On the parametric resonance of container ships

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Ocean Engineering 34 (2007) 1021–1027 www.elsevier.com/locate/oceaneng

On the parametric resonance of container ships Metin Taylan Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey Received 15 November 2005; accepted 19 April 2006 Available online 20 December 2006

Abstract This work deals with parametric resonance which poses a great danger especially for container ships sailing in following or head seas. Important parameters that are effective in roll resonance are pointed out. For this purpose, a containership is taken as an example to analyze its stability in longitudinal waves based on the method worked out by American Bureau of Shipping (ABS). Unfavorable sailing conditions such as heading and speed, which directly depend on the environmental conditions, have been determined for this particular ship. These conditions may be reported to the master to guide him to keep his ship out of parametric resonance zones. Numerical details of the procedure have been worked out and provided as well. r 2006 Elsevier Ltd. All rights reserved. Keywords: Parametric resonance; Extreme motions; Container ships; Longitudinal waves

1. Introduction Ship motions in extremes seas have been studied by many researchers for many years. Thus, prediction and effects of ship motions in waves have been known to naval architects for a long time. It is obvious that, ship rolling in beam waves may be crucial since it leads to capsizing (Taylan, 2003). However, ship stability in following, heading and quartering seas may be equally dangerous because of the phenomenon called parametric resonance (Belenky et al., 2003; Belenky and Sevastianov, 2003; Kan, 1987; Paulling, 1961; Sanchez and Nayfeh, 1990). Parametric roll resonance can be defined as the resonance of the ship at which encountering frequency of the ship and the waves is approximately twice the natural frequency of roll motion when sailing in longitudinal waves. Once it is established and sustained, it leads to excessive motions and even capsizing of the ship. Especially container ships are very susceptible to parametric resonance due to the nature of their cargo (Friesch et al., 2004). Nowadays, very large container ships more than 10,000 TEU are built due to increasing demand Tel.: +90 212 285 6410; fax: +90 212 285 6508.

E-mail address: [email protected]. 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.04.007

from the market. It is projected that container ships with even more capacity will be designed and built in the years to come. Studies show that in the last 15–20 years, there have been so many accidents of loss of container cargo reported as a result of parametric resonance. A very recent incident in the North Pacific which resulted in heavy loss of containers from a post-Panamax container ship raised attention towards the importance of the problem once again. Although the phenomenon has been known for over 50 years, the severity of it urged IMO, 1995 (MSC/ Circ.707) and some classification societies to take necessary preventive actions. IMO put new rules into effect. Recently, American Bureau of Shipping (ABS) prepared a guide for the assessment of parametric roll resonance in the design of container ships. Moreover, they are preparing to assign an optional class notation for safety performance of these types of vessels (ABS, 2004). There are certain environmental and physical conditions that should simultaneously overlap in order parametric resonance to occur. These conditions may be summarized as; the ship must be sailing in head, quartering or following seas, the wavelength should be approximately around the range of the ship length, the encountering frequency may be twice the natural frequency of roll. In addition, character of ship’s initial stability in longitudinal waves

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M. Taylan / Ocean Engineering 34 (2007) 1021–1027

Nomenclature B ship breadth L length between perpendiculars dm draft amidships Dm depth amidships d(xi xCj) draft of ith station with jth position of the wave crest y(xi xCj) half-breadth of ith station at draft dCi, BM(xCj) metacentric radius for calculated ith position of the wave crest GM metacentric height GMmin minimum metacentric height in waves GMmax maximum metacentric height in waves GMa difference between GMmin and GMmax

plays an important role in parametric resonance since there is a vast difference in the righting energy of the ship when it is on wave crest or wave trough. 2. Ship stability in following or head seas Ship stability is somewhat different in following or head waves than in beam waves. When a ship is on a wave crest, its waterplane area reduces when compared to still water situation. Conversely, the waterplane area increases when the ship is in wave trough. In both cases, it is assumed that wave crest or wave trough is located near amidship. This in turn affects, instantaneous restoring energy in rolling motion and eventually total stability qualities. One can no longer treat metacentric height of the ship as constant. Metacentric height becomes the function of wave crest position along ship length that is, function of time. This complicates the evaluation of ship stability. Standard ship stability curve (GZ) is dependent on wave crest position along the ship length and must be determined instantaneously.

