The effects of hull form variations on the roll damping of warships

THE EFFECTS OF

HULL FORM VARIA TIONS ON THE

ROLL DAMPING OF WARSHIPS

received his B.S. from the University of Toronto in mathe-matics and physics in 1977, his M.A. from Princeton Uni-versity in applied mathematics in 1978, and his Ph.D. in mathematical physics in 1982, also from Princeton. He joined Defence Research Establishment Atlantic in 1982 as a member of the Ship Dynamics Group, where his interests have been in roll prediction methods, slamming, and rudder-roll stabiliza-tion. He is the author of ten technical papers, and a member of the International Association of Mathematical Physicists.

A B S T R A C T

The results of a study to identify warship hull form char-acteristics which maximize roll damping are described. The two main hull form characteristics which influence the roll damping of warships at the cruise speed are the separation of the appendages from the centre of gravity, and the form of the bilge. The influence of other hull characteristics is rauch less important. To maximize roll damping for a given set of ap-pendages, the hull should be designed with a small bilge radius, and with the appendages located as far as possible from the centre of gravity. A n example is given of two hulls with identical values of (g), B / T , C B , CW, and Cp with significantly different roll responses. These parameters should be chosen to optimize the pitch and heave characteristics of ships, and the bilge radius should be minimized to reduce roll.

N O T A T I O N Ajk added mass coefficient

B ship beam

B A viscous roll damping o f appendages other than

bilge keels

BBK bilge keel component o f roll damping B E eddy-making component o f roll damping B H skin friction o f the hull

Bjk damping coefficient

B44 roU damping coefficient

B44 nondimensional roll decay coefficient, 0 W0B44

^44 = — — — 2C44

B44F roll damping due to dynamic l i f t on the

append-ages

B44V viscous roll damping

B44W wave-making roll damping '

b bilge keel breadth

C B block coefficient

CG centre of gravity

CLQ l i f t curve slope

Cp prismatic coefficient

Cp non-dimensional pressure coefficient, Cp = P*

Vl e(row^4)^

Cp maximum value o f the pressure coefficient Cp

Cp minimum value of the pressure coefficient Cp Cw waterplane coefficient

C44 roll restoring coefficient

Fn Froude number

F L amplitude of the lift force on an appendage F4 roll exciting moment

G M metacentric height I4 roll moment of inertia

K G distance f r o m the keel to the centre o f gravity

Lx sectional length

M N instantaneous roll resisting moment due to the

normal force on the bilge keels

M S instantaneous roll resisting moment due to the

hull surface pressure @ length/displacement ratio

n unit outward normal to the hull

P* difference between the hull surface pressure and the hydrostatic pressure

P + dimensional pressure corresponding to C^ P" dimensional pressure corresponding to Cp

ro distance f r o m the centre of gravity to an append-age root

To vector f r o m the centre o f gravity to an appendage root

R bilge radius

S appendage surface area T ship draft

U ship speed

yo distance f r o m a chine to the centreline zo distance f r o m a chine to the wateriine

a flare angle at the wateriine jS angle o f attack

e deadrise angle

r]2 sway displacement ri2 sway velocity

i/2 sway acceleration

r]4 roll angle rj4 roll amplitude 7]4 roll velocity ri4 roll acceleration r]6 yaw angle i]6 yaw velocity

rje yaw acceleration

Q mass density of water 0) angular frequency

wo roll natural frequency X vector cross product

I N T R O D U C T I O N

The relationship between hull f o r m coefficients and the seakeeping performance o f ships is now well estab-lished. Schmitke and Murdey [1] used model data f r o m

the National Research Council of Canada Series for Fast Surface Ships and the results of a computer gen-erated parametric study to identify hull form coeffi-cients which are important in determining vertical plane responses and discussed seakeeping and resistance trade-offs. Bales [2] performed seakeeping calculations for twenty destroyer huUs and used regression analysis to develop a model which relates hull geometry to a sea-keeping figure of merit.

References [1] and [2] limited their attention to the vertical plane responses. Of the lateral plane responses, roll is the most important from the seakeeping point of view. Bales assumed roll could be adequately controlled by proper appendage design, while Schmitke and Murdey cite earlier work by Schmitke [3] which con-cluded that the influence of hull form on rolling is of secondary importance.

