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A Study of Wave-Induced Vibrations (ist report)
Koji Kagawa,* Member
Katsuo Ohtaka,* Member
Mitsuyoshi Onoue*
Summary
Wave-induced vibrations (springing) have been studied both theoretically and experimentally. The model experiments have been carried out using a tanker model of 7 meter length. Higher harmonic excitation in regular waves (which means the exciting force with the frequency multiple of the fre-quency of encounter) and the response to it have been confirmed both in theory and experirnenL Also it has been shown that the natural frequency of the 2-noded hull vibration and the ship speed play an important role in the response of springing. The necessity of the future work is emphasized especially in the field of estimation of hydrodynamic exciting force in the region of short wave length. in order to refine the numerical accuracy.
1. Introduction
Wave-induced vibration has become one of the important problems of recent large ships. Goodman1>
treated the subject by means of the strip method and the linear random vibration theory. He con-sidered that the phenomenon is the resonance of the 2-noded hull vibration with the waves whose frequency of encounter is equal to the natural frequency (called as "ist resonant encounter" here-after). Gunsteren studied the problem nearly in the same way as Goodman. Whereas Kumai3' developed the theory proposed by Watanabe5 in which the wave-induced vibration is treated as the resonance, of the 2-noded vibration with the waves whose frequency of encounter is 1/n of the natural frequency where n is integer (called as "n-th resonant encounter" hereafter). Recently Tasai dis-cussed the phenomenon from the different points of view71. Also Matsurnoto et al.81 studied the wave-exciting forces.
In the present paper the theory takes into account of n-th resonant encounter as well as the ist resonant encounter.
Model experiments to investigate the exciting force in regular waves and the response in regular and irregular waves have carried out by means of 7 meter tanker model,
. '
2. Theoretical sftdy
The wave exciting force is considered as follows.
4F=(my)+Ny+4Fa
4F=wave exciting force per unit length of ship m=added mass per unit length
ta
a\
y=vertical velocity of water=
- Vs) V
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* Nagasaki Technical Institute, Technical Headquarters, Mitsubishi Heavy Industries, Ltd.
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N=hydrodynamic damping due to wave making 4Fa=variation ôf buoyant force
'-r--r
rVg=shjp8peM,
t=tlme T
x=body coordinate along ship length
Since the first term of (1) is due io the momentum change of water, the exciting force is reduced to the following equation.
am.
6m(m)=my+-j-yVs--y
(2)
(3)
if the nonlinearity of hydrodynamic force is neglected, (3) coincides with the equation derived, by Goodman.
and 4Fn=2pgby, then
4F=mi?+(N_V4!)Y+2pgby
(4)
where p=density of water, b=half beam,
g=acceleration of gravity
In the present paper, however, the effect of nonlinearity of the last two terms of (3) are taken into account in the same manner as Kumai, i.e., the added mass and half beam vary according to the elevation of water (draft). So the higher harmonics of the frequency of encounter are taken into ac-count.
The effective exciting force is obtained by the following equation.
F=r'4FXdX
(5)
where X=normal coordinate (normal mode) L2= length of ship
For the qualitative discussion, the simplified calculation procedure hasbeen derived adding following assumptions.
1) Ship form is approximated by the ellipse for the cross section and cosine curve for water line.
The Ist and the 2nd harmonics of frequency of encounter are considered. Normal mode is assumed by the following equation.
1= 8.475(X/Lj)'+7.539(/L)2_.O.355
The effect of ship speed on the vertical velocity of water is neglected, namely
y=i
The results of the simplified calculation are as follow.
a) The wave exciting force in head sea is given by the followingformulae.
Fl=Hwwe2LsBlGzlExp{_Ç(.!.k)1}
'i+CoJ
(6)
r
i
t
A Study of .Wave-Iiidùced Vibrations (ist report) 361
which is the same as that of encounter.
Fasingle amplitude of the 2nd harmonics of wave exciting force, a frequency of which ¡s two times of that of encounter.
.M=circular frequency of encounter of ship and wave
H=wave height L,=length of wave encountered B=breadth of ship d=draft of ship
G3, G3=functions of L/L,
C0=correction factor by buoyant force and hydrodynamic damping force
It is to be noted that, whereas the ist harmonics of exciting force, FL, are proportional to the wave height, the 2nd harmonics are proportional to the wave height squared. In the same manner it can
be shown that the n-th harmonics of exciting force is pro. i
portional to the wave height to the power n.
.i'
The behaviour of the function G0 is shown in Fig. 1.
/
G2 increases with L/L, in an oscillatory manner, and the'
'/
"/
enveloping line is proportional to (L/L,)3. G3 is alike.
//
/
Since L of the resonant encounter increases with increasing
->
I'
ship speed and/or decreasing natural frequency, G2 and G3,and consequently excitation, are on the increase with higher
/
ship speed and/or lower natural frequency.In addition to these functions, the terms including LO
exponential, which represent the effect of subsurface, has also the same tendency. This characteristics of springing concerning ship speed agrees with the results measured on
as
board by Bell and Taylor".
