### Tedìrde Hosdio1

WAVE-INDUCED ANTISYMMETRIC RESPONSE OF A FLEXIBLE sHIDeift

IN AN IRREGULAR SEAWAY

I. Introduction

Unified dynamic theories have been given for both symmetric [1,21 and antisymmetric [2,3] responses

of flexible hulls travelling in a sinusoidal seaway. From

such theories it has been shown that the mode shapes

and natural frequencies of the dry hull and the

re-sonance frequencies of the wet hull play important parts in determining the form of the responses at any position in the hull. These responses can be found by considering the response amplitude operators (RA O)

of the structure when it is excited by regular sinusoidal waves of unit amplitude.

The theory of symmetric response has received the

more attention and has been extended so as to en-compass transient responses of a hull in waves as well

as the spectral responses to a known irregular seaway [2,4,5].

The antisymmetric theory, being the more

compli-cated [6-8] and requiring detailed structural data (which is rarely available), still requires refinement in matters of detail rather than of general approach.

The indications are that, for a container ship, results depend very sensitively on the allowance made for warping stiffness [2,3] and it is therefore highly de-sirable that the allowance for, and data on, warping

should be refined.

In this paper, the container ship [2,3] is assumed to

travel at 26 knots in bow seas (i.e. with a heading angle

of x 1350). The antisymmetric response amplitude

operators are calculated for lateral shear force, lateral

bending moment and twisting moment at different

points (i.e. 1/4, ¡/2, 31/4) along the hull. Comparisons are made to illustrate the influence of warping. The calculated data are used to derive response spectra for

the ship operating in an irregular long-crested seaway.

Statistical properties defined in terms of the moments of the individual spectra are thus derived. Where con-ventional techniques permit it, a further comparison is made between the response spectra and statistics

*) Department of Mechanical Engineering, University Collgc London, England.

by

R.E.D. Bishop, W.G. Price and P. Temarel

Summary

Calculated results are presented for a flexible container ship travelling at 26 knots in a long crested irregular

bow sea. Representative mean square spectral densities are given for lateral shear force, lateral bending moment and twisting moment. The influence of allowing for warping stiffness is examined and comparisons are made with results obtained using conventional theory based on the assumption of hull rigidity.

of the flexible hull on the one hand and those based on the assumption of hull rigidity as in conventional

analysis on the other. 2. Basic theory

The responses of the flexible hull to regular sinus-oidal waves of unit amplitude and frequency of

en-counter we are governed by the equation [21

### A(w)(t) +B(ccjj)(r) +Cp(t) Zei()et.

The square matrices A, B, C contain terms of both

structural and hydrodynamic origin and are neither symmetric nor positive definite; they depend on the

frequency of encounter through the hydrodynamic

contributions. The hull is idealised as a non-uniform

free-free beam and the hydrodynarnic data are derived

from an appropriate strip-theory [9,1 0J. The column matrix p(t) contains the generalised principal coor-dinates of the dry hull audi is a cohtnn matrix of the

generalised wave excitation forces due to waves of

unit amplitude approaching the ship at some arbitrary

heading angle x ( 1800 for head seas). All the matrices

are of order N, determined by the number of principal

coordinates used in the analysis.

The responses of lateral shear force !"(,t), lateral bending moment M(x,t) and twisting moment T(x,t)

at any position x from the stem are

N =

_{V()p.}

r3 '
N
= _{M,.(X)p}

### ee

r3 J### T(x,t) = E T(x)p

where V,(x), Mr(X) T(X) are the characteristic

func-tions of shear force, bending moment and twisting

moment respectively corresponding to the rth

prin-cipal mode of the dry hull. It will be noted that

Vr(X)=O=Mr(X)= T,(x)

The response amplitude operators of these responses

may be derived in the usual manner. Thus the shear force operator associated with the frequency of

en-counter w is N

= _{E V(X)PrI}

Similar relationships exist for the other two responses.

