On the experimental results of ship motion, longitudinal bending moment and slamming pressure

On the Experimental Results of

Ship Motion, Longitudinal Bending Moment and Slamming Pressure

by Tamotsu Nagai ' Michio Kadota Takaichi Fukuda * and Kenji Dokai ''

our. Soc. of Naval Architects of Japan No. iZO, 1966

IS SC

1967

National Defense Laboratory * _{Maizuru H eavy industries, Ltd.}

Introduction

Studies on the ship behaviour among waves have recently improved so much by the statistical approach, so that we might foresee some quantitative results about the ship behaviour even at the stage of initial design. Sofaras our knowledge is concerned, the data about the destroyer which are available at the initial design stage, are considered very few.

In order to examine the motion of destroyers in rough seas and guess previously scme motion of them, the sea tests which covering different sea states were carried out using two sister ships.

The items measured are pitching and rolling angles, acceleration, heaving motion, hull girder stress, local stress on the bottom panel

and slamming pressure on the bottom of for ebody; and all of these recorded data are used for statistical as well as spectral analyses by applying time series procedureS

Theoretical calculations, on the other hand, under the cross flow hyposesis are pursued in order to obtain the response amplitude

operator s ot pitching angle, heaving motion, bending moment and

relative motion, etc. Then the power spectra of them are also calculated by estimating wave spectra of the experimental water areas.

Comparing with both calculated and analysed results, we obtained the facts which distributions of variation in pitching angle,heaving motion and hull girder stress etc. follow to Rayleigh's law, allowing from

67% to 90% confidence limit and slamming pressures to truncated exponential distribution; and also from the power spectra the pitching angle, heaving motion and acceleration show good coincidences from the practical view-point, but on the contrary bending moments some discrepancies especially at the point o1 maximum magnitudes.

Number of occurrences of slamming impact and the distribution of slamming pressure are theoretically calculated under several

assumptions about the threshold velocity and the bottom shape, and

their results are compared with recorded data showing some discrepancies for number of occurrences, but for the distribution of slamming pressure good coincidences.

1. Destroyer Used for Tests,Test Instrumentation and

Environmental Condition

The destroyers used for tests are two sister ships A and B. of which principal dimensions are shown in Tabte 1 and snip conditions at the time of each sea tests on the Japan Sea in Table Z, Sea tests of ship A were held on January, 1964, and those of ship B on Dec. 1965.

Test instrumentation of ship A is mainly arranged for the observation of motion and strength, but that of ship B for the observation of slamming phenomena, strength and motion of ship as shown in Figs. i and 2.

Under the constant speed, the sea tests of ship A were carried out during about 10 minutes by keeping each angle of heading of oncoming waves relative to ship from zero through 360 with 4Y interval.

These tests were repeated by changing ship speed. In case of ship B tests of both heading and following of on.oming waves relative to ship was carried out during about half an hour by changing ship speed from

12 knots through 24 knots.

As tests of ship A were lasting over 12 hours, therefore the effect of variation of environmental condition to the recorded data was considered inevitable. Environmental condition of ship B was less severe than in

case of ship A, The recorded data of both ships only in case of

heading of oncoming waves were used for analyses under two different environmental conditions.

2. Theoretical Background

I)2)3)

Applying the strip method developed by Fukuda by dividing the ship body into 20 sections, the response amplitude operators in case of heading of oncoming waves relative to ship were calculated using the nondimensional weight distribution curves as shown in Fig. 3, for

regular waves of

-i

0.75, 1.00, 1.50,2.00 and ship speeds of

=0-'0.6. For calculation of virtual mass and damping coefficient tue method developed by Tazai 4) was used.

Following the expression given in reference (2), the equations of pitching and heaving motions are shown below.

F =

CoS&Je

t -

J

t

-

E

cos (c0t * o(s)

N1=TC0SCJet

5Sfl()el

-10cos(t +ß)

_/

* Numbers in bracket refer to the References at the end of this paper. In Eq ( 1) a, A, d and D are coefficients being correlated to the mass and

moment, respectively;b, B, e and E to damping;c, C, g and G to the force of stability;F and M are the exciting forces. All of these coefficients explained above are determined by using the weight distribution curve, Bonjean-curve, lines, virtual mass and damping coefficient

diagrams.