The response of springing to the sinusoidal exciting
g..1\
.1
,.
force can be derived by the modal analysis. Using the re lationship between the amplitude and the stress amidships
LI#Lá
of a large tanker, the following expression is obtained for Fig. i Relation between L/L, and G3 the ist resonant encounter in the ballast condition.
.=8.91
X i0'C5(B/d)'1(D/L,)(B/L3)(i
0.004728 N2V, )5 (7)where u=spnnging stress amidships Cb=blOck coefficient
D=depth of ship -' 4= displacement of ship
- N2=2-noded natural frequency of hull vibration
The effects of natural frequency and displacement on the stress amidships are shown in Fig. 2. lt
can be emphasized from this figure that the effect of natural frequency on the springing response is quite large.
In the detailed calculation, on the contrary to the simplified one, the lines of ship are precisely considered. The effect of ship motion is also taken into account approximately by replacing wave amplitude with the relative displacement of ship and wave. An example of the calculated stress for a 180,000 t.dw. tanker is shown in Fig. 3, where the nth resonant encounter means that the frequency of encounter is one n-th of natural frequency of hull vibration and the springing stress is caused by the n-th harmonics of wäve exciting force. The wave height has been assumed by combining Beau fort number with ISSC wave spectrum. The relation between significant wave height, meañ period .and Beaufort number are those by Roll shown in Table 1. From Fig. 3, it can be seen that the springing
362 i37
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N, ()
Fig. 2 Relation between natural frequency, displacement and springing stress
O.'
IenTh of wave encountered
Fig. 3 Detailed calculation of springing stress of a 180,000 t.dw. tanker
Table 1. Relation betceen significant wave height, mean period
and Beaufort number
-s
L
f I -6 7 8 9 10 11 2.9 3.75 4.9 6.2 7.4 8.4 7.25 7.8 8.3 9.0 9.5 10.0¡'1 stress caused by the higher hermonics of exciting force would not be so large as that caused by the
ist harmonics except the 5th resonant encounter in the sea state of Beaufort number 8, which sug. gesta that the springing becomes large in the rough sea.
3. Model experiments
The length of the model is 7 meter and the lines are nearly the same as those of a 180,000 t.dw. tanker. The model has the 4.4 meter long parallel part of aluminum plate and wooden both ends. Some of the experiments were carried out with the integral model. After that the model was cut so as to lower the natural frequency in order to give resonant condition with the wave encountered.
Results of experiments are summarized as follow.
a) Distribution of wave-induced pressure on the ship bottom
The model was put standing still and was subjected to the regular incident waves. The fluctuating pressure was measured along the length of the ship and one of the results are shown in Fig. 4 (ist harmonics) and Fig. 5 (2nd harmonics). From Fig. 4 it is seen that the measured pressure amplitude is very smaller than the calculated one and varies along the length, with its maximum at the bow
and minimum at the midship part. The athwartships distribution of the measured pressure shows its minimum at the center line. The measured pressure distribution along the length is different
B. No. 5
Hj1(m) D(sec)
2.15 6.5
ej
-Lve
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wave eeTh L35. wave Seht 4.72c. fm9u.ç,afwwaaI.rLViH -Ive.--a--II
uII,
IIIiii1I
AWILVAVAVAWAVAYAVAYA£WAWAWAWAW4
yyaYYffAV
tad .. ffs,wa. w.aead .H,. pa.t ad(a) distribution along the center line of bottom (a)
IalIad peeve dit fvtive aweedpewediftve pei*adean distribution along the. center line of bottom.
u Fig. 6 Time history of resonant encounter of flexible model in regular wave
Fig. 4 Distribution of wave induced pressure of Fig. 5 Distribution of wave induced pressure of model (ist harmonics). model (2nd harmonics).
from that assumed in the strip method and so it is necessary to modify the distribution of wave ex-citing force in the theory so as to comprise the experimental results.
In this experimentthe 2nd harmonics could be measured, the results of which are shown in Fig.
5. This has a flatter longitudinal distribution than the ist harmonics and the athwartships distribution
is nearly similar to the pressure due to added virtual mass in the hull vibration.
b) Response in regular waves .
The flexible model was towed in regular waves and the springing response was measured. The examples of the time history are shown in Fig. 6 (a)(c). Fig. 6 (b) and (c) show the higher harmonic resonance exists as expected by the theory. Fig. 7 shows the stress response to the ist harmonics of exciting force (ist resonant encounter), which exhibits humps and hollows similar to the function
A Study of Wave-Induced Vibrations (ist report) 363
(b) distribution across the bottom (b) distribution âcross the bottom
a.
