LOW FREQUENCY DRIFTING FORCES ON MOORED STRUCTURES
IN WAVES Ir. J.A. Pinkster and Dr. Ir. J.P. Hooft
Netherlands Ship Model Basin
Paper for presentation at the 5-th
International Ocean Development Conference, Tokyo, September 25 - 29 1978.
/
7z-LOW FREQUENCY DRIFTING FORCES ON MOORED STRUCTURES IN WAVES
I . J.A. Pinkster and Dr.Ir, J.P. Hooft
Netherlands Ship Model Basin
ABSTRACT
A recently developed method, based on three-dimensional potential theory, to compute the mean wave drifting forces on a free floating structure,is extended to include low frequency oscillatory components. The component of the low frequency force due to the second order ve-locity potential is included using a simple approximation which is compared to some two-dimensional cases for which exact solutions are
known.
The results of computations of the total mean and low frequency surge force on a barge in head seas are presented.
The computation method is used to estimate the influence of a wave feed-forward control signal on the nett mean and low frequency surge force in head waves on a dynamically positioned barge. INTRODUCTION
The mean and low frequency second order wave drifting forces acting on structures moored in waves are usually of interest from the point of view of mooring loads and horizontal motions. Investigations into the nature of the wave drifting forces have generally had as back-ground the need to include these effects in the total load on struc-tures moored by anchor lines or cables (1) (2)
(3) (4).
Such investigations were initially prompted by the fact that the observed horizontal motions of vessels moored in irregular waves and the associated mooring line forces showed low frequency compo-nents which could not be explained by linear wave theory alone. Van Oortmerssen
(5)
demonstrated that the low frequency horizontalmotions of a vessel moored to a jetty in irregular waves were in many cases directly attributable to the second order wave drifting force thus underlining the necessity for reliable means of deter-mining this part of the total wave forces.
The influence of the second order wave drifting forces is not only restricted to the horizontal plane. This is, for instance, indicated by Numata et al
(6)
in _connection with the steady tilt ofsemi-submersible vessels in regular waves.
The presence of low frequency components in the vertical motions of large volume structures with small water plane area during model tests in irregular waves infers that non-linear hydrodynamic effects play a large part here also.
Up to now, however, the majority of the investigations concerning the mean and low frequency wave drifting forces is concerned with forces and motions in the horizontal plane. In (7) Faltinsen and Lf6ken reviewed the available methods for calculating the horizontal drifting forces on fixed and floating objects. A comparable study on the methods to calculate the vertical wave drifting forces has not been presented yet.
This paper deals with a method to calculate the mean and low frequency second order wave drifting forces which is applicable to all six
degrees of freedom.
The method is based on direct integration of all contributions to the second order forces over the instantaneous wetted surface of the hull of the structure.
The equations for ,the second Order forces and moments are evaluated using a computer program based on threedimensional potential theory which includes the effect of waterdepth:
LOW FREQUENCY SECOND ORDER FORCES IN IRREGULAR WAVES
In irregular waves the wave elevation in a point may be written as
follows:
(t) = E ci Cos (w.t+E.)
a. i=1
The square of the wave elevation is:
N N
(t) =-
EC.C.
i=1 j=1 1 J
The lbw frequency part of the-Square of the wave elevation it:
The low frequency second order wave drifting forces are related
to
the square of the wave elevationand
may be writtenas
folloWs:N N
(2)
E.
j
ECCP. Cos {(w.-w )t
i
As indicated by the subscripts i and j, Pij
and.Qu
are quantities which are a function of two frequencies wi and wj.Themeansecorldorderforceisfouridbyputtizigui..(0.:
E C.,2
P.
i=1 1 11
With wi # w. the P-- and Q-- values also give the in-phase and
out-J ij ij
of-phase components of the oscillating part of the force when the in, coming waves consist of two regular waves with frequencies_wi and
J
w. respectively. This is known as a regular, wave group. The fre-quency of the group is
(.0-1-w..J N N E E C Cos
f(
. w.)t + (E.-E.)1-1
-j i=1 j=1 i=1 j=1 N N + E I C.C. Q1. i=1 j=1second order transfer function
for
the part of the force whichis in-phase
with the low frequency part of the wave height squared.out-of-phase part of
the
second order transferfunction.
