Stability O Eng

8/7/2019 Stability O Eng

http://slidepdf.com/reader/full/stability-o-eng 1/31

Ocean Engineering 26 (1999) 1389–1419

www.elsevier.com/locate/oceaneng

Stability of small fishing vessels in longitudinalwaves

M.A.S. Nevesa,*, N.A. Pe´

rezb, L. Valerioa

aDepartment of Ocean Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, BrazilbInstitute of Naval and Maritime Sciences, Austral University of Chile, Valdivia, Chile

Received 19 November 1997; accepted 21 January 1998

Abstract

The dynamic stability of fishing vessels in longitudinal regular waves is investigated, bothanalytically and experimentally. In particular, the influence of stern shape on the parametricstability of fishing vessels is studied. Vessels TS and RS have very similar main characteristics,

but their sterns are different. Although their linear responses are comparable, both analyticaland experimental investigations indicate substantial differences in their dynamic stability inlongitudinal regular waves. Strong resonances are found for the vessel with the deep transom.The analytical method takes into consideration the effects of the heave and pitch motions andwave passage and shows good agreement with experimental results. Stability limits areobtained for different conditions and are used as an aid in the discussion of the results obtainedin the tests when relevant parameters are changed, such as wave amplitude and frequency,metacentric height and roll damping moment. © 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Ship stability; Roll motion; Parametric resonance

Nomenclature

a restoring term in Mathieu equationAij added mass or inertia in i mode due to j modeAw water plane areab0(x) breadth at the water lineBij damping in i mode due to j mode

* Corresponding author. E-mail: [email protected]

0029-8018/99/$ - see front matter © 1998 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 2 9 - 8 0 1 8 ( 9 8 ) 0 0 0 2 3 - 7



1390 M.A.S. Neves et al./Ocean Engineering 26 (1999) 1389–1419

B(˙

) damping momentB44 linear roll damping coefficientB444 second order roll damping coefficientC (,z, , ) total restoring moment in roll modeC ij restoring moment in i mode due to j modeC 444 third order roll restoring coefficientC 44z second order roll restoring coefficient due to heave motionC 44 second order roll restoring coefficient due to pitch motionC 44 second order roll restoring coefficient due to wave passagee0 non-dimensional amplitude of parametric excitationF 0i external force or moment in i modeg gravityhw

wave heightI i mass moment of inertia with respect to i axism ship massM wo amplitude of external wave action in roll modeq amplitude of parametric excitation in Mathieu equationu damping term in Mathieu equationU ship speedW wave frequencyW n roll natural frequencyx amplitude of motion in Mathieu equation

z heave motionZ g vertical position of centre of mass phase between excitation and wave amplitude of parametric excitation due to wave passage1,2 components of parametric excitation due to wave passage roll angle pitch angle wave amplitude phase of the parametric excitation wave length

density i phase of external force or moment

1. Introduction

The capsize of an intact ship is a phenomenon which by its very nature involvesexcessive motions. Predictably, this is a complicated problem, involving, in general,complex non-linear couplings in six degrees of freedom, and some simplificationsmust be considered in the formulation of prediction techniques (Oakley et al., 1974;

Kuo and Odabasi, 1975; Bishop et al., 1981). These excessive motions may be pro-duced by many different factors and some dangerous situations for ships at sea havealready been identified. One of these is the development of the so-called parametric






2. Equations of motion

Let Oxyz be a right handed coordinate system (Fig. 1) with axis Oz passing throughthe centre of mass G, Ox pointing forward and coinciding with the calm water sur-face. As defined in Fig. 1, surge, sway and heave are the translational motions, roll,pitch and yaw are angular motions with respect to axes x, y and z, respectively.

