to 7 show the obtained steady solutions (amplitude and phase lag), both exact and perturbative, corresponding to the set of parameters identified with the procedure given above. The stability analysis of the multiple solutions at the wave frequencies where experimental evidence of bifircation has been obtained, has been carried out in the V dP plane. The DOA of the steady solutions are given in Fig. 7-8 respectively. In Fig. 9 the strong dependency of the equivalent
o~~~I~~~~~~~j 3.2
3.6
4.0
4.8
4.4
S.2
5.6
6.0
Angular frequency (radls)
Fig. 1
Experimental steady rolling amplitudes at constant wave steepness Sw =1/30 for a 1:50 scale model of destroyer. 45
F-
T
40 r
JI[
I
35
11
em:
Ilf ~
~
-
I
-..~
15
10
1"
1
'"
..L
-'-
'I
1
o 30S
3.6
3.7
3.8
3.9
4.0
Angular r",quency (radls)
Fig. 2
Magnification of Fig. 1 in the bifurcation frequency range. Two distinct frequencies show multiple steady rolling amplitudes. 50
40 30 20
Oil
10
~ '§ :
effective wave slope a 0>1< according to eqn (7) is highlighted.
·10 ~20
·30 ·40 20
40
60
Time (s)
Fig. 3
80
100
Time hystory of roll when (0 =3.6424 rad / s . The jump between the two steady oscillations has been obtained by a shock excitation on the model.
45
:::--r-r
~
~ ::0
,
,
I I£
E 35
,
,
!
r
t:
,'30
T
,I -
.L
I
I
I
r--..
'-
25
]-
20
~
15
I .....
'-
~
, \
~ .~
~
~
I
I
..
~
I
INwncrial.oIulim I
-.......::::
I
~~
1PerturbmW. solutionJ
L.Y
I
10
3
I I
I
o 15
16
Time (s)
Time hystory of roll when (i) =3.6720 rad / s . The model has been restrained during the first oscillations. The jump between the two steady oscillations has been obtained by a shock excitation on the model.
Fig. 4
Fig. 7
17 18 Angular frequency (radls)
3.9
4.0
Magnification of Fig. 5 in the bifurcation frequency range.
45
T
40
,~
35
~
30
1
Z5
1 ~
\'\
\",\
,
i\ ) \
I
20 15
10
~ E~~
H
I 1
/~
Bockboruo cwv. Ptrtu.tb&tiVd 10IutionJ
~ t:,.
~. j
j
o 3.2
3.6
4.0
~
4.4
........
4.8
1'--.-
5.2
t---! 5.6
6.0
-40
Angular frequenoy (radiI)
Fig. 5
·20
·10
0
(0
20
30
40
50
u(deg)
Exact ( ) and perturbative (- ) steady rolling solutions with the identified set of parameters.
Fig.8
0 ·20
·30
Domains of attraction (DOA) of anti· resonant (A) and resonant (R) oscillations at (i) =3.6424 rad / s. ( . solutions of eqn (10); - - - - separatrix; tJ experimental data).
E --=:--"
-40
..."
.
·3
0.2
0-
then c = oo(roll) ) (Ix +J x ) (Ix +J x ) of vanishing stability in still water. The solution
A __
where angle
_~_~~2_4c
_~+~~2_4C
~~--I
8=
8 =.-t where 1 and L
is near zero in order to have enough time for capsize. However, at
..?: = 2, L
to achieve
Ct) e
also
= 0 the
required Fn is 0.564. For many commercial vessels such Froude numbers are rather above their operational
.
"-
range. Even a Froude number near Fn=0.399 (which corresponds to -
=
L
exactly equal to 1.0 and can be
taken as a lower limit for Ct)e 0) is still too high. On the other hand, if nonlinear surging occurs, the ship should remain for sufficient time around the crest of the wave even though its nominal Froude number might lie at a considerable distance from zero. It is known that given a sinusoidal wave of specific length and height, the large-amplitude-surging type of response appears at a nominal ship speed that is well below the wave celerity c, and subsequently in a region where we is away from zero. Large amplitude surging is likely to lead into surf-riding. However it should be remarked that surf-riding takes place in the region of the trough and as such, it does not pose a direct capsize threat by pure loss. So the very condition that linear theory nominates as the single most dangerous for pure-loss seems to be 'immune' of this capsize mode! The real threat that is associated with the transition to surf-riding is in fact broaching. The linear approach may be reasonably valid however up to a wave steepness where surf-riding cannot arise. But then it is unlikely that in such, not particularly steep, waves the restoring capability of the ship will be reduced so dramatically that it can generate pure-loss. Quite often the minimum steepness that can give rise to negative restoring at the upright condition lies within the range of steepness where surf-riding can exist. In summary, it seems to be unwise to neglect the effect of this nonlinearity. Since the surge nonlinearity is 'felt' through the increasing importance of higer order harmonics in response, the equation that needs to be studied is Hill's-like with the following specific form:
(15)
In Fig.5 we show the effect that the nonlinearity of surge can have on the shape of the instability boundaries.
The effect of rudder and yaw-roll coupling The importance of this effect for parametric instability and capsize is unknown at this moment. Several ship types are known to exhibit however coupling of this nature, including containers, ro-ro ferries and fishing vessels. The basic mechanism is that yaw induces roll that, in tum, causes more yaw. Also, an alternating roll moment is induced on the hull directly by the rudder as it oscillates to maintain on average the desired course of the ship. This moment is "felt" in roll if it represents a fair percentage of the righting moment of the ship. In quartering seas ships can perform considerable yawing motions that depend also on the method of steering, Spyrou, 1997. Due to this yawing the encounter frequency rather than being constant, is in fact a periodic function of time. It becomes obvious that deeper study of this mechanism entails the use of a multi-degree mathematical model.
REFERENCES
BARR, R.A., MILLER, E.R., ANKUDINOV, V., LEE, F.C. (1981) Technical basis for maneuvering perfonnance standards. Technical Report 8103-3, Hydronautics, Inc., submitted by the United States to the International Maritime Organization (IMO). BISHOP, S.R. AND CLIFFORD, MJ. (1994) Non-rotating orbits in the parametrically excited pendulum. European Journal of Mechanics, 13,581-587. BLOCKI, W. (1980) Ship safety in connection with parametric resonance of the roll. International Shipbuilding Progress, 27, 36-53. BYANT, PJ. AND MILES, J.W. (1990) On a periodically forced weakly damped pendulum. Part 3-Vertical forcing. Journal of the Australian Mathematical Society, Series B, 32, 42-60. DUNWOODY, A.B. (1989) Roll of a ship in astern seas - Responses to GM fluctuations. Journal of Ship Research,33,4,284-290. FEAT, G. AND JONES, D. (1981) Parametric excitation and the stability of a ship subject to a steady heeling moment. International Shipbuilding Progress, 28, 263-267. GRIM, O. (1952) Rollschwingungen, Stabilitat und Sicherheit im Seegang. Schiffstechnik, 1, 1, 10-21. HAMAMOTO, M., UMEDA, N., MATSUDA, A. AND SERA, W. (1995) Analyses on low cycle resonance of ship in astern seas. Journal of the Society of Naval Architects of Japan, 177, 197-206. HAMAMOTO, M. AND PANJAITAN (1996) Analysis of parametric resonance of ship in astern seas. Proceedings, Second Workshop on Stability and Operational Safety of Ships, Osaka, November, 3646. HA YASHI, C. (1964) Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton. HUA, J. (1992) A study of the parametrically excited roll motion of a ro-ro ship in following and heading waves. International Shipbuilding Progress, 39,420,345-366. IBRAHIM, R.A. (1985) Parametric Random Vibration, John Wiley & Sons, New York. KAN, M. AND TAGUCHI, H. (1992) Capsizing of a ship in quartering seas (Part 4. Chaos and fractals in forced Mathieu type capsize equation). Journal of the Society of Naval Architects of Japan, 171, 83-98 (in Japanese). KERWIN, J.E. (1955) Notes on rolling in longitudinal waves. International Shipbuilding Progress, 2, 16, 597-614. McLACHLAN, N.W. (1947) Theory and Application of Mathieu Functions, Oxford. MINORSKY, N. (1962) Nonlinear Oscillations, Van Nostrand, New York. NA YFEH, A.H., MOOK, D.T., MARSHALL, L.R. (1973) Nonlinear coupling of pirch and roll modes in ship motions. Journal ofHydronautics, 7, 4,145-152. NAYFEH, A.H. AND MOOK, D.T. (1979) Nonlinear Oscillations, Wiley-Interscience. NAYFEH, A.H. (1988) Undesirable roll characteristics of ships in regular seas. Journal of Ship Research, 32, 2,92-100. NA YFEH, A.H. AND OH, LG. (1990) Nonlinearly coupled" pitch and roll motions in the presence of internal resonance: part 1- Theory. International Shipbuilding Progress, 37, 420, 295-324. PAULLING, J.R. AND ROSENBERG, R.M. (1959) On unstable ship motions resulting from nonlinear coupling. Journal of Ship Research, 2. PAULLING, J.R. (1961) The transverse stability of a ship in a longitudinal seaway. Journal of Ship Research, 4, 37-49. SETNA, P.R. AND BAJAJ, A.K. (1978) Bifurcations in dynamical systems with internal resonance. ASME Journal of Applied Mechanics, 45, 4, 895-902. SKALAK, R. AND YARYMOVYCH, M.L (1960) Subharmonic oscillations of a pendulum. Journal of Applied Mechanics, 27, 159-164. SOLIMAN, M.S. AND THOMPSON, J.M.T. (1992) Indeterminate sub-critical bifurcations in parametric resonance. Proceedings of the Royal Society of London, Series A, 438, 433-615. SPYROU, K.J. (1996) Dynamic instability in quartering seas: The behaviour of a ship during broaching. Journal of Ship Research, 40, 4,46-59. SPYROU, K.J. (1997) Dynamic instability in quartering seas: Part ID- Nonlinear effects on periodic motions. Journal of Ship Research, 41, 3 ,210-223.
UMEDA, N., HAMAMOTO, M., TAKAISHI, Y., CHIBA, Y., MATSUDA, A., SERA, W., SUZUKI, S., SPYROU, K. AND WATANABE, K. (1995) Model experiments of ship capsize in astern seas. Journal of the Society of Naval Architects of Japan, 179,207-217. WEINBLUM, G. AND ST. DENIS, M. (1950) On the motions of ships at sea. Transactions, Society of Naval Architects and Marine Engineers, 58, 184-248. ZAVODNEY, L.D. AND NA YFEH, A.H. (1988) The response of a single degree of freedom system with quadratic and cubic nonlinearities to a fundamental parametric resonance. Journal of Sound and Vibration, 120, 63-93. ZAVODNEY, L.D. AND NAYFEH, A.H. (1989) The response of a single degree of freedom system with quadratic and cubic nonlinearities to a principal parametric resonance. Journal of Sound and Vibration, 129,63-93. ZAVODNEY, L.D., NAYFEH, A.H. AND SANCHEZ, N.E. (1990) Bifurcations and chaos in parametrically excited single-degree-of-freedom systems. Nonlinear Dynamics, 1, 1, 1-21.