Cos (W t+E ) COs (W.t+E;)
J =J
Sin {(!jj t + (6 _E-j )}
F..
=j ..2+
in which
2
1,3 ij
Forij the following relationship exists:
F. F.
;j ji
The second Order transfer function it derived based on potential
theory
in the following sectionsEQUATIONSFOR THE SECOND ORDE14 WAVE FORCES ON FLOATING VESSELS .-Assuming that the. fluid motions may be descritea by a Velocity
po-tential 0, the hydrodynamic forces acting on a vessel are found by integration of the fluid pressures over the wetted part of the hull:
= - II p
n dSis according to 'Bernoulli's equation:. The following relationships exist:
P-. * 6
Q3: ..
= -Q..
7The low frequency force
in
a regular wave grOup consisting Ofregu-lar waves with frequency, amplitude and phaSe 4.)- 1 and w2., 4 2' E2 retmectiVely i$: (2). 2 2 F " P + P +-11 1 2 21 Nu) +
-c2))
+ (Q
) Sinf(
-w2)t +(e
,E )}For the
sake of
brevity, use will be made of the amplitudes of the deeona order transfer functions.Theseare defined as follows:
P = -Pg x3 - "t LP 1q7DI 12
and. is the total velocity potential including all orders:
0 = CO( ) )
13
Weare concerned here with the second order wave fOrces. The second order force referenced to .a fixed system of co-ordinate axes which coincides with the body axes in the mean position of the body follows
from (See ref,- .(-81)!
9
-(2) = f C (1 wL r - ffr-iPlf7D(1 - 8 _ (1) n dl + 2 I
-134)t(2) - P kidSA similar expression is deduced
for
the second order moment:I c
WL -- 0 )1 nidS -(1 (1 p . V )} (X x ft) dSThe expression shown here differ from earlier expressions for the second order force and moment in so far, that they have to be evalua-ted on the surface of the body and give insight in the mechanism by which the waves and vessel interact to produce the second order force. We will see this upon closer inspection of the expression for the
second order wave force and restrict ourselves to the surge force in irregular head waves.
The expression for the surge force becomes:
( x5(1) nld F1
2)
=f
1.,(1.) WL- II
T1
101) (2). 8o, - p -(1)The above expression contains three components; The first part is a
line integral around the static water-line of the vessel of the square of the relative wave height. The relative wave height is the wave height as measured from the vessel. This contribution arises from the fact that near the surface the pressure in the waves can be approximated by the hydrostatic pressure. The hydrostatic pressure increase at the mean water-line on the hull is proportional to the relative wave height and the additional area on which the pressure acts is also proportional to the relative wave height. This results in an inwardly directed force proportional to the square of the re-lative wave height which has the form of the first part of the above equation.
The second contribution to the second order wave drifting force is a
consequence of the choice of the axes to which the second order force is referenced. The expressions 14 and 15 apply to force components parallel to and about a fixed system of axes with
X3 axis vertically
15 2)
upwards. The surge force given in expression 16 is along a horizontal Xi axis and not along the moving longitudinal xl body axis. As a
result of this, first order vertical hydrodynamic and hydrostatic
forces acting along the vertical body axis G-x3 will give a longitudi-nal second order force component according to:
F
13
( )
where:
Taking into account that the vertical force force which equals:
F3(1)
= m . R (1)3
this contribution becomes: (1)
m x
.5 3
The third contribution to the second order wave drifting force stems from the integration of all second order hydrodynamic pressures over the mean submerged part of the hull. These pressure contributions
follow from the second order part of the Bernoulli equation 12.