The mathematical model employed in this study to describe the non-linear rollmotion is of the form:

(I x A44)¨

B(˙

) C (,z, , ) M wocos(Wt ) (1)

The damping moment is assumed to be given by:

B(̇) B44̇ B444|̇|̇ (2)

The restoring moment is derived under the assumption that vertical motions dueto small amplitude waves are small, such that relative vertical displacement at apoint of the length of the ship may be taken as the sum of three effects, heavemotion, pitch motion and far-field wave profile. The roll restoring moment in wavesis then assumed to be given by:

C (,z, , ) C 44 C 4443 C 44zz C 44 C 44 (3)

where C 44z, C 44 , C 44 represent time-dependent variations of hull restoring character-istics due to heave and pitch motions and unit amplitude wave passage. For thecoordinate system defined above and as derived in Appendix B:

C 44 GM (4)

C 44z g1

2 L

b20(x)

dy

dz |0,x

dx zgAw (5)

Fig. 1. Coordinate system and definition of modes of motion.



1393M.A.S. Neves et al./Ocean Engineering 26 (1999) 1389–1419

C 44 g1

2 L

b20(x)

dy

dz |0,x

xdx zgAwxf (6)

C 44 gcos(wt ) (7)

and where:

√ 21 2

2

tg−121

with 1 and 2 given by:

1 L

1

2b20(x)

dy

dz |0,x

zgb0(x)cosw2

gxdx (8a)

2 L

1

2b20(x)

dy

dz |0,x

zgb0(x)senw2

gxdx (8b)

The heave and pitch motions are obtained as the solutions of the set of coupledlinear equations defined as:

(m A33)z¨

B33z˙

C 33 A35 ¨

B35 ˙

C 35 F 03cos(Wt 3)

A53z¨

B53z˙

C 53z (I y A55) ¨

B55 ˙

C 55 F 05cos(Wt 5) (9)

The reader should refer to the Nomenclature section at the beginning of the paperand Appendix B of this paper for the list and definitions of symbols given above.

As the present study is limited to longitudinal waves, M W 0 0, that is, it isassumed that there is no external excitation in the roll mode of motion. By dividingall terms in Eq. (1) by (I x A44) the equation of roll motion assumes the form:

¨

(b44 b444|˙

|)˙

(c44 c4442 e0cos(Wt )) 0 (10)

where e0 the amplitude of parametric excitation in the roll motion equation containscontributions from heave and pitch motions and wave passage effect, as indicatedpreviously.

3. Description of the experiments

The tests were conducted at the ship model basin of the Austral University of Chile, where regular waves can be generated by means of a flap-type wave generator.The experimental results presented in this paper correspond to results obtained from




two different test programmes conducted by Pe´

rez (1985); Pe´

rez and Sanguinetti(1995).

The two models were constructed to scale 1:30. Hull particulars are given inAppendix A. The experiments performed up to now are all for the case of zero speedof advance. In the experiments, weight distribution, wave amplitude and frequencywere varied in order to:

1. perform roll decrement tests at different natural frequencies;2. investigate the sensitiveness of the two similar hulls to parametric stability in the

first two regions of resonance of the Mathieu diagram.

Roll decrement tests were performed for initial angles near 15°. Distinct oscillatoryfrequencies were obtained by adjusting weights on board. Models were tested bothwith and without bilge keels.

The procedure in the parametric tests was to position the ballasted model longitudi-nally in the tank, loosely prevent it from drifting by a thread, and, by generatingregular waves in one extreme of the tank, roll angular displacements were recordedwith the model free to oscillate. No artificial impulse, inducement or bias was appliedto the model in order to help the roll motion to start to develop. Starting from rest,the model was excited in heave and pitch by the incoming waves. In unstable con-ditions, the transfer of energy from these modes to the roll mode allowed roll motionto set up. In stable conditions the model remained oscillating in heave and pitch,with no tendency to develop roll motion. The tests in parametric stability were mostly

conducted for tuning the frequency in the W

2W n condition. A few tests wereconducted at the W W n condition.In order to investigate the influence of the increased damping in the stabilization

process, each model was tested in turn with and without bilge keels.Bilge keel dimensions and arrangements were identical for the two models, fitted

along half of the model length, with a constant breadth of 15 cm (to ship scale).Fig. 2 shows two views of the arrangement of bilge keels fitted to the TS hull.