The solutions of (1) arc given by

ç =

ç

sn

(2)

co5jt

S=COSet+ß),

C3)

and the bending moment amidships is

qì=mcozOJet nz

n

(4)

In Eqs (2), (3) and (4) Ç0, and 772.Ø are the response amplitude

operators of heaving and pitching motions and bending moment amidships, respectively.

Several examples of the calculated , and 77Z _{are shown in} Figs.4 through 8 in the non-dimensional _{form together with the test} model already published and the cas.e of reference (5) within the region of livear relation. _{Judging from these figures it seems to us there are} no differences in case of heaving motion among them except that given by Kroukovsky. _{In case of pitching motion, the results obtained from} destroyers seem to be less than those _{by other ships, but the magnitudes} of differences are so small that we may neglect them. In case of bending rnometit shown in Fig. 8 the experimental _{results given in reference (3)} were inserted, by which we can judge there are comparatively good

coincidences between them if we consider the facts which they are the mean values of obtained magnitudes at the fore-and abtbody.

For ships paving principal uimensions shown in i. aule ¿, by using curves such as obtained in Figs 4 through S we can generally calculate the response of ship even at the stage of initial design.

In Figs 9 through 11 the cross curves of Figs 4 through 8 are given. In Fig. 12 the responseamplitude operator of relative displacement is given by using Eq. (13) which will be given later.

The number of slams and the _{probability dencity function are already} given by Tick _Ochi '7) 8)

Following the references (7) and (8) the number of slams per unit time,

_/\f

_{is given by}

r

Zcr

E-r

A/ /

/E±r

'''SLT2Trj

E

where and _{are the cumulative energy densities of relative} displacement and relative velocity _{between wave and ship bottom,}

respectively; d is the draft at a specific location;Zr( er) the threshold velocity.

And the probability density _{function of slamming}

_{pressure, f(p)}

isgivenby

/

/

2CE.ErJ

Zr

In Eq( 6) C is the coefficient determined by the form of bottom hcll at the instant of impact, p the slamming pressure at the bottom _{which is}

given by

-p =2CZ,

-a r(cr) is the threshold _{pressure given by}

=2C

EzT and _{in Eq( 6) may be calculated by using the results}

obtained from the strip method as shown later.

3. Results Obtained from Tests and Their

Comparison with Those of Theoretical Calculation

3. I _{Statistical Analyses of Ship Motion and Banding Moment}

Using recorded data the distribution factors of number of occurrences and of cumulative occurrences were obtained for ship motion and bending

moment. Several representative examples are shown in Fig. 13 through 1 7. From the histograms of ship motion for short term recording it is generally known that their distributions are mostly coincident to the Rayleigh's distribution, therefore we tried also to plot the Rayleigh's distribution, p(x) on each figures in Fig.13 through 17 using the

2 cumulative energy density E which is the mean of (double amplitude)

obtained from each oscillograph such as

-E=

(q)

and

Comparison between Eq( 10) and most of histograms _{shows almost} coincidences. On the other hand, as the precise magnitude of E will be obtained from the area of spectrum such as

E=8(X:

f)

84

z

(io)

(17) therefore Eq( 11) is also inserted into Figs _{1 3 through 1 7, which show} reasonable coincidences within the practical use.

Jasper 9) tried to check quantitatively the _{compatibility between} both histogram and Rayleigh's distributions. _{Following his method of} check we tried also to check their compatibility, resulting to allow from 67% to 90% confidence limit _{as shown in Figs 18 and 19.}

On the other hand, as Longuet-Higgins shows in his paper io) t:hat if the Rayleigh's distributions _{are satisfied there exists the fixed} correlation among the arithmetic _{mean, one-third mean, one-tenth} mean of many wave heights and the cumulative energy density E; and therefore after calculating these _{means from histograms and by plotting} them versus _{then we have Figs 20 and 21 showing the good}

coincidences between the line of mean of curves and the coefficient by -Longuet-Higgins. Judging from these phenomena _{explained above almost} all of distribution data obtained from tests show to follow Rayliegh's distribution, but on the contrary the relation between _{the maximum} magnitude and _{doesn't show good results as shown in Fig. 22 in} which it doesn't always approach to the theoretical result _{even if the} number of variation, N increases, by which prediction _{of the maximum} magnitude in case of short term distribution seems to us difficult.