Fn 0.4
Fn
0.5 0.3
0.4
(1,2)
(1.5, I)
0.3
0.2
0.2 0.1.
o
C-____~______~____~____~
1.
4
3
2
1
1..5
2
2.5
3
3.5
4
AIL · 0' l'• The condition of exact resonance in F Ie>' yaw arises for certain combinations of AIL and Fn. Each curve is defmed by the pair (ro 0 I , n) .
Fig: 2: The loci of exact resonance for rolL
50
1.2 40
h 1
30 0.8
20
0.6
1.0
o~--~-~--~------~--~----~
5
10
15
20
25
30
35
4(roo /roei
Fig. 3: Numerically derived Strutt diagram, ~=0.025 S·I, (Q:) = 0.133 s'l
5
10
1.5
20
25
30
35
4(roo/roe)2
Fig. 4: The instability domains on the plane of h versus 4( roo / roe)2
13
o
6
4
2
8
10
12
y=cos(2T) + O.2Scos(4T)-O.25
1.6 1.4
h 1.2
II i!i
1
0.8
, I
0.6 0.4 S
10
lS
20
2S
30
3S
4( 000 / OJei
Fig. 5: The effect of asymmetry
SOME RECENT ADVANCES IN THE ANALYSIS OF SHIP ROLL MOTION B. Cotton, J.M.T. Thompson & K.J. Spyrou Centre for Nonlinear Dynamics and its Applications University College London Gower Street, London WCIE 6BT, UK
ABSTRACT In an effort to place our previous investigations of ship roll dynamics within physically based limits, we extend a numerical steady state analysis to higher frequency forcing. Working with a simple nonlinear roll model, a number of different phenomena are discussed at above resonant frequencies, including sub-critical flip bifurcations and a second resonance region. We then discuss a highly generalised approach to roll decay data analysis that does not require us to predefine damping or restoring functions. The problem is approached from a local fitting standpoint. As a result the method has potential for further extension to more complex models of damping as well as restoring force curves.
INTRODUCTION Previous studies of beam sea roll models [1, 2, 3] have focussed on the resonant region, where linear theory would predict capsize to be most likely. Here, we explore the steady state dynamics at higher frequencies of forcing and discuss some new features of the control space. In particular we discuss capsizing wave slopes at high forcing frequencies. Interestingly, the capsizing slopes are of similar magnitude to those at resonance. The derivation of accurate representations of damping functions as part of a ship roll model is highly desirable in the study of roll dynamics. Roll damping functions, however, are extremely difficult to obtain by theory or experiment. The tendency has been to remain with simple linear or low order nonlinear velocity dependent models [4, 5]. To test the validity of such approaches we must be able to obtain damping functions from experimental data efficiently and accurately. However the difficulty in separating parameters in any such analysis has hindered improvement on existing ideas. Here we approach the problem from a local fitting standpoint using linear approximations to reconstruct a globally nonlinear curve. Although the approach discussed is applied over all the data, separating angle and velocity dependent terms remains a serious problem. We conclude by briefly discussing some ideas for improving our ability to deal with these difficulties.
HIGH FREQUENCY FORCING During the design of roll experiments it is necessary to ascertain the forcing parameter ranges over which our nonlinear oscillator model is valid. In particular we need to consider two limits;
the maximum wave slope and frequency. The former is a consequence of the nature of waves and simple to evaluate. The latter is a more subtle problem related to the fact that the beam of a ship must be small compared to the wavelength for the model to be applicable. Firstly we write our roll equation as, 18"
+ B(8') + m9GZ(8)
= IAkw 2 sin(wj'r)
(1)
where the prime denotes differentiation with respect to real (unsealed) time, T, I is the rotational moment of inertia about the centre of gravity (incorporating any added hydrodynamic mass), 8 is the roll angle relative to the wave normal, B(8') is the non-linear damping function, GZ(8) is the roll restoring force, Ak is the wave slope amplitude (A is the wave height and k the wave number) and wI is the wave frequency. We also write Wn as the natural frequency of linearised ship motions. We then utilise a simple non-dimensionalised model for roll motion, the Helmholtz-Thompson equation [2, 6] (2) x + j3i; + x - x 2 = F sin wt where, in terms of (1), our two parameters are F = Akw 2 /8v and w = wJlwn with x = 8/8v. We also introduce the parameter J = Ak/(2(8v) = F/w 2 which is a scaled measure of wave slope based on a linear capsize analysis, [7]. Here, 8v is the angle of vanishing stability and ( the effective linear damping coefficient. We also set j3 = 2( = 0.1. The first limit is a consequence of the nature of water waves. For a steepness above H / A ~ 1/7 the wave will break and the use of a simple sinusoidal forcing is no longer valid. Thus, with wave slope Ak = 7r H / A, we can write, (3) The model assumes that the ship tries to follow the motions of the water particles in the wave and does not interfere with the pressures in the wave. This is only valid when the beam of the ship is small compared to the wavelength. We can thus write a minimum wavelength, Amin, permissable in terms of the beam, b (4) Amin = €b where we take, as a first estimate,
€ ~
6. This in turn gives us a maximum forcing frequency (5)
leading to
max
wmax = w _1_ =
u;
_9_ (6) 27r€b where Wn and Tn are the natural roll frequency and period of the ship. Note that this second limit is due to the approximations of our roll model whereas the first is a feature of wave behaviour. Tn
Wn
Substituting in two real ship values (a purse seiner [8] and a container [9]) for beam dimension and natural frequency we. find, Ship Purse Seiner Container
8v [degrees]
40
-
[s] 7.47 19.4
Tn
b[m] 7.6 25.4
wmax 1.4 1.9
Jmax
3.2
-
Therefore as a first step we extend previous steady state analyses to frequencies up to w ~ 2 with the additional 0>=1.9 limit Jmax ~ 3. Using numerical techniques we are able to plot the development of steady state oscillations whilst varying wave amplitude (or slope). This process is repeated for a range of forcing frequencies. For below resonance frequencies it has been shown [10] that for (2), as F is increased, escape (corresponding to capsize) occurs with a jump from a fold bifurcation. Above resonance the system escapes from a chaotic orbit after a period doubling cascade. For the latter case the flipZ initial flip bifurcation is often taken to be a sufficiently accurate indicator of capsize in the control space. Figure 1: Schematic example of a high fre-
(---------+~1·
quency capsize mechanism
Figure 1 shows a schematic example of a discontinuous jump found at higher frequency forcing. The solution path shows restabilisation after a sub-critical flip bifurcation onto a period 2 oscillation. Here we would see a sudden increase in roll amplitude. In this case the flip bifurcation is not a good estimate of capsize. With further increase in F, the system undergoes a period doubling cascade to chaos, before escaping. Note that the fold Y and the subsequent flip Z are bifurcations of the period 2 oscillation. We illustrate. the high frequency bifurcations in a control space diagram, figure 2. The steady state capsize line show the wave slope at which capsize occurs when J is increased in small steps from zero. The ragged nature of this line is primarily due to the computational approximations required in the numerical procedure. 3
r---~~-~-'------r------r------~-----.------~r-o
steady state capsize
L
2.5
2
J
...- 1.8 must be considered a highly dangerous
phenomenon. Of further interest is the existence of an effective second resonance region at w ~ 1.8 which shows qualitative similarity to the 'wedge' at resonant frequencies. At this second resonance capsizability of the model (as measured by J rather than F) is comparable to that at resonance. Note that the use of the scaled wave slope, J, rather than the amplitude, F, gives the correct emphasis to capsize in this higher frequency region. A simple design formula (based on a linear analysis), [2], predicts capsize at J = 1, which is a reasonable lower bound in the above case. For higher damping, this J = 1 formula is found to be more accurate.
ROLL TIME SERIES ANALYSIS We have recently been considering whether we can extract the damping and restoring curves from simple roll decay data. In general, given a roll decay time series we can take two basic approaches to fitting our nonlinear model to the data; global or local. A global approach predefines a polynomial to describe the damping (or restoring) functions. The predictions of such a model can thus be fitted to the data over some number of roll cycles. A local method does not require the pre-definition of these functions and instead fits local linear approximations over small sections of the data. These local approximations are then combined to reconstruct a global, nonlinear fit. Here we present the basic method and discuss its failings as well as their possible solutions. The first step is to model the time series so that we can obtain estimates for its derivatives. At time 'ri the time series will have some value fh Using the surrounding points we can also approximate ei and Oi. We may employ a number of different methods to do this. Here we employ a Savitsky-Golay filter [11} that we have succesfully used to obtain double derivatives from experimental roll decay data. We again use our roll motion model (1) and assume that we can write the two functions B(B) and GZ(O) as locally linear. We can now write our equation of motion locally as,
I8 + Bo + BIB + rng().. + /-to)
= 0
(7)
and
GZ(O)local = ).. + /-to B(B)local = Bo + Ble
(8)
(9)
If we write Bo + rng).. = C, we are left with three unknowns (BI' /-t, C) and thus require three equations to find these unknowns.
Therefore we simply need to sample the time series at three nearby points. Nearby here means that they must be close enough in phase space such that our local dynamical model is valid. This gives the local slopes for B(e) and GZ(O) and the constant C. Since we cannot easily separate C we instead specify GZ(O) = 0 and B(O) = 0, and integrate over our local slopes to reconstruct the restoring and damping curves. We then scan through our time series selecting three consecutive points every step and solving the equations to obtain locally fitted parameters over a wide range of phase space. We then reconstruct the curves by integrating over the local slopes.
EXAMPLES AND IMPROVEMENTS As an example we have taken some numerically generated data from a model with known restoring and damping functions (the symmetric escape equation, [2], which is similar to (2) but with a restoring force of x - x 3 ). Here we have reconstructed damping and restoring simultaneously. Figure 3 show the reconstructed GZ curve.
0.4 0.35 0.3 0.25 0.2
.
a>
0.15 0.1
"
0.05
Note that for this method velocity and an-0.05 gle dependent parameter separation remains -0.1 a problem (the equations we are solving to -0.15 0 0.2 0.4 0.6 0.8 1 find B l , I-" and C become ill-conditioned and 9 much of the data series proves unusable for this method. Therefore we have applied the Figure 3: Reconstruction of restoring force curve for the method carefully over parts of the data set for symmetric escape equation, the reconstructed points are shown with the original curve which it succeeds. In figure 4 we plot a reconstructed nonlinear damping curve. Here the restoring function was pre-specified and the damping taken to be dependent only on velocity. Therefore parameter separation was not a problem and all of the data was used. The routine has also been applied to some experimental roll decay data and was found to perform well in the presence of limited precision and noise. This experimental data was from a low angle decay test and so the restored functions were very close to linear. It was found that calculations of natural frequency using the reconstructed G Z gave results accurate to within 1% of the measured values.
