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OCR for page 157
Nonlinear Motions and Whipping Loads
of High-Speed Crafis in Head Sea
S.-K. Chou, F.-C. Chin, Y.-~. Lee (National Taiwan University, China)
AssTiACT
In this paper, the nonlinear motions and wave
loads including whipping effects of high-speed
crafts traveling in head sea are investigated
theoretically and experimentally. The analysis
is performed on an existing large-sized high-s-
peed craft, following a modified nonlinear
strip method and treating the ship's hull as an
elastic beam from the viev-point of hydroelast-
icity. The ship's hull is regarded as an Euler
beam or Timoshenko beam and the structural
response is represented by modal superposition
method and finite element method g separately.
The elastic backbone model testing technique is
adopted to carry out the experiments for measu-
ring vertical bending moments acting on ship's
hull . Through the comparison with experimental
results, the validity of the present calculati-
on method is confirmed, and through serial
calculations,the influence of structural rigid-
ities on wave loads are also clarified.
INTRODUCTION
Ship motions and wave loads of a displacement
type ship in small amplitude waves can be
estimated satisfactorily by the linear strip
theory t-3. Dynamic behaviors of a displacement
type ship suffered serious slamming in rough
seas can be investigated by a nonlinear strip
theory developed by Yamamoto, Fujino, and
Fukasawa4, taking account of nonlinearities
caused by hydrodynamic impact, the ship hull's
shape and configurations. Dovever, when a
high-speed craft travels even in the moderate
sea condition, nonlinear characteristics of
ship motions and wave loads get significant
because of high-speed traveling in waves.
Several years ago, Chin, one of the authors,
and Fujino5~8 developed a practical method,
which is in principal based on the conventional
Ordinary Strip Method synthesis but modified to
be able to evaluate nonlinear hydrodynamic
impact forces as well as dynamic lift in waves,
for calculating vertical motions and wave loads
of a high-speed craft which travels in regular
head sea, and its validity was verified by
comparing the computed motion and wave loads
ls7
with experimental results performed by using a
ship model of hard chine type as well as a ship
model of round bilge type. Recently, it was
confirmed further that this method also can be
applied even to estimated vertical motions of
fishing vessels in head sea With accuracy
enough for practical user.
In the studies on nonlinear motions and wave
loads of a high-speed craft mentioned at the
above, the ship's hull is treated as a rigid
body. However, in general, the size of high-sp-
eed craft has increased significantly8-~°, and
the occurrence of not only local damage due to
serious slamming, but also whole structural
damage caused by the subsequent whipping of the
hull should be possible in case of a high-speed
craft of large size. fence, it becomes importa-
nt to investigate the influence of hydro-elast-
ic interactions on the structural responses of
a large-sized high-speed craft in Haves-.
In this paper, the authors investigate the
nonlinear characteristics of ship motions and
whipping loads of high-speed crafts theoretica-
lly and experimentally. The analysis is perfor-
med on an existing large-sized high-speed
craft, following the above-stated modified
nonlinear strip method basically, but extended
to treat the ship's hull as an elastic beam,
from the view-point of hydroelasticity. That is
to say, the ship's hull is regarded as an Euler
beam or Timoshenko beam and the structural
response is represented by modal superposition
method and finite element method, separately.
The experiments are carried out by using an
elastic backbone model. In order to generate
pronounced whipping loads acting on model,
rigidity of the elastic backbone is selected
more flexible than that should be scaled down
directly from the actual ship. Comparing the
results of serial calculations of different
structural representation methods with results
obtained in elastic backbone model experiments,
the influence of wave length, wave height and
advance speed on wave loads are discussed.
Furthermore, the influence of hull vibration ,
which is related to flexural rigidity and shear
rigidity of hull structure, are also examined.
OCR for page 158
2.THEOlY
In the previous formulation of nonlinear
vertical rigid-body-motions of high-speed
crafts running in head sea, nonlinearities of
hydrodynamic forces acting on the ship hull are
assumed to be exclusively due to the time-
variation of submerged portion of the hull.
This approach is followed basically in this
paper, except the ship hull's girder is discre-
tized into Timoshenko beam for considering both
bending and shear deformation, and the formula-
tion of finite element method is used to take
both low frequency and high frequency vibration
into consideration. The formulation of the
present method is highly similar to that descr-
ibed in detail in the referencess'~'7.Therefore,
for convenience sake, the basic concept of the
method will be subsequently described briefly.
2.l Coordinate System and Incident Wave
The coordinate systems and the sign conventi-
on of translational and angular displacements
used hereafter are shown in Fig.~. A space-fix-
ed Cartesian coordinate system O-lYZ is introd-
uced so that the X-Y plane coincides with the
still water surface and the Z-axis directs
downward. The ship advances in the negative ~
direction at a constant speed V. Another coord-
inate system o-xyz is ship-fixed with origin o
located at the center of gravity of the ship
and the x-axis parallel to the base line of the
ship.
ri is the initial trim, and rs and (s are the
increments of trim and sinkage due to steady
running In calm water respectively, While ~ and
~ denote the variation of pitch angle and heave
displacement in waves respectively. The counter
clockwise rotation around the y-axis and down-
ward heave displacement are regarded as positi-
ve.
The incident wave (w is described as
(W=(acos (AX COST+ sinr+wet)
in the ship-fixed coordinate system. For small
I, the wave profile can be approximated by
(W=(acos(~x+~et)' and the sub-wave profile
can be similarly approximated by
cosh~(h-zal
(c=(a cosh~h---= C°S(~X+wet)
where (a is the wave amplitude, ~ is the wave
number, h is the water depth, me is the encoun-
ter frequency defined by ~e=~+~V and ~ the wave
frequency. za is the instantaneous draft at
section x and expressed by
z~=d-x~tan(rs+~+~`s+~+w<-(w)/cosr (~)
where d is the sectional draft of the ship
without forward speed in calm water
~ is defined as r=ri+rs+8
WV is the displacement induced by elastic vibr-
at~on.