GMm mean value of metacentric height Vs ship speed, kn Vpr parametric roll speed, kn V ship speed, m/s VCB(xCj) vertical position of center of buoyancy at ith position of the wave crest TW wave period o0 natural frequency of roll in calm water om natural frequency of roll for the mean value of GM oa natural frequency of roll for the change of metacentric height oW wave frequency l wavelength m roll damping

Solution of the Mathiue equation may be found in many references and depends strictly on the values of p and q. Thus, the solution may be periodic, increasing or decreasing in nature. As it is known, in calm water, roll damping decreases roll amplitudes owing to the ship generated waves, eddies and viscous drag. Likewise, damping is very important in the growth of parametric resonance (Taylan, 2000). Depending on the critical values of roll damping, parametric resonance may or may not take place. This fact can simply be explained by the energy balance between damping and change of stability. Another important aspect of parametric resonance is the amplitude of the motion which depends on the character of the stability curve (GZ curve). In longitudinal waves, GM is no longer constant but depends on the roll angles except for small values of roll (8–101). On the other hand, roll period (frequency) is a function of GM. Therefore, after certain angles of roll, conditions for parametric resonance cannot be established and the growth stops.

3. Parametric resonance and roll motion 4. Importance of ship speed and heading When a ship is sailing in longitudinal waves, during its oscillatory motion, if it comes to a point where encountering frequency is twice the natural roll frequency, it starts experiencing significant roll amplitudes because of the change in its stability between wave crest and wave trough positions. This phenomenon is described as the parametric resonance, which can simply be explained by the Mathiue equation of the form: d2 f þ ðp þ q cos tÞf ¼ 0, (1) dt2 where p is a function of the ratio of forcing and natural frequency and q the parameter that dictates the amplitude of parametric excitation.

In terms of parametric resonance, the relative direction of ship’s heading with respect to waves is very critical (Shin et al., 2004). Following and head seas are the most critical directions in this respect. Encounter frequency which may be defined as the frequency that ship passes through wave troughs and crests with or frequency of change of ship stability, is another important factor for a ship moving in waves. Chain of events between the suitable combination of ship speed, direction, metacentric height and wave characteristics leads to parametric roll resonance. Frequency of encounter depends on the wave frequency which is also related to the wavelength.

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5. Roll motion and susceptibility criteria Basically, the procedure requires computation of stability of the ship in waves, and then the amplitude of the parametric resonance. Susceptibility criterion is utilized to check whether the situation indicates any vulnerability. Finally, if any susceptibility is detected, the severity of parametric roll is then calculated by a numerical procedure. It should be kept in mind that, the severity of parametric resonance found as a result of this procedure is a warning sign for the designers and operators. For more accurate and conclusive findings and data, model tests and supporting computer simulations need to be considered. The ship is said to be susceptible to parametric resonance if the following inequality holds (ABS, 2004): 0:25  0:5q  0:125q2 þ 0:03125q3  q4 =384ppp0:25 þ 0:5q.

ð2Þ

Then, damping criterion needs to be checked: qffiffiffiffiffiffiffiffiffiffiffiffiffi o0 m oqk1 k2 1  k23 , oE

(3)

where k1 ¼ 1  0:1875q2 , k2 ¼ 1:002p þ 0:16q þ 0:759, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2  16 þ q4 þ 352q2 þ 1024p k3 ¼ . 16q

   

ð4Þ

if k341, the damping criterion is not satisfied, if k3o1, and Eq. (2) is not satisfied, the ship may not be susceptible to parametric roll, if both Eqs. (2) and (3) are satisfied, the severity criterion has to be applied, if the inequality (2) does not hold, then the vessel may not be susceptible to parametric resonance.