364
G3 in Fig. i; At the present, however, the precise location of humps is difficult to assess since there is some inevitable inaccuracy. Therefore only the envelope of these humps are discussed hereafter. Fig. 8 shows the comparison of the envelopes of the response in the ist resonant encounter. The calculation has been carried out for two cases. One is under the condition that the wave exciting force is distributed in the same manner as predicted by the strip method. The another is under the assumption that the lengthwise distribution of the exciting force is similar to that in the model ex-periment as shown in Fig. 4. Though the inclination of the envelope of response is steeper in the experiment, the latter calculation gives more agreeable result with the experiment.__
Regarding the higher harmonics of exciting force, it has been mentioned from theory that the n-th harmonics is pro-portional to the wave height to the power n. Fig. 9 (a)(e)
show the magnitude of response versus wave height. Though
there are some scattering due to the errors in measure-ments, the overall tendency of experiments confirms the
theoretical result. - -3? 10 -
j
t :s,,Je cpflf,,d, vik 3tT,33 w.yeb& 35 1.3 .3 V.("4$Fig. 7 Relation between wave length, ship speed and stress of flexible model
-0.5 LO Lw(m)
0.05 DI b.Ø Ienfh of Wave eunfered
Fig. 8 Comparison between mea-sured and calculated accelera-tion of flexible model
(c) Response in irregular waves
The above-mentioned model was towed in irregular waves of four different spectra and the stress and acceleration response was measured. Owing to the restriction of space, however, only the brief
summary of the results is given here, the details being left to the next report. r
The springing response in irregular waves seems to follow Rayleigh distribution since some stati-stical values, such as maximum, mean, 1/3 highest mean etc. of amplitude is proportional to tempo-rary- rms.
Generally speaking, the springing and wave bending coexist in irregular waves. The relation
g,'=G,1'+c, holds where èa, c and u3 are rms's of the combined, wave bending and springing
stresses respectively. .-
..
.In the wave spectrum corresponding to milder sea state, the stress response is dominated by springing. As the spectrum becomes rough sea, the stress due to wave bending increases and, in ad-dition to springing, there is a slight symptom of whipping, though the slamming or panting can not
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Fig. 9 Relation between wave height and n-th harmonic response of
flexible model -03 0330 03 S02 2 001 J 0050.5 00502 SCI Ql (43 4th p002024 (025th roso,02*020200(03 H..(cm) 2 2 lO 02 .10
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V.0 L. :4to ii.' 0.77 010 0.034 & H..() be realized by visual inspection. In view of this, it would be tentatively supposed that the springing would be replaced with whipping if the sea state becomes rough.4. Conclusive remarks
The simplified calculation shows that the wave exciting force, consequently the response of spring-ing. increases with increasing ship speed and/or decreasing natural frequency exhibiting humps and hollows, and that the n-th higher harmonic excitation is proportional to the wave height to the power a. The detailed calculation shows that the higher harmonic excitation becomes larger with increasing Beaufort number. Though these characteristics of springing coincide with the results of model ex-periments and the experience, the following items request the future works for the precise estimation of the response of springing.
a) Establishment of the refined theory .for hydrodynamic exciting force, including study on variation of wave structure along the length of ship.
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Confirmation of the higher harmonic excitation in the actual ship. The treatment of the higher harmonic excitation in the random vibration.
Evaluation of added mass in the higher harmonic excitation, which is tentatively treated as equal to that for the hull vibration in the present paper.
Estimation of the damping in hull vibration.
Consideration of the effect of whipping in rough sea.
Study of wave spectrum especially with regard to waves of short. length.
The authors express hearty thanks to Prof. Yamamoto and to members of Ship Structure Committee of West Japan for useful discussions. The authors wish to thank Mr. Takahashi and Mr. Ish.ibashi for their cooperation in model experiments. Also the authors wish to thank Miss Noma and Mrs. Matsu-moto for their preparing the computer program and assistance.
References
Goodman, R.A.: Wave- excited main hull vibration in large tankers and bulk carriers, RINA
(1970).
Gunsteren, F.F.: Springing, Wave induced ship vibrations, ISP (1970).
Kumai, T.: Wave induced force exciting hull vibration and its response, JSNA, West Japan, No.
44, (1972). S
Kumai, T., and Tasai, F.: On the wave exciting force and response of whipping of ships, Euro-pean Shipbuilding, No. 4, (1970).
Watanabe, Y.: On a cause of occurrence of whipping, (1968), published in a collection of posthu-mous manuscripts, (1973).
Bell, A.O. and Taylor, K.W.: Wave excited hull vibration, Shipping World and Shipbuilder,
(1968).
1) Tassi, F.: On the calculation methods of wave induced vibration of ships, JSNA, West Japan, No. 48, (1974).
8) Matsumoto, K., Kadoh, H. and T. Kozono: Study on Exciting force of. Springing Vibration, J. Soc. N.A. Kasai, Japan, (1974).