Hi. ~ or ~-,.;^~~-x
t
. ~
Fig.! Coordinate system
2.2 Sectional External Forces
The velocity of wave particles relative to
the ship's hull can be divided into two compon-
ents, Ur and Vr, which are parallel to x- and
z-axes respectively. By assumption of orbital
velocity component of wave particle in X-direc-
tion is negligible and high order term dropped.
Ur and Vr are approximately expressed by
U=-Vcosr
Van ( (e-() C O S Jinx REV S inT=\Jv
(2)
Meanwhile, the two components of relative
velocity for a ship running in calm water are
expressed by
UO=-V ~ cos ( [i+Ts)
Vo=-V sin(ri+rs)
(3)
In deriving the hydrodynamic forces, the
state of steady running in calm water is consi-
dered as the initial reference condition. From
this point, the z-direction relative velocity
Vr is expressed as follows
Vm (fir-Vo) ~Vo
=(`e-~)cos7+xt~wv-visinr-sin(ri+rs))
-Ysin(ri+rs)
(4)
thus Or is separated into one steady term TO
associated with running in calm water and the
remaining oscillatory term Vr-VO due to waves.
2.2.1 Sectional Force due to Change of Fluid
Momentum
-
Denoting the heave added mass of a tranverse
section located at x for oscillatory motion
with pSztx,t), and the sectional heave added
mass for steady running in wave and in calm
water with pSztx,t) and pSzO(x), respectively,
then the sectional hydrodynamic force due to
the time variation of fluid momentum, when a
ship is traveling in waves, can be described as
follows
is8
OCR for page 159
f m=a~{pSz (x, t ) ( Vr-V a) +pS z (x, t ) V O-pS z O (x) V 0} (5 )
In equation (5), pSz(x,t) and pSz(x,t) are
both evaluated by the instataneous submerged
portion of the ship section while running in
Haves. The last term pSzO is evaluated by the
submerged port ion of the ship sect ion while
running in calm water.
The operator ;~ can be expressed as
=~U~¢Vcosr~, thus the sectional force
due to change of fluid momentum can be decompo-
sed into the following five components
~ T
f m=f ma+f mj +f mj +f imp+f i mp ( 6 )
Where
f ma =pSz fix, t ~ EVE
fmj =-Vcosr-8PS (x,t) . ~Vr-VO~
f no =-Vcos rt dpS`'o (x, t ~ _ Ups z O (x) ~ V
f imp= - x ~ (fir-To)
f ~mp=§P5dx~x~t).Yo
(7)
The physical meaning of each term in the
r.h.s of equation (6) are as follows
fma: sectional hydrodynamic inertia force
fmj: hydrodynamic force due to longitudinal
variation of sectional heave added mass
associated with vertical oscillatory
velocity
*
fmj hydrodynamic force due to longitudinal
variation of sectional heave added mass
associated with vertital steady velocity
component of constant forward speed.
limp: sectional impact force due to time
variation of sectional heave added mass
associated with the vertical oscillatory
velocity
limp sectional impact force due to time
variation of sectioal heave added mass
associated with vertical steady velocity
of constant forward speed.
2.2.2 Sectional Damping Force
In a similar manner, the sectional damping
force fr is expressed as follows
* *
f~pNz(x,t) (Vr-Vo)+pNz(x~t)vo-pNzo(x)vo (8)
where the sectional heave damping coefficient
pNz(x,t), pNz(x,t) and pNzo(x) have the physic-
al meanings analogous to those of sectional
heave added mass pSz(x,t), pSz(x,t) and pSzO(x)
, respectively.
2.2.3 Restoring Force and Froude~rylov Force
The sum of sectional restoring force and
Froude-Krylov force can be approximatly expres-
sed as follows
fs=-pg{A(x,t)-Ao(x))cosr (9)
where p is the fluid density and 9 is gravitio-
nal acceleration AD is the sectional area of
the portion under the undisturbed still Hater
surface and A(x,t) represents the sectional
area of the portion under the undisturbed
effective incident wave surface by considering
Smith Correction.
The sectional external force in total is
obtained by summing the force components stated
above, ie.
fz=fm~fr~fs
2.3 Equation of lotion
(10)
The displacement and rotation angle of a
ship's section, which includes the vibration
component as well as the rigid-body-motion
component, are denoted by ~ and ¢, respectivel-
y. Then the bending strain ex and shear strain
can can be expressed by
ax=
7=~¢
'-,~
Fig.2 Displacement representation
(11)
The kinet ic energy T. strain energy V, and
Work done by external forces W can be describ-
ed as follow
T= ~ lp(~2dX
(12-~)
Where ,~ denotes the sectional mass of ship's
hull
V= Vbi Vs
(12-2)
where Vb, denotes the strain energy due to
bending deformation, is expressed by
Vb==lEI(~) dx
(12-3)
Vs. denotes the strain energy due to shear
deformation, is expressed by
59
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VS=-~-IGAw(~-~) dx (12-4)
EI and GAW are sectional flexural rigidity and
shear rigidity , respectively
W=-~-liz~dx (12-5)
In this manner, variational approach can be
introduced to derive the Eamilton's Principle
o.s
o.o
[( T-V+ ~-~0 (13) _o.s
where D denotes the dissipation function and
[D is described by
be= fib ~ ~ Vb+ As ~ VS ~ ¢= ~ W=W ( 13 - 1 )
where fib and As are the structural damping
coefficients corresponding to bending and shear
deformation, respectively.
The total displacement of a ship's section,
a, can be expressed by a linear combination of
N coordinate functions Wj as follows
Mode Shape Function /:
//
~_` /~\,,';;% ,//,/,/i
it, / \ \~,,~/ `, ~ / /
S. S.
Fig.3 lode shape functions of 3rd-Sth mode
X, X~ X3 _
1 1 1 > X
O t/2 l
w = ~ Wj(X)-~(t) (14) ov, ~ ~v2: ~v3
J_
_ we wv2 wv3
where qj's are the generalized coordinates, and
j=1,2 denote the rigid body motions correspond-
ing to heave and pitch, j>3 are related to
vibration components.