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Severity of parametric roll resonance can be determined by solving the roll equation below numerically by using a suitable program: _ þ o2 f ðf; tÞ ¼ 0. € þ 2mo0 f f 0

(5)

6. Numerical example for a container ship In this paper, the procedure which is explained in ABS (2004) to determine dangerous and safe zones of operation for a particular container ship has been utilized and some steps are even simplified. For the theoretical background and details, the readers are referred to ABS (2004). A midsize container ship whose main dimensions are given below is chosen as an example for the numerical calculations: Length between perpendiculars Breadth Depth Design draft Vertical center of gravity Metacentric height

LBP B D d VCG GM

¼ 168.0 m ¼ 28.0 m ¼ 16.0 m ¼ 10.0 m ¼ 12.0 m ¼ 0.596 m

Body plan of the ship is given in Fig. 1. The wave conditions are determined such that they would represent the worst condition for parametric resonance, Table 1. Regular waves are considered and wavelength of the design wave is taken equal to the ship length. GM values of the container ship are computed depending on the position of the wave crest being at various positions along the ship’s length. The results of these calculations are plotted and given in Fig. 2. In order to apply the susceptibility criterion, the data which are composed of wave and stability characteristics of the vessel given in Table 2 can be used. Sample results of the

Fig. 1. Body plan of the container ship.

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1024

susceptibility criterion given in Table 3 may be found by using the procedure explained in Section 5. As can be seen from the Table, since both susceptibility inequalities are positive, severity of roll resonance needs to be investigated. GZ curves for different positions of the wave crest along the ship’s length are supplied in Tables 4.1 and 4.2. Other supplementary input information regarding forward ship speed is shown in Table 5. Severity check involves solution of the second order differential equation with nonlinear damping: € þ 2mo0 f _ þ o2 f ðf; tÞ ¼ 0. f 0

(6)

Table 1 Wave characteristics Wave length (equal to ship length) l Wave height (by linear interpolation) Wave period Circular wave frequency, ow

168.0 m 12.536 m 10.369 s 0.606 rad/s

Eq. (6) may be converted into a set of first-order equations by a suitable change of variables. Numerical solution of the set of first-order equations in the matrix form can be established by any of the known techniques. The set of equations is given as follows: ;

j þ2dj þ f ðf; tÞ ¼ 0, ;

f j ¼ 0.

ð7Þ

Time-dependent restoring term of the roll equation depends on the characteristics of the GZ curve: f ðf; tÞ ¼

ðfÞ GZðjfj; tÞ. GM0

(8)

In Eq. (8), it is seen that the restoring term is the function of roll angle and time. Thus, it can be manipulated by a step-bystep solution covering all values on the 3-D GZ curves using a computer program or approximated by a third or higherorder polynomial to simplify the case. In this work, the latter have been used and the GZ curves were approximated as a third order polynomial as (Taylan, 1999, 2002) GZ ¼ a1 f  a3 f3 .

Coefficients a1 and a3 can be determined by the characteristics of GZ curve as follows:

4.5 4 3.5

a1 ¼ GM, 2GM 4Afv a3 ¼  4 , f2v fv

GM, m

3 2.5 2 1.5 1 0.5 -100

(9)

0 -50 0 50 Position of wave crest along ship length, m

100

Fig. 2. GM values at various locations of wave crest along ship length.

ð10Þ

where jv is the angle of vanishing stability and Aj v the area under the GZ curve up to jv. Then, the above-mentioned equation of roll motion can be solved easily by a fourth-order Runge–Kutta algorithm using the data in Table 6 as input. The calculations have been carried out for damping values of m ¼ 0.03, 0.05, 0.075 and 0.10 and all range of speed values including V1 and V2 that may cause

Table 2 Parameters of susceptibility criterion Parameter

Symbol

Formula

Minimum GM (m) Maximum GM (m) Parametric forcing amplitude Mean GM (m) Amplitude of stability change in longitudinal waves expressed in terms of frequency (rad/s) Mean value of stability change in longitudinal waves expressed in terms of frequency (rad/s) Forward speed most likely for development of parametric roll (kn) Encounter frequency (rad/s) GM in calm water (m) Natural frequency of roll in calm water (rad/s) Critical damping coefficient Parameter of susceptibility criterion Parameter of susceptibility criterion