In spite of the conclusion of Reference [3], there is still interest in determining the relationship between hull form and roll response. Recently, Walden and Kopp [4] computed ship motions for 17 waiship hull forms and performed a regression analysis in an attempt to link hull form parameters with the roll response. They concluded that the dominant influence on roll re-sponse is metacentric height, G M , and that it was not possible to provide guidance on other parameters, since the effects could be positive or negative depending on the value of G M .

In this paper, the influence of hull form variations on roll motions is reexamined from the point of view of identifying hull form characteristics which maximize roll damping.

THEORETICAL BACKGROUND

Defence Research Establishment Atlantic (DREA) has recently updated the methods used to compute the viscous roll damping component in ship motion pro-gram SHIPMO [5],[6],[7J. Roll damping coefficients predicted by the new version of the program, SHIP-MOB, were compared with full-scale trial data for a number of warships, and the correlation between theory and experiment was usually fair to good, given the sig-nificant experimental error inherent in full scale trials.

SH1PM03 computes the lateral ship motions using the coupled sway-roll-yaw equations as derived by Salvesen, Tuck, and Faltinsen [8]. The equations are transformed to a stability axis system as described by Schmitke [9]. The coordinate system translates with the mean velocity of the ship with the origin located at the equilibrium position of the centre of gravity. The x-axis points to the bow, and the y-axis to port, both being parallel to the undisturbed free surface, and z is measured vertically upwards.

The roll equation is given by

A24Ï/2 + B24»j2 + ( A 4 4 -I- 14)^4 + B44»)4 + C44>?4

+ A46i;6 + B^eijfi = F4e'"', (1) where 7j2, i)4, and •qe denote sway, roll and yaw,

respec-tively. The Ajk and Bjk are added mass and damping

co-efficients respectively, U is the roll moment of inertia,

C44 is the roll restoring coefficient, F4 is the roll exciting

moment, and to is the angular frequency.

It is assumed that the roll damping coefficient B44 can be written down as the sum of three terms

B44 = B44W + B44F + B44V, (2) where B44W is the wave-making damping, B44F is the

damping due to dynamic l i f t , and B44V is the viscous roll damping. It is further assumed that the viscous roll damping can be represented by the following sum

B44V = BBK + B E + B H + B A , (3) where BBK denotes the bilge keel contribution. B E

denotes the hull eddy-making damping, B H denotes the skin friction of the hull, and B A denotes the viscous effect of appendages other than bilge keels.

The sectional wave-making damping is computed in SHIPM03 using either the conformal mapping method or the Frank close-fit method [10]. The dynamic lift component is computed using the method of Schmitke [9]. The bilge keel component is computed using the Kato method [11]. Depending on the form of the sec-tion, the sectional eddy damping at zero forward speed is computed using either the method of Ikeda, Himeno, and Tanaka [12] or that o f Tanaka [13]. The effect of forward speed on the eddy damping component is taken into account using the correction factor o f Ikeda et al. [12]. The skin friction component is computed using the Kato method [14] combined with the forward speed cor-rection of Tamiya and Komura [15]. The viscous effect of appendages other than bilge keels is obtained at zero speed by treating the appendages as oscillating flat plates. A t positive forward speed, the eddy damping correction factor of Ikeda et al. is used. This correction factor was derived for ship hulls and not flat plates, but given the relatively small size of B A , this procedure is sufficiently accurate for practical purposes. Interference effects between the various components o f roll damping are neglected except for those between fins and bilge keels. The methods and algorithms used to predict the viscous roll damping in SHIPM03 are described in more detail in Reference [7].

This division into components is a convenient method of calculating roll damping, but physically such a divi-sion may be impossible. For example, it is clear that the addition of bilge keels to a ship changes the eddy-mak-ing dampeddy-mak-ing by provideddy-mak-ing sharp edges f r o m which most eddies are shed; however, this need not interfere with the application of the method. In model tests, the roll damping due to bilge keels can be determined as the dif-ference between the roll damping of a ship model with and without bilge keels. Empirical formulae for bilge keel damping can be developed on that basis, and total roll damping computed as the sum of the various com-ponents. It should be emphasized that the methods used to predict (he viscous component are empirical; hence, they are potentially limited to hulls which fall within the range of the data base.