4. Calculation of ship hydrodynamic data

A 3-D panel method developed by Inglis and Price (1980) was used in the potentialflow calculations of the added masses and damping coefficients in the heave, rolland pitch modes, and exciting forces and moments in the heave and pitch modes inlongitudinal waves. These coefficients have been published elsewhere (see Neves,1981; Neves et al., 1988). As a consequence of the similitude of the submerged formand inertia coefficients of the two hulls, linear responses in heave and pitch arealmost the same. Comparative amplitude and phase diagrams in the frequencydomain for longitudinal waves for zero speed of advance are shown in Fig. 3 andFig. 4 for the heave and pitch motions, respectively.

In order to incorporate viscous effects in the roll damping moment, use was madeof the procedure proposed by Ikeda, described by Himeno (1981). With referencebeing made to Eq. (2), the linear coefficient takes account of wave generating and




Fig. 2. Bilge keels arrangement and side view of TS hull.






method. This was motivated by the fact that beam-to-draft ratios for the hulls investi-gated in this paper are slightly higher than those considered in Ikeda’s empiricalassessment of eddy damping. Fig. 5 gives the equivalent roll damping coefficientfor the two hulls with and without bilge keels for a roll amplitude of 15°. The resultsindicate that the TS hull (with less rounded forms) is more damped than the RShull, either with or without bilge keels. As mentioned above, in order to check theapplicability of Ikeda’s method to the ships under investigation in this paper, rolldecrement tests were performed, in which initial angles in roll near 15° were givento the ship models. The Froude method (Spouge, 1988) was used to reduce the data.Results from these roll decrement tests are given in Fig. 6.

Roll decrement test results given in Fig. 6 are in general in good agreement withthe semi-empirical results given in Fig. 5. The same trends are observed. Effectively,test results also give a higher damping effect for the TS hull compared with the RShull when the two models are not fitted with bilge keels. With bilge keels, the TShull is more damped than the RS hull for frequencies above W 1.25 r/s. Fittingthe models with bilge keels introduced a great amount of damping that tended toincrease with the frequency of oscillation.

Figs. 7 and 8 show damping moments plotted against time for the two ships intwo typical conditions. These figures illustrate the good agreement found when rolldecrement test results are compared with results from calculations based on the Ikedasemi-empirical model. In general, for the metacentric height range of values used inthe parametric experiments described later on, the results from the roll damping

analysis performed may be summarized as:1. Damping for the RS hull without bilge keels is well described by the Ikeda

method.2. For the TS hull without bilge keels, the Ikeda method produces slightly lower

damping than that obtained by the experimental procedure.

Fig. 5. Damping coefficients for 15° roll amplitude, calculated using the Ikeda method, for the two

vessels, with and without bilge keels.




Fig. 6. Damping coefficients for the two vessels obtained from roll decrement tests, models with and

without bilge keels.

Fig. 7. Comparison of roll damping moment for the TS hull.

3. For roll amplitudes within the 10–15° range, the Ikeda method slightly underesti-mates the damping effect introduced by the bilge keels in both models. Yet, thenet effect of introducing bilge keels to the ship models seems to be well describedby the semi-empirical method.

4. It is concluded that the semi-empirical method described by Ikeda produces arealistic damping assessment for the two hulls considered in this study, in good

agreement with experimental results.5. Damping levels for the two hulls are comparable, but in general the TS hull is

more damped than the RS hull.




Fig. 8. Comparison of roll damping moment for the RS hull.

6. Roll motion analysis

Numerical integration of Eq. (10) was performed using a fourth-order Runge–Kutta algorithm. The coefficients employed were those described previously in Sec-tions 4 and 5. Results obtained from these numerical integrations are now comparedwith the roll time series registered in the experimental programme for the modelsexcited by longitudinal regular waves.