GMmin GMmax GMa GMm oa

Fig. 2. Fig. 2. GMa ¼ 0.5(GMmaxGMmin) GMm ¼ 0.5(GMmax+GMmin) oa ¼ ð7:854 GMa1=2 Þ=B

0.125 3.933 1.904 2.029 0.387

om

1=2 om ¼ ð7:854 GMm Þ=B

0.400

Vpr

Vpr ¼ 19.06|2omow|/o2w

oE GM o0 m p q

oE ¼ ow+0.0524 VS o2w o0 ¼ (7.854 GM1/2)/B p ¼ ðo2m  ðmo0 Þ2 Þ=o2E q ¼ o2a =o2E

Result

10.027 0.799 0.596 0.217 0.100 0.249 0.235

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Table 3 Sample results for susceptibility criterion Parameter

Formula

Result

Left boundary of inequality (2) Right boundary of inequality (2) Susceptibility inequality (2) outcome Coefficient k1 of damping criterion Coefficient k2 of damping criterion Coefficient k3 of damping criterion Boundary of damping criterion inequality (3)—damping threshold value Effective damping Susceptibility inequality (3) outcome

0.250.5q0.125q2+0.03125q3 0.25+0.5q Positive k1 ¼ 10.1875q2 k2 ¼ 1.002r+0.16q+0.759 k3 ¼ q216+(q4+352q2+1024r)1/2/16q qk1 k2 ð1  k23 Þ1=2 mo0/oE Positive

0.126 0.367

Table 4.1 GZ curves for different position of wave crest Heel angle Distance from FP to wave crest (m) 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 GM (m)

13.44

26.88

40.32

53.76

67.2

0 0 0 0 0 0 0.327 0.255 0.186 0.143 0.101 0.026 0.596 0.490 0.374 0.287 0.207 0.064 0.800 0.693 0.556 0.433 0.315 0.099 0.933 0.853 0.705 0.562 0.406 0.149 1.002 0.953 0.821 0.679 0.496 0.224 1.013 0.991 0.893 0.778 0.581 0.297 0.975 0.976 0.905 0.823 0.635 0.349 0.898 0.916 0.859 0.803 0.642 0.363 0.781 0.812 0.760 0.718 0.592 0.330 0.622 0.659 0.611 0.591 0.482 0.255 0.420 0.464 0.434 0.423 0.324 0.128 0.196 0.247 0.221 0.206 0.118 0.050 0.055 0.000 0.031 0.050 0.130 0.267 0.332 0.275 0.309 0.332 0.407 0.52 3.425 2.811 2.143 1.645 1.180 0.355

80.64 0 0.024 0.049 0.067 0.094 0.125 0.143 0.143 0.122 0.076 0.002 0.109 0.271 0.465 0.697 0.281

Table 4.2 GZ curves for different position of wave crest Heel angle Distance from FP to wave crest (m) 94.08 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 GM (m)

107.52

120.96

134.4

147.84

161.28

0 0 0 0 0 0 0.015 0.008 0.046 0.129 0.226 0.322 0.035 0.027 0.105 0.260 0.448 0.601 0.052 0.055 0.190 0.401 0.652 0.806 0.077 0.091 0.283 0.537 0.798 0.948 0.111 0.151 0.387 0.644 0.900 1.035 0.128 0.219 0.497 0.723 0.969 1.064 0.129 0.273 0.569 0.767 0.990 1.046 0.109 0.285 0.569 0.770 0.956 0.982 0.062 0.257 0.515 0.716 0.866 0.875 0.020 0.176 0.425 0.615 0.720 0.722 0.142 0.042 0.284 0.464 0.535 0.528 0.302 0.135 0.098 0.257 0.313 0.307 0.494 0.347 0.132 0.008 0.052 0.056 0.714 0.584 0.394 0.271 0.236 0.222 0.195 0.143 0.585 1.490 2.573 3.495