The viscous componeiUs are, oi" course, nonlinear. These terms are inciudec! in the linear model by defining ociuivalenl linear terms which dissipate the same amoimt of energy in a roll cycle as Ihe nonlinear viscous terms.

ROLL D A M P I N G OF WARSHIP HULLS In this section, Ihc roll damping of seven warship hulls of frigate/deslroyer size is computed, and the im-portance of the various components is discussed. The results will be presented in terms of the nondiinensional roll decay coefficient 844 defined via

where wo denotes the roll natural frequency.

Loading conditions have a great influence on the roll response. In order to identify the effects of hull fonn on roll damping, sirnilar loading conditions were used for all of the hulls, KG was assumed to be thirty per cent larger than the draft, G M was taken as 8 per cent of beam, and the roll radius of gyration in water was as-sumed to be 40 per cent of beam. It was not possible to normalize the displacement in the same way. Displace-ments typical of normal operating conditions were chosen for all of the ships.

The roll damping was calculated at the roll natural frequency of each hull. The viscous terms are all aitipli-lude dependent. The roll amplitude was taken to be 7.07 degrees, corresponding to an RMS roll of 5 degrees. The relative importance of the various components changes with forward speed. A t low speeds, the viscous com-ponent is most significant, whUe at high speed, the dy-namic lift component is dominant. A t the cruise speed, both the viscous and the dynamic lift components are important. The forward speed for the calculations in this paper was 18 knots, which represents a typical cruise speed for warships.

Table 1 shows the calculated roll damping coefficients for the seven hulls, designated A to G. For each hull, the total roll damping is shown, along with the various com-ponents. A l l of the hulls have bilge keels: hulls D and E have one pair of passive fins, while hulls F and G have two pairs o f passive fins. A sample bodyplan for one of the warships (Hull A ) is shown in Figure 1.

It is clear from Table 1 that for warship hulls at the cruise speed, the only significant components of roll damping are the dynamic l i f t component and the bilge

Tnble I: Cnlculnled Roll Damping CoefficieiKs for Wnrslilps

Hull B44 B44\V B44F BDK U E B H B A A 0.096 0.004 0.040 0.051 0.000 0.001 0.000 B 0.129 0.006 0.081 0.041 0.000 0.001 0.000 C 0.165 0.006 0.076 0.082 0.000 0.001 0.000 D 0.119 0.005 0.093 0.020 0.000 0.001 0.000 E 0.117 0.004 0.069 0.043 0.000 0.001 0.000 F 0.117 0.003 0.075 0.038 0.000 0.001 0.000 G 0.118 0.003 0.084 0.030 0.000 0.001 0.000

keel componeni. The components due to eddy-making damping and the viscous effects of appendages other lhan bilge keels are essentially zero at this speed, but arc of significance at lower speeds (Fn < = 0.2). The skin friction component is only important at the model scale, An analy,sis of (he factors affecting the magnitudes of the lift and bilge keel components of roll damping will now be given.

The dynamic lift component is computed by summing the dynamic lift due to the skeg, the rudder, the fins, and the propeller shaft brackets. Interference between the hull and the appendages is neglected. The dynamic lift of the hull itself is assumed to be adequately approx-imated by the dynamic lift of the skeg. Schmitke has shown this to be a reasonable assumption for single-screw merchant ships with large skegs [16]. In general, this approximation may lead to conservative estimates of this component of roll damping.

The three factors which influence the magnitude of the lift component are the dimensions, the orientation, and the location of the appendages. The effect o f vary-ing the appendage size is straightforward, and indepen-dent of the hull form. Vaiying the onentation of an appendage changes the angle between the roll lift force and the vector i^o connecting the centre o f gravity, CG, with the appendage root. To maximize the effectiveness of a given appendage, the normal to the surface of the appendage should be perpendicular to To, This effect is again independent of hull form.