Very good agreement was obtained in the comparisons, as shown in Fig. 9 andFig. 10 for the RS hull, and Fig. 11 and Fig. 12 for the TS hull. The comparisonsgiven in Fig. 9 and Fig. 10 are for RS hulls without bilge keels in low metacentricheight conditions; the tuning in all cases is near the W 2W n condition and corre-sponds to unstable roll motions. It is important to notice that the waves are quitehigh and steep. Fig. 11 gives results for roll motion for the TS hull in waves withhw 0.90 m. For the low metacentric height defined for this condition, motions

Fig. 9. Comparison of numerical integration with experiment for the RS hull, GM = 0.27 m.






Fig. 12. Comparison of numerical integration with experiment for the TS hull, GM = 0.85 m.

7. Mathieu diagrams

The linear variational equation of Eq. (10) may be expressed in the form of adamped Mathieu equation:

x 2ux (a u2 16qcos2t )x 0 (11)

Limits of stability corresponding to Eq. (11) may be determined and the effect of damping and metacentric height on the various regions of parametric resonance maybe assessed (Valerio, 1994). Fig. 13 shows the effects of three different levels of damping coefficient, u 0, u 0.035, and u 0.070, on the limit curves corre-

Fig. 13. Mathieu diagram for three damping levels: u = 0.000, u = 0.035, u = 0.070.




sponding to the first three regions of parametric resonance. It is seen that by raisingthe damping level, the first region (W 2W n) is practically unaffected, while thesubsequent regions are more and more affected by damping. As a consequence, thethird region of parametric resonance practically disappears for small levels of damp-ing.

Stability limits for a damped Mathieu equation are also affected by the naturalfrequency. Fig. 14 shows the limits of stability near the W 2W n and W W nregions of resonance for two values of metacentric height, GM 0.35 m and GM

0.85 m for the TS hull. It may be noticed that the higher the metacentric height,the higher the limits are. That results from the frequency dependence of the rolldamping coefficient, as given in Fig. 5. The implication is that at the first region of resonance the damping level is much higher than at the second region of resonance.

8. Stability analysis

Considering that the mathematical model describes very well the dynamicsinvolved in the parametric destabilization process of the ships, Figs. 15–20 havebeen prepared with the purpose of helping in the interpretation of the results. In eachof these figures, graphics (a) and (b) show the time series obtained at the wave basinand graphics (c) show the corresponding points plotted in the Mathieu diagram.

Fig. 15(a) shows a case of strong instability for the TS hull, with bilge keels, in

the range W

2W n, with GM

0.32 m. In less than six cycles the roll angle reachesapproximately 40°; a very dangerous condition, meaning a real risk of capsize. Forthe RS hull to undergo such intense destabilization, it was necessary to reduce themetacentric height to GM 0.27 m and to remove the bilge keels. Yet, the resultinginstability requires more than eight cycles to reach roll angles of the order of 28°.This result is given in Fig. 16(a), and demonstrates that the TS hull is much more

Fig. 14. Limits of stability for the TS hull for two values of metacentric height.




Fig. 15. Transom stern hull with bilge keels, GM = 0.32 m.




Fig. 16. Round stern hull without bilge keels, GM = 0.27 m.




Fig. 17. Transom stern hull without bilge keels, GM = 0.35 m.




Fig. 18. Transom stern hull, without bilge keels, GM = 0.48 m.














insignificant. This is confirmed in the Fig. 16(c) plotting, which shows point 2 welloutside the unstable area of the Mathieu diagram. Comparing Fig. 16(a) with Fig.17(b), it is seen that both ships reach approximately 30°, but with the TS hull doingso in six cycles, whereas for the RS hull it takes nine cycles. It should be noticedthat in this comparison both ships are without bilge keels and the TS hull is moredamped than the RS hull. Metacentric height for the RS hull is smaller and waveheight is much higher. This is clearly another demonstration that the TS hull is muchmore sensitive to parametric destabilization than the RS hull. Observing point 1 inFig. 16(c) and point 2 in Fig. 17(c), it can be noticed that both points are very closeto the limit curve. Point 2 in Fig. 16(c) is located in the stable region. The waveheight is high (hw 2.0 m), but as shown in Fig. 16(b), no amplification of rollmotion occurs. Comparison of points 1 and 2 in Fig. 17(c) and their respective timeseries in Fig. 17(a and b) gives the effect of increasing wave height for the TS hullwithout bilge keels. Fig. 18(a and b) again shows similar results for the TS hull atfixed GM and damping level. A small increase in wave height results in a correspond-ing lower point 2 compared with point 1 in Fig. 18(c).