94.08 0 0.015 0.035 0.052 0.077 0.111 0.128 0.129 0.109 0.062 0.020 0.142 0.302 0.494 0.714 0.195

0.990 1.046 0.168 0.240 0.027

parametric resonance. Two examples from the outcome of these solutions are given in Figs. 3 and 4 for different ship speed and damping values. Having repeated the calculations for a number of speed and damping values in the following waves, the results are presented in Table 7 (Celebi, 2005). Amplitudes beyond 151 draw the boundaries of dangerous zones for the ship. Conversely, any roll amplitude values less than 151 show safe zones of operation. 7. Conclusions Ship motions in longitudinal waves are as dangerous as the motion in beam waves or any other wave direction. The phenomenon called parametric resonance may be experienced when sailing especially in following and head seas. The mechanism and causes of it have been studied by many researchers and the results have been known to naval architects for many years (Umeda, 1990, 1999; Umeda and Yamakoshi, 1993; Sanchez and Nayfeh, 1990). Primary causes of the phenomenon can be listed as: the case at which natural period of rolling is about twice the encountering period, flare shaped forms which are more susceptible to parametric resonance, and wave height and ship heading. Container ships are the ones that suffer greatly from parametric resonance. There have been so many accidents of container loss reported over the years. However, remedies are also available towards avoiding parametric roll resonance and eventually container loss both in design and operation stages (Umeda et al., 2003). The present study illustrates an example of how parametric resonance can be avoided by determining zones of parametric resonance during the course of voyage based on the method outlined by ABS (2004). For a selected midsize container ship, the results are given in Table 7 for various ship speed values and damping coefficients. The cases marked ‘‘unlimited’’ in the table are the heading, speed and damping combinations that lead to unlimited rise of roll amplitudes and could not be defined by the numerical solutions and should be kept away from. If the above calculations are repeated for different ship headings

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1026 Table 5 Sample results for forward speed calculations Parameter

Symbol

Formula

Results

Minimum GM value (m) Maximum GM value (m) Mean GM value (m) Mean frequency (rad/s) Natural frequency in calm water (rad/s) Forward speed 1 (kn) Forward speed 2 (kn)

GMmin GMmax GMm om o0 V1 V2

Table 4.2 Table 4.2 GMm ¼ 0.5(GMmax+GMmin) 1=2 om ¼ ð7:854 GMm Þ=B o0 ¼ (7.854 GM1/2)/B Vpr ¼ 19.06|2o0ow|/o2w Vpr ¼ 19.06|2omow|/o2w

0.143 3.495 1.819 0.378 0.217 8.973 7.825

Table 6 Data required to solve roll equation Parameter

Symbol

GM value in calm water (m) Natural frequency of roll in calm water (rad/s) Roll damping coefficient Time step (s) Time range (s) Initial roll angle (deg.) Initial roll velocity (deg./s)

GM o0 d Dt TR f0 _ f 0

Formula

Result 0.596 0.217 0.011 0.830 402.284 5.0 0.0

o0 ¼ (7.854 GM1/2)/B d ¼ 0.05o0 Dt ¼ 0.314/om TR ¼ 152.2/om

Table 7 Parametric roll amplitudes (in degrees) V (kn)

Fig. 3. Solution of roll motion for Vpr and m ¼ 0.1.

