Long term predictions of vertical and horizontal accelerations for a ship in ocean waves

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Long-Term Predictions of Vertical and Horizontal Accelerations for a Ship in Ocean Waves*

by

A. Shinkai** and J. Fukuda**

Introduction

Many computer programs are now available for the purpose of analysing the hull structure strength including the local, transverse and longitudinal strength. In view of the general importance of local and transverse strength analysis for large oil tankers, ore carriers and bulk carriers, accurate informa-tions are required about the local and transverse internal loads on the hull structure due to bulk cargo such as oil, ore and grain as well as the external hydrodynamic loads on the hull surface, which are both induced by the ship motions in waves. As the first stage of evaluating the internal dynamic loads due to oil, ore and grain, the magnitude of vertical and horizontal accelerations induced by the ship motions is a

matter which requires very close analysis.

Recently, in Japan, the systematic research works on the local and transverse loads including external and internal

loads, which are induced on transverse ship sections, have been continued by the Research Committee SR-131 under the sponsor-ship of "The Shipbuilding Research Association of Japan". As a part of those research works, the long-term predictions of

ver-tical and horizontal accelerations have been made for an ore carrier of 247 meters length and 117,570 tons deadweight.

Im-portant results of the predictions are presented here.

Deift

* _{The full paper (in Japanese) will be published in}

Trarisac-tions of the West-Japan Society of Naval Architects, No. 50, Aug. 1975.

** Department of Naval Architecture, Faculty of Engineering, Kyushu University.

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2

Predicting Methods

Response Functions of Vertical and Horizontal

Accelera-tions

By the aid of the rnddified strip theory, the heave, pitch,

sway, yaw and roll motions of a ship in regular oblique waves

can be obtained as follows [1] ,

=

ÇoCOS(Wet_)

=

cCO5Wet ± s5et

=

cos(wt-c)

ccet ±

_sS1et

n =

ncos(uit-c)

= flcC0et f

= 1pocos(wetc)

cc0et * 1SiflWt

O = O cos(w t-c ) = O cosw t + O sinw t

o e O c e s e

By using the solutions of ship motions in the form of (1),

the vertical and horizontal displacements of _{a point p(x,y,z)}

cari be obtained as follows,

Z =

-

_{(x_xG)}

= Z cos(w t- ) = Z cosw t + Z SiflW t

o e Z c e s e

Y

= n

+ (xxG)tP -zO

= Y000S(WetEy) = Y COStA)

t

+ YSiflwt

Accordingly, the vertical and horizontal velocities and

also the vertical and horizontal accelerations of _{a point (x,y,}

z) can be obtained as follows, by ignoring the _{component of the}

gravity acceleration,

=wZcos(wt-c +ir/2)

eo

_e _Z (4)

=wYcos(ojt-c +ir/2)

eo

_e

_Y

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=

_eo

2 cos(u.

t - c

+ ir) ₍₆₎ e Z = 2

eo

cos(w

t - c

+ ir) ₍₇₎ e Y

where the coordinate system as defined in _(lJ is adapted.

Predictions of Vertical and Horizontal _{Accelerations in}

Short-Term Seaways

By using the calculated response amplitudes of vertical

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where

[A(w,6+ï)] : response amplitude of acceleration

w : circular frequency of a component wave

6 : average heading angle against the average wavé

direction

angle between the average wave direction and a

component wave direction

H average wave height estimated visually

WT = 211/T, T : average wave period estimated visually

c) Long-Term Predictions of Vertical and Horizontal Accelera-tions

The expected long-term probability that the peak value of vertical or horizontal acceleration exceeds a certain level can be evaluated by assuming that the ship nwigates always with a constant heading angle 6 in a sea route and by

consider-ing all headconsider-ing angles with equal probability in that _{sea route.}

It follows that, respectively,

where

R2 = (2/11)

[11/2 /)

111/2 JO

or horizontal acceleration together with the I.S.S.C. wave

spectrum adapted to the short-terni sea condition, the variance

of vertical or horizontal acceleration can be evaluated based upon the linear superposition theory. It follows that

[f()]2[A(w,6+y)]2cos2ydwdy (8) [f(w)]2 =

O.11H2w(w/wT)5exp[_O.44(w/wT)]

Q(6) =

¡f

exp [-a/2R2]p(H,T)dHdT 1211 Q = (1/211) / Q(6)d6 Jo (10)

p(H,T) : long-tel-m probability of occurence for the sea

condition of the average wave height H and the average wave period T

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4

Predicted Results

Short- and long-term predictions of vertical and horizon-tal accelerations have been carried out for the ore carrier

"KASAGISAN-MARU" (247 meters length, 117,570 tons deadweight). Important results of the predictions are illustrated in Figs.