The influence of increased damping may be observed by comparing Fig. 15(b)with Fig. 17(b). In Fig. 15(b) the TS hull is fitted with bilge keels, GM 0.32 mand hw 2.4 m. The roll angle reaches approximately 32° in nine cycles. In Fig.17(b) the same ship, without bilge keels, but with a larger GM and lower waveheight (hw 0.9 m) reaches 30° in only six cycles. The introduction of the bilgekeels contributes to reduce the distance in the Mathieu diagram from the plotted

point to the limit curve, thus decreasing the intensity of the amplification of themotion. But, as the hull is strongly subjected to parametric excitation in frequenciesaround the tuning W 2W n, the effect of fitting bilge keels is quite limited in thisrange of frequencies.

Another interesting comparison is between Fig. 17(b) and Fig. 19(b). Both casescorrespond to the TS hull without bilge keels, excited by waves of equivalent height.But the metacentric height in the two cases is very different. The wave frequencyin Fig. 19(b) is high (W 2.08 rad/seg), corresponding to a large parametric exci-tation e0 (see Fig. 3), but large metacentric height (GM 0.85 m). The resultingunstable motion is less strong than the motion given in Fig. 17(b), and another view

of that can be seen in the location of point 2 in Fig. 19(c), which lies very close tothe curve of the stability limit, compared with point 2 in Fig. 17(c), located wellinside the unstable region. As a consequence of the large metacentric height con-sidered in Fig. 19 (GM 0.85 m) the limits of stability are relatively high, speciallyfor the W W n zone, as shown in Fig. 19(c). The wave tested in the conditionshown in Fig. 19(c) is a very steep one, (hw /14.8). The steepest wave generatedin this study was that corresponding to Fig. 10. In that case, hw /10.6.

Clearly, an index to the intensity of the parametric instability of a damped systemwith internal excitation is the distance of the plotted point to the curve of stabilitylimit in the damped Mathieu diagram; this distance defines the amplification of

motion.This is also applicable to points 1 and 2 plotted in Fig. 20(e), representative of

the motions shown in Fig. 20(a and b), respectively. These two conditions have large




values of the parameter q (q 0.55 and q 0.72, respectively), much higher thanthose presented in Fig. 16(c), Fig. 18(c) and Fig. 19(c). Nevertheless, the time seriesin Fig. 20(a and b) have both a very slow amplification, due to the fact that theunstable region is narrow near the W W n tuning, and the points are necessarilynear the limit of stability. The wave in Fig. 20(b) is reasonably smooth (hw /26.2).Yet, even for very steep waves, no significant resonance occurs in the W W n zoneof stability.

9. Hull form and parametric stability

As the tests and numerical simulations have indicated quite different roll behaviourfor the two hulls, it was found relevant to investigate numerically the different termsin the parametric excitation and to see how these could be related to hull form.

Fig. 21 shows the curves of parametric excitation amplitude e0 (divided by c44—see Eq. (10)) for the two hulls, for different frequencies and wave amplitude

1.0 m (real scale). The figure shows curves for two different values of metacentricheight. Parametric excitation is in all cases much higher for the TS hull, when com-pared with the other hull, for the whole frequency range considered, when both shipshave the same metacentric height. In Fig. 22 the relative importance of each compo-nent that determines the amplitude of parametric resonance for the TS hull withmetacentric height of 0.85 m may be observed. That is, Fig. 22 shows the influence

of the heave motion, wave passage and pitch motion for the TS hull. For smallfrequencies, up to W 1.0 rad/seg, the heave motion effect is cancelled out by

Fig. 21. Amplitude of parametric excitation for the two hulls in the frequency domain for two distinct

levels of metacentric height.