If the velocity potential of the fluid motion is defined relative to a fixed system of co-ordinates and the body is carrying out small
amplitude motions about the mean position, it can be shown that the second order hydrodynamic pressure in a point on the submerged hull becomes:
(2)
= _
(see Figure 1)
= first order vertical force = first order pitch angle.
(1)
2)_ipk7,15(1)12 -(1)
I f3()c
in
Which: 0 = first order approximation for the total Velodity potential = 0 ( 1 ) + 0 ( 1 ) 0.(1) (1 ,w(1) (f) ' ll). 1) .'
.i'd '.' -111associated with respectively the undisturbed incoming waves, the diffraction potential and the body motion'
-potential.
0 = second order approximation for the total velocity
potential consisting of contributions,froM the incoming waves, diffraction and body motibns.
For vessels or floating structures at zero forward speed, the
compo-nents of the second order wave drifting forces and moments which de-pend on products of first order quantities may be calculated using computer programs based on three-dimensional potential theory (8), (9). The components which depend on knowledge of the second order
poten-tial
0(2)
present a bigger problem. Faltinsen and L$ken (10) havequantified the contribution due to
0k2)
for the two-dimensional case by solving, up to second order, the potential problem posed by thenon-linear free surface condition.
first order velocity potentials
18
19
20
For the general three-dimensional case no exact solutions have been presented. As indicated by Newman (11) and also inferred by results obtained by Faltinsen and Liken, it appears that for many practical cases the contribution due to the second order potential will be small. This conclusion is justified when regarding the low frequency wave
drifting forces on structures which have very low natural frequencies due to their large effective mass in relation to the stiffness of the mooring systems. As a consequence of the low damping at the natural
frequency, the wave drifting force components with the same frequency as the natural frequency are of importance.
It can be shown that, as the frequency approaches zero, the contri-bution to the drifting forces due to the second order potential vanishes leaving only contributions which are due to products of first order quantities.
Results obtained by Bowers (12) on the low frequency surge motions of a barge in irregular head waves indicate that, as the natural surge frequency is increased by increasing the stiffness of the mooring system, the influence of contributions to the low frequency drifting force related to the second order potential increase also. The general conclusions to be drawn at this time is that for systems which have very low natural frequencies the second order potential will not be very important but for systems with higher natural fre-quencies it may be necessary to include a contribution due this potential.
Systems with low, natural frequencies in the horizontal motions are for instance vessels moored to single point mooring systems in deep water. In such cases natural periods in the order of several hundreds
of seconds are not uncommon. Higher natural frequencies are found in, for instance, the vertical motions of semi-submersible type struc-tures which have natural periods of 20-40 seconds or in the horizontal motion of vessels moored to jetties. Due to the stiffness of the
mooring systems natural periods of the horizontal motions of 30-100 seconds occur in such cases.
-APPROXIMATION FOR THE CONTRIBUTION OF THE SECOND ORDER 'POTENTIAL From the afore-going it will be clear that at least an indication of the magnitude of the contribution of the second order potential to the low frequency wave drifting forces in relation to other components is desirable.
To this end an approximative method can be used based on linear
potential theory which is applicable in both two and three dimensions. The approximation is based on the assumption that the major part of
the low frequency second order force due to the second order potential is the wave exciting force component due to the contribution of the
undisturbed incoming waves to the second order potential. See ref. (15). In this method the first order wave exciting force in waves with
wave number k equal to the wave number k.-k., associated with regular wave groups occurring in irregular waves, is transformed to a second
order wave exciting force by simply multiplying the first order wave exciting forces by a factor depending on the values of the frequen-cies w. and w. of the regular wave components of the regular wave group involved
The result is the second order wave exciting force on the structure with frequency w.-w. of the regular wave group.
j
This method of approximating the contribution of the second order potential is possible due to the similarity of the components of the second order potential associated with the undisturbed incoming waves and the first order potential associated with a regular wave.