We must now note that the _{case of local phenomenon such as the} stress measured on the bottom pend of the forebody _{doesn't follow} Rayleights distribution at all.

3. Z External Force

In order to get the spectrum of ship motion theoretically from the response amplitude operator we have to prepare the spectum of wave. For this purpose, the perfect observation of sea conditions was not only carried out with the preparation of as many weather charts as possible during quite long days before tests, but also the observation of sea area, where sea tests will be held, by using the observation ship was completely done from one day before the day of tests. In our case the direct

measurement of wave was not pursued.

No problems are there about the distance of wind blow, and hence judging from both the space of the Japan Sea and the observed wind

velocity, when we can predict the wind velocity and the duration of

blow from the weather charts and by comparing them with the observation data obtained during tests, we can determine approximately the external forces. Fortunately, in case of ship A wind of ¿6 knots and in case of

ship B wind of 20 knots were blowing forcoTnparatively long period, and hence we can guess these cases were on perfectlydeveloped waves. Under this assumption the wave was calculated, showing scarcely no predominant differences between them and observed data. From the above phenomena, as for external forces the spectrum given by Neuman«'jf the perfectely developed wave was used for analyses, although the effect of small discrepancy of predicted wind velocity to the ship motion can not be disregarded.

3. 3 Slamming Pressure

When the ship goes on in rough seas we usually hear so often such big sound as induced by hammer on the bottom panel at the forebody and feel as if the ship is completely in slamming.

Judging from the data of water pressure recorded on bottom panels, however, the number of occurrences of slams is much less than the number predicted from our feeling. For example, during half an hour

recording in ship B we have the number of slams 21 from data recorded by wave height measuring apparatus compared with the number of encounter with waves about 370, corresponding to the number of slams 120 by our

feeling ori board.

Although this number of occurrences of slams so much varies

depending upon the scale of waves, ships speed and the angle of oncoming waves to ship, the number of slams seems to us less than that guessed by our feeling.

The distribution of impact load was of the triangular _{or sine form} having peak from 1 kg/cm to 5.8kg/cm, and its duration were

considered to be approximately 10 milli-secondsto 60 milli-seconds. It is known ') 8) that histogram of the number of slams _versus pressure peak follows the truncated exponential distribution, and slamming occurs when the ship bottom appears above the _{sea and only} when the relative velocity between ship bottom and the surface of the sea becomes higher than the limit, say threshold velocity.

From recorded data histograms are obtained to follow the exponentia. distribution approach as shown in Figs 23 and 24, from which one -terth mean and one-third mean were calculated by using Eq( 12) given below:

=2 (() +

2.fOE),

(Z

33OE)

Saying tha't° _{ZD(c-, follows Froude's low, Ochi gives the several ship}

examples about -value, and on the other hand Fukuda gives

Zr(cr)O.12fLfor the destroyer

type. Assuming O.l2/7

he constant

throughout all sections of ship and using Eq( 12),C's values were calculated and plotted in Fig. 25, in which tan& is the slope angle of bottom hull at the location of water _{pressure gage.}

In Fig. 25 the magnitude obtained by model test of mariner _{type when} O is plotted, and results to approximately a line. In our calculation given later, the line obtained in Fig. 25 is used. As generally known, coefficient C determined such as above is not precise at all, but unfortunately no sufficient data to determine C _{are there in our hand,} so that we may need hereafter to check for rigorous determination of C,

In order to obtain the distribution of slamming _{pressure and} the number of slams, we need to _{prepare the distributions of relative}

displacement and relative velocity between the ship bottom and the wave surface at each location where the water _{pressure was recorded.}

The response amplitude operator of the relative displacement in case of heading is given by