Fig. 22. Components of the parametric excitation for the TS hull, GM = 0.85 m.

the wave passage effect, and parametric excitation is dominated by the influence of pitch motion.

It should be noticed that the linear responses in heave and pitch for the two hullsare, in practice, the same (see Fig. 3 and Fig. 4) and under comparable values of damping levels and wave excitation, the TS hull, with its pronounced transomarrangement is much more unstable than the RS hull. It can be deduced from thiscomparative analysis that the distinct levels of parametric excitation found for thetwo hulls are due to the different stern arrangements. In fact, as can be seen in Fig.23, the lines plan for a vessel with a transom stern leads to a non-symmetrical longi-

Fig. 23. Longitudinal distribution of breadth and flare at water line for the two hulls.




tudinal distribution of sectional breadth and flare at the water line, resulting, forinstance, in a large C 44 coefficient, as given in Eq. (6). This second order formcoefficient is three-times larger for the TS hull than for the RS ship when metacentricheight is GM 0.85 m. Actually, it is observed that it is the longitudinal distributionof flare that really makes the difference for the hulls under study, since the longitudi-nal distribution of area is not very different from one hull to the other. In fact, theround-stern hull, having a more ‘smooth’ longitudinal distribution of flare, is lessexposed to internal excitation in roll.

10. Conclusions

The parametric stability of fishing vessels in longitudinal regular waves has beenexamined both experimentally and numerically, with special emphasis on the influ-ence of stern shape on the amplification of the motions.

The conclusions that emerge from this investigation appear to be coherent andrealistic. They suggest, for instance, the following sensible results:

A square stern may have a pronounced destabilizing effect. It may play animportant role in allowing high levels of parametric excitation.

The mathematical model was capable of describing even very intense rolling

motions. The distinct contributions of heave and pitch motions and wave passageeffect were disclosed.

Model experiments and numerical simulations produced very strong instabilitiesleading to large roll angles in few cycles for the TS hull with low metacentricheight in the W 2W n zone of resonance. These results provide evidence thatparametric resonance may be responsible for the capsize of some small fishingvessels.

No significant resonance occurred in the W W n zone of the Mathieu diagram.

The experimental results given above are relevant not only to validate the numeri-

cal simulations, but also as a contribution to an area in which experimental evidenceis scarce. Clearly, progress in the field of ship stability in waves must depend uponthe acquisition of much more experimental data, since there is a serious lack of experimental studies with ship models and collection of full-scale empirical dataregarding the behaviour of fishing vessels in waves.

Acknowledgements

This work was partially supported by CNPq and CAPES of Brazil and CONICYTof Chile. The authors express their thanks for this financial support.




Table 1

Principal particulars of ships

Denomination RS TS

Length (m) 24.36 25.91

Length between perpendiculars 21.44 22.09

(m)

Beam (m) 6.71 6.86

Depth (m) 3.35 3.35

Draught (m) 2.49 2.48

Displacement (tons) 162.60 170.30

Water plane area (m2) 102.50 121.00

Trans. radius of gyration (m) 2.62 2.68

Long. radius of gyration (m) 5.35 5.52

Length of bilge keel (% Lpp) 25–75 25–75Width of bilge keel (m) 0.15 0.15

Appendix A

Particulars of ships

The main characteristics of the ships used in this paper are listed in Table 1,followed by Fig. 24 which shows their line plans:

Appendix B

Roll second order hydrostatic restoring coefficients

The hydrostatic restoring moment in roll for small angles may be expressed as:

C (z,, ) gGM (B1)

Fig. 24. Body plans of tested vessels (TS and RS).




where:

GM I T

zb zg (B2)

is water density, g is gravity, is displaced volume, GM is metacentric height,I T , is inertia of area with respect to axis x, zb is vertical coordinate of centroid of submerged volume, zg is vertical coordinate of ship centre of gravity.

The geometric hull characteristics given in Eq. B(1) may be assumed to varyperiodically when the ship motions are induced by regular waves of small amplitude.