0.09 0.08 0.07 0.06
~
"
0.05 0.04 0.03 0.02 0.01
,
We can improve our ability to deal with the parameter separation problem by employing singular systems analysis [12], to provide us Figure 4: Reconstruction of a linear plus cubic damping with more information on how and where the curve with specified GZ method fails. Treating the fitting as a matrix inversion problem we can rewrite our set of equations as, 0.1
0.2
0.3
0.4
0.5
0.6
(10) or
Ax=b
(ll)
By expressing the problem is such a way, we are able to utilise Singular Value Decomposition (SVD) which can be used to both solve for x and also provide information on separability of the parameters. When the data does not distinguish well between two or more parameters then A becomes ill-conditioned and this can be detected with SVD [ll].
The solution is obtained by decomposing A and then back-substituting given b (it is similar in application to solution by standard matrix decomposition methods). If A is ill-conditioned then SVD will provide the best approximation to a solution in the least squares sense. Thus we are able to go further than is possible with the simpler approach. A further reason for employing SVD is that we can add additional rows to A and solve for x with a reduced likelihood of ill-conditioning. We can do this by simply selecting more nearby data points to provide local roll equations. A still more powerful addition is to include further rows representing energy balance equations for the sampled data points.
CONCLUSIONS A steady state bifurcation analysis of a simple roll model has been extended to higher forcing frequencies. We have discussed a number of new phenomena, with particular reference to capsize mechanisms. The higher frequency region has been shown to bear qualitative similarities to that around resonance and we have identified a second resonance region. Capsizing wave slope at frequencies around w ~ 1.8 is found to be comparable to that at resonance, although the feasibilty of such conditions occuring must be considered. Furthermore we have shown that the usage of the flip bifurcation as a capsize estimate must be made carefully in this high frequency regime. Secondly, we have applied a local fitting method to numerically generated roll decay data and succesfully recovered a nonlinear damping function. The method has been extended to the simultaneous reconstruction of restoring and damping curves, but in this case parameter separation problems remain. The basic difficulty is the separation of velocity and angle dependent terms over the whole data series. We have discussed the application of Singular Systems Analysis to improve our ability to deal with this problem and sketched out how it may be applied.
REFERENCES [1] J.M.T. Thompson. Chaotic phenomena triggering the escape from a potential well. Proceedings of the Royal Society London, 42, 195-225, 1989. [2] J.M.T. Thompson. Designing against capsize in beam seas: Recent advances and new insights. Applied Mechanics Reviews, 50, 307-325, 1997. [3] L.N. Virgin. On the harmonic response of an oscillator with unsymmetric restoring force. Journal of Sound and Vibration, 126, 1988. [4] J.F. Dalzell. A note on the form of ship roll damping. Journal of Ship Research, 22 (3), 1978. [5] M.R. Haddara and P. Bennet. A study of the angle dependence of roll damping moment. Ocean Engineering, 16, 411-427, 1989. [6] E. del Rio, A. Rodriguez-Lozano, and M.G. Velarde. Prototype Helmholtz-Thompson nonlinear oscillator. Review of Scientific Instruments, 63, 4208-4212, 1992. [7] A.G. Macmaster and J.M.T. Thompson. Wave tank testing and the capsizability of hulls. Proceedings of the Royal Society London, 446, 217-232, 1994.
[8] N. Umeda, M. Hamamoto, Y. Takaishi, Y. Chiba, A. Matsuda, W. Sera, S. Susuki, K. Spyrou, and K. Watanabe. Model experiments of ship capsize in astern seas. Journal of the Society of Naval Architects of Japan, 177, 207-217, 1995. [9] S. Takezawa, T. Hirayama, and S. Acharrya. On large rolling in following directional spectrum waves. In Fourth International Conference on Stability of Ships and Ocean Vehicles, volume 1, pages 287-294, University of Naples, Italy, September 1990. [10] J.M.T. Thompson. Nonlinear Mathematics and its Applications, chapter 1, pages 1-47. Cambridge University Press, Cambridge, 1996. ed(Aston, P.J.). [11] W.H. Press, S.A. Teukolsky, W.T. Vettering, and B.P. Flannery. Numerical Recipes in C, 2nd Edition. Cambridge University Press, Cambridge, 1992. [12] D.S. Broomhead and G.P. King. Extracting qualitative dynamics from experimental time data. Physica D, 20, 217-236, 1986.
Summary of Session 3: Application of Non-linear Systems Dynamics to Ship Stability Discussion Leader: Prof. Armin Troesch (University of Michigan, USA) Paper 1: Developing an Interface between the Nonlinear Dynamics of Ship Rolling in Beam Seas and Ship Design, by K. Spyrou, B. Cotton, and 1M. T. Thompson Paper 2: The Role of Mathieu's Equation in Horizontal and Transverse Motions of Ships in Waves: Inspiring Analogies and New Perspectives, by K. Spyrou Paper 3: Some Recent Advances in the Analysis of Ship Roll Motion, by B. Cotton, 1M. T. Thompson, and K. Spyrou Paper 4: Nonlinear Roll Motion and Bifurcation of a RO-RO Ship with Flooded Water in Regular Beam Waves, by S. Murashige, M. Komuro, and K. Aihara Paper 5: Intact Ship Stability in Beam Seas: Mathematical Modeling of Large Amplitude Motions, by G. Contento and A. Francescutto The first two papers presented analytical stability analyses for roll (regular beam seas) and yaw (regular following seas), respectively. In both cases, the dynamic model was reduced to a single degree of freedom, allowing for a careful and rigorous examination of nonlinear stability. The beam seas roll equation examined the different evolutions of transient and steadystate capsize, by using a Melnikov analysis or harmonic balance. The authors speculated as to whether the two different mechanisms leading to capsize involved the same parameters. Questions were raised as to the meanings or definitions of "transient" and "steady-state capsize". Yaw instability was modelled using a Mathieu equation with autopilot. Sensitivity of yaw-roll coupling to surge and rudder motion was discussed. The workshop participants questioned the meaning of system instability and how this related to vessel capsizing. The third paper presented a methodology by which experimental roll decrement test results could be analyzed to yield roll damping and restoring moment curves. The procedure showed good comparisons between estimated damping/restoring functions and numerically generated data. The effect of noise on the estimates, as would be present in real decay curves, was unknown. The fourth paper examined the effect of water-on-deck on rolling. A bifurcation analysis was performed on a coupled roll and flooded-water system in regular waves. The analysis, which demonstrated system sensitivity to parameter changes, showed that both small harmonic and large sub harmonic motions can coexist in parameter ranges similar to those of previously conducted experiments. A video presentation and answers to questions suggested that the surface of the entrapped water remained nearly straight (e.g. flat) with minimal sloshing taking place. While the amount of sloshing depends Summary of Discussions in Sessions 1-5, 3rd International Workshop on Theoretical Advances in Ships Stability and Practical Impact. Hersonissos, Crete, October 1997
5
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upon the quantity or depth of flooded water (presumably more water - more sloshing), the lack of sloshing in the experiments does appear to justify in some cases the modelling assumption of treating the water as an additional single degree of freedom, represented by the flooded water slope. The model experiments had a mass of water to mass of model ratio of approximately 10%. The fifth paper examined experimental uncertainty in parameter estimation when experiments are conducted near frequency ranges where bifurcations are present. MUltiple steady periodic responses were first estimated and then experimentally demonstrated for a scale model of a destroyer. In that critical frequency range, the analysis showed that the identification capability of linear damping and excitation was good, while the capability of separating the total damping into linear and nonlinear damping components was poor. Generally, the workshop participants felt that nonlinear system analysis is desirable and that the applications were becoming more directed towards practical concerns. It was felt that a dictionary of terms, in particular rigorous definitions of capsizing, surf riding, and broaching would be useful. Another question related to the usefulness of defining resonance for systems that exhibit large nonlinear behavior. An illustration of the confusion that results from multiple meanings was shown when it was pointed out the stability of solutions of differential equations does not necessarily have any relation to the stability of a ship and its tendency or resistance to capsize. It was recommended to expand the single degree of freedom model to include other modes of motion such as heave and sway. It was felt that for systems with bias (e.g. wind heel or cargo shift), sway would be a consideration. Many of the reduced models represent two dimensional motions, similar to vessels in beam seas. It was noted that most actual capsizings were related to three dimensional motions and the usefulness of studying the relatively limited two dimensional case was questioned. Participants responded that beam seas motions can lead to down flooding or cargo shift, actions that did result in capsize. In addition, the parameter space in which capsize can occur is generally unbounded and a multivariable or multiparameter environment subspace (e.g. one based upon a two dimensional model) can aid in identifying critical parameter ranges in which a more complete three dimensional analysis would be attempted. In conclusion, the value of nonlinear systems analysis has been and will continue to be in the support of model tests and in the support of developing simulation tools.
Summary of Discussions in Sessions 1-5, 3mInternational Workshop on Theoretical Advances in Ships Stability and Practical Impact, Hersonissos, Crete, October 1997
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SESSION 4:
Interface and Overlaps in the Seakeeping, Manoeuvring and Stability of Ships
Discussion Leader: Prof. Shigeru Naito (Osaka University, Japan) Presenter 1:
Prof. Aposto1os Papaniko1aou (NTUA, Greece) Methodologies for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability
Presenter 2:
Prof. Katsuro Kijima (Kyushu University, Japan) The EjJect ofRoll Motion on Ship Manoeuvrability
Presenter 3:
Prof. Tsugukiyo Hirayama (Yokohama Nat. Univ., Japan) On the Capsizing ofa Fishing Boat in Head Seas
Presenter 4:
Dr. Naoya Umeda (NRlFE, Japan) Sensitivity of Broaching-to with Respect to Seakeeping and Manoeuvring
METHODOLOGY FOR THE EVALUATION OF LARGE AMPLITUDE SHIP MOTIONS IN WAVES AND OF DYNAMIC STABILITY Apostolos D. Papanikolaou 1
ABSTRACT The formulation of a mathematical model for the simulation of large amplitude ship motions and of capsize of a damaged ship at zero forward speed in waves is presented. A numerical solution (algorithm), developed for the purpose of systematic evaluation of the so obtained theoretical model, is outlined and results from an application of the method to a Ro-Ro passenger ship are presented and discussed. Finally, common characteristics and differences between the presented large amplitude, non-linear, ship motions theory and the related dynamic ship stability theory are addressed and discussed in the light of common theoretical approaches to the ship stability problem.
flood water mass
fJ
e, rp, t{f (J
r OJ
NOMENCLATURE
A; B;/a)
F F~i(t)
F~,i (t)
I G
Gx'y'z'
Kg(r)
Mc,MG ms I
infinite frequency added mass coefficients (i, j = 1, ... , 6) damping coefficients (i, j = 1, ... , 6) sum of external forces diffraction forces (i = 1, ... , 6) radiation forces (i = 1, ... , 6) intact ship inertia matrix intact ship centre of gravity. body-fixed co-ordinate system kernel function (i, j = 1, ... , 6) sum of external moments about points C and G respectively intact ship mass
w'
inertial co-ordinate system co-ordinate transformation matrix time flood water centre of gravity. position vector of G with respect to the inertial co-ordinate system position vector of W with respect to the inertial co-ordinate system wave heading in the inertial coordinate system wave heading in the body-fixed coordinate system Euler angles (e roll, rp pitch and t{f yaw) wave frequency time lag angular velocity vector expressed in the inertial co-ordinate system angular velocity vector expressed in the body-fixed co-ordinate system
INTRODUCTION The present paper derives from current research at the Ship Design Laboratory of NTUA on the damage stability of Ro-Ro passenger ships in waves, in view of recent regulatory developments of IMO (SOLAS 95, Regional agreement, Reg. 14) to allow the
Professor, Head of Ship Design Laboratory-NTUA
A P anikolaou "Methodology for the Evaluation of Large Amplitude Shi~ Motions in Waves and of Dynamic Stability", int Wo;kshop on Theoretical Advances in Ship Stability and Practlctl Impact, Crete, October 28-29, 1997.