= z/= z csúJt

si-t CJt =

oS (jt

z== Ç- (zZ) -

co

Z= Ç

- sn

and therefore by combining with the calculated _{results by the strip method} explained in Chapter 2, we can obtain the _{response amplitude operator.}

Tue cumulative energy density of relative displacement is given by

E =2

_z.7-

EZrJ 7(w) dc)

*22.

o

and that of relative velocity is given by

=

_{2ff : [Z:0}

_î

) d

(/5)

Therefore after Ez,E

_{were obtained by Eqs (14) and (15), the}

number of slams and probability density will be determined by Eqs (5) and (6) as shown in Figs 23,24 and 26, from which it is recongnized that probability distribution shows comparatively _{good coincidences.}

The number oi occurrences of slams is different _{depending upon} the position of water pressure _{gage. The theoretical result of number}

of slams per second shows the _{great increase in the forward direction,} say, changing from 0.2L to 0.IL s shown in Fig 26, when compared with the result obtained by inserting wind _{velocity 20 knots and}

(/2)

(14-)

(/3)

Froude' s number 0. 41 in one case of the test conditions, which shows the half or one-third magnitude compared with the theoretical results at 0. t 5L or 0. IL position. The cause of this great difference may be considered not only due to the errors exist among coefficient used in calculation, but also virtually increased threshold pressure induced by the steep angle of bottom shell surface at the position of pressure gage and the high position of location from the keel line.

The recorded impulsive prcssure at each pressure gage such as A, B, C, D, and E position arc shown in Fig. ¿7 when Froude' s number is 0. 41,

and neglecting small phase lags which exist among these pressure peaks at each position. Altogh it is generally known that the position of the maximum pressurer and their distribution are always changing from time to time depending upon the local ship response in waves and the slope angle of bottom shape when impact, in our tests the pressure at B position was almost higher than that at A position.

3. 4 Ship Motion and Bending Moment

In order to get the power spectrum from recorded data about pitching, heaving, acceleration and longitudinal stress on upper deck amidships, we applied the spectrum analysis by time series procedure

¡4)

to tI'e obtained oscillograms such as given below

-

-

ÌXX

2V)

=;

2T N N

it

527: ¿

x-x/

N

xt

x2

t =

By using Eq( 1 6 ) we made the cor relograms, calculating time

series x1

x2

, x- -

- -, x,,up to N450 and 60 at 0. 5 second

interval, and then inserting into the following equation

+

Tí

(i

6)

we obtained the spectrum. Multiplying it the dispersion such as giver below:

'V

,

(/8)

we have the power spectrum.

On the other hand, by using the response amplitude operator

obtained by the strip method and the wave spectrum we have theoretically the power spectrum of ship motion such _{as given below} e

fA(e)

_(/9)

Séreral representative examples _{obtained from both experimental} and theoretical treatments explained above are shown in Figs ¿8

through 31.

As shown in Figs 28 and 29 the power spectra of pitching angle of A ship in case of V//7.ì _{0.43 and B ship in case of}

_V//i

_=0.41

show comparatively good coincidences, _{but in case of}

_v/'j

_{0. 31 Fig 30} doesn't show good coincidences. The power spectrum of'bending

moment amidships shows big discrepancies of frequencies as shown in

Fig 31.

The reason of this is due to the errors induced by the effect of local phenomena already included in recorded data obtained by strain-gage method. In order to get the longitudinal _{strain precisely, we shall have} to use another type of displacement _{meter hereafter.}

In case of heaving motion and accelerations the power spectra are obtained, showing also such comparatively good coincidences as already shown in Figs 28, 29 and 30 in _{case of pitching motion.}

4. _Conclusion

In this paper the comparison between recorded data and calculated results by the strip method is carried _{out about the ship motion,}

longitudinal bending moment and slamming _{pressure obtained in rough} seas under the different environmental conditions, _{by using the two} sister destroyer-type ships.

The following results _{were obtained:}

(1) The comparatively good coincidences were shown in the distribution of spectra about pitching angle, heaving motion and accelerations.