For longitudinal waves, the restoring moment may be expressed as a function of variations in the vertical relative displacements of the ship, as indicated in Fig. 25,where is the free surface instantaneous elevation.

For a generic wave position, the sectional beam may be expressed as a functionof local hull derivative, computed at average water line as:

b b0 2dy

dz |0,x

(B3)

such that the roll restoring moment in longitudinal waves becomes

C () g(I T () M b() zg(n)) (B4)

where:

M b

zb

Changes in inertia, area centroid and submerged volume are given by:

I T L

1

12b0(x) 2

dy

dz |0,x

3dx (B5)

M b L

zbA0 1

2b0(x)2

2

3

dy

dz |0,x

3dx (B6)

Fig. 25. Variation of sectional beam with relative vertical displacement.




L

A0 b0(x) dy

dz |0,x

2dx (B7)

where the integrals are to be calculated along the ship length.Considering only the linear terms in Eq. B(5), B(6), B(7), the roll hydrostatic

moment is then approximated by:

C C 44 gL

1

2b20(x)

dy

dz |0,x

zgb0(x)dx (B8)

where C 44 is the restoring term in calm water.Assuming that vertical motions due to small amplitude waves are small, the verti-

cal relative displacements may then be assumed to be the sum of heave ( z), pitch( ) and wave passage ( ). Thus, the component due to heave may be expressed as

C z gL

1

2b20(x)

dy

dz |0,x

zgb0(x)zdx (B9)

or

C z C 44zz (B10)

where:

C 44z gL

1

2b20(x)

dy

dz |0,x

dx zgAw (B11)

z z0cos(wt 3) (B12)

Analogously, the component of parametric excitation due to pitch motion may beexpressed as:

C C 44 (B13)

where:

C 44 gL

1

2b20(x)

dy

dz |0,x

xdx zgAwxf (B14)

0cos(wt 5) (B15)

and the component due to wave passage may be expressed as:

C gL

1

2b20(x)

dy

dz |0,x

zgb0(x) ¯ cosw2

gx wt dx (B16)




or:

C ¯

gL1

2 b2

0(x)

dy

dz

|0,x zgb0(x)

cosw2

gxcoswt senw2

gxsen wt dx

C ¯ gcoswt

L

1

2b20(x)

dy

dz |0,x

zgb0(x)cosw2

gxdx

senwt L1

2 b2

0(x)

dy

dz

|0,x

zgb0(x)senw2

g xdx (B17)

where:

C C 44 ¯

(B18)

C 44 gcos(wt ) (B19)

and:

√ 21 2

2

tg−121

1 L

1

2b20(x)

dy

dz|0,x zgb0(x)cosw2

gxdx

2 L

1

2b20(x)

dy

dz|0,x zgb0(x)senw2

gxdx

At low frequencies sine terms tend to zero, whereas cosine terms tend to one,such that at low frequencies the ship goes with the wave and the heave contributionto the parametric excitation tends to cancel out with the wave passage effect, that is,

w→0⇒senw2

gx→0;cosw2

g→1

or:

→L

1

2b20(x)

dy

dz|0,xdx zgAw⇒C 44 → C 44z




Fig. 26. Vertical relative displacement.

Fig. 26 illustrates the fact that vertical relative displacement for small motionsmay be approximated by

z x ¯

cosw2

gx wt (B20)

Assuming that the heave and pitch motions may be taken as independent of rollmotion, they may be assumed as simple harmonic motions defined at the same fre-quency as wave encounter frequency. So, the amplitude of parametric excitationcomposed of three components may be approximately given as:

0 gL

1

2b20(x)

dy

dz |0,x

zgb0(x)dx (B21)

where:

acoswt bsenwt

a zcos 3 x cos 5 cosw2

gx

b zsen 3 x cos 5 cosw2

gx

where 3 and 5 are phases in the heave and pitch motions, respectively.

References

Bishop, R.E.D., Neves, M.A.S., Price, W.G., 1981. On the dynamics of ship stability, RINA Paper No. 10.