The first order potential associated with a regular wave is as follows: (
0.2)
A.. j2.
-ai (k.-k.) tgh (k--k.) h-j
Since first order wave pressures and exciting- forces are propdrtional to the value of g this means that the first Order wave exciting
fOrces P(1)
found for the selected k value and the normal Value of now become(1' 2) gij
F. =
13
In order to complete the transformation the first Order wave ampli-tUde Ca MUSt be altered so that the amplitudes of both potentials shall be equal or:
Ca gij
= Alj .-w,)
Cosh k 3+h Cosh
k h
The components_ Of
the.low
frequency part of. the second order potential associated with the undisturbed incoming waves have the following formCos (kx+wt+E) Cosh (ki-kj
)(X
+h) 3 Cos {(k -k )x+ Cosh (k.-k.) ht
+(.E -6
)1 22-j-In which A-- is a coefficient depending on w., w. and the waterdepth
1.7 j
h and is given by Bowers (12).
The similarity in both potentials is evident.
We will now show how the wave exciting force due to potential components as given by equation 22 can be determined from the first order wave
exciting force due to first order potential components as given by equation 21.
The aim is to transform the first order potential so that it is the exactly the same as the second order potential. Since forces are directly related to the potential this means that the forces may be transformed in the same way (This assumes that the diffraction
DO-tentials are also equal which generally will not be quite true, see (15)) For this we first put the wave number k equal to k.-k. which implies that the wave lengths in relation to waterdepth ana tie dimensions of the structure will be the same in both cases.
Secondly we alter the value of g so that the frequency w in equation 21 will be equal to the frequency w.-w.in equation 22.
Since
0(1)
is a first order potentiil io which the well knowndis-persion relationship applies, the value of g which must be used follows from:
-)2
21 23 224 25From which it follows that
(w.-w.)
1 j 1,1
gij
-The value of Fij(1,2) are now transformed as follows to a second order
- (2)
force Fij per unit
13
.C.: Ca f.. 13 .) A.. 0 F 1,1 -(1'2) . .. gii 13which combined with equation 14 gives:
-(2) A.
ij
(w.-w1 j ) )F.. . F
The above equation thus states that in
order
to estimate the second order .low frequency wave drifting fore or moment due to the compo-nents of the second order potential .P2) it is_only necessary to determine the first order wave exciting force F(1) for .a wave number value k equalto
the WaVe number ValUe k.-k. of the second order potential component and then multiply the 14sults by the real coefficientfij where:
Aij (w -w.)
i
COMPARISON BETWEEN THE EXACT RESULTS AND THE APPROXIMATION
It can be shown that this method of approximation gives exact results in two simple cases and gives a reasonable approximation for a third, more practical, case.
The first case concerns the second order pressure due to the second order potential in undisturbed irregular waves in a point X3 = -a below the still water level. The second order pressure is:
(2) ,(2)
,-13
For the component given in equation 22 the amplitude of the pressure is: (2) = Alj(wi-Cosh (k.-k.)(-a+h) 1 Cosh (k..,k j) h
For the first order potential the pressure follows from:
( 1 A ( 1 ) = -p 27 28 29 30 31 32
The amplitude of the pressure using the first order potential of equation 21 and unit wave amplitudea and wave number k = k.-k.
j
(1) pg Cosh (k. k.)(--a+h)1 .
Cosh (k.-k-) h
1
Using
the coefficient f-- given in equation. 29, gives thefollo-wing approximation for the second order pressure
(2)
p.. = p A..( . -w.) C. .
1-3 13 1 3 1
Cash (k.-k. )(-ath)
Cosh (k.-k.) h
13
which equals the exact value given in equation 31.
The reason for this is that other contributions to the exact value which are neglected in the approximation namely, those due to dif-fraction and body motions are in this case zero.