To maximize the lift damping of a given appendage, the distance ro separating the appendage from the centre of gravity should be as large as possible. The rolling velocity of a ship moving with forward speed, U , in-duces an angle o f attack, /3, on the appendage, given to the first order by /3 = ro«^4/U, where 54 is the roll amplitude. The amplitude of the hft force F L on a given appendage is given by

F L = Vl eU^SCLa/S, (5)

where q is the mass density of water, S is the appendage surface area, and C L » is^the hft curve slope. It follows from Equation 5 that F L is proportional to ro. The moment of this force about the roll axis is obtained by multiplying by the moment arm, ro; hence, the lift damping component is proportional to rS.

As indicated above, the bilge keel component of damping is computed in SHIPM03 using the empirical formulation of Kato [11]. Kato found that the

effective-Figure 1. Bodyplnn of Hull A .

ness of the bilge keels depended on bilge keel area, breadth and aspect ratio, Reynold's number, the dis-tance ro from the CG to the bilge keel root, draft, KG, and the form of the bilge. The dependence of bilge keel damping on the bilge keel dimensions and the Reynold's number is independent of the hull form and will not be discussed here.

Increasing the separation between fhe bilge keels and the CG increases the effectiveness o f t h e bilge keels. The viscous damping force on the bilge keel depends on the square of the local rolling velocity; hence, the bilge keel damping moment is proportional to ro- Increasing KG and the draft T also increases the effectiveness of the bilge keels, but the dependence of bilge keel damping on KG and T is not as strong as the dependence on ro.

As first pointed out by Bryan [17], the bilge keel damping consists of two components, one due to the normal force on the bilge keel, and the second due to the pressure on the hull surface generated by the bilge keels. The dependence of bilge keel damping on the form of the bilge is due to this second component. The most im-portant factor is the bilge radius, R, defined as the radius of curvature of the hull surface at the location of the bilge keel, but the damping is also influenced by the deadrise angle and the flare angle at the wateriine.

In the method of Ikeda et al, [12], the two com-ponents of bilge keel damping are computed separately. Figure 2, which is reproduced from Reference [12], shows the hull surface pressure distribution induced by bilge keels in a model test. The results are shown in terms o f the non-dimensional pressure coefficient, Cp, defined via

Cp = P* '/2e(roco)j4)

(6)

where P* is the difference between the hull surface pres-sure and the hydrostatic prespres-sure.

The dependence of bilge keel damping on hull geome-try will be illustrated in an idealized situation, Ikeda et al. adopt a more sophisticated approach, but to develop some physical intuition, it suffices to consider a simple case. Figure 3 illustrates a hard-chine section of beam B, draft T, deadrise angle e, and flare angle at the wateriine

Cp r MEASURED •• 0.199 4 2 0 -2 -4 b= 0.009 m vl>MEAsy T % -0.295 7 '1 12 -2 - 4

Figure 2. Measured hull surface pressure distribution (from Reference |12|).

a. Let b be the bilge keel breadth, and denote by yo and Zo the distances from the chine to the centreline and wateriine, respectively. From geometry

and yo Zo = T l - T tan a 1 - tan a tan e T - y t a n e 1 - tan a tan e (7) (8)

Assume that the bilge keels induce a linear pressure distnbution on the hull surface as illustrated in Figure 4, The maximum pressure coefficient is denoted by Cp, the minimum by Cp and the pressure goes to zero at the wateriine and on the centreline of the ship.

The instantaneous roll resisting moment due to the hull surface pressure, Ms, will now be evaluated. Denote by P"^ and P" the dimensional pressures corre-sponding to Cp+ and Cp, respectively; Let r be the vector from the CG to an arbitrary point on the hull, as shown in Figure 4. Let P* denote the surface pressure at this point, and let ft be the unit outward normal to the hull, Ms is obtained by integrating |r X P*n| over the hull surface, where X denotes the vector cross product. The result is

Figure 3. Hard-chine section for hull surface pressure example.

1

m

ty:

m

Figure 4. Idealized hull surface pressure dLslribulion for hard-chine section.