3. The seaway and its statistical properties

The mean square spectral density of the shear force response (say) at position x is given by the familiar input-output relationship of random process theory

### [li]. It is

= lV(XWe)I2

### "e'

where (we) is the wave encounter spectrum of the

seaway. We shall use an ITTC wave spectrum, i.e. a

spectrum of the form

,,) r.._exp(_B/w4)

where A = 8.1x103g2, B =3.1 1/h3 and h113 is the significant wave height measured in metres. The corres-ponding encounter spectrum is

### WeO)/I

### 1 ---

_{g}cosxl with

we =

### w - -

CoSX gwhere co is the absolute frequency of the waves and U is the mean forward speed of the ship. These spectra

-the wave spectrum and tile encounter spectrum

### - are

illustrated in Figure 1 for a significant wave height h113 =5m, U 13.38 rn/s and x = 135°.

The nth statistical moment, in0(x), of the shear

a
.T.T.C. SEASPECTSE*1
4.00'
3.50
3.00-2.50
2.00
"
.50-
'.00-.50
0. _{ü.}
5.00
.20 .40 'EJ '.50 .00 '.20 .40 I'. 60

force response spectrum at position x is given by

### m(x)=7 wvv(X,e)0'(e

for n = _{0,1,2...Various statistical indices ma be}

found on the basis of such moments [2,11]. Thus the

average period of wave uperossing is
T(x)_{= 27rI{mO(x)/m2(')},}

and it will be noted that this quantity varies along the length of the hull.

4. Computations 4.1. General description

The principal dimensions of the container ship to

be considered are [3]

¡=281.Om, B=32.26m, T=l2.2m and

A=67 150 tonne f.

Results will be presented, giving antisymmetric

responses at positions x = ¡/4, 1/2, 31/4 measured from

the stem of the vessel travelling at 13.38 rn/s in long

crested bow seas (i.e. with x =135°).

The hull was divided into 50 sections for both the structural and the hydrodynamic calculations. Two

dimensional hydrodynamic properties were determined using multiparameter conformal transformations [12]

together with the strip theory of Salvesen, Tuck and

Faltinsen [91. The effects of bilge keels and of the rudder were ignored.

In calculations for the dry hull the modal damping

factors were taken as

in the absence of any reliable data on the subject.

Calculations were made in which warping stiffness

Figure 1. (a) Wave spectrum and (b) encounter spectrum used in computations. An ITTC representation is used with h113 5 m,

U 13.38 rn/s andx 135°.
was excluded
8NCÛUNTEA
2.00
.80
.60
1.40
.20
1.00
L _{.80}
.60
3
.40
e
.20
0. 00
0.

(so that C1 =0) and included (C1 0).

SPECTRU1 U _{3.38 -I.} _{r} 35.0 6.3.

I,.. 5.00

00 i',øO ".50 too 2'SO '.00 .50 4'.00

### P3-0.010,

P7 =0.024,### P4-0.012,

8= 0.030, p5 =0.015 P9 =0.037### P6-0.019

a _.25L N-6 C S-' E z 3 X ,. -ad/s

denote C00 . C=O resp.

b x=.25L ,N6

a. rod/s

denote CO C=O Rqdbodyresp.

= .25 L N-6

u. rod/s

denot e CeO C=O Rigid body resp.

Figure 2. Shear force, transverse bending moment and twisting momsnt response am-plitude operators for x = 1,4 in a ship travel-ling at 1335 rn/s in regular sinusoidal bow

Figure 3. Shear force, transverse bending moment and twisting moment response am-plitude operators for x = 1/2 in a ship travel-ling at 13.38 rn/s in regular sinusoidal bow waves, i.e. x = 135°. a xa.50L .N6 b 400 300 -E 200-C 800 700 600 E 500 E y. 3: 400

### -

300 3 x=.50 ,N6 200 A I00 o,. rad/s denote C0 . C=0 . Rigidbodyresp. a = .50 N6 a a A a A Do o os a o a. radisdenote C,0 . CaO Rigid body resp.

o. radis

denote C,0 . Cr0 resp.

Table i

The natural frequencies of the dry hull and resonance frequencies of the wet hull with and without allowance for warping stiffness

modal dry hull

index

r

### C10

C1=0### wej'

¡IXFigure 4. Transverse bending moment res-ponse amplitude operators for x = 1/4 and

x= 1/2 evaluated making allowance for all

modes up to and including that of order

N= 9.

