Theoretical motions of two yacht models in regular head seas on the basis of damping coefficients derived for wide V-forms

STEVENS NSTTUTE OF TECH

DAVIDSON LABORATORY

CASTLE POINT STATION HOBOKEN. NEW JERSEY

Theoretical Motions of Two Yacht Models in Regular Head Seas on the Basis of

Damping Coefficients Derived for Wide V-Forms

by Paul Kaplan and Jacobs

NOLOGY

Lab.

_{y. Scheeps1ouwku,4}

Technische HogeschooI

Deift

ARCHIEF

-This work was carried out for the Office of Naval Research under contract Nonr 263-10.

A _{(D. L. Project FX 2057)}

introduct ion

The linearized theory of ship motions in regular head or following

seas which was developed by Korvin-Kroukovsky in references [i.] and (2]

was applied in the latter paper to the calculation of mottons of eight

widely different ship models.

Comparisons between the computed motions

and motions observed in towing-tank tests over a range of model speeds

and at several wave lengths showed generally satisfactory agreement in

the cases of several typical commercial ship forms.

The agreement was

excellent in the case of a destroyer model. Of the eight, this model

came closest to being wall'sidad at the LWL, and in this sense conformed

to the assumptions made in the linear motions theory and in the theoretical

computations of virtual mass and damping coefficients.

These were for

Lewis' (3] analytically defined sections resembling normal ship sections.

'The virtual mass coefficients were taken from Lewis [3] and Prohaska (4)

and the damping coefficients from Grim [5].

The analytical procedure was found to be inadequate in the case

of a sailing yacht characterized by large slopes of sides at the LWL

throughout its length and by pronounced bow and stern overhangs and a

cutaway forefoot (see Fig. i).

Failure was almost complete in estimating

amplitudes and phases of pitching and heaving motion, as also in indicating

trends with speed, both for the original yacht model and for its lengthened

counterpart.

The original model

of 370, was the stubbiest of the

1699B, with a displacement-length ratio

eight studied.

The lengthened yacht model

1699D (see Table i. for particulars) had the same section forms spaced

wider apart and dimensioned so that the displacement-length ratio was that

of the destroyer, 60.

The failure tri the case of this model indicated

that it was the section slopes and end overhangs, rather than the fineness

ratio, which violated the assumptions of the theory.

These characteristics of tie yacht form point to a possible

non-linearity in the restoring forces and moments and corresponding

cross-"coupling terms of the equations of motion.

However, it was not certain

whether it was this nonlinearity or the inexactness of the virtual mass

-and damping coefficients which was responsible for the discrepancies

between the calculated and observed motions of the yacht.

Although the

N-593

-1-Lewis forms include what appear to be V-sections, unlike the yacht's sections these are tangent to a vertical at the LWL. In ref erence.[2)

the sections

are assumed to be Lewis sections in all cases because no

theoretical or experimental information on wide V-sections was then available

in practical

form.

New Damping Coefficients

Since, on the basis of thin ship theory, , the ratto of the

amplitude of the waves caùsed by an oscillating section to the amplitude

of the

heaving motion of the section, has been computed for sections having large slopes at the LWLO The results are presented in reference

[6) which includes curve of A va. dimensionless frequency B*c2/2g for parameters of beam!.draft ratio B /H and section coefficient

whére B' is the section beam at the LWL, H the draft and S the area to the LWL. Thé..;basic section forms chosen for

illustration

in

Ref.

(6] are those given by Haskind [7].