= (P^ - P-) X

3 cos e 3 cos a

yoT sin e 2

zoB sin a

\vliere Lx is the sectional length. For small flare angles and deadrise angles (a, e < = 10 degrees) such as us-ually occur on waiship hull forms, Equation 9 can be approximated by the expression

Ms Lx

= - (P +

12 P-)(B'-+ 4 T 2 - 7 ( a + £)BT), (10) where a and e are expressed in radians.

The instantaneous roll resisting moment due to the normal force on the bilge keels, M N , will now be consid-ered. The pressures on the two sides of the bilge keels are P+ and P". Assuming that the pressure distribution is uniform over each side o f the bilge keels, the result is

M N Lx

^ = 2 b ( P - - p- ) V y g + zg. ( U )

For small a and e, this becomes M N

b(P+ - P-) V B ^ -I- 4T= - 4(« + 6 ) B T . (12)

' - ' X

From Equations 10 and 12, the flare angle and the dead-rise angle may be seen to reduce both components o f bilge keel damping.

To estimate the relative importance o f the two com-ponents of bilge keel damping, the ratio o f Equations 10 and 12 can be computed. To first order, the result is

M S ^ V B ^ + 4T^ ƒ, 5(a + e ) ( B / T ) l

M N 12b (B/T)2 + 4 (13)

For the warships considered in this paper, the average falue o f this ratio is 1.4. Thus, for hard-chine sections, ;he hull surface pressure component o f bilge keel damp-ng is larger than the normal force component.

igure 5. Effect of hull surface pressure distribulioii on n ection mth rounded bilges.

For sections with rounded bilges, such as usually occur on warships, the hull surface pressure component of bilge keel damping is no longer so large. Figure 5 illu-strates a typical section. The presence o f bilge keels again alters the pressure distribution on the hull surface, but in this case, the normal to the hull surface is almost parallel to the radius vector, so that I F X P*n| is small where the hull surface pressure is the largest. For ex-tremely rounded sections, the hull surface pressure com-ponent of bilge keel damping is essentially zero. For the hard-chine section considered above, the hull surface pressure component was larger than the normal force component. This implies that the effectiveness o f the bilge keels is greatly influenced by the bilge radius, and that decreasing R will result in large increases in roll damping. The influence of R on the effectiveness of tiie bilge keels was first noted by Bryan [17].

The above discussion shows that the two main hull form characteristics which influence the roll damping o f warships at the cruise speed are the separation o f the appendages from the centre of gravity, and the form of the bilge. The influence o f other hull characteristics is much less important. To maximize roll damping for a given set o f appendages, the hull should be designed with a small bilge radius, and with the appendages located as far as possible from the centre o f gravity. A n example o f such a hull is given in the next section.

H A R D CHINES FOR WARSHIP H U L L S To illustrate the effect of appendage location and form o f the bilge on roll damping, a hard chine variant of Hull A was denved. A bodyplan for Hull A is shown in Figure 1. The new hull will be called HuU H , and is illustrated in Figure 6. The two hulls have identical waterplanes and area cui-ves. In order to keep the same area curve, the deadrise angle and the waterhne flare angle had to be increased, which tends to decrease roU damping; however, the greatly reduced bilge radius wiU be seen to override this effect. Both huUs were fitted with identical bilge keels o f breadth 0.61 metres and length 41.8 metres. The bilge keel trace for HuU A is shown in Figure 1. The bilge keel o f HuU H was fitted along the chine hne. The separation between the CG and the bUge keel is slightly larger for HuU H than for HuU A . The skeg o f HuU A is faired into the hull. HuU H was fitted with an unfaired skeg o f equal planform.

A potential flow calculation was performed for Hull H using a first-order panel method as described by

Figure 6. Bodyplan of Hull H .

Mackay and Hally [18], It was found that the chine loca-tion was roughly along a streamline, and that the hydro-dynamic behaviour of the hull was similar to that pre-dicted for other warship hulls, at least in its gross characteristics. It is expected, of course, that the pres-ence of the chine would increase the vortex shedding along the bilge, an effect that is not modelled in the potential flow calculation. It is also expected that Hull H would have higher resistance than Hull A , but the difference has not been quantified.