The second case concerns the horizontal low frequency wave drifting force, due to the second order potential, acting on a vertical wall in deep water. It can be shown that the approximation is also equal to the exact result in this case. The reason for this is that the first order incoming waves and the first order outgoing waves are identical (total reflection) and the total second order potential consists of a contribution associated with the undisturbed incoming waves and a contribution due to the outgoing diffraction waves.
Since the approximation gives the exact value for the second order potential associated with the incoming waves it also gives the exact value for the second order potential associated with the out-going waves and hence the approximation is also the exact value. The third ekample concerns the two-dimensional case of a free floating cylinder in beam waves as presented by Faltinsen and LSken (10).
These authors solved the second order problem exactly and gave results on the contribution of the total second order potential to the low frequency second order sway force in regular wave groups in deep water. The method of approximation presented here was ap-plied to the same case using results given by Vugts (13) on the first order sway force in regular beam waves.
The coefficient fij of equation 29 becomes for deep water:
(wi2-us,2)
f
-13
4g
The results are presented in the form of the amplitude of the low frequency second order force due to the second order potential for
arangeofc0/Tibinationsofwiandw.which are the frequencies of two
waves making up a regular wave group. The exact results are given in Table 1 and the approximated results in Table 2.
Comparison of the results show that near the line wi = w. the ap-proximation is good but for larger differences between wi and wj
the approximation is considerably higher than the exact value. Further study will be required to determine the reason for the large differences which occur at higher difference frequencies. At the present, however, it may be tentatively concluded that the method of approximation gives the right order of magnitude to the low frequency forces due to the second order potential for diffe-rence frequencies which are not too large. For the cylinder in
33
34
:beam seas large differences between the exact results
and
the aptroXi-mation occurred for values of the non-dimensional differencefre-quency greater than about 0.1.
RESULTS OF CALCULATIONS OF THE TOTAL SECOND ORDER FORCE.
Finally some results of calculations of the total mean and low fre-quency second order surge force on a -barge type vessel are presented. The dimensions of the barge Were as follows:
..Length 150 t Waterdepth 50 in
Breadth 50 m Draft 10 m.
A body plan is given in Figure 2. The first order wave loads and motions in regular waves as well as the mean drifting forces of
equation 4 and equation 5 were calculated using a three-dimensional potential theory computer program and compared with measurements. The results have been extensively treated in (8).
In Figure 3 the calculated mean wave drifting force in surge direction is cotpared with measurement for head waves.
In this paper the calculated results are extended by the low fre-quency wave drifting forces in regular wave groups. The calculations are again based on equation 4 and equation 5. The contribution due to the second order potential has been calculated using the method outlined in this paper.
The results of calculations are presented in Table 3 in the form of the amplitude of the total second order surge force in regular wave groups consisting of two regular waves with unit wave amplitude and waVe frequencies of wi and w..
The mean drifting forces in egular waves of unit amplitude are found on the diagonal where wi = w. The negative sign of the quan-tities on the diagonal denotes tha the mean force in head waves is
in the negative X1-direction.
The results in the table should be interpreted as follows: If the regular wave components making up the regular wave group have fre-quencies, for instance, of 0.8 rad./sec. and 0.9 rad./sec. respec-tively and the waves have amplitudes of 0.5 in and 0.75 m. Then the total second order surge force is:
F1(2) = -26.0.52 - 27.07.52
+ 20.5.0.7532
Cos (0.9-0.8)t= - 21,7 + 24 Goa 0,1t tons 36
The mean surge force in irregular head waves with spectral density S(w) follows from:
(2) co
F1 = 2 f S (w) .
o
The spectral density of the low frequency surge force follows from:
co
I s
(w) s(w,w)
dw w+u) .]