Fig. 4. Solution of roll motion for V ¼ 9.5 kn and m ¼ 0.1

between 01 and 3601, a polar plot of dangerous and safe zones may be drawn. For the given sample container ship, resonance amplitudes in Table 7 reveal that for the ship speed range between 7.5–17 kn, the amplitudes are quite high and

3.0 5.0 6.5 7.5 V2 ¼ 7.825 V1 ¼ 8.973 9.5 Vpr ¼ 10.027 13.0 15.0 17.0 20.0

m 0.030

0.050

0.075

0.100

9.167 Unlimited 6.875 Unlimited 24.637 18.908 Unlimited Unlimited 7.047 Unlimited 20.054 14.037

7.162 28.075 6.589 35.523 20.626 13.751 Unlimited Unlimited 6.875 Unlimited 17.762 13.178

6.589 7.735 6.016 16.616 15.756 12.032 Unlimited 32.659 6.589 Unlimited 15.183 12.319

4.584 5.787 5.500 7.563 9.282 10.772 10.886 30.367 6.303 35.523 12.605 11.746

unacceptable. These types of speed values should be avoided. It should be imperative to prepare such polar plots for every ship and provide those to masters to change their speed and direction to stay away from troubling combinations. In order to develop such polar plots, comprehensive computer programs such as LAMP of ABS (2004) are needed. But, it has been shown in this work that with even a simple numerical equation solver, dangerous zones may be determined by simplifying approximations to time dependent GZ function. Masters should be equipped with necessary tools and knowledge to avoid such casualties. Along with design and operational solutions, roll stabilizers should not be left out for mitigating excessive roll.

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References ABS Guide for the Assessment of Parametric Roll Resonance in the Design of Container Carriers, 2004. American Bureau of Shipping. Belenky, V.L., Sevastianov, N.B., 2003. Stability and Safety of Ships. Risk of Capsizing, vol. 2. Elseveir, Amsterdam. Belenky, V.L., Weems, K.M., Lin, W.M., Paulling, J.R., 2003. Probabilistic analysis of roll parametric resonance in head seas. In: Proceedings of STAB’03 Eighth International Conference on Stability of Ships and Ocean Vehicles, Madrid, Spain. Celebi, N., 2005. Parametric roll resonance of ships. Graduation Project, ITU. Friesch, J., Jensen, G., Mewis, F., 2004. Hydrodynamic design challenges for very large container ships. HANSA 9, 74–77. IMO, MSC/Circ. 707, 1995. Guidance to the master for avoiding dangerous situations in following and quartering seas. Kan, M., 1987. Surging of large amplitude and surf-riding of ships in following seas. Japanese Society of Naval Architects Journal 162, 49–62. Paulling, J.R., 1961. The transverse stability of a ship in a longitudinal seaway. Journal of Ship Research 4. Sanchez, N.E., Nayfeh, A.H., 1990. Nonlinear rolling motions of ships in longitudinal waves. International Shipbuilding Progress 37 (11), 247–272.

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Shin, Y., Belenky, V.L., Paulling, J.R., Weems, K.M., Lin, W.M., 2004. Criteria for parametric roll of large containerships in head seas. In: SNAME Annual Meeting. Taylan, M., 1999. Solution of the nonlinear roll model by a generalized asymptotic method. Ocean Engineering 26, 1169–1181. Taylan, M., 2000. The effect of nonlinear damping and restoring in ship rolling. Ocean Engineering 27, 921–932. Taylan, M., 2002. Overall stability performance of alternative hull forms. Ocean Engineering 29, 1663–1681. Taylan, M., 2003. Static and dynamic aspects of a capsize phenomenon. Ocean Engineering 30, 331–350. Umeda, N., 1990. Probabilistic study on surf-riding of a ship in irregular following seas. In: Proceedings of STAB’90: Fourth International Conference on Stability of Ships and Ocean Vehicles, Naples, Italy. Umeda, N., 1999. Nonlinear dynamics of ship capsizing due to broaching in following and quartering seas. Journal of Marine Science and Technology 4, 16–26. Umeda, N., Yamakoshi, Ya., 1993. Probability of ship capsizing due to pure loss of stability in irregular quartering seas. Naval Architecture and Ocean Engineering, Society of Naval Architects, Japan 30, 73–84. Umeda, N., Hashimoto, H., Vassalos, D., Urano, S., Okou, K., 2003. Nonlinear dynamics on parametric roll resonance with realistic numerical modeling. In: Proceedings of STAB’03 Eighth International Conference on Stability of Ships and Ocean Vehicles, Madrid, Spain.