The roll damping coefficients calculated for the two hulls are shown in Table 2. The speed, frequency, and roll angle are the same as for Table 1. In performing the calculations for the hard chine hull, the bilge radius was taken to be 3 feet, rather than zero, to avoid going out-side the range of the-data base of Reference [11]; there-fore, it is expected that the calculated bilge keel com-ponent o f roll damping is conservative. The total roll damping for the hard chine variant is eighty per cent larger than that o f the original hull, and the bilge keel component is more than twice as large.

So far, the discussion has been limited to roll damp-ing. To show the significance of this increase in roll damping, the RMS roll response of Hulls A and H in sea state 5 was computed using SHIPM03. The seaway was represented by a Bretschneider spectrum [19] with a sig-nificant wave height of 3.25 metres, and a modal wave period of 9.7 seconds. Both unidirectional and short-crested seas with a 90 degree spreading angle were con-sidered.

The RMS roll responses in long- and short-crested seas are plotted to a base of heading angle in Figures 7 and 8, respectively. A heading angle of 180 degrees rep-resents head seas. A t the worst heading, the RMS roll of Hull H is 27 per cent less than the roll of Hull A in uni-directional seas, and 21 per cent less in short-crested seas,

Hull H has a substantially larger operational envelope than Hull A as a result of its superior roll performance. Reference [4] considers a hull to be fully operational only i f the RMS roll is less than four degrees. Referring to Figures 7 and 8, it can be seen that Hull H has im-proved tactical options in quartering seas compared with Hull A .

It is concluded that hull form has a significant effect on the roll response of warships, and in particular, that the bilge radius is a key hull form parameter for roll. This finding is different from the conclusions of Ref-erences [3] and [4], which did not consider bilge radius. The hull form parameters considered in Reference [3] were the length/displacement ratio, ( M ) , beam/draft ratio, B / T , block coefficient, C B , and water plane co-efficient, Cw. Reference [4] considered B / T , Cw, and

Table 2: CHlculiited Roll Damping Coefficients for Warship A and ils Hard-Cliine VarianI

Hull B44 B44VV B,MF BnK Br: B H B A A 0 . 0 9 5 0.004 0 . 0 4 0 0.051 0 . 0 0 0 0.001 0 . 0 0 0 H 0 . 1 7 6 0.0O4 0 . 0 4 0 0.131 0 . 0 0 0 0.001 0 . 0 0 0

the prismatic coefficient, Cp. (Other factors not related to hull form were also considered in these references.) The effect of these parameters on the roll response was found to be small, and it was recommended that hull form parameters be optinuzed for vertical plane re-sponses. Hulls A and H have identical values of all of the,se parameters, and yet have different roll character-istics. It is concluded that the bilge radius has a more important effect on the roll response than the above parameters.

References [3] and [4] emphasized the benefits to be obtained from active stabilization. In Reference [3], Schmitke found that at a forward speed of 20 knots, active stabilization could reduce the roll at worst head-ing by more than 50 per cent. This is a larger roll reduc-tion than can be achieved through any changes in hull design. The effectiveness of fin and rudder stabilizers is strongly influenced by the forward speed, and for Froude numbers less than about 0,2, the bilge keel com-ponent of roll damping becomes dominant. A t low speeds, the changes to hull form discussed here will result in significant roll reduction, even for hulls with active stabilization. 8 . 0 8 . 0 1 1 1 _ 1 8 0 ° = H E A D S E A S 0 6 0 / \ H U L L A RM S ROL L ( 4 . 0 20 / / H U L L H X V

/

Figure 7 . Roll response of HuUs A and H In sea slate 5 —

long-creslcd seas. 8 0 8 0 1 1 1 1 8 0 ° = HEAD SEAS __^60 -0 -J " \ H U L L A

«

Figure 8 . Roll response of Hulls A and H in sea slate 5 —

shorl-cresled seas. 60 Naval Engineers Journal, Seplember 1987

C O N C L U S I O N S

I has been shown (hat the two main hull form char-Mistics which influence (he roll clamping of war-)s at the cruise speed are the separation of the ap-dages from the centre of gravity, and the form of the e. Significant increases in roll damping, and uclions in roll response at the worst heading can be ieved by designing hulls with small bilge radii, and 1 the appendages located as far as possible from the

m example has been given of two hulls with identical erplanes and area curves, but significantly different responses. It is concluded that the bilge radius has a :e important effect on the roll response than @ l , r, Cb, C W , and C p , These parameters s h o u l d b e sen to optimize the pitch and heave characteristics of )s, and the bilge radius minimized to reduce roll,

A C K N O V / L E D G E M E N T

; is a pleasure to thank D r . David Hally for valuable u-'iions about the hydrodynamics of the hard-chine , and for performing the potential flow calculations.