Proc~rd
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physical modelling of the damage stability of Ro-Ro passenger ships in waves as an alternative to the so-called 'water on deck' regulatory concept. In the light of these developments, it becomes evident that the availability of proper computer algorithms, allowing simulation of the capsize of a damaged ship in waves and providing the necessary flexibility and efficiency to address systematically alternative design measures, is of great importance in order to improve the survivability of the ship and to ensure compliance with SOLAS regulations.
An advanced numerical integration method is implemented, based on the extrapolation technique, and used for the numerical integration of the formulated non-linear differential equations. The employed method proved to be very fast and accurate. A DEC 3000 Alpha workstation is used for the development and systematic numerical evaluation of the developed algorithm.
2. MATHEMATICAL MODEL 2.1 Co-ordinate Systems
Based on previous work at Ship Design Laboratory of NTUA in the field of linear and non-linear ship motions (Zaraphonitis and Papanikolaou [11]), a six degrees of freedom mathematical model of ship motions in waves, at zero forward speed, has been formulated and solved numerically in the time domain, allowing the simulation of ship motions and the prediction of capslZlng under specific environmental conditions. The non-linear, 6 DOF equations of ship motions, accounting for the effect of flooding, have been exactly formulated based on large amplitude rigid body dynamics. However, in order to simplify the solution of the equations of motion, the mass of the flood water is assumed to be concentrated at the centre of volume occupied by the fluid. A semi-empirical water ingress / outflow model accounting for the damage opening and the effective pressure head is used for the estimation of the flow of water into and out the damaged compartments. Radiation and wave diffraction forces are calculated from hydrodynamic coefficients evaluated by a 3D computer code in the frequency domain, applying the impulse response function concept. Froude-Krylov (undisturbed incident wave) and hydrostatic forces are calculated by direct pressure integration over the instantaneous wetted ship body surface.
Four co-ordinate systems' will be used to express the equations of motion. Let OXYZ be an inertial co-ordinate system, with OZ vertical and positive upwards and Gx'y'z' a body-fixed co-ordinate system with G located at the centre of gravity of the intact ship. We introduce also a co-ordinate system OX'Y'Z' which is always parallel to the body-fixed coordinate system and Gxyz which is always parallel to the inertial co-ordinate system.
X' X
Fig. 1. Co-ordinate systems When the ship is at rest, point 0 coincides with G and all co-ordinate systems coincide with
A Papanikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc. 3 n1 Int. Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997. 2
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each other. The instantaneous position of the ship is uniquely defined by the position vector XG of point G with respect to the inertial coordinate system and the three Euler angles (yaw, pitch and roll). Let P be a point in space and X its position vector with respect to the inertial co-ordinate system. Let X', x and x' be the position vectors of point P with respect to systems OX'Y'Z', Gxyz and Gx'y'z' respectively. These vectors can be transformed to each other using a co-ordinate transformation matrix R:
(1) The full mathematical expression for the coordinate transformation matrix R is given in appendix A. In the following, all vectors or matrices expressed with respect to OX'Y'Z' or Gx'y'z' will be marked with an ('), while one or two dots over a variable or function denotes first or second time derivative.
F
(2)
For the angular motion the expression is:
(3) where F and Me are the sum of all external forces and moments (about C) applied to the dynamic system consisting of the intact ship and the flood water, expressed in the inertial co-ordinate system. Let W be the centre of gravity of the flood water and X w its position vector, expressed with respect to the inertial co-ordinate system. Equation (2) can take the form:
(4) where ms is the mass of the intact ship and mw is the mass of the flood water. It can be easily proved that:
ffLpX x XdV =msXa x Xa +R(Iw')+
2.2 Equations of Motion We consider the complete dynamic system consisting of the intact ship and the flood water. In order to simplify the derivation and solution of the motion equations of the above dynamic system, the mass of the flood water is assumed to be concentrated at its centre of gravity. This is a rather reasonable assumption, since as already discussed in previous work (see e.g. D. Vassalos [10]), the effect of sloshing is expected to be weak. Sloshing can induce considerable dynamic effects when the excitation frequency is close to the natural frequency of the flood water. But the possibility of a resonance is rather small, since the roll natural frequency of Ro-Ro ferries is usually very low.
.:. = dtd fl'rJv PXdV
(5)
+mw(Xa +xw)x(Xa +iw) where Xw is the position vector of W, expressed in Gxyz. Let MG be the sum of all external moments about point G, also expressed in Gxyz .
(6) Introducing equations (5) and (6) in (3), after some manipulation the equation of angular motion can take the following form:
(7)
According to Newton's second law: A Papanikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc. 3rd Int. Workshop on Theoretical Advances in Ship Stability and Practicallrnpact, Crete, October 28-29, 1997.
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2.3 Exciting Forces
calculated in the frequency-domain by the three-dimensional computer program NEWDRIFT (Papanikolaou, [6]). The integral of equation (9) for the Kernel functions is calculated numerically using Filon's method. Due to the very fast decay of the Kernel functions, the integration of the convolution integral in equation (8) is truncated at an appropriate upper limit.
The external forces and moments consist of the following parts:
A quadratic roll damping model account for viscous effects.
a. Froude-Krylov and Hydrostatic forces Froude-Ktylov and hydrostatic forces and moments are calculated by direct numerical integration of the incident wave pressure and hydrostatic pressure respectively over the instantaneous wetted surface. Integration is extended up to the instantaneous free surface, taking into account the ship's motions and the free surface elevation due to the incident wave. The distortion of the free surface due to the diffraction of the incident wave system and due to radiation is omitted.
c. Diffraction forces Diffraction forces and moments are approximated by the linear superposition of the elementary diffraction forces associated with each of the component waves composing the encountered wave train:
The details of the derivation of the above equations can be found in Zaraphonitis [12]. In appendix B, equations (4) and (7) are transformed in a form more suitable to be used in a numerical integration scheme.
b. Radiation forces Radiation forces and moments are associated with the motion of the ship. Ignoring the nonlinearity of the problem and following the so-called Cummins procedure ([ 1]), they are calculated from the added mass and damping coefficients of the ship: 00
F;.Jt) = -Ai; X~j
- f Ki/r)X~Jt -r)dr, o
i,j=I, ... ,6
(8)
IS
used to
N
F;i(t)= Re LF;D ((In,fitKn(XaYa,t) , n=l
i= 1, ... ,6
(10)
where F/ ((J n' fit) IS the frequency-depended diffraction coefficient of mode i, C(XaYa,t) is the instantaneous wave elevation at point G of wave component n and fit is the relative wave heading (fit =fi -If! ).
2.4 Water Ingress Model
The rate of inflow or outflow of flooding water mw is calculated by integration over the surface A of the opening:
and
(11)
(9)
dQ is expressed by a semi-empirical formula (Hutchinson, [2]):
A;
is the infinite-frequency added mass where coefficient and By ( (J ) is the frequency-
'I
dQ = gK sign( Hout - ) Hin V Hout
-
H in IdA
(12)
depended damping coefficient of the ship, A P~anikolaou, "Methodology for the Evaluation ofl-arge Amplitude Ship Motions in Waves and of Dynamic Stability", Proc.3 lnt Workshop on Theoretical Advances in Ship Stability and Practicallrnpact, Crete, October 28-29, 1997. 4
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where K is an empirical weir flow coefficient, H aut is the height of the external free surface and H in is the height of the internal free surface.
Let mn be the nth order moment of the continuous wave spectrum:
2.5 Natural Seaway Modelling Two different approaches are used for the modelling (realisation) of the incident wave spectrum by a finite number of harmonic waves. According to the first approach, we introduce a lower and an upper limit for the wave frequency (j min and (J max' The continuous incident wave spectrum is discretized in a number of N hannonic wave components with frequency: (J =(J
n
.
mm
+ /lil(J
discretization calculated by the two approaches (amplitUdes and corresponding frequencies of the individual harmonic waves).
(13)
and amplitude: (14)
(16)
The comparison of the calculated moments of order n = -1, ... ,4 for the continuous wave spectrum and for the discretized wave systems is also presented in figures 2 and 3. It can be seen that for n;:::2 the first approach gives better results, which is expected since there are more waves in the high frequency range. On the contrary, the second approach gives better results for the moments of negative order. Let T_I be the average period of the continuous wave spectrum.
where:
(17a) il(J
=
(J
max
-(J.
N
mm
(15)
Following the second approach, the area under the incident wave spectrum curve between (J min and (J max is subdivided in N parts of equal area ds. The incident wave spectrum is decomposed in N harmonic wave components of equal a = .J2ds and frequency amplitude corresponding to the centre of the nth elementary part. In both cases, the phase angles of the regular waves are randomly distributed in the range 0 to 277:.
Let TI be the period corresponding to the average frequency of the continuous wave spectrum:
~=2TC(JOJS(OJ)dOJ ~J=2TCmom o ~
(17b)
l
The calculated values of T_I and TI for the continuous spectrum and for the discrete wave systems resulting from the above two approaches are given in the following table:
The wave energy of the discretized wave systems resulting from the above two approaches, obviously equals the wave energy of the incident irregular seaway. In figures 2 and 3 a JONSW AP spectrum with significant wave height Hs =4.0m and peak period Tp =8 sec is presented, along with its A P~anikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc.3 Inl Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997. 5
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simulation scheme are presented in morc in D. Spanos et al. [7].