2.3.1 Nodal Superposition Method Form_lation
Por modal superposition analysis, the mode
shape functions wj are obtained by lyklestad'-
St3 method for a free-free Euler beam, the
first 4 mode shape functions corresponding to
vibration deformation, j=3-S are illustrated in
Figure 3.
Equations of motion derived from Equation
(13) by applying Galerkin method can be expres-
sed in matrix form as follows
tdij]{qj)~tCij]{qj)+tKij]~={fi) (~5)
where the assumption ¢=~ is used and detailed
expression of generalized mass, damping and
stiffness matrices, :~],~C],tK] and generalized
force function {f3 are summarized in the Appen-
d~x A.
2.3.2 Finite Element Method Formulation
In finite element method, the generalized
coordinates It's correspond to nodal displacem-
ents.
The vibration components of w and ¢, denoted
by TV and ~v, can be approximated in terms of
{qv), ie.
|vVll = ~N]{qv~i (16)
§vJlx=xj-xj+~
{qv), the degrees of freedom at node point
Fig.4 Degree of freedom in element
within the j-th element, are
{qv~j = ~qwvt
~qevJ
(16-1)
and the corresponding interpolation functions
tN] are set to be quadratic forms
END = tg:
Where
~wv]
{qwv~j = Wv21
Nv3] j
(16-2)
riv;
(16-3) {qov}j= lpv2 (16~)
[NV]=[1-36+262 46~2-~+262 0 0 0 ] (16-5)
[N~]=[O O 0 1-3~+262 46~2 _~+262 ] (164)
X-X j = x (16-7)
The resultant equations of motion in terms of
total degrees of freedom combined by different
elements can be expressed similarly in follovi-
ng matrix form:
[11 ] {q }+ [C ] {q}+ [K ] {q}={f } (17)
The detailed expressions of various elements
included in the coefficient matrices t11 ],
tC ], OK ], and force vector {I } are summariz-
ed in the Appendix B.
160
OCR for page 161
a. NUdERICAL SOLUTION
The numerical values of instantaneous sectio-
nal hydrodynamic coefficients are required for
various sections of the ship to calculate the
dynamic responses of a ship in waves.
The evaluations for these coefficients are
performed under the following assumptions:
Blithe sectional hydrodynamic coefficients
for heave motion in the z-direction are assumed
to be equivalent to those in the Z-direction.
(2)The sectional hydrodynamic coefficients
corresponding to oscillatory motion are evalua-
ted at encounter frequency He for the part due
to rigid body motion, and those at infinite
frequency are used for the part due to vibrati-
on.
(3)The sectional hydrodynamic coefficients
corresponding to steady forward velocity are
evaluated at infinite frequency under the
high-speed condition.
(4)The sectional hydrodynamic coefficients
are evaluated for the instantaneous submerged
portion under the undisturbed wave surface.
The nonlinearities of hydrodynamic forces
related to the time-varying sectional hydrodyn-
amic coefficients and hydrodynamic impact
forces are treated in such a manner as describ-
ed subsequently.
The sectional hydrodynamic coefficients at
several different prescribed drafts of a secti-
on are computed by Frank close-fit methods for
each transverse section. Those hydrodynamic
coefficients for different drafts are expressed
by a polynomial of n-th order as a function of
the instantaneous sectional draft. taking use
of such a polynomial expression, the sectional
hydrodynamic coefficients are evaluated at each
time step during numerical integration of
equations of vertical motion.
If some section is clear of water at.a
certain time step, then during the following
re-entry stage, water surface pile-up is consi-
dered according to Wagner's wedge impact theor-
yt~ that is to say, the instantaneous draft is
assumed to be r/2 times of that under undistur-
bed Wave surface. This consideration will be
disregarded when the pile-up water surface
cross over the chine. Furthermore, it is assum-
ed that the hydrodynamic impact force is able
to be disregarded when the ship section is
detaching from the Hater. The validity of this
assumption is confirmed by the results of the
forced oscillation test performed by Yamamoto,
Fujino and Ohtsubotfi. The bottom impact and
flare impact are evaluated with different
schemes of computing the rate of change of the
added mass.
The structural damping coefficient fib and ~
are set to be of same value and can be express-
ed by
pb=ys= r ~ (18)
where ~2V is the natural frequency of 2-node
vibration and ~ is the corresponding logarithm-
ic decrement.
For the numerical integration of the equatio-
ns of motion, Newmark-4 method with =/4 is
used, and the discrete time increment it adopt-
ed for time integration is 1/500 of the encoun-
ter period.
From the view point of the stability of
numerical integration, the encountered wave
amplitude grows up gradually to steady state
during the calculation and the dynamic respons-
es are recorded only after stationary state
motion is reached.
Wave loads can be estimated either by the
integration of fz- ~ along the ship's length
or by the evaluation from differential formula
in terms of W. the discrepancy between these
two methods is insignificant and the former
method gives more consistent results. Hence,
the calculation results presented in this paper
are all obtained by applying the integration
evaluation, exclusively.
4. EXPERIlENTS AND NDlERICiL PREDICTION
4.1 Elastic Backbone Model
In order to investigate the sectional Have
loads along the ship's length and verify the
validity of the numerical prediction method
described in the previous sections, elastic
backbone model testing techniques' t7 has been
selected for experiments. ~ model in scale
1:14.S of an existing 44.5 meter high-speed
craft of hard chine type is used. The principal
particulars and body plan are shown in Table ~
and Figure 5, respectively. it square station 1
to 8, the model made of wood, is divided into 9
segments which are connected with a backbone
composed of 2 aluminum alloy (6063) beams as
shown in Figure 6. The bending and shear rigid
Unit mm
3
4
5
1 ~
1~-o
11 1
2
o
o
1<- 3750 - ~
'A ;~ /~
Fig.5 Body plan of Boat-4450
Length Overal 1 L
Breadth ( I) B
Depth ( ~ ) D
Draft ( 38~) d
Displacement W
Longitudinal Position of C.G. LOG
__ _
Longltudinal Gyradius Yk
~-
44.50 m
7 Fin m
. . ~
_
3.50 m
1.58 m
220.0 ton
2.05 m aft 3E
- 27.1 AL
.