Table. I illustrates the influence of warping on the dry

hull natural frequencies and wet hull resonance

fre-quencies. The highest modal index taken in the

cal-culations of wave induced antisymmetric _{response}
was generally N = _{6, though results are included for}

whichN 9.

A comparison was made between predictions made

on the basis of rigid and flexible body theories. This relates to lateral bending moment and twisting

mo-ment at the three positions along the hull. The rigid body results were obtained using program UCLARM

[1], which is a variant of program SCORES [131.

4.2. Response amplitude operators

Figures 2 and 3 illustrate the three response

am-plitude operators at x = 1/4 and 1/2 respectively. The
frequency range _{o.e covers the first three resonances}

for C1

### *

O and five resonances for C1 = 0. In thesecalculations the highest modal index used was N=6.

The differences between the predictions for C1 = O and

C1

### *

_{O are clearly visible especially in the regions of}

resonance.

a

### wead/S)

1/AThe influence of the dry hull mode shapes is shown by the different magnifications at the resonance at the two positions along the hull. For example, the curve for C1 * O and x =1/2 shows a marked resonance

of the shear force at 2.05 rad/s although no such

resonance is apparent at x= ¡/4. This is due to the

re-lative magnitudes of V3 (1/2) and V3 (1/4); whereas

V3 (l/2)

### *

0, the vaine of V3 (1/4) is small_{[31.}

Comparison of the results for lateral bending mo-ment based on the assumption of rigidity show limited

agreement with those of the dynamical theory outside the regions of wet hull resonance. The discrepancies arise because the former, more rudimentary analysis is unable to account for dynamic effects. A large

dis-crepancy occurs around w = 1.0 rad/s and this may be

reduced for the C1

### *

O case by increasing the modalsummation to N = _{9 as shown in Figure 4. Although}

the dry hull natural frequencies and wet hull resonance frequencies of these three additidnal modes differ greatly from 1.0 rad/s, the modes concerned

con-tribute significantly to the RAO. In fact it has been shown previously [7] that the componentiTí7(x)p7(t) makes the largest contribution of the additional terms

= .25 . N9 o

### C.

_{:=c}2 0 0 0.17 0.10 0.17 0.10 3 . 2.23 1.93 2.05 4.82

_{1 .94}4.47 4 3.89 2.07 3.42 9.40 1.94 4.47 5 7.61 3.14 6.74 21,67 2.88 7.46 6 10.85 3.90 9.24 31.30 3.44 9.47

wet hull resonance

b E E z 3 X 806 700 600 500 400 300 200 100 X = .501

### N9

. ,ad/sdenote CEO . 0=0 . Rigidbodyresp

and does so because mode r =7 has two nodes in both

bending and twist and falls in a sequence of twist

do-minated modes.

A comparison of the results for the response opera-tor of twisting moment with those obtained with the conventional rigid body theory reveals that there is reasonable agreement in the region of roll resonance (We = 0.17 rad/s) but that large discrepancies then begin to appear. This is especially true in the region of

= 1.0 rad/s, the dominant region of the wave

en-counter spectrum shown in Figure 1. An increase in

the summation index N proves to have little effect now, either for C1 =O or C1 * 0.

The existence of such a discrepancy may be attri-buted to the low natural frequencies of the hull (see Table 1) and the dominance of twisting in the modes [3,7]; this twisting produces a very substantial con-tribution to the twisting moment, mainly through the

component T3(x)p3. This is something which the rigid

body theory is unable to cope with and, since the

cor-Figure 4. continued.

responding natural frequencies are low, the encounter spectrum magnifies its influence very considerably. It is evident that conventional techniques are

danger-ously optimistic in such circumstances.

4.3. Statistical properties

The response spectra in Figures 5 and 6

corres-pond to the response amplitude operators shown in

Figures 2 and 3 respectively, for the sea spectra of Figure 1. The discrepancies discussed previously between predictions based on the flexible and the rigid

hull theories are both magnified and distorted by the shape of the wave encounter spectrum. Again the

dis-crepancy in the lateral bending moment response

spec-tra can be reduced to some extent by taking N=9 in

the modal summation, as shown in Figure 7.