(Z) In case of bending moment, however, we couldn' t get good

coincidence, and hence need further _{investigation about this phenomenon.} 3) Although several unknowns _{are still there, the spectum of impulsive} pressure induced by slamming follows trunca.ted _exponential

distribution and shows comparatively _{good coincidences between the} calculated distribution. _{The number of occurrence s of slams is far less}

in the recorded data than in the theoretically predicted results, say, the latter has two or three times larger magnitudes than the former at

O. 15L through O. iL.

(4) Almost all of the recorded data about pitching angle, rolling angle and bending moment show the Rayleigh distribution allowing from 67% to 90% limit, but the maximum magnitudes of them have quite large dis crepancies.

5) Judging from the facts that several results obtained in the response operators show reasonable good tendencies, the curves shown in this paper might be considered available to the initial design.

r,

References

(1> Kroukdvsky, K. : Pitching and heaving motions of a ship in

irregular wave, T.S.N.A.M.E.,1959

Fukuda, Jun-ichi rOn the midship bending moments of a ship in regular waves, Jour. Socof Naval Architects of Japan. Nos. i lo, 111, 1961 and '62

Fukuda, Jun-ichi, J. Shibata,H. Toyota and A. Hoshikuma Theoretical evaluation of bending moments of a destroyer in

regular waves, Jour. Soc.of Navel Architects of Japan, No. i 12, 1962 Tazai, Fukuzo :Virtual mass and damping force about ship pitching and heaving motions, Jour. Sócof Naval Architects of West Japan, No.21,1961

Takezawa, Seiji :A study on the large bulbous bow of a high speed displacement ships, Jour. Soc. of Naval Architects of Japan, No.

110, 1961

Tick, L.J : Certain probabilities associated with bow submergence and ship slamming in irregular seas, Jour, of ShipResearch, Vol.

2,No. 1, 1958

Ochi, M. K. :Prediction of occurrence and severity of ship slamming at sea, Fifth Symposium on Naval Hydrodynamics, O.N.R. U.S.A and Skip Modeltanken, Norway, Bergen, Norway, 1964

Ochi, M.K. :Random inpact loads due to ship slamming in rough seas, First Conf. on Ship Vibration, Acoustic and Vibration,

D.T.M.B., 1965

Jasper, N.H. :Statistical distribution patterns of ocean waves and of wave induced ship stress and motions with engineering

application, T.S.N.A,M.E.,1956

Longuest-Higgins :On the statistical distribution of the heights of sea waves, Jour, of Marine Research, 1952

Pierson, W.J., G.Neumann andR.W. James :Practical methodfor

observing and forcasting ocean waves by means of wave

spectra and statistics, U.S. Navy Hydrodynamic Office, 1955 St. Denis, M., W.J. Pierson On the motions of ships in confused

seas, T.S.N.A.M.E. ,

Vol. 61, 1953

(1 3) Fukuda, Jun-ichi, J. Sliibata :Some problems associated with the ship motions in rough reas and the effects of ship length on them,

No. 30, Jour. Soc. of Naval Architects of West Japan, 1965

14) Yarnanouchi, Yasufumi :On the analysis of ships' oscillations as a time series, No, 99, Jour. Soc. of Naval Architects of Japan, 1956

'z-TABLE i PARTICULARS OF COMPARED SHIPS

TABLE 2 CONDITIONS AT SEA TESTS

Ref No. ( 1 _{) (} 5 ) ( 4 ) Test Ship

Lpp (rn) 1.740 106 115 91 B (m) 0.1853 10.7 12 10.4 d (m) 0.0634 3.62 4 3.59 J

(t )

0.0111 2,139 2,890 1,785 Cb 0.55 0.507 0.511 0.524 K 0. 241 0. 32 * 0. 242 0, 242 LIB 9. 40 9. 90 9. 58 8. 74 Note * in water SHIP A SHIP B FORE (m) 3.360 3,150 DRAFT MIDSHIP (rn) 3. 600 3, 345 AFT (m) 3.820 3.540 TRIM (rn) 0.460 0.390 (0. 506%L) (0. 428%L) DISPLACEMENT (t) 1,779 1,654

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