2 ,w+P) dw 371U
where:
F1 (w,w)
C'IC2
"IC2
As indicated by equation 4 and equation 5 the total second order forces consists of a number of contributions, five in total. In (1)1) it was indicated that for ship or barge shaped vessels the contri-l-ution due to the relative wave height is in most cases dominant. This is again illustrated in Fig. 3 in which four components of the
total mean surge force shown in Figure 2 are shown. The fifth con-tribution which is due to the second order potential is zero in this case. The contributions shown in Figure 3 are:
Relative wave height
2
-iPg f n1 dl
II: Pressure drop due to velocity squared
II -1Pli(1)12 i. dS 4o
0
Pressure due to product of gradient of first order pressure and first order motion
, -(1)
if -P kx .V0t (1))n1 dS
S.
IV: Contribution due to products of angular motions and inertia
.forces,
(1) ;(1)
3
From Figure 3 it is seen that the relative motion contribution I is indeed dominant and that the other contributions only reduce the effect of this contribution somewhat.
The value of the contribution due to the second order potential is shown in Table 4. In this table the amplitude is given of this com-ponent which can be compared with the total forces given in Table 3. The results of Table 4 show that the contribution to the mean force
is again zero and that the force tends to increase for larger dif-ference freauencies. Comparison with the results in Table 3 reveals that the second order potential contribution compared to the total is generally not very large in the range of practical difference frequencies which for this type of vessel is generally not higher
WL r
7-; Mean drifting force coefficient
in
regular Navesshown on the diagonal of Table. 3.
amplitude of the low frequency drifting force with frequency i given in Table 3 in the values beside
the diagonal.
spectral density of the irregular waves.
contribution:
than about 0.1 rad./sec.
CALCULATION OF THE EFFECT OF WAVE FEED-FORWARD
In (14) the fact that the contribution of the relative wave height term tends to be dominant was used to reduce the low frequency ho-rizontal motions of a dynamically positioned ship in waves.
Experimentally obtained results indicated that the relative wave height measured continually at a number of points around the line could be used to create a thrust control signal (wave-feed-forward) which significantly reduced the low frequency horizontal motion in irregular waves. An example of the results obtained with a scale 1 to 82.5 model of a loaded 200,000 dwt vessel is given in
Figure
5.
The essential part of the method depends on the instantaneous evaluation of the following equation:
= -ipg
c 2 ndlWL
which is the contribution due to the relative wave height.
The low frequency part of the above component is computed continually and used as a thrust control signal for the propulsion units thus
counteracting directly part of the low frequency wave drifting force and yawing moment acting on the vessel.
As seen from Fig.
4,
this contribution is in the order of double the total force given in Figure 3. The thrust control signal generated by equation 43 should therefore be reduced by about a factor 2 in order to cancel, as much as possible, the low frequency wave drif-ting forces by the thrust forces of the propulsion units. This means that the nett force on the vessel is reduced and hence the low fre-quency horizontal motions will be considerably reduced compared to the case without this wave drifting force compensation.In Table
5,
the nett mean and low frequency surge wave drifting force on the barge is given for the case that the relative wave height con-tribution is reduced by a factor 2 and then used as a thrust controlsignal aimed at compensating the total mean and low frequency force acting on the vessel. These values are given in Table 3. It is
assumed that the reaction of the propulsion units to the thrust con-trol signal is instantaneous.
Comparison of the results in Table 5 with those given in Table 3 shows that the control signal based on the relative wave height effectively reduces the nett force as has also been found experi-mentally in (14). This example serves also to illustrate the use that can be made of computations of the mean and low frequency wave drifting forces using the method of direct integration of pressures. CONCLUSIONS
In the aforegoing, a simple method to estimate the contribution to
the low frequency second order wave drifting forces due to the
second order potential was presented. For cylindrical cross sections the agreement between the approximation and exact two-dimensional results appears to be fair in the region of not too large difference frequencies or frequencies of the wave groups. At this time, however, this method appears to be the only way of estimating for
three-dimensional cases, a contribution which, for many practical situa-tions of moored structures, is probably small compared to contri-butions due to products of first order quantities. The approximation will give best results when the contribution to the second order
43
potential of the first order diffraction waves and waves due to body motions are negligible. This requirement is probably satisfied more by vessels such as semi-submersibles than by ordinary ship or barge
shapes.