Table ~ Principal particulars of Boat - 450
OCR for page 162
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Representative terms from entire chapter:
bending moment
_,B EI
IHI I IN I IHI I F1 Id IHI IHI I IHI ~ IHI I ~g o 8
' ~I:~
AS 1 2 A 3 4 5 s 6 78 \ FP 0.7
J Backbone
, ~
~ ~ ~ - 1: r i -my ~ ~
detail of A A ~.2scm2 131 1 g7g Section B-B ton/m
\ - ~31 1 Ir1 3?9 6.0
(/ tB
on modes) while in some comparison cases, 4
mode shape functions calculation called "IODE
4") and even only the first two rigid-body-mot-
ions mode shape functions calculation (hence
called "RIGID") are also performed.
(b)Finite element calculation called "F.E.~."
or "EULER")
In this kind of calculation, if Timoshenko
beam element formulation is used to evaluate
the dynamic response of the ship hull's struct-
ure, the notation "F.E.~." is adopted. The
calculation in terms of Euler beam element
formulation by neglecting the &hear deformation
is called "EULER" for distinction.
5. COlPARSION BETWEEN NDIERICAl PREDICTION AND
EXPERIMENTAL RESULTS
Figures 9 to 18 illustrate the nondimensiona-
lized peak-to-peak bending moment distribution
along the ship's length under various wave and
speed conditions. In Figures 9 to Il. the
experimental results for the case of Fn=0.35
which may be considered as a typical speed of
"non-planning" condition are shown together
with the results predicted by the two kinds of
numerical computations, namely, DIODES " and
"F.E.~.". Both of the predicted values by modal
superposition calculation "IODE-6" and finite
element calculation "F.E.~." agree well with
the experimental results. In the cases of
Fn=0.70 and i.0 which may be considered as a
typical speed of "semi-planning" and "planning"
condition respectively, shown in Figures 12 to
18, the predicted values by "IODE-6'r are satis-
factory, except for the cases of relatively
short waves, in which it tends to underestimate
the wave loads acting on the fore-bodies.
Nevertheless, the discrepancy in fore-bodies is
improved significantly by the "F.E.~." calcula-
tion. As seen in these figures , it can be said
that both of "MODE-6" and "F.E.~."calculations
give reasonable results, and the agreement
between the predicted responses and experiment-
al results seems satisfactory enough for the
practical point of view.
Figure 19 illustrates the forward speed
dependence of bending moments at various square
stations 4 to 7 in the selected wave condition
of A/L=~.5 and Hw/l=l/4o. In Figures l9 (e) and
Figure l9(b), the nondimensionalized peak to
peak bending moments obtained from experiments
are plotted together with predicted results by
"NODE-6" and "F.E.~." calculations, respective-
ly. As shown in these figure, the discrepancy
in the trend between the "Iode-6" prediction
and the measured responses tends to be signifi-
cant in the speed range of Fn=0.70 to 1.0.
However, the predicted values obtained by
"F.E.~." calculation and experimental results
show qualitatively similar trends in full speed
range, and their agreement in values is also
remarkable.
In order to manifest the validity of the
present nonlinear prediction of responses , the
time histories of bending moments at square
station 1 to 8 as well as C.G. acceleration and
bow acceleration obtained by "F.E.~." calculat-
ion are shown together with the measured ones
0.06 .
A_
0.05
-
0.04
0.03
0.02
0.01
Fn = 0 .3 5 Hlr/A Esp. Mode-6 F.E.1L
A/L = 1.125 1447 D
o.ooo. o~z.lo 3.10 4.10 5.10 6.0 7.0 8.0
/,,''
A,
/,,''
i,'
8
ff~.
/,~' "" ~
"A ho
it\\
" \\
" \\
"\\
" \ \
"\ \
~ ~ \
I I I i-- ~
on inn
Fig.9 Longitudinal distribution of vertical
bending moment (Fn=0.35,)/~=1.125)
0.06
-
$ 0.05
be
Q
-
0.04
0.03
n no
~ . ~ ~
~ 0.05
Not
0.04
0.03
c~ 0.02
PA
o
0.01
0.000 o
163
Fn = 0.3 5 H~/A EYP. Mode-6 F.E.M.
\/L = 1.50 1~40 0
_
0.02
0.01
o.oo o. O
/ ~ ~
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1 0.0
S. S.
Fig.iO Longitudinal distribution of vertical
bending moment (Fn=0.35,1/L=~.5)
Fn = 0 .3 5 | Her/)< Exp.Mode-6 F.E.M.
\/L = 1.70 1 1440 a__ _
1
1.0 2.0 3.0 4.0 5.0 o.Oi.D 8.0 9.0 1 0.0
S. S.
Fig.ll longitudinal distribution of vertical
bending moment (Fn=0.35,1/L=~.7)
0.06
0.06 _
_ -
m 005 ~
N~
06
Q
0.04 _
0.03 _
0.02 _
(4
~0.01
~^,\m
Fn = 0 .7 0 Hlr/A Exp. Mode-6 F.E.M.
A/L = 1.125 1549 0 ~
~8
//, ~ \ \
/,'
//'
//,'
I'o
~,,~ _ /.' ~i~\.
0.0 1.0 2.0 3.0 4.0 5.0 6.07.0 8.0 9.0 10.0
S. S.
Fig. 12 Longitudinal distribution of vertical
bending moment (Fn=0 . 70 ,1/L=1 . 125)
0.07
0.06
~_
0.05
0.04
0.03
~ 0.02
P~
o
0.01
0.00,
Fn = 0 . 7 0 Hw/A Esp. Mode-6 F.E.M.
\/L= 1.50 1/26 ·
1/31 ~
1/40 0 --
1/50 0 - -
0 0
,P- _
/-~ `,_` \ ~
~ 1 1 1 ·d
v.u 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
S. S.
Fig. 13 Longitudinal distribution of vertical
bending moment (Fn=0.70,I/L=~.5)
_ -
m 0 05
bD
0.04
Fn = 0 .7 0 H'r/A Exp. Mode-B F.E.M.