1 _{4. Those illustrations show the predicted results of}

ver-tical and horizontal accelerations induced on the longitudinal axis through the ship's centre of gravity, which are produced by the heave, pitch, sway and yaw motions but not by the roll motion. The wave statistics of the North Atlantic Ocean sum-marized by Walden [2] are utilized for the long-term predict-tions.

According to the predicted results, there were found the

following trends of vertical and horizontal accelerations.

].) Generally the vertical accelerations _{are larger than the}

horizontal accelerations.

The vertical accelerations induced on the ore carrier

"KASAGISAN-MARU" are found to be very large in _{seaways of the}

average wave period of 10 _{12 seconds, and the horizontal}

ac-celerations are found to be considerabily large _{in seaways of}

the average wave period of 6 8 seconds.

The vertical and horizontal accelerations _{are found to be}

largest at the fore and after ends and _{to be smallest at}

mid-ship. Magnitude of the vertical _{or horizontal acceleration is}

almost equivalent at the fore and after ends.

The vertical accelerations are found _{to be largest in head}

sea conditions and to be smallest in following _{sea conditions.}

They decrease with decreasing of ship _{speed in head and bcw sea}

conditions but increase with decreasing of ship _{speed in}

follow-ing and quarterfollow-ing sea conditions.

The horizontal accelerations are found to be largest in

beam sea conditions and to be smaalest _{in head and following}

sea conditions. They decrease a little with decreasing _{of ship}

speed.

Further investigations on the vertical and _horizontal

ac-eìerations will be continued in order to make _{clear the}

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References

[1] 3. Fukuda, R. Nagamoto, M. Konuma and M. Takahashi :

"Theo-retical Calculations on the Motions, I-lull Surface Pressures

and Transverse Strength of a Ship in Waves" The Memoirs of

the Faculty of Engineering, Kyushu University, Vol. 32, No.

3, 1973.

{2] H. Walden : "Die Eigenschaften der Meereswellen im

Nord-atlantischen Ozean" Einzerveröffentlichungen Nr. 41, Hamburg, 1964.

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-J N

3

2 o 105L (FP), Fr -015 -I80° 35 90 45 -0 3 -J N 12 I0 (MWSHIP) Fr -0.15 SI80° 135 90 45 o Li0 -J N 2 X='-0.5L (AP), Fr -0.15 Fig. i

Standard Deviations of Vertical

Acceleration Induced on the Ship

in

Short Crested Irregular Seas

8=1800 135 90 45 : O o V L/Ae 3 6 12 8 6 4 1814 IO T (SEC) 16 2 -g-- VL/A0 3 8 6 1814 IO T (SEC) 4 T (SEC) 6 2 1814 IO

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o o ---S -/_ ° -- --_--- -- -S-S-. 3 16 12 8 6 4

-3

O

/

---N

/---Fig. 2 Standard Deviations of Horizontal Acceleration

Induced on the Ship in

Short Crested Irregular

Seas X-0 (MIDSI-IIP) 8-180° -J --S X--O.SL (AP) 8l800 Fr -OIS 35

I

Fr-O.I5 :

---:

90 : 45 1814 IO T (SE C) ¡814 IO

-2 16 ¡2 8 6

-3

-J

I

12 O C-0.5L (FP) 8I8O° Fr - 0.15 35 90 45 o

S,

--'

//'

St

-

.--/

-S-I 3 16 ¡2 8 6 ¡8 14 lO T(SEC)

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-AP

0.5

WHOLE YEAR, Q-IO8, &=180° (HEAD SEAS)

AP 2 2-7. o: Fr-O _I FP FP o: Fr = O x: 0.10 .0 o: 0.15 0.5

WHOLE YEAR, QIO, ALL HEADINGS

Fig. 4

Horizontal Accelerations Predicted with the Exceeding Probability of i08 in the North Atlantic Ocean for Beam Sea Condition and for All Headings

+: 0.05 x: 0.10 o: 0.15 AP 2-FP Fig. 3

Vertical Accelerations Predicted with the Exceeding Probability

of

10_8

in the North Atlantic Ocean for Head Sea Condition and for