One of the subjects left out of consideration in this paper is the applicability of the mean wave drift forces in regular waves for estimating the low frequency oscillatory wave drifting forces in irregular waves.
Faltinsen and
Liken
(10) concluded from the calculated values of the low freauency wave drifting force in beam waves on a two-dimen-sional cylinder that in many cases the mean wave drifting force could be used for estimating purposes.Preliminary results presented here (see Table 3) on the surge force in head waves show that this assumption is generally valid if the natural surge frequency of the moored vessel is lower than about 0.1 rad./sec., although in Table 3 the amplitude of the low fre-quency surge force in wave groups with a frefre-quency of 0.1 rad./sec. consisting of regular waves with frequencies of 0.7 rad./sec. and
0.8
rad./sec. shows a value of 36 tons/m2 as compared with between 22 and 26 tons/m2 on basis of the regular wave hypothesis. After further verification of the computed results by means of model tests more attention will be paid to this subject.REFERENCES
Hsu, F.A. and Blenkarn, K.A.: "Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas".
Paper OTC
1159,
Houston,1970.
Remery, G.F.M., Hermans, A.J.: "The slow drift oscillation of a moored object in random seas. Soc. of Petroleum Engineers Journal,
1972.
Rye, H., Rynning, S., Moshagen, H.: "On the slow drift ogcillations of moored structures", Paper OTC
2366,
Houston,1975.
Arai, S., Nekado, Y., Takagi, M.: "Study on the motion of a moored vessel among irregular. waves".
J.S.N.A. Japan, Vol. 140, Dec.
1976.
Van Oortmerssen, G.: "The motions of a moored ship in waves". N.S.M.B. publication No.
510, 1976.
Numata, E., Michel, W.H., and McClure, A.C.: "Assessment of sta-bility requirements for semi-submersible units". Trans. SNAME,
1976.
Faltinsen, O.M. and Liken, A.E. "Drift forces and slowly varying forces on ships and offshore structures in waves-". Norwegian Maritime Research No. 1,
1978.
Pinkster, J.A., Van Oortmerssen, G.: "Computation of the first and second order wave forces on bodies oscillating in regular waves".
Second International Conference on Numerical Ship Hydrodynamics, Berkeley,
1977.
Pinkster, J.A.: "Low frequency second order wave forces on vessels moored at sea".
Eleventh Symposium on Naval Hydrodynamics, University College, London,
1976.
Faltinsen, O.M. and Liken, A.E.: "Slow drift oscillations of a ship in irregular waves".
Applied Ocean Research No. 1,
1978.
Newman, J.N.: "Second order, slowly-varying forces on vessels in irregular waves"._
International Symposium of Marine vehicles and
structures in waves.
Bowers, E.C.: "Long period oscillations of moored ships subject to short wave seas". Paper presented to R.I.N.A., August, 1975.
Vugts, J.H.: "The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface".
Report No. 194, Laboratorium voor Scheepsbouw-kunde, Delft, 1968.
Pinkster, J.A.: "Wave feed-forward as a means to improve dynamic positioning".
Paper No. OTC 3057. Offshore Technology Con-ference, Houston, 1978.
Pinkster, J.A.: "Second order low frequency wave exciting forces on structures in irregular waves".
(To be published). NOMENCLATURE
frequency
in
rad./sec.-random phase angle
amplitude of component with frequency w.
1
a large number,
in-
and out-of-phase components of the second order low frequency forceamplitude of second order force
F.