)~/L = 1.70 1551 0
_ - _
~ 0
0.03 - /~
/ / ~0 n '. \o
0.02
a)
o
0.01
0.00 0
~' o - '"~
~1 1 1 1 1 1 ~
_.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1 0.0
S. S.
Fig. 14 Longitudinal distribution of vertical
bending moment (Fn=0.70,1/L=~.7)
_%
m 005
0.04
0.03
0.02
c)
o
0.01
P~ ~ ^^
.
Fn = 0.70 H'r/A Exp. Mode-B F.E.M.
\/L = 2.00 1S45 O
.~
/,' 8
:,,~a
,/''B
"~\
"\ \
"~4
'`~\\
u.uu 0. ~ 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10 .0
S. S.
Fig. 15 Longitudinal distribution of vertical
bending moment (Fn=0 . 70, )/L=2 . 0)
n n7
. _ .
0.06 _
~_
m 005
~o
0.04
0.03
0.02
0.01
000 -
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Fn = 1 . O O H'r/~` EYP. Mode-B F.E.M.
~/L = 1.50 1S540 o°
/
1: ~
_~'
/ 1 1 1 1 1 1 1
~ . \
'>\
'\\
1
s. s.
Fig.16 Longitudinal distribution of vertical
bending moment (Fn=~.O,I/~=l.S)
0.06
0.05
bO
Q
0.04
0.03
0.02
o
~0.01
P~ nnn
. . _
Fn = 1.00 H,r/A Exp. Mode-B F.E.~.
A/L = 1.70 1/3Gt 0 - - _
_
~o
/-_ \
/'~ ~
'o\
. '~
\~
I t
.-- o. ~t 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 l 0 0
S. S.
Fig.17 Longitudinal distribution of vertical
bending moment (Fn=~.0,1/~=~.7)
164
o.o6r
Fn = 1. O O H'r/A Esp. Ilode-B F.E.~.
A/L = 2.00 1~40 O°
-
~ 0.05
an
0.04 _
0.03 _
~ 0.02
PA
0.01
0.00,
i' Pro"
,~/ ~
,~ ~
1 1 1 1 ~ 1 1
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
S. S.
Fig.18 Longitudinal distribution of vertical
bending moment (Fn=l.O,}/L=2.0)
0.07
0.06
A_
~ 0.05
g
-
0.04
0.03
0.06
-
m
by
Q
0.04
0.05
0.03
0.02
P.
o
0.0 .
0.00 t
in both cases of Fn=0.70 and 1.0 in Figures 20
and 21, respectively.
Although, the time histories of bending
moments obtained by "F.E.~." calculation show
slight difference of shape at hogging conditio-
n, from that obtained by measurements, as seen
in Figures 20(a) and 21(a), the predicted time
histories of bending moments and accelerations
agree qualitatively well with the measured
ones. The plausible reason of the discrepancy
in time histories at hogging condition, in
which the bow sections emerge from the water
surface, between the predicted and the measured
bending moments is that the incoming wave
surface is assumed to be undisturbed even when
the ship travels in waves at a high speed,
namely, the effects of spay while planning
occurred are not taken into consideration.
6. EFFECTS OF STRUCTURAL RIGIDITIES
.
A SL 1.5 B~p. IlODB-i,
·
~ 0.02
4)
O.01
/:
///
·/~/!
ire/
o/
.
000 , 1 1 1 , 1 1 1 , ~ ,
0. ~ 0.2 0.4 0.6 0.8 1.0 1.2
Fn ~ Vs/~L ~
Fig.19(a) Effect of ship speed on vertical
bending moment ("IODE-~" calculation)
0.o7r 1
Hw; 2 ~- 1/40 S S. 4
S.S.6
S.S.7
_
_ :~.'
_ .
11 , 1 , 1
0.0 0.2 0.4
0.6 0.8 1.0 1.2
Fn ( Vs/~L )
Fig.l9(b) Effect of ship speed on vertical
bending moment ("F.E.~." calculation)
Exp. F.E.5I.
o
.
·
~ ,/
,,~
~ / ,
,',/ orb. ° /
~ try \A A'
As clarified in the previous sections, it can
be said that vertical wave loads, in which
whipping loads are included , acting on a
high-speed craft traveling in head sea, can be
predicted by the present "F.E.~." calculations
with accuracy enough for the practical use.
F7 0 70 Exp. F.E.M.
Hw/~: 1/40
~,~ ~
~ ~;S.S.~
S.S 1
I ~ t Full Scale
2 SEC
Fig.20(a) Time histories of bending moment
by experiment and calculation
(Fn=0.7~/L=1.5 7 Hw/~=1/40)
165
Fn : 1.00
A/L 1.50 Exp . F.E .M
~ ~ \: V ~
Bow
C .G .
2 SEC Full Scale
Fig.20(b) Time histories of bow acceleration
and acceleration at C.G.
(Fn=0 ~ 7, A/L=1 · 5,Hv/~=1/40)
Fn : 1.00
A/L: 1.50 Exp. F.E.M.
Hw/~: 1/40
AS Ace: A-;
:~ \\ ~ IS.S 6 ~
A 1.,~ ~ if' V;
~ ': ':I'~S.S7~: \/ ~
(\~ is.sV2\/\~
~S.S~
0
.,4
p:
0
, ' 1
~ , 0
up
I ~t Full Scale
2 SEC
Fig. 21 (a) Time histories of bending moment
by experiment and calculation
(Fn=~.O,)/L=1.5,Hw/~=~/40)
Fn : 0.70
A/L 1 50 Exp. F.E.M
,_ · ~ : ~
~ c ~
ID
~C G.
0
P
I ~ t Full Scale
2 SEC
Fig.21(b) Time histories of bow accelertion
and acceleration at C.G.