Table 2. Average Wave Periods
Cont. Wave Spectrum Approach A Approach B
T_I 7.29048 7.29656 7.28516
TI 6.86722 6.86722 6.86722
Since in the first approach the frequencies (Ji of the harmonic wave components are equally spaced in the interval [(J min' (J ma.x] , the resulting wave system is periodic by 2rc/f3.a. Therefore, in order to simulate a genuine irregular seaway for a sufficiently long time, the number N of the harmonic wave components should be very large. On the other hand, since the frequencies (Ji of the harmonic wave components calculated by the second approach are not equally spaced, the resulting wave system gives a closer simulation of an irregular seaway concept even with a few wave components. Thus, the second approach is considered to give a better representation of a proper incident irregular seaway realisation.
3. NUMERICAL SOLUTION
For the time being, the computer program on a DEC-3000 workstation computer, WEtl Alfa microprocessor. Simulation time is z! 15 times slower than real time for the Go' one compartment flooding and for an in(" wave train consisting of 20 wave COmpO!i We expect to achieve real time simulati:" the f " ALPHA PW 433 MHZ worksta
The basic aspects of the adopted numerical procedure for the calculation of the exciting forces and for the implementation of the
!,
(
4. DIS:::::USSION OF RESULTS Simulation records for the motion of ,.; existing Ro-Ro vessel, in service between l' ~ Greek mainland and the Aegean Sea isb,:;;, are presented. The main characteristics of Ii':; Test Ship are presented in Table 1.
Table 1. Test Ship Main Characteristics LBP B T D MAlNDECK DUPPERDECK
The resulting system of differential (actually: integra-differential) non-linear equations is integrated numerically in the time domain, using an advanced integration method based on the extrapolation scheme described by Stoer and Bulirsch [9]. This method proved to be very fast and accurate, especially for this specific type of problems. The relatively large time step of advance, that characterises the method, allows a significant reduction of the number of calculations of the right hand side of the equations, and hence considerable saving in computing time. For the implementation of the numerical integration, an appropriate computer algorithm has been developed, using a constant time step of advance.
(1,
LIGHT SHIP DISPLACE.MENT KG
142.00 m 22.80 m 6.40 m 8.00 m 12.90 m 7884 t 11354 t 9.874 m
In figure 4 the discretization of the vessel in surface panels is presented. 2x177 panels where used to describe the vessel's hull (bottom, sides and upper deck). The incident wave is described by a JONSW AP spectrum with Hs =4.0m significant wave height and ~ =8 sec peak period (fig. 3). The initial wave heading fJ is equal to 90°. The second approach is applied for the modelling of the continuous spectrum with a wave system consisting of 20 harmonic waves.
A. Papanikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc. 3'd lnt Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997. 6
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The simulation of the ship's motion IS performed for two cases. At first the ship is considered intact, floating freely at the free surface at zero forward speed. In the second case, one ship compartment of 66m in length, located 3m ahead of the intact ship's LCG and extending from side to side and from the main Deck to the upper Deck, is considered flooded. The flood water mass is set equal to 10% of the intact ship's displacement and is kept constant throughout the simulation (no water inflow or outflow is considered). In fig. 5 the simulation records for both cases are presented. In the first row, the free surface elevation at the centre of gravity of the intact ship is presented (point C), followed by the results for the heave, roll and pitch motion. In the next four rows, the results for the second case (ship with flooded compartment) are presented. The fifth row shows the free surface elevation at point C. Note that, although the incident wave system remains the same in both cases, the free surface elevation given in the first row differs from that in the fifth row. This is because, in the inertial co-ordinate system, point C is moving in different ways in the two cases, following the motion of the ship. In figures 6 to 11 the same results are presented in the form of face diagrams.
After all, it is the opinion of the author that the presented simulation model will be a valuable tool in the process of designing a Ro-Ro vessel, enabling the designer to analyse the impact on damage stability of different design solutions and to maximise the survivability of the vessel, before proceeding to the experimental investigation.
6. ACKNOWLEDGEMENTS The author wishes to acknowledge the support to the present research by the Greek Secretariat General for Research and Technology (code ITENE~ 1995) The study is also supported through technical information provided by the Greek Shipowners Association for Passenger Ships, the Union of Greek Coastal Passenger Shipowners and the Hellenic Chamber of Shipping.
7. REFERENCES l.
2.
accumulation studies by the SNAME ad hoc Ro-Ro safety panel', Workshop on
5. CONCLUSIONS A mathematical model and the corresponding numerical solution procedure for the simulation of large amplitude motions and capsize of a damaged ship is presented, followed by numerical results from the application of the method for a typical Greek Ro-Ro vessel. Further work is now underway in NTUA-SDL towards the refinement of the mathematical model and the computer algorithm in order to increase the accuracy and speed of the algorithm. In the near future, a series of systematic experiments in the NTU A towing tank is scheduled in order to experimentally fully validate the accuracy of the method.
Cummins, W. E., 'The impulse response function and ship motions', Schiffstechnik, vol. 9, no. 47, pp. 101-109, June 1962 Hutchinson, L., 'Water on-deck
3.
Numerical & Physical Simulation of Ship Capsize in Heavy Seas, University of Strathclyde, 1995 de Kat J. 0., Paulling, J. R., 'The
simulation of ship motions and capsizing in severe seas', Trans. SNAME, vol. 97, 4.
pp. 139-168, 1989 de Kat, J. 0., 'Large amplitude ship
motions and capsizing in severe sea conditions', Ph.D. Dissertation, Dep. of
5.
Naval Architecture and Offshore Engineering, University of California, Berkeley, July 1988 Letizia, L., Vassalos, D., 'Formulation of
a non-linear mathematical model for a damaged ship subject to flooding', Proc.
A Papanikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc. 3'd Int. Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997.
7
22/10/97
6.
of the Sevastianov Symposium, Kaliningrad, May 1995 Papanikolaou, A. D., 'NEWDRIFT: The
one is uniquely defined by the position vector XG and the set of the three so-called Euler angles: roll (e), pitch (tp) and yaw (Ijf).
six DOF three dimensional diffraction theory program oj NTUA-SDL jor the To obtain the body-fixed co-ordinate system calculation oj motions and loads oj from the inertial one, the later is supposed to arbitrarily shaped bodies in regular be translated to Gxyz and then rotated by an waves', NTUA-SDL, Internal Report, angle Ijf about the yaw axis, then by an angle tp 7.
Athens 1988 Spanos, D., Zaraphonitis,
Papanikolaou, A. D., 'On a 6DOF G.,
about the new pitch axis and finally by an angle about the new roll axis.
e
mathematical model jor the simulation of The transformation matrix between the inertial ship capsize in waves', to appear in the and the body-fixed coordinate system is given 8.
Proc. of the 8th Int. Congress on Marine Technology, Istanbul, November 1997. Spanos, D., 'Theoretical-numerical
by: coStpcoSljf
modelling oj large amplitude ship motions and oj capsizing in heavy seas', Dr. Eng. Thesis, Dep. of Naval Architecture, NTUA, in progress. 9. Stoer, B., Bulirsch, R., 'Introduction to numerical analysis', Springer-Verlag, New York, 1980 10. Vassalos, D., 'A realistic approach to
assessing the damage survivability of passenger ships', Trans. SNAME, vol.
R
=
sinesintpcoSljf
cosBsintpcoSljf
- cosBsinljf
+ sinesinljf
sinesintpsinljf
cose sintp si nljf
+ cosecoSljf
- sinecoSljf
sinecostp
cosBcostp
costpsmljf
- sintp
(18)
102, pp. 367-394, 1994 11. Zaraphonitis, G., Papanikolaou, A. D.,
The derivation of(18) can be found in [12].
Structures, vol. 6, 1993 12. Zaraphonitis, G., 'Formulation oj the
w= Rw '
'Second order theory and calculations oj Let wand w' be the angular velocity vector motion and loads oj arbitrarily shaped 3D expressed with respect to the inertial and the bodies in waves', Journal Marine body-fixed coordinate systems respectively:
equations ojmotionjor a damaged ship in waves', NTUA, Ship Design Laboratory Internal Report, 1997
APPENDIX A COORDINATE SYSTEMS TRANSFORMATION When the ship is at rest, point 0 coincides with G and all co-ordinate systems coincide with each other. When the ship is moving, the position and the orientation of the body-fixed co-ordinate system with respect to the inertial
(19)
It can be proved (see [12]) that:
w'=B[e
rp
Vir
(20)
where:
o
- sintp ] cose sinecoStp - sine cosecoStp
A Papanikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc. 3'd Int. Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997. 8
(21)
22/10/97
APPENDIX B
(26a)
EQUATIONS OF MOTION In order to proceed with the numerical integration of the equations of motion, equations (4) and (7) must be transformed to a more appropriate form.
[~4
~5
~6
r =[e
rp
r
If
~i=~i-6' i=7, ... , 12
(26b) (26c)
and
From eq. (4) we derive:
(26d) The equations of motion (eq. 22 and 25) can take the form:
From eq. (7) it can be proved that:
Mo = R(Ictl+w'x (Iw')) +
mw Xw x F + ms+mw
+ msmw Xw x iw + msmw Xw x ms +mw ms +mw
(Xo + iw)
~i=~i+6' i=l, ... , 6 and
(27a)
(23) (27b)
and from eq. (20):
Finally, from eq. (11) and (12) we can derive: (27) (24)
Inserting (24) in (23) manipulation we derive:
I· Bm
J -B~
and
after
some
~ M~ -w'x (Iw'):::-~vi sin~l-
sinO+Bvi : : : sinO l-eifJ cose-eifJ sine cosrp-ifJifJ cose sinrp
(25) Let
t
be a 13 -dimensional vector, with:
A P~anikolaou, "Methodology for the Evaluation of Large Amplitude Ship Motions in Waves and of Dynamic Stability", Proc.3' Int. Workshop on Theoretical Advances in Ship Stability and Practical Impact, Crete, October 28-29, 1997. 9
22/10/97
TABLE 1:
Ship Main Characteristics
L BP
142.00
m
B
22.80
m
T
6.40
m
DMAINDECK
8.00
m
DupPERDEcK
12.90
m
LS Weight
7884
tons
Displacement
11354
tons
KG
9.874
m
Fig. 1 - Test Ship, discretized by 2x177 panels.
Period [sec] 30
25
20
15
Period [sec]
10
2,,-----.-----.-----.-----.-----.-.
30
20
25
15
10
3
01
"'-
~ 2
"
---.-~:-:-I"'---....,
;>
"-
OL-__L -_ _ 0.2
0.4
~
__
0.6
~
__
0.8
~
'"
"- "-
~
,
=a
'-.
'
........
---
_ __ L_ _- L_ __ L_ _
1.0
1.4
1.2
1.6
~
1
~
1.8 0.4
Frequency, [rad/sec]
0.8
4
JONSWAP Wave Spectrum Hs = 4.0 m , Tp = 8 sec Discretized by 20 wave camp.