0 total Velocity potential including all orders
a small parameter (1)
0 , first and second order approximations to the total potential
specific mass of water acceleration of gravity (1)
matrix containing first order oscillatory angular motions, see ref. (8)
direction cosine, positive outwards from body
dl length element along the waterline mass of body in air
instantaneous wetted surface mean wetted surface
7?1)
first order linear motion vector of a point on
So dS surface element of S or
So
,position vector of surface element in body axes
wave number waterdepth
X3 vertical co-ordinate of a point positive if above the mean surface
a amplitude of a regular wave
-(1)
first order wave exciting force or moment
gid transformed gravity constant
spectral density of irregular waves frequency of second Order force
r )
'r first order relative wave height at a point along the
waterline including effect of undisturbed waves, dif-fraction and body motions
length of the barge
draft of horizontal two-dimensional cylinder with breadth equal to 2.d.
matrix containing mass of body
matrix containing moment of inertia
Table 1. Second order potential contribution to drift forces on a
cylinder in beam waves: Exact values according to Faltinsen and 1,(6ken.
F pgLC .C2
Table 2. Second order potential contribution to drift forces on a
cylinder in beam waves Approximation
as
given in this paper:1
0.59 0.72 0.84 0.95 1.12 1 g 0.59 0 0,02 0.01 0.02 0,13 0.72 0.02 0 0.03 0.01 0.10 0.84 0.01 0.03 0 0.03 0.04 0.95 0.02 0.01 0.03 0 0,06 1.12 0.13 0.10 0.04 0.06 0 woR 0 0.59 0.72 0,84 0.95 1.12 0.59 0 0.02 0,09 0.16 0.26 0.72 0.02 0 0.03 0.10 0.21 0.84 0,09 0.03 0 0.03 0.15 0.95 0,16 0.10 0.03 0 0.08 1.12 0.26 0,21 0.15 0.08 FpgLC C2
63 2 0.5 0.6 0.7 0.8 0.9 15 1.0 1.1 0.5 8 17 20 0.6 0.7 0.8 14 13 8 6 11 114 0 0 14 12 0,9 15 1.0 17 14 /4 0 1.1 20 13 114 12
Table 3. Computed total second order surge force_on a barge in head
seas. tfirn2 \\\1\\ (1)2 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.5 0 2 14 47 46 25 71 0,6 2 17 48 53 19 74 0.7 14 17 -22 36 41 20 49 0.8 47 48 36 -26 32 31 24 0.9 46 53 41 32 -27 31 26 1.0 25 19 20 31 31 -29 28 1.1 71 74 49 24 26 28 -29
Table Second order potential contribution tb sur e force on a
Table 5. Nett second order surge force left after application of wave drift compensation on a barge in head seas.
X3 (1) F3 IX3 &O) 0) F3 x5
Fig. 1- Cohtribution of pitch ang.le and heave
force
to the second order surge force.F tf/m2 C2
wl
W2\
0.5 0,6 _0.7 0.8 0,9 1,0 1.1 0.5 0 9 6 28 43 7 61 0.6 6 22 37 10 52 0,76.
6 +14 9 14 11 20 0.8 28 22 9 0.9 43 37 14 7 1.0 7 10 11 1,1 61 52 20 7 73.6 2 4 oo
N"I*L7
5 METRES 1.2 .x3 x3 COMPUTED MEASURED15w
Lg
LENGTH 150 METRES BREADTH 50 METRES DRAFT 10 METRES 5 METRESFig. 2. Body plan of barge including facet and control point distribution.
3'O 4.5
,I I
'Fig.. 4.
Components of computed mean surge drift force in head waVeszWAVE
All.AL.aliAl.
WITHOUT WAVEWAVE-FEED-FORWARD
---- WITH WAVE -FEED-FORWARD
50 100 TIME (sec.)
Fig. 5. Sway and yaw motions of a 200 KDWT D .P . vessel in irregu-lar bow auartering seas. Significant wave height
4.9
m.1,5