Bloki, W., 1980. Ship safety in connection with parametric resonance of the roll. International Shipbuild-

ing Progress 27 (306), 36–53.

Campanile, A., Casella, P., 1989. Form stability reduction among waves for series 60 hulls. Ocean Engin-

eering 16 (5/6), 431–462.

Froude, W., 1863. Remarks on Mr. Scott Russell’s paper on rolling transactions. INA 4, 232–275.




Himeno, Y., 1981. Prediction of ship roll damping-state of art, Department of Naval Architecture and

Marine Engineering, The University of Michigan, Report No. 239, September.

Inglis, R.B., Price, W.G., 1980. Calculation of the velocity potential of a translating pulsating source,

RINA Paper W2.Kerwin, J.E., 1955. Notes on rolling in longitudinal waves. International Shipbuilding Progress 02 (16),

597–614.

Kuo, C., Odabasi, A.Y., 1975. Application of dynamic systems approach to ship and ocean vehicles

stability. Proceedings of the International Conference on Stability of Ships and Ocean Vehicles, Glas-

gow.

Fang, M.-C., Lee, C.-K., 1993. On the dynamic stability of a ship advancing in longitudinal waves.

International Shipbuilding Progress 40 (422), 177–197.

Morral, A., 1979. Capsizing of small trawlers. The Royal Institution of Naval Architects, Joint Evening

Meeting, Glascow and Rina Spring Meetings, London.

Nayfeh, A.H., 1973. Perturbation methods. John Wiley & Sons Inc.

Neves, M.A.S., 1981. Dynamic stability of ships in waves. Ph.D. Thesis, Department of Mechanical

Engineering, University College London, University of London.Neves, M.A.S., Perez, N.A., Sanguinetti, C.O.F., 1988. Analytical and experimental investigation on the

dynamic stability of fishing vessels in regular waves (in Spanish). Ingenierı´a Naval No. 638–639,

Spain.

Neves, M.A.S., Valerio, L., 1994. Parametric stability of fishing vessels. Proceedings of the Fifth Inter-

national Conference on Stability of Ships and Ocean Vehicles, Florida.

Neves, M.A.S., Valerio, L., 1995. Transom stern vs. round stern: comparison of stability in waves of two

small fishing vessels. XIV IPEN: Pan-American Congress of Naval Engineering, Lima, Peru, June.

Oakley Jr. O.H., Paulling, J.R., Wood, P.D., 1974. Ship motions and capsizing in astern seas. Tenth Naval

Hydrodynamics Symposium, M.I.T., June.

Paulling, J.R., 1961. The transverse stability of a ship in a longitudinal seaway. Journal of Ship Research

04 (4), 37–49.

Paulling, J.R., Rosenberg, R.M., 1959. On unstable ship motions resulting from nonlinear coupling. Jour-nal of Ship Research 03 (1), 36–46.

Pe´

rez, N., 1985. Development of some experimental techniques in regular waves (in Portuguese). M.Sc.

Thesis, Federal University of Rio de Janeiro, RJ, Brazil.

Pe´

rez, N., Sanguinetti, C., 1995. Experimental results of parametric resonance phenomenon of roll motion

in longitudinal waves for small fishing vessels. International Shipbuilding Progress 42 (431), 221–234.

Sanchez, N.E., Nayfeh, A.H., 1990. Non-linear rolling motions of ships in longitudinal waves. Inter-

national Shipbuilding Progress 37 (41), 247–272.

Skomedal, N.G., 1982. Parametric excitation of roll motion and its influence on stability. Proceedings of

the Second International Conference on Stability of Ships and Ocean Vehicles, Tokyo, Oct.

Spouge, J.R., 1988. Non-linear analysis of large-amplitude rolling experiments. International Shipbuilding

Progress 35 (403), 271–320.

Valerio, L., 1994. Parametric stability of ships in longitudinal regular waves (in Portuguese). M.Sc. Thesis,Federal University of Rio de Janeiro, RJ, Brazil.

Documents

Stability O Eng