Based on these values of new damping coefficients

in

heave and pitch, b and B ràspectively, and croSs-coupling coefficient

components e2 and 'E2 , due to dissipative

damping, have

been computed 'for the two yacht models by the _{method described in the appendix of}

reference [2). These are compared in Figs. 2 and 3 with the coefficients derived from Grim's values of . The new damping force and moment

coefficients are about one-half the earlier. The cross-coupling

isttive-dath'

i:'tr

been reduced to practical non-existences

Since the trend with frequency ò of the coefficients b and B derived from thin ship theory are similar to those

_derived

_{from Grim's} coefficients,' the effect of the generally lower level should be only' to increase the maximum predicted motion amplitudes. _{On the other hand,} the very diferent cross-coupling terms obtained _{on the bastÉ of the} thin ship development for V-forms should change the predicted motion

phases with respect to the waves,

and the sp ed at which maximum amplitude

occurs. ' '. '

N-593

-2-Revised Predictions of Motions In

The new values of b,B,e2 and E2 based on reference [6] have been substituted in the

equations

of motion of reference [2] and the theoretical pitching and heaving mottons of the two yacht models have been recomputed. The results, in

the form of

double amplitudes of heave and pitch and phase

Ïagsof maximum heave and maximum pitch after wave

node

amidships, are plotted in Figs. 47 as dash lines. _{The theoretical calculations of} reference [2], based on Grim's damping forces, are represented in these

figures by

solid lines and the experimental data of reference [8] by

circles.

As these figures show, use of the damping coefficients derived for V-'soctions with large slopes of sides at the LWL results in much closer agreement with experimental measurements. _{The correlation between} calculated and measured phase lags is _{now particularly good. The new} calculations also indicate correctly the speeds for maximum motions. The revised amplitude predictions, although at least as close and

for

the most part closer to the experimental values than the earlier cal. culations, are still too low. Also the exaggerated narrow peak in

heaving

at synchronism which occurred in tests of the lengthened yacht

model is still not indicated by the calculations, but the trend of _the new theoretical curves does

shów an

increasing ampi. if icat Ion.

Discuss ion

Although the new damping coefficients have improved the estimates of pitching and heaving motion in regular head seas

for

the two yacht models, the agreement between calculated and experimental amplitudes

is still not as good as for the other 6 models investigated _{in reference} [2]. Other factors are evidently involved.

The computation of the damping forces ta based on the assumption of two-dimensional flow. However, corrections

for

_{three-dimensional} flow indicated by Havelock [9] and Vossers [lo] (see Figs. 16 and 17 of reference [2]) would not change the damping

in

heave in the vicinity of synchronism and could increase the damping

in

pitch as much as 40°/o

Regular Head Seas

N-593

-3-in the case of the orig-3-inal model and 10°/o -3-in the case of the lengthened

yacht.

This would lower

the maximum predicted pitching amplitude and worsen the correlation with experiment.

The virtual mass coefficients employed in the calculations are those derived for the Lewis sections. Virtual mass coefficients for wide V-forms are not available. However, these are expected

to be not

too different under the assumptions of the linear theory.

It may be that

the non-linearities are important in this case, but

the usual

effect of including non-linearities is to limit the motion

amplitudes, i.e. to reduce

the

amplitude predicted

by

use of linear theory. AlSo an exSmination of the motion recorda did not show any appreciable evidence of higher harmonics or subharmonica that would usually be associated with nonlinear phenomena. Thus, some other explanation is necessary, and an obvious one is provided when cons ider

ing the exciting force and moment due to the waves. These terms are based upon developments for a semi-circular

hull

with vertical tangents at the LWL (see reference [2]). Modifications with the virtual mass

coefficients appropriate to Lewis sections allow the expressions to

_be

applied to the class

of fairly full forms

that have been

successfully

treated in reference (2)). However, it is

very

possible that the

yacht

model with sloping sides will

have a different representation for the

wave excitation

effects, as

it did for damping. On

the basis of

this

possibility it

appears that a combined experimental and theoretical program devoted to the subject of the vertical force and pitching _moment due to waves on the restrained yacht model should be carried out. An

investigation of this type, which was proposed on the basis of the pre-sent results,: has been

carried out at the

_{Davidson Laboratory as a}

mestertS

_{thesis (U].}

There ta no question, however, from the results presented

in this

report, that use of appropriate hydrodynamic coefficients ta _{a great} step forward in improving the

theoretical predictions0

N-593

-4-*By

_{Cruising Club Ruiez Rated LWL}

_{0.3 LWL + 0.7 (LWL104) where}

LWL104 is the length on the waterline at a draft of 1.04 x the

load draft.