\::!
~
2
S "l
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.6
1.8
JONSWAP Wave Spectrum Hs = 4.0 m , Tp = 8 sec Discretized by 20 wave camp. 3
Continue - Discrete mO 0.9898 0.9898 ml 0.9056 0.9056 m2 0.8956 0.8963 m3 0.9762 0.9778 m4 1.1878 1.1906 Tm 6.8681 4.6542
en
1.4
,-----------~----------------------,
3
=i:l
1.2
Fig. 3 - Roll RAO by NEWDRIFT.
,-----------~----------------------.
'0 01
1.0
Frequency, [rad/sec]
Fig.2 - Heave RAO by NEWDRIFT.
4
0.6
1.8
2.0
Continue - Discrete mO 0.9898 0.9898 ml 0.9056 0.9056 m2 0.8956 0.8934 m3 0.9762 0.9645 m4 1.1878 1.1496 Tm 6.8681 6.8681
'0 01 --.. '" "0
(Rolling Motion)
• (Ix:r +Jr.J¢+ K~¢+ K¢2~¢I- Jrc iJriI- YvVzG+ YI" rilzG -mxzGUriI-myzGV -myxG Ij/zG
=K~.AsG,¢,B, Ij/) +~p4RhRU/fa sinaR case> (Heave and Pitch Motion)
•
= •
(m + m z )
W + Z IV W + Z o· e + Z d B + Z B e
-
m yZ G ¢ 2
Z~·.K((G,¢,e,Ij/)+ mg
(lyy + Jyy)e + MiJ + MBB + M/vW + MwW + mxzaU + Jxx¢ti
= M~.K(sG,¢,B,Ij/) As external forces by waves, only Froude-Krylov forces were taken into account, because specified 6
sea state was relatively long wave comparing to ship length. This assumption seems realistic looking at the results of numerical simulation compared with experiments. For rolling moment, instantaneous wetted surface by wave and heeling angle was taken into account for more precise calculation. Added mass and damping coefficient are obtained from NSM and Motora's Charts(1959). Derivatives for maneuvering equation of motion was quoted from that of similar ship by Karasuno (1990), but some tuning was done for expressing her basic characteristics in still water experiments.
4.2 Results of numerical simulation For confirming the calculation, comparison was made with experiments in still water. Results were all shown in full scale. Fig.16 shows the results by zig-zag maneuvering like check helm. Double amplitude of rudder angle is about 35 degrees and the period of ruder motion is about 20 seconds and five times larger than the
natural period of roll.
Broken lines are
by experiments.
For
numerical
simulation ,rudder angle time history from experiment was used. Numerical simulation show a little deviation in heading angle, and smaller rolling angle, but it will be said that even though relatively large and quick movement of rudder angle, excited roll angle is very small. Furthermore it will be said that this numerical simulation is relatively reliable. Fig.17, show the comparison with experiment in slightly oblique regular wave. Initial heading angle was about 206 degrees, so about 25 degrees shifted from head wave. The rudder angle was fixed at zero degree. This case, roll is excited and numerical simulation show relatively good coincidence with experiments. This also show that the assumption of Froude-Krylov force was appropriate. Wave period, GM vahle and ship speed were corresponded to the specified value of the ship capsized, and the wave height of 1.85m is near to that of specified, but large rolling tend to capsizing can not be seen. Fig. 18 show the comparison with experiment in turning in regular waves. Initial heading is head sea, and then moved rudder to 35 degrees like step function. Phase shift of calculated rolling or pitching angles from experiment are seen because of heading angle time history is different, so the value of corresponding heading angles are written in this figure. If we look at roll angle at the same heading angle, both calculation and experiment show good coincidence. This case, excited roll angle is large but not enough to capsize. Around the specified sea condition, numerical simulation seems reliable ,so we used our numerical simulation code for the evaluation of capsizing process estimated from experiments. 7
In heading regular waves, parametric roll oscillation will occur by the change of GM in wave trough or crest as Fig.19. For the case of relatively long wave like this time, this change become small and wave period do not fulfill the parametric resonance condition. Another possible phenomena is excitation of asymmetric roll by the coupling of shifted weight or shifted shipping water and heaving. Fig.20 is the results of asymmetric roll by shifted weight. Initial heel is 9 degrees(starboard down) ,and this corresponds to the effect of shipping water. This time, also the good coincidence between calculation and experiment can be seen, but not enough to capsizing. Next, according to the estimated process to capsizing, we simulated shipping water effect and net shifting effect by the transverse movement of a weight on the deck. Initial heel by shipping water is set about 9 degrees and the heel angle that the weight start to move was set as the same as that of experiment (20 degrees). Of course, this simplification cannot reappear the dynamic effect of shipping water precisely. In the Fig.21, 7 seconds wave correspond to the specified condition but capsizing did not occur, because further weight shift did no occur. But in the case of 5 seconds wave, large roll motion excited and weight shift occurred, and finally capsized. Wave condition that capsize occurred is not that of specified, but this wave condition was the same as capsized condition in physical experiment. The main reason will be that the model condition was not the same as that of specified from actual ship. As the same results was obtained comparing to experiment, we carried out systematic calculation and obtained the critical combination of H(m) and GM(m) that tend to capsizing. This result is shown in Fig.22. Experimental condition means that the gyradius was used as that of experiment. Arrow point was confirmed in experiment. From this ,for specified condition, capsizing will occur in smaller wave height than that of experimental condition. Estimated GM at capsized ship was 1.45 m, and so if wave period is 6 sec, critical wave height become about 2m ,and this coincide with that of specified by the information of weather station with wave sensor .. From these physical and numerical simulation ,estimated process to capsizing seems to be confirmed to some extent.
5. Conclusion In this paper, we studied about the possible process of capsizing of a fishing vessel in head waves, both by physical and numerical simulation including the effect of maneuver. Summarizing this study, we can introduce the following conclusions.
8
(1)
Comparing to large vessel, the influences of shipping water on deck and succeeding cargo
shift on ship motions are large to small vessel. So, considering those effect, we could estimate a possible capsizing process of the given ship. (2)
Concerning to this ship, shifted deck house arrangement to port side and some prevention to
freeing port capacity initialized the heel of starboard down. So, the similar fishing vessel have some dangerous tendency. (3)
For numerical simulation, we adopted the so called Horizontal-Body-Axis System for
constructing equation of motion both including maneuvering and seakeeping motion. This was introduced by Hamamoto et al (1993). From the numerical simulation, rapid maneuver did not cause large rolling for the given ship. (4)
In our numerical simulation, shipping water on deck and net shift was simulated by the
weight shift. From this simulation, we could confirm the capsizing process estimated from experiment, and also presented the critical wave height tend to capsizing at the given GM, by numerical simulation. Acknowledgment: The authors want express their gratitude to Mr. K. Miyakawa and T. Takayama who managed difficult experiments, and to the graduated student Mr. M. Fukushima and those other students who contributed to this experiment. Furthermore to associate prof. N.Ma who gave daily advise and support to the students. References Motora,S.(1959): On the Measurement of Added Mass and Added Moment of Inertia for Ship Motions (partl-3) , Journal of The Society of Naval Architects of Japan, Vo1.105 (part 1,pp.8392),106 (part 2,pp.59-62, Part 3, pp.63-68) (in Japanese)
Hirayama, T. (1983): Experimental Study on the Probability of Capsizing of a Fishing Vessel in Beam Irregular Waves, Journal of The Society of Naval Architects of Japan, Vol.154, pp.173-184
Hirayama, T. et.al.(1985): On the Capsizing Process of a Fishing Vessel in Breaking Waves, Journal of The Kansai Society of Naval Architects, Japan, No.196,pp.19-30 (in Japanese)
Karasuno,K., et al.(1990): Physical-Mathematical Models of Hydro-or-Aero-Dynamic Forces Acting on Ship Moving in Oblique Direction, MARSIM & ICSM'90, pp.393-400(1990)
Hamamoto,M & Kim, Y.S.(1993): A New Coordinate System and the Equations Describing Maneuvering Motion of a Ship in Waves, Journal of The Society of Naval Architects of Japan, Vo1.173, pp209-220 (in Japanese)
Hirayama, T., et. al. (1994): Capsizing and Restoring Characteristics of a Sailing Yacht in Oblique and Breaking Waves, Journal of The Kansai Society of Naval Architects, Japan, No.221,pp.1l7-122 (in Japanese)
Hirayama,T. et.al.(1997): Study on Capsizing Process and Numerical Simulation of a Fishing Boat in Heading Waves, Jouranal of The Society of Naval Architects of Japan, Vo1181,.pp.169-180 (in Japanese)
9
Table 1. Principal Dimensions SHIP ITEMS
1/23 MODEL
Specified Value
Value 1
Loa
(m)
29.220
1.270
Lpp
(m)
23.000
1.000
B
(m)
5.900
0.257
D
(m)
2.150
0.0934
___lmJ __ 1·2Q.
Wave Angular Frequency
IS (nd/sec)
o
10
Wave Angular Frequency
IS (rad/sec)
Directional Distribution
Directional Distribution
Wave Direction
Power Spectrum of Wave
(deg)
Wave Direction
(deg)
Fig.II Measured directional wave spectrum used for experiment
1.with shipping water
2.net shifting
3.4.5.6.7.8.process to capsizing
I.
h.
I.
Fig.13 Photograph from Video Film .. Model ship capsized in directional spectrum wave.