Table 2 gives the root mean square values of the

response (i.e. .,/m0,), while Table 3 shows the average uperossing periods at the three positions along the hull.

Table 2

Rms values of responses at three positions along hull. (The rigid body results are independent of the modal index N)

antisymmetric theory response 1/4 1/2 31/4 N=6 N=9 N=6 N=9 N=6 N=9 Shear C1r0 2.03 3.04 1.82 1.97 2.26 2.97 force (MN) C1 0 2.88 2.88 2.62 2.65 3.06 3.04 Twisting C1 *0 62.27 63.42 101.61 103.11 39.10 32.44 moment (MNm) C1 =0 92.16 92.16 139.84 140.46 61.10 59.51 Rigid body 19.14 12.25 8.53 Bending

### C *0

94.22 114.28 167.63 254.42 88.00 123.46 moment C1 =0 93.47 103.65 235.15 239.52 93.79 95.57 (MNm) Rigid body 144.90 272.39 118.37a _{=.25L} .N6
I6
4
w
z
3
X
b
21000
¡8000-fi,
500&-E

### z

120009000 -3 60003000 -52000 48000 ¿.4QQQ 40000 ° 36000 - 32000 28000 24000 20000 16000 - 12000### 4

8000 4000 o e. rad/sdenote CaO . C=0 . Scores resp.

.25 N.6

a. radis

denote CaO . C,=0 . Scores resp.

C

### x=.25t N6

e. rad/s

denote CaO . C,O . Rigidbodyresp.

Figure 5. Shear force, transverse bending moment ai,d twisting moment spectra for

x 1/4 in a ship traveliing in a long crested

Figure 6. Shear force, transverse bending moment and twisting moment spectra for x = 1/2 in a ship travelling in a long crested

bow seaway of significant height h3 5 m.

a

### =.5Ol ,N6

Q) E### z

>:: 3 X (Q E### z

3 X 36000 30OO 28000 24000 20000 16000 -2000 = 8000 4000 -e. red/sdenote CAO . CaO resp.

g

w. red/s

denote CAO . 0=0 RigdbodyresP.

C

### A.50L

,N6 180000 loop00 140000 120000 100000 80000 60000 40000 20000 w. red/sdenote CaO . C.=O . Rigidbodyresp.

i (s 10- 8-A a A A A 6

### z

o oO 4 AO D o A A A A 3 D A b .50 L . N-6 X 2### i

b (J) E

### z

X 40000 20000 180000 160000-40000 120000- 10000080000 -- 60000-der.ot e C+0 . C0 Rigid body resp.

s= .501 N'-9 A A A A A w. rad/s

denote CaO . C,=0 . Rigidbodyresp.

Table 3

Average uperossing period for the responses (The rigid body resultsare

independent of the modal index N)

Figure 7. Transverse bending moment spectra for x ¡/4 and x = ¡/2, evaluated

takingN 9. average uperossing period (s) for: theory ¡/4 112 31/4

### N6

N-9 N=6 N=9 N-6 N-9 Shear force### C10

5.59 5.87 4.63 4.69 5.35 5.54 C1=0 4.56 4.55 4.26 4.23 4.02 4.06 Twisting### C10

5.01 4.64 4.18 4.26 5.04 4.39 moment### C0 4.26

4.22 4.18 4.12 4.02 4.09 Rigid body 5.94 5.67 5.52 Bending### C10

5.38 5.63 5.58 6.03 5.07 5.67 moment### C10

4.70 4.59 4.23 4.23 3.99 4.08 Rigid body 6.10 6.24 5.66 a (r. E### z

3 X 52000 48000 -44000 40000 36000 32000 28000 24000 20000 16000 - 120008000 4000 -00### ...25L ,N-9

+## _Jf

A. radis 2.5. Conclusions

Computed results are presented for a container ship

travelling in a seaway. The ship may be assumed to be

either flexible or rigid and the correspondíng resuJts are contrasted. Further comparisons are given illus-trating the marked influence of warping stiffness in

the analysis of the flexible ship.