(Fn=t · 0, l/L=1 · 5 ,Hw/l=1/40)
Fn : 0.70 ~ Bow Bending Moment
/L 1~50 Acceleration at S.S.7
Hw/~: 1/20 ~
RIGID ~<
MODE-4 ~,~
MODE-6 \ <
EULER
F. E . M. j
Actual Ship
,=
A
o
·_4
a)
m
o
1
cd
on
Fig.22 Time histories of accelertion at C.G. and bending
i moment at S.S.7 by various calculations
(Fn=0 · 7, A/L=! · 5, Hv/~=~/20)
166
0.030 ~
0.040
no30
0.020 _
A_
~ 0.010
m
06
a O.ooo
-
-0.010
-0.020
-0.030
-0.040
Fn : u.tu
A/L: 1.50
~ I-I/: 1/20
Ho.~
RIGID
MODE-4
)dODE-6
B
F.R Y.
A.P'~ /
'_ 1
\\ ~/
\\ 'a
~ -.
it'
Fig.23 Longitudinal distribution of vertical bending
moment by various calculations
(Fn=0.7,1/L=1.5,Hw/l=~/20)
In this section , various kinds of calculati-
on stated in section 4.3 are applied on the
actual high speed craft to investigate the
effects of structural rigidities on wave loads.
in order to illustrate the effects more clearl-
y, the computation performed at a severe wave
condition of l/L=~.5 and H~/l=1/20 with forward
speed of Fn=0.70, are shown in Figures 22 and
23. Figure 22 shows the time histories of bow
acceleration and bending moments at square
station 7, and Figure 23 shows the nondimensio-
nalized bending moment peak values distribution
along ship's length obtained by various calcul-
ations. It can be seen in these figures, by
comparing with "EULER" calculation, the "RIGID"
calculation, in which structural rigidities are
considered to be infinite and no vibration can
be recognized, may underestimate the sagging
moment significantly. Furthermore, by comparing
with "F.E.~." calculation, the "EULER" calcula-
tion, in which shear rigidity is assumed to be
infinite and no shear deformation can be recog-
nized, may underestimate the hogging moment
remarkably. Therefore, the "F.E.~. calculation
by treating ship hull's girder as an Timoshenko
beam seems to be necessary for predicting the
wave loads acting on it at severe condition.
Furthermore, in order to illustrate the
rigidities' dependence of wave loads, "F.E.~."
calculation is applied on the actual high speed
craft with variations of flexural rigidity and
shear rigidity separately, at the same conditi-
ons which is stated above , namely, I/L=1.5 and
Hw/~=1/20 with forward speed of Fn=0.70. Figu-
re 24 shows the flexural rigidity's dependence
of nondimensionalized bending moment peak
values at square stations 5, 7 and 8, while the
shear rigidity is kept to be original value of
the actual ship. It can be seen in this figure,
decreasing the flexural rigidity may reduce the
0.020
^ 0.010
$
m
o0
lo. o.ooo
-0.01 0
-0.020
F.E.U | I lain : 0. (0
S S 5 A/L: 1.30
S.S.8 --- lo/): 1/20
-- ~ ~
0 ~' 1 10 10 1 10 ~ Log(EI/EIs )
_ _
l o Sag
\
/
/
Fig.24 Effect of flexural rigidity on vertical
bending moment (Fn=0.7,1/L=1.5,Hw/l=1/20)
0.030 r
0.020
~ 0.010
$
of
Q 0.000
-
-0.010 ~
I Say
-0.020
-0.030 '
S.S.5
S.S.7
S.S.B - - -
F.E.H in
1 I/L 1 50
LIIW/N 1/r O |
I , 1O 10 ' 1O ~ Log(GAw/GAws )
Fig.25 Effect of shear rigidity on vertical
bending moment (Fn=0.7,1/~=1.5,Hw/l=~/20)
sagging moment acting on midship section signi-
ficantly. Similarly, Figure 25 shows the shear
rigidity's dependence of nondimensionalizied
bending moment peak values at square stations
5, 7 and 8, while the flexural rigidity is kept
to be original value of the actual ship. It can
be seen in this figure that although decreasing
the shear rigidity may reduce sagging moment of
midship remarkably, the hogging moment of
midship may be increased, and the sagging as
well as hogging moments acting at sections of
fore-bodies may be increased significantly.
However, it can be said that for the prediction
of wave loads acting on the actual high-speed
craft, neglecting shear deformation may undere-
stimate the hogging moment, but has no signifi-
cant effects on sagging moment.
167
CONCLUSIONS
From the present investigation into nonlinear
motions of large-sized high-speed craft in head
sea and whipping effects included wave loads
acting on it, the following conclusions may be
drawn:
(1) Through the comparison between numerical
prediction and elastic backbone model testing
results, the present "F . E. M. " calculation metho-
d, which is principally based on a modif fed
nonlinear strip method, and following the
Timoshenko beam elment formulation, can be
applied to estimate nonlinear motions and wave
loads including whipping effects of a high-spe-
ed craft in head sea with accuracy enough for
the practical point of view.
(2) Through serial calculations of different
structural representation methods, the influe-
nces of neglecting the effects of vibration
related to flexural deformation or shear defor-
mat ion on the accuracy f or predict ing the
vertical wave loads of a high speed craft can
be summarized as f allows:
( i) The prediction, which neglecting the
effects of vibration related to flexural defor-
mation, may underestimate the sagging moments
along the ship's length significantly.
(ii) The prediction ,which neglecting the
effects of vibration related to shear deformat-
ion, may underestimate the hogging moments
along the ship's length significantly.
(3) Through serial calculations on various
structural rigidity of hull's structure on
vertical wave loads acting on a high-speed
craft can be summarized as follows:
( i ) By decrees ing the f lexural rigidity, the
sagging moment at mid-ship section which depen-
ds on impact strongly is reduced.
(ii) By decreasing the shear rigidity, saggi-
ng moment at mid-ship section is reduced signi-
f icantly, but at the same time, the hogging
moment of mid-ship section and the sagging as
well as hogging moments of forward ship's
sections may be increased.
ACKNOWLEDGEMENTS
The f inancial support from the National
Science Council of Republic of China under the
grant NSC-77 0403-E002~8 is gratefully acknow-
ledged. The authors wish to acknowledge the
encouragement and the helpful discussions of
Prof. M. Fujino, University of Tokyo. They also
would like to express their cordial thanks to
fir. C. H. [i of the National Taiwan University
for his cooperation in carrying out the experi-
ments. The computation was carried out by CDC
CYBER 2.3 in the computer center, National
Taiwan University.