Minus GM
NelShift (Dynamical)
Fig.14 Possible process to capsizing in head waves. Thick lines show the estimated process
,~Earth
Fixed Axes Axes x'
"
Fig.15 Coordinate System describing Horizontal Body Axes
Table 2. Definition of variables for equation of motion m m, ml' m, fa
ITT Ia Ja J .. Ja
U V
W 4>
e
tP Xc Zc (c p
AR UR f.
aR
a
T t R XF.K YF.K ZF.K KF.K MF.K NF.K
g
lR hR
HEADING & RUDDER ANGLE 4~~~~~~~~~~
mass ofahip added mass (x direction) added mass (ydirection) added mass (zdirection) moment of inertia (rotation about xl moment of inertia (rotation about.n moment of inertia (rotation about ZJ added moment of inertia (rotation about xl added moment of inertia (rotation about J? added moment of inertia (rotation about z\ velocity (x direction) velocity (y direction) velocity (zdirection) roll angle pitch angle heading angle length from G to effective point of my (x direction) length from G to effective point of m, (z direction) coordinate of G on earth fixed system direction) density of fluid area of rudder effective attack velocity of rudder slope of lift coefficient of rudder effective attack angle of rudder rudder angle I propeller thrust behind ship thrust deduction fraction resistance of ship Froude-Krvlov force (x direction} Froude-Krylov force (ydirection) Froude-Krylov force (zdirection) Froude-Krylov moment (rotation about xl Froude-Krylov moment (rotation about xl Froude-Krvlov moment (rotation about xl acceleration of gravity length from G to center of pressure on rudder (x direction) lenfrth from G to center of pressure on rudder (zdirection)
~2 OJ) d)
"0
~
- - Calculation ----- Experiment
«(
- - Calculation ----Experiment"
I
I
(sec) Fig.I6 Simulation of heading and rolling angle using experimental rudder motion as input. V=4 knots in still water
()
,yyp:y~ Hw=O.71 m
I
I
WAVE Hw=1. 85
em)
~ :B3!S~~~&Yl
-1
-~O
PITCH
PITCH
3
ROLL
-E~::;~~l
22 Or-r-,...,.....,....,.-.r-r-r-.....,.=;:H$E:;=A#;D~IN~G'-.-.-........-.-.,....,....,...,..,....,
~ 21 r-----~~~~~~~-=~~_=~~ d) 20 2- 19 181~~~~~~~~~~~~~~~
o (sec)
Fig.I7 Measured and simulated ship motion in regular head waves (Tw=7sec,Initial Heading Angle=206deg,Rudder Angle=O,V=4knots)
Hw=O.71m
Hw=1.85m
_l~~~~~~~~~~~Wu~
-1 \k-'--'-'--'--'--I (deg)
20
(a)
Fn=1.4
(a)
0.01
10
o
5
time(sec) 10
-0.01
Q> (deg)
20
(b)
20 Q> (deg) W=5.31kg
GZ(m) 0.02
Fn=1.6
10
(b)
GZ
c\e at Fn=9
0.01
o
20 Q> (deg) W=5.31kg
Q> (deg)
20
-0.01 GZ(m) 0.02 GZ curve at Fn=O
Fn=1.8
10
o
(c)
0.01
time(sec) 10
Fig.I: Time histories of roll motion of ShipB-45 at trim angle of 2° measured by free rolling test, without initial heel
20 Q> (deg) W=5.31kg
-0.01
Fig.3: GZ curve of ShipB-45 at trim angle of 2°, center of gravity O.074m, ship weight 5.31kg for several advance speeds :(a)Fn=1.4 (b)Fn=1.6 (c)Fn=1.8 (m) 0.1
GZ(m) -e0.04 -E]. -S;J-
trim O· 2· 4·
0.02
-0.1
20 heel angle Q> (deg)
0.1 (m) -0.02
Fig.2: Body Plan of ShipB-45
Fig.4: GZ curve of ShipB-45 at center of gravity O.04m, ship weight 5.31kg and Fn=1.6
6
800
."
...
~
. ."
..
...... :
,"".' ..
',..
....
'1',
'.
"
r ......
;
(2)
GMo / B < 0.02-0.03',
which is characteristic for
cranky ships, this relation is realized with a high degree of probability. When the value of apparent frequency roe differs from
irerO
the possibility of realization
of different unstable solutions of the Matieu equation is not the same. For small roll damping values V ¢o and small disturbance levels ~ GM/GMo
the width of unstable regions is
proportional correspondingly to (~GM/GMo)n, and the depth of stability modulation necessary for unstable roll evaluation (the threshold of parametric roll excitation) appears to be proportional to the I-st or 1I2 degree of roll damping coefficient.
In particular, the
excitation threshold for the main parametric resonance (n= l) is determined by condition [I] . (3)
For monohull ships without
bilge keels the nondimensiond linear roll damping
coefficient 2u ¢o is within the limits of 0.05-0.lO, therefore condition (3) seems to be easier realized than the static instability condition (2) and manifests itself in a rather wide range of the parameter Ll GM/GMo values. Not only the above mentioned results which determine the crankiness presence or absence and the frequency regions where parametric role may occur are known nowadays, but
6
the \.'Jiculation tcL'hniqucs to determine the amplitudes of such rolling motion are dcv\:'lopcd.
which give an idea of crankiness degree and its danger [5, 13,21,22]. J. Kenvin [21] calcttlated the rolling motion amplitudes in the main parametric
resonance regime on the basis of Mathieu equation and has taken into consideration the
_W_W....
nonlinear character of roll damping by means of binomial formula use with linear and
quadratic terms for resistance law. Specialists from Poland [13] considered the nonlinear _.~....,I.lzilng"@!It._~~.hJ~~]ti~~~.~~
the nonlinear in damping and restorjng moment roll equa.tiop. numerically . haying JakenJh~
'"
~
• • • ...........
,'
.'
~"
•••
••
:.'.
.....
•
"_,"..
•
..
I ••• ••
I'
-of
":
,"
,"
'.',;
.:'
•
'.
'~".o"
...... ,
.....
.'
~. . . .
"
•• ,
... .
~tlJ.~iliW· al,teratipn' iB seaway· mto consideration; 'and 'g6t. satisfaetry agreement WitH the test ..-
(see Fig.I). G. Vilensky [5] established general analytical solution of nonlinear roll eq.uation
fot the case of shIp sailing in regular'foliowi~g'and q'uarteri~g ~a~~. li;re th~'stabil'ity'~urve' ,
form and its modulation were expanded successfully into thrigonometrical series, and the disturbing wave moment and static wind moment were taken into consideration.
Sea state 7 Metacentric height
heading angle
=0.3 m
X=Oo
: I
I I
i
Maximal heel Capsizing
50'.55' 40°.50'
I
30"+40'
I
20
10
10
20
speed, knots Fig.2. Relation between maximal heeling angles during the parametric roll
(¢YoQr)
and ship's speed and course angle max
X to the wave:
GM = 0,3 m - metacentric height; HI/3 = 6,5m - significant wave height. Calculated research [5] and model tests in seakeeping basin demonstrated that under the conditions of purely following seaway the parametric roll with frequency We is significantly
7
kmer than the roll \\hich occurs with frequency Well. However, the parametric e\citation \.vith frequency
COe
in the stern quartering waves can be summed up with the resonance effect
of the exciting moment. This case of combinational resonance (see Fig.2) doesn't coincide with kno\.vn solutions of Mathieu equation and may lead to dangerous heeling angles ( - 60° ). The essential part of zero harmonic (a constant component) is characteristic for this mode of ........... 4~~
~~WI"iPiij~jiiMf~._.J;lh\~... •111, • • tpill. .. " . " •• _",,"*'
\vithout the wind [5]. " . . ........ ···.}'{~~~nt ~eiperi!TI~~taf ·.~~d. ~al~'latjo~ res~ich ·.bY,.means of"ana~iica~"meti'tQd:~5] : •
..
.....
•
...... ~
. . , ",
','
_'
•
• • ',,..
1"
.'
.....
I"
...
':"'...
• • • • ,'
executed in the Krylov Research Institute confirmed the known facts, that rolling motion. •
••••
'.
to
....
"
.,
......
,
t . . . .. " . . '
. ',' .' . . . . .
' .. " "'Par"an1etrica'l1y 'excit'ed iil' following seaway" can be' developed right up to the capsizing. It was found that with the relation
increase and the coefficient V¢o decrease maximal
inclinations or crankiness of a ship increases, the range of apparent frequencies of encounter at which the mentioned roll regimes exist widens, and the rate of their amplitUdes growth increases. Known opinion has been confirmed that the parametric resonance in the regime of We/2 is not dangerous in head seas. In this case it arises with a sufficiently high stability and
consequently relatively small ~GM/GMo, high natural frequencies
n,
and occurs with
small amplitudes or doesn't occur at all. On the contrary, rolling motion that arises in the main parametric resonance regime in following seaway is as a rule several times higher in amplitudes than the usual one caused by the exciting moment and serves as an indication of dangerous ship crankiness. As an illustration for above said Fig.3 demonstrates the results of a three-meter multipurposed bulkcarrier model test under unfavourable loading case connected with container transportation on the upper deck ( GZ max curve - 65° and GM o
=0,67 m).
= 0,35 m,
vanishing angle of stability
The tests were carried out to evaluate a ship's crankiness with
various modifications of the constructional elements and model loading, and also in order to work out the recommendations for limitations of crankiness during sailing in purely following waves of sea state wave
(HI!3
=6,5 m).
The experimental data correlate quite well with the
maximal roll amplitude values in the main parametric resonance regime calculated with consideration of Kerwin's recommendations [2l}. Fig.4 demonstrates the variation of 8
:::H--+-so~
,"
_
! .. ' ' . '
',I'. ," •• ',
'.' ... : ~.'''' '.
1
,'';'
;.- • • • . 1.....0
CAPSIZI:-::
25
l'
D
I
.
..
3
Recent Stability Regulations on Existing and New Ships Impact on Overall Ship Safety
by
Sigmund Rusaas Det Norske Veritas and JohnSpouge DNV Technica
\\OSL 19\DTP306\R&D\RORO\Imp_saf.doc
Page 1
SUMMARY
Followed by the recent tragedies by the "Herald of Free Enterprise" and the "Estonia", much effort has been laid down by the regulatory bodies to enhance the stability regulations, most notably the "SOLAS 90" standard and the "Stockholm Agreement". This paper discusses the impact of these standards on the overall ship safety by using an example ship before SOLAS 90, after upgrading to SOLAS 90 and according to the Stockholm Agreement. The comparison is based on calculation of the subdivision index using the proposed framework from the Joint North-West European research project on Safety ofPassengerlRo-Ro vessels. The paper also presents the main conclusions from that project, and discusses areas where further improvements may be necessary.
INTRODUCTION
Most efforts to improve safety of ships have focused on a specific aspect which is believed to be important, such as improved criteria for stability following damage after a collision. In principle, all such efforts are valuable, although inevitably some are more effective at improving safety than others. Work within the Joint North-West European research project on Safety ofPassenger/Ro-Ro vessels (hereafter called the NWE project), have provided documentation of major risk factors applicable to PassengerlRo-Ro vessels and their relative importance to each others. One of the main conclusions from that project is that the single most important risk factor for a Passenger/Ro-Ro vessel is collision followed by flooding and rapid capsize. The work also concluded that the risk for flooding the vehicle deck through the bow doors have been significantly reduced as a result of recent rule developments, leaving collision as the main remaining risk. This supports recent work to improve damage stability requirements.
HOW TO ASSESS IMPROVED SAFETY AS RESULT FROM IMPROVED DAMAGE STABILITY REGULATIONS.