N593

= 5*

TABLE 1.

Model Pajcü1ars

Model Ni.unber 1699B

i699D

Rated

LWL*,

ft.

4.28

5.71

Beam, ft.

1.22

0.709

Draft, ft.

0.690.

0.417

Displacement (w), lb.

5048

24.5

B1ck Coefficient

0.23

0.23

A/0.OIL)3

370

60

Radius of gyration in air, ft.

1.07

1.37

NatûtieLi pitching period,

afloat

calm water, Sac.

0.81

0.63

Natal' heaving period,

REFERENCES

Korvtn-Kroukovsky, B. V., "Investigation of Ship

Motions in

Regular

Waves", Trans. SNAMB, 1955.

(2) Korvin-Kroukovsky, B. V. and Jacobs, WR., "Pitching and Heaving Motions of a Ship in Regular Waves", Trans. SHAME, 1957.

[3) Lewis, F. M., "The Inertia of the Water Surrounding a Vibrating Ship",

Trans. SNAME, .929.

(4] Prohaska, C. W.J "The Vertical Vibration of Ships"b

The Shipbuilder and Marine Englne»Builder, Oct .Nov. 1947.

[5) Grim,

"Berechnung der durch Schwingungen eines Schiffakrper

erzeugten

Hydrodynamischen Krfte",

Jahrbuch

der Schiffbautechnischen

Gese11schaft

1953.

(6] Kaplan P., and Jacobs, W. R., "Two DimensionalDamping Coefficients

from Thtrp'Ship Theory", Davidson Laboratory

Note No. 586,

April 1960.

(1] Haskind, H. D., "Two Papers on the Hydrodynamic Theory of Heaving and

Pitching of a Ship" (English Translation), SHAME T and R

Bulletin

No. 1.12, April 1953.

(8] Nuinata, E. añd Lewis, E.

V., "An Experimental Study of the Effect

of Ectreme Var tat ions in Proportions and Form

on Ship Model

Behavior in Waves", Davidson Laboratory (Experimental Towing Tank)

Report Ño. 643, December 1957.

(g) Haveock,

T. Ho,

"The Damping of

Heave and Pitchi a Comparison of

Two»dime.nsionel and

Three-dimensional

Calculations", Trane. INA,

i 956.

(io) Vossers, G.,

Discussion of

Haveleck's paper,

Trans.

INA, 1956.

Dalzóli, J. F., "An Experimental Investigation of the Heaving Forces

and Pitching MomentS on a Restrained Yacht Model in Regular Waves",

Master'a Thesis, 1960, Stevens Institute of

_Technology.

N-593

6-Fiq. I

40

30

20

IO

b (HEAVE)

B (PITCH)

-

FROM R E F. FOR VEE J SECTIONS

e

-E

(CROSS COUPLING)

2 2 GRIM

- e

_{2 - - 2}

_{REF. 6} I _i I I _{_.._______._:::::i--}

i

I 6 7 8 9 IO II 12 13

o)

FIG.

2 - TWO DIMENSIONAL

DAMPING

COEFFICIENTS

COMPUTED FOR

YACHT MODEL 1699 B.

FROM GRIM FOR LEWIS'

30

20-IO

r-B b

B (PITCH)

b (HEAVE)

-e

2

-E2 (CROSS COUPLING)

FROM GRIM FOR LEWIS SECTIONS FROM REF.

6 FOR VEE

SECTIONS - GRIM

-e

₂

E

₂

_REF.6

r I I I 5 6 7 8 9 IO II 12 I-,

w

FIG.

3 - TWO

DIMENSIONAL

DAMPING

COEFFICIENTS

COMPUTED

FOR

o o s

j

PITCH

s 2 3

.

4

200

I 00

MODEL SPEED,

FT./SEC.

O

j

O I 2

PHASE

LAGS

HEAVE

,V0LD

PERIMEN T o CALCULATIONS

PITCH

s ___

o s

FIG. 4- MOTIONS

OF