R E F E R E N C E S

Schmitke, R . T . and D . C Murdey: "Seakeeping and Resistance Trade-Offs in Frigate Hull Form Design," Thirteenth Symposium on Naval Hydrodynamics, Tokyo, Japan, 1980, pp. 455-478.

Bales, N.K.: "Optimizing the Seakeeping Performance of Destroyer-Type Hulls," Thirteenth Symposium on Naval Hydrodynamics, Tokyo, Japan, 1980, pp. 479-504, Schmitke, R . T . : "The Influence of Displacement, Hull Form, Appendages, Metacentric Height, and Stabiliza-tion on Frigate Rolling in Irregular Seas," S N A M E Star Symposium, 1980, pp. 203-217.

Walden, D . A , and P . J . Kopp: "Influence of Hull Form Parameters on Roll Motion," D T N S R D C Report SPD-1143-01, 1985,

Schmitke, R . T . and B . T . Whitten: " A F O R T R A N Pro-gram to Predict Ship Motions in Waves," D R E A Tech-nical Memorandum 81/C, 1981.

[6] Oiaham, R, and O. Millar: "SH1PM02: An Updated U.ser's Manual for the SHIPMO Computer Program Incorporating Measured Sea Spectra and Wave Loads," D R E A Technical Memorandum 84/0, 1984.

[7] Graham, R.: "SHIPM03: Improved Viscous Roll Damp-ing Predictions for the SHIPMO Computer Program," D R E A Technical Memorandum 86/212, 1986.

(8) Salvesen, N., E . O . Tuck, and O. Faltinsen: "Ship Mo-tions and Sea Loads," rnins. SNAME, Vol. 78, (1970) pp. 250-287.

[9) Schmitke, R . T . : "Ship Sway, Roll, and Yaw Motions in Oblique Seas," Tims. SNAME, Vol. 86, (1978) pp. 26-46.

[10] Frank, W.: "Oscillation of Cylinders In or Below the Free Surface of Deep Fluids," NSRDC Report 2375,

1967.

[11] Kato, H . : "Effect of Bilge Keels on the Rolling of Ships," Memories of Ihe Defence Academy, Japan, Vol. 4, (1966) pp, 369-384.

[12] Ikeda, Y . , Y. Himeno, and N. Tanaka: " A Prediction Method for Ship Roll Damping," University of Osaka Prefecture, Department of Naval Architecture Report 00405, 1978.

[13] Tanaka, N.: " A Study on the Bilge Keels (Part 4 — On the Eddy-Making Resistance to the Rolling of a Ship Hull)," Jour. Soc. of Naval Arch., Japan, Vol, 109, (1960) pp. 205-212.

[14] Kato, H . : "On the Frictional Resistance to the Rolling of Ships," Jour. Soc. of Naval Arch., Japan, Vol. 102, (1958) pp. 115-122.

[15] Tamiya, S. and T , Komura: "Topics on Ship Rolling at Advance Forward Speed," Jour. Soc. of Naval Arch., Japan, Vol, 132, (1972) pp, 159-168.

[16] Schmitke, R . T . : " A Note on Lift-Generated Roll Damp-ing of Ships with Full Midship Section and Large Skeg," Proceedings of the Nineteenth A , T , T . C . , 1980, pp. 327-336,

[17] Bryan, G , H , : "The Action ofthe Bilge Keels," T , I , N . A „ Vol. 62, (1900) pp. 198-219.

[18] Mackay, M. and D . Hally: "The Calculation of Potential Flow around Ship Hulls," D R E A Technical Memoran-dum 85/204, 1985.

[19] Bretschneider, C . L . : "Wave Variability and Wave Spec-tra for Wind-Generated Gravity Waves," Beach Erosion Board, Corps of Engineers, Technical Memo 118, 1959.