Significant differences are found between results

obtained with the 'rigid' and 'flexible' theories. The

discrepancies lie in the region of the dominant wave

encounter frequency at about We = 1 .0 radIs. Whereas

the discrepancy could be reduced substantially for the lateral bending moment by allowing more contribu-tions in the modal summacontribu-tions, this leads to little im-provement of agreement where twisting moment is

concerned. Any increase in the structural damping

coefficients of the flexible ship theory or modification

of the roll damping in the rigid body analysis influen-ces only the magnitudes of the resonaninfluen-ces and the discrepancies in this frequency region remain.

The explanation of these discrepancies is to be

found in the nature of the vessel itself. This ship has low natural frequencies when dry and low resonance frequencies when wet and the lowest of these latter

coincide with the dominant wave encounter frequency.

The rigid body analysis cannot make allowance for this. This means that conventional rigid body theory fails for ships, like the one under discussion, with low

natural frequencies.

The 'rigid ship' theory produces low results. These can be matched for a flexible ship, but only if it

pos-sesses high natural frequencies when dry, and resonan-ce frequencies when wet - frequencies lying well

above the region of dominant waves in the encounter

spectrum. This was indicated by modifying the

proper-ties of the ship in question by making it more rigid,

i.e. by neglecting the actual flexibility of the ship.

The conclusion to which this investigation leads is

that conventional 'rigid ship' theory may fail

hopeless-ly to predict the responses of a perfecthopeless-ly normal hull in a seaway. While, to be sure, the 'flexible ship'

ap-proach certainly needs improvement in points of

### detail - and again attention

is drawn specially towarping stiffness - this is a problem that confronts naval architects. No amount of 'improvement' is likely to render the rigid ship approach trustworthy.

References

Bishop, R.E.D., Price, W.G. and Tam, P.K.Y., 'A unified dynamic analysis of ship response to waves', Trans. RINA, 119, 1977, 363-390.

Bishop, R.E.D. and Price, W.G., 'Hydroelasticity of ships', Cambridge University Press, 1979.

Bishop, R.E.D., Price, W.G. and Temarel, P., 'A unified dynamic analysis of antisymmetric ship response to waves', Trans. RINA, 122, 1980, 349-365.

Bishop, R.ED., Price, W.G. and Tam, P.K.Y., 'On the dynamics of slamming', Trans. RINA, 120, 1978, 259-280. Belik, O., Bishop, R.E.D. and Price, W.G., 'On the slam-ming response of ships to regular head waves', Trans. RINA, 122, 1980, 325-337.

Bishop, R.E.D., Price, W.G. and Temarel, P., 'Antisym-metric vibration of ship hulls', Trans. RINA, 122, 1980,

197-208.

Bishop, R.E.D., Price, W.G. and Temarel, P. 'Wave-induced antisymmetric response of a flexible ship', Num Anal, of the Dynamics of Ship Structures, 1979, 129-146. (Euro-mech 122, AIMA, Paris).

Bishop, R.E.D, Price, W.G. and Temarel, P., 'Antisym-metric response of a flexible ship in a seaway', lnt.

Con-ference on Recent Advances in Structural Dynamics,

Southampton University (ISVR), 1980,417-426.

Salvesen, N., Tuck, E.O. and Faltinsen, O., 'Ship motions nd sea loads', Trans. SNAME, 78, 1970, 250-287.

Vugts, JI!., 'The hydrodynamic forces and ship motions in oblique waves', Netherlands Research Centre TNO,

report lSOS, 1971.

Price, W.G. and Bishop, R.E.D. 'Probabilistic theory of ship dynamics', Chapman and Hall, London, 1974.

Bishop, R.E.D., Price, W.G. and Temarel, P.. Hydrodyna-mic coefficients of some swaying and rolling cylinders of arbitrary shape', International Shipbuilding Progress, 27,

1980, 54-65.

Raff, A.I., 'Progam SCORES - ship structural response in waves', Ship Structures Committee Report No. 230, 1972.