1. Krovin-Kroukovsky, B. V., "Investigation of
Ship [lotions in Regular Waves" Trans., Society
of Naval Architects and Marine Engineers,
Vol.63 (1955)
2. Tasai, F. and Takagi, M., "A Theory on
Ship Dynamic Responses in Regular Waves and its
Prediction Method ", The 1st Symposium on
Seakeeping, Society of naval Architects of
Japan (1969) (in Japanese)
3. Salvesen, N. ,E.O.Tuck and Faltinson, O.,
"Ship lotions and Sea Loads", Trans., Society
of Naval Architects and Marine Engineers,
Vol.78 (1970)
4 . Yamamoto , Y ., Fuj ino 11. and Fukasawa, T .,
"Iot ion and Longitudinal Strength of a Ship in
Read Sea and Effects of Nonlinearities (1st,
2nd, 3rd Reports) ", Journ. Society of Naval
Architects of Japan, Vol.143(1978), Vol.144
(1978), Vol.145(1979) (in Japanese)
5. Fuj ino, M. and Chiu, F. C ., "Vertical
motions of high-speed Boats in Head Sea and
Wave Loads", ~ of Naval Architects
of Japan, Vol .154 (1983) ( in Japanese)
6. Chiu, F. C . and Fuj ino , 111., "Nonlinear
Prediction of Vertical lotions and Wave Loads
of High-speed Crafts in Head Sea", Internation-
al Shipbuilding Progress, Vol.36, No.406 (1989)
7. Chiu, F. C. and Fuj ino, 1., "Nonlinear
Prediction of Vertical lotions of a Fishing
Vessel in Head Sea", Journal of Ship Research,
(to be published, Accepted: July, 1989)
8. Kaneko Y. and Baba, E ., Structural Des ign
of Large Aluminium Alloy High-speed Craft,
London, Royal Institution of Naval Architects
(1982)
9. Kaneko Y., Takanashi, T. and Kihara, K.
"A Proposal for Design Load on Structural Hull
Girder and Bottom Structure of Large High-speed
Craft ( 1st , 2nd Reports) " , Trans ., West-Japan
Society of Naval Architects, Vol.70 (1985),
Vol.72 (1986) (in Japanese)
10. Wang C.T., etc. "Iodel Test on the 4450
Boat", NTU-INA Tech. Rept .213, Institute of
Naval Architecture, National Taiwan University
(1985)
11. Kanedo, Y. and Takahashi, T., "Comparison
between Nonlinear Strip Theory and Nodal Exper-
iment on Wave Bending Moment Acting on a Semi-
displacement Type High-speed Craft", Trans.
West-Japan Society of Naval Architects, Vol.71
(1986) ( in Japanese)
12. Chiu, F . C ., Lee , Y. J . and thou , S . K ., "A
Consideration on Vertical Wave Loads Acting on
a Large-sized High-speed Craft", Journ., Socie-
ty of Naval Architects of Japan, Vol.163 (1988)
13. Frank,W. and Salvesen, N., "The Frank
Close-f it Ship lotion Computer Program", NSRDC
Report No . 3289 (1970)
14. Wagner, H., "Uber Stoss-und Greitvorgange
an der Oberf lache von Flus s igke iten", Z . A . 1 . 11 .,
Band 12, Heft 4 (1932)
15. Yamamoto,Y., Fuj ino, 11., and Ohtsubo, H.,
"Slamming and Whipping of Ships among Rough
Seas", Numerical Analysis of the Dynamics of
Ship Structures, EUROlECH 122, ATMA, Paris
(1979)
16. Bishop, R.E.D.,"Myklestad's Method for
Non-uniform Vibration Beam", The Engineer, Dec.
14 (1956)
17. Takahashi, T., and Kaneko, Y., "Experimen-
tal Study on Wave Loads Acting on a Semi-displ-
acement Type lligh-speed Craft by Means of
Elastic Backbone Models, ~iah-sueed Surface
Craft Conference '83, London (1983)
68
7. CONCLUSIONS
From the present investigation into nonlinear
motions of large-sized high-speed craft in head
sea and whipping effects included wave loads
acting on it, the following conclusions may be
drawn:
(~) Through the comparison between numerical
prediction and elastic backbone model testing
results,the present "F.E.II. " calculation metho-
d, which is principally based on a modif fed
nonlinear strip method, and f allowing the
Timoshenko beam elment formulation, can be
applied to estimate nonlinear motions and wave
loads including whipping effects of a high-spe-
ed craft in head sea With accuracy enough for
the practical point of view.
(2) Through serial calculations of different
structural representation methods, the influe-
nces of neglecting the effects of vibration
related to flexural deformation or shear defor-
mation on the accuracy for predicting the
vertical wave loads of a high speed craft can
be summarized as follows:
(i) The predication, which neglecting the
effects of vibration related to flexural defor-
mation, may underestimate the sagging moments
along the ship's length significantly.
~ ii) The prediction Which neglecting the
effects of vibration related to shear deformat-
ion, may underestimate the hogging moments
along the ship's length significantly.
(3) Through serial calculations on various
structural rigidity of hull' s structure on
vertical wave loads acting on a high-speed
craft can be summarized as follows:
(i) By decreasing the flexural rigidity, the
sagging moment at mid-ship section which depen-
ds on impact strongly is reduced.
(ii) By decreasing the shear rigidity, saggi-
ng moment at mid-ship section is reduced signi-
f icantly, but at the same time, the hogging
moment of mid-ship section and the sagging as
well as hogging moments of forward ship's
sections may be increased.
ACKNOWLEDGE]lENTS
The f inancial support from the National
Science Council of Republic of China under the
grant NSC-77-0403-E002~8 is gratefully acknow-
ledged. The authors wish to acknowledge the
encouragement and the helpful discussions of
Prof. 11. Fujino, University of Tokyo. They also
would like to express their cordial thanks to
fir. C. H. Li of the National Taivan University
for his cooperation in carrying out the experi-
ments. The computation was carried out by CDC
CYBER 2.3 in the computer center, National
Taiwan University.