One problem with the recent damage stability upgrades is that they are all deterministic, i.e. they deal with a limited set of assumed damages only. These assumed damages are specified by a given length (the "SOLAS damage length"), and a penetration ofB/5. If all these damages complies with a predetermined set of criteria, then the ship is considered "safe enough". No effort is made to study what happens if the damage extends beyond these limits. Damage statistics indicate that in a given collision with water ingress there are approximately 50% chance that the damage is longer than the "SOLAS damage", and 50% chance that the penetration is deeper than B/5. In the worst case the deterministic regulations will only cover abt. 25% of the total probability outcome from a collision with water ingress. A study of one or two of the deterministic damage cases therefore does not give any useful measure of the safety level, no matter how sophisticated this study is carried out. This is the main reason for my scepticism to model tests for demonstrating compliance with any deterministic criteria. The safety level may only be assessed using a fully probabilistic approach, combining the following aspects:
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Page 2
Frequency of collisions and probability of flooding, based on average experience for Passenger/Ro-Ro vessels. Probability of ship sinking given flooding. Expected number of fatalities given sinking. Each of these aspects are discussed below.
FREQUENCY OF COLLISION AND FLOODING The frequency of collisions involving Passenger/Ro-Ro vessels in NW Europe during 197894 is estimated as 1.4 x 10-2 per ship year (NWE-report REP-T09-003A V.5.9). This includes all cases where the ferry touches another vessel, even if the damage is minor. The probability of a collision producing flooding through the side of the vessel is evaluated as follows, based on collision experience with PassengerlRo-Ro vessels in NW Europe during 1978-94 (NWE-report REP-T09-003 Fig. 6.1): - Probability of collision occuring while ferry is under way: - Probability of collision causing serious damage: - Probability of ferry being the struck vessel: - Probability of flooding given that the ferry is struck in a serious collision:
0.94 0.29 0.50 0.25
These gives a combined flooding probability of: 0.94 x 0.29 x 0.5 x 0.25
= 0.034 per collision
The overall flooding frequency in collisions is then: 1.4 x 10-2 x 0.034 = 4.8 x 10-4 per ship year. This neglects flooding through the bow which might result if the ferry strikes another ship, but this is not likely to cause the ferry to sink.
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Page 3
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0.060000
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0.002181
0.002726
0.000491
0.003598
0.126759
0.001363
0.001363
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1.19E-04
1.19E-04
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Probability Frequency per collision per year
PROBABILITY OF SHIP SINKING GIVEN FLOODING This probability may be found by a probabilistic damage stability calculations, giving the "Attained Subdivision index (A)". In the NWE project an updated procedure is derived, which assesses this probability in a more realistic manner than before. In particular the effect of water on deck is now included in a realistic way, together with a number of other updates. This procedure is then used to assess the probability of ship sinking given flooding according to the following standards: SOLAS 74178 - these were the regulations in force up to 1992, and stipulates only requirements to margin line, GM and max heel after damage. Most existing ships are still operating according to this standard. SOLAS 90 - upgradings to take account of residual damage stability, i.e. area, range and height of the residual GZ curve. Provisions for dealing with heeling moments caused by passenger movements, wind and launching of life-boats are also included. "Stockholm Agreement" - Provisions to take account of trapped water on the Ro-Ro deck. In order to assess the probability of sinking, subdivision index for an example Passenger/RoRo was made according to the standards necessary to reach the requirements of the different regulations: 1. SOLAS 74178 The arrangement is of the traditional type with open deck and centre casing. The worst damage case requires a metacentric height of 1.98 m in order to comply with the criteria. The corresponding subdivision index was found to be 0.54 2. The vessel upgraded to meet the SOLAS 90 requirements. In this upgrading three bulkheads on the Ro-Ro deck was installed, and the metacentric height was slightly increased, to 2.2 m. The subdivision index now reached 0.74. 3. The vessel upgraded to meet the Stockholm agreement. In order to meet these regulations, additional sponsons will have to be installed, increasing the metacentric height to 2.74 m. The corresponding subdivision index now reaches 0.80. Table 1: Overall Survival probabilities for different regulations: Regulation SOLAS 74178 SOLAS 90 Stockholm Agreement
Arrangement Open deck 3 WT bulkheads 3 WT bulkheads + sponsons
\\OSL19\DTP306\R&D\RORO\lmp_saf.doc
GM 1.98 2.2 2.74
Subdivision Index 0.54 0.74 0.80
Page 5
FAT ALITIES IF THE SHIP SINKS The average proportion on board who are killed when a Passenger/Ro-Ro vessel sinks has been estimated from world-wide experience (NWE-report REP-T09-003A V.12.I). The percent fatality results show the effect of water depth and speed of capsize, as follows: Slow sinking in shallow water: Slow sinking in deep water: Rapid capsize in shallow water: Rapid capsize in deep water:
0.2 % fatalities 2 % fatalities 23 % fatalities 72 % fatalities
In the absence of useful data on the speed of capsize, it has been assumed that 50 % of events are rapid (typically within about 10 minutes) and 50 % are slow sinkings (typically delayed for several hours but ending with a rapid capsize). Actual experience supports this (European Gateway was a rapid capsize; Saitobaru was a slow sinking). It has to be mentioned that collisions are assumed to occur in deep water. This assumption is based on the Example ship's route and crossing traffic, but is less valid for NWE ferries in general (e.g. European Gateway was in shallow water).
The average proportion of people on board who are killed in an accident is estimated as: (72%+2%)/2 = 37 %
With an estimated average number of people onboard the example ship of 1000, the average number of people killed when a ship sinks are: 1000 x 0.37 = 370
FATALITY RISK RESULTS FOR THE EXAMPLE SHIP The overall fatality risk when satisfying the different regulations can now be estimated by combining the flooding frequencies, survival probabilities and the average fatalities as follows: SOLAS 74178: 4.8 x 10-4 x (1-0.54) x370 = 0.082 fatalities per ship year SOLAS 90: 4.8 x 10-4 x (1-0.74) x 370 = 0.046 fatalities per ship year Stockholm Agreement: 4.8 x 10-4 x (1-0.80) x 370 = 0.035 fatalities per ship year These risk results are in fact societal risks expressed as annual fatality rates, equal to longterm average numbers of fatalities per year, arising purely from collisions in which the ship sinks due to flooding. To place these risks in context, the other fatality risks on the ship are estimated to amount to approximately 0.1 fatalities per ship year. This includes fires, groundings, other accidental floo.ding, and collisions causing fires. The effect of the extra bulkheads on other flooding events (may be negligible) and fire (which may be significant) are here omitted for simplicity .
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Page 6
This shows that on the example ship SOLAS 90 has already reduced the risk by: (0.082-0.046)/0.182 = 20 % The Stockholm Agreement could achieve a further reduction of: (0.046-0.035)/0.146 = 8 %
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Page 7
Comparison of Fatality Risks
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Slide 11
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Tlte European Association of Classification Societies - EURACS By Mr. M. Hutlter (BV) and Mr. M Dogliani (RlNA)
Final Announcement ofthe 3rd International Workshop on Theoretical Advances in Ship Stability and Practical Impact and Related Events, CRETA MARIS, Hersonissos, Crete, October 1997
APAP
6
29/12/97
-
The European Association o/Classification Societies (EurACS)
EurACS and its involvement inEUR&D M. Ruther (BV), M. Dogliani (RINA)
Open Stability Forum - Crete 27 October 1997
The European Association o/Classification Societies (EurACS)
WHAT IS EurACS? III
III
The Association of the 5 Classification Societies in the EC BV GL LR RINA DNV Which represents: 126 millions GRT of sea going ships 2 millions GRT ships on order plus inland waterways ships in the EC as well as oversea. TO SUPPORT THE EC INTERESTS Open Stability Forum - Crete 27 October 1997
1
The European Association 0/ Classification Societies (EurACS)
EurACS objectives • Improve standards of safety at sea and pollution prevention. technical of standards • Co-ordinate Members. • Promote uniform interpretation of international conventions. • Promote the idea of classification in respect of R&D activities FOR SAFER EUROPEAN WATERS Open Stability Forum - Crete 27 October 1997
The European Association o/Classification Societies (EurACS)
EurACS Members activities
• Setting up technical standards. • Inspections, assistances to enable industry to meet the standards. • Publication of survey status of classed ships. • R&D work for shipping and offshore.
ASSURING CONFRONTATION OF PHILOSOPHIES AND TECHNIQUES Open Stability Forum - Crete 27 October 1997
1
The European Association o/Classification Societies (EurACS)
EurACS: R&D involvement II II
-
EurACS members have their own R&D structure. As an Association, EurACS is involved, since 1994, in the European Commission R&D activities: membership at the High Level Panel of the Maritime Industry Forum (MIF); chairmanship of MIF's R&D Coordination Group (R&DCG); coordination of the development of the EU R&D Masterplan; chairmanship of the Industry Interface to the EU "Maritime" Task Force; advisory to the Commission for developemnt of the V
Open Stability Forum - Crete 27 October 1997
The European Association o/Classification Societies (EurACS) R&DCG (set up by MIF in December 1994) Chairmanship
EurACS
Classification Societies
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COREDES
Shipbuilders (CESA R&D
MIF Panel 2
Marine resources (Offshore, renewable energies, aquaculture, fisheries, deep sea activities)
Members
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Since the creation of the Task Force "Maritime Systems of the Future", the R&DCGbecame the core of its Industry "Mirror Group"
Open Stability Forum - Crete 27 October 1997
I
The European Association of Classification Societies (EurACS)
R&DCG Mission To co-operate with the EC in identifying and avoiding unnecessary overlappings between programmes and projects To provide the EC with a co-ordinated input for the preparation of the FP 5 To initiate and maintain a global Maritime Industries' approach to R&D To meet these goal, and in agreement with the Commission, the R&DCG decided to develop the European Maritime Industries' R&D Master Plan (MP)
The final version of the MP was delivered and approved during the MIF Plenary Session of July 1996
Open Stability Forum - Crete 27 October 1997
The European Association of Classification Societies (EurACS)
What is the Master Plan
•
An organised review of the Maritime Industry needs
•
A strategic document reflecting Industries wishes
•
A common R&D framework for all maritime actors
•
A dynamic clustering tool to be followed and periodically updated lit is NOT a list of proposed or recommended projects
Open Stability Forum - Crete 27 October 1997
1
The European Association o/Classification Societies (EurACS)
I Sector 1 M an'time T ransfort ch run ' 0
I Area 1
·· ·
MasterPlan
Sector 2
I Area 2
·
Area 1
Area 2
Area 3
Area 4
Area 5
·· • R&D Priority areas o Reconunendations
Open Stability Forum - Crete 27 October 1997
The European Association o/Classification Societies (EurACS) Structure of the Master Plan Sector 1 The Maritime Transport Chain of 2000+ 1. Designing, building & maintaining • Competitivity Improvement • Safety • Environment sustainability • New vessels for new shipping 2. Operating o Maritime logistics • Safety improvements • Environment sustainability For each sector:
Sector 2 Marine resources 1. Offshore oil & gas 2. Operating 3. Fisheries & aquaculture 4. Fresh water & minerals 5. Seas pace utilisation
... R&D needs are described and organically grouped ... R&D priorities are given
Open Stability Forum - Crete 27 October 1997
I
The European Association o/Classification Societies (EurACS) MP & networking
Priority area
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