REFERENCES
i. Krovin~roukovsky, B.V., "Investigation of
Ship [lotions in Regular Waves" Trans., Society
of Naval Architects and Marine Engineers,
Vol.63 (1955)
2. Tasai, F. and Takagi, ]1., "A Theory on
Ship Dynamic responses in regular Waves and its
Prediction Method ", The 1st Svmnosium on
Seakeeping, Society of naval Architects of
Japan (1969) (in Japanese)
3. Salvesen, N. ,E.O.Tuck and Faltinson, O.,
"Ship [lotions and Sea Loads", Trans., Society
of Naval Architects and Marine Engineers,
Vol.78 (1970)
4. Yamamoto , Y ., Fuj ino 11. and Fukasawa, T .,
"[lotion and Longitudinal Strength of a Ship in
lead Sea and Effects of Nonlinearities (Ist,
2nd, 3rd Reports) ", Journ. Society of Naval
Architects of Japan, Vol.143(1978), Vol.144
(1978), Vol . 145 (1979) (in Japanese)
5. Fuj ino , ]1. and Chiu, F . C ., "Vertical
motions of high-speed Boats in lead Sea and
Wave Loads", Journ. Society of Naval Architects
of Japan, Vol.154(1983) (in Japanese)
6. Chiu, F. C . and Fuj ino , 1111., "Nonlinear
Prediction of Vertical lotions and Wave Loads
of ligh-speed Crafts in Head Sea", Internation-
al Shipbuilding Progress, Vol.36, No.406 (1989)
7. Chiu, F . C . and Fuj ino , 11., "Nonlinear
Prediction of Vertical [lotions of a Fishing
Vessel in Head Sea", Journal of Shin Research,
(to be published, Accepted: July, 1989)
8. Kaneko Y. and Baba, E., Structural Design
of Large Aluminium Alloy High-speed Craft,
London, Royal Institution of Naval Architects
(1982)
9 . Kaneko Y., Takanashi , T . and Kihara, K .,
"A Proposal for Design Load on Structural Hull
Girder and Bottom Structure of Large Bigh-speed
Craft (ist , 2nd Reports) " , Trans ., West-Japan
Society of Naval Architects, Vol.70 (1985),
Vol.72 (1986) (in Japanese)
10. Wang C.T., etc. Model Test on the 4450
Boat", NTU-INA Tech. Kept .213, Institute of
Naval Architecture, National Taiwan University
(1985)
11. Kanedo, Y. and Takahashi, T., "Comparison
between Nonlinear Strip Theory and Model Exper
iment on Wave Bending Ioment Acting on a Semi
displacement Type ~igh-speed Craft", Trans.
West-Japan Society of Naval Architects, Vol.71
(1986) (in Japanese)
12. Chiu, F.C., Lee, Y.J. and thou, S.K., "A
Consideration on Vertical Wave Loads Acting on
a Large-sized High-speed Craft", Journ.~ Socie-
ty of Naval Architects of Japan, Vol.163 (1988]
13. Frank,W. and Salvesen, N., "The Frank
Close-fit Ship lotion Computer Program", NSRDC
report No. 3289 (1970)
14. Wagner, H., "Uber Stoss-und Greitvorguge
an der Oberflache von Flussigkeiten", Z.A.~.!
Band 12, Deft 4 (1932)
15. Yamamoto,Y., Fujino, 1., and Ohtsubo, I.,
"Slamming and Whipping of Ships among Rough
Seas", Numerical Analysis of the Dynamics of
Ship Structures, EUROlECD 122, ATBA, Paris
(1979)
16. Bishop, R.E.D.,"lyklestad's Method for
Non-uniform Vibration Beam", The Engineer, Dec
14 (1956) ~
17. Takahashi, T., and Kaneko, Y., "Experimen-
tal Study on Wave Loads Acting on a Semi-displ-
acement Type Digh-speed Craft by leans of
Elastic Backbone Model", Di~h-sueed Surface
Craft Conference '83, London (1983)
169
APPENDIX A
lij=J (~+pszj)Viwjdx
Cij=J,iEIWiWjdx+J( ~pSZi IpNzj)WiWjdx
~Vcos~JpSzj(WiWi-WiW~)dx
-Vcosr[pSziWiWj]A
Kij=JEIWiW~dx-Vcos~J( ~ pNzj)WiW~dx
-V2cos 2rJpSzjW,iWJdx
+v2cos2~[pszjwi'~]A
f i=cosr{pSz~ Widx+Vcos~JpSzieWidx
' + J (~ pNZ) (eWidx-VcOsr [pSzieWi] A
-pgJ (A-io)`idx}
-Vsin(ri+rs) {Vcos~J (pSZ-pSzo)w/dx
+J(pNz-pNzo+ ~ )Widx-Vcos7(pSz-pSzO)Wi]A}
(19)
(20)
(21)
(22)
APPENDIX B
The elemental coefficient matrices [! ]j
[C tj [K ]j and force vector {f }j associated
vit j-th element are given by
* 1J T
[l ]j=(~+#Sz)jlO [Nw] [Nw]dg
[K ]j=(EI)jlo [Ne]T[Ng]d~
+(0iW)jlO ([Ne]-[Nw]) ([Ne]-[Nw])dg
(23)
-Vcosr{( ~ pNz)jl i[Nw]T[Nw]d: (24)
~Vcosr(pSz);(lO [Nw]T[Nw]d(-[Nw] [Nw]l~j)}
[C ] j= ( ~bEI) j| o [Ne] T [Ne] d~
+(7sGAw)jlo([Ng]-[Nw])T([Ne]-[Nw])d~
+( ~ PIZ)ii i[Nw]T[Nw]d~ (25)
+(Vcos~pSz);~{|O([Nw] [Nw]-[Nw] [Nw])d~
n1
-[Nw] [NW]I1;}
*
{f }j=cosr{[pSz ~ ( ~ pNz) ie-P9(~-~°)
-Vsin(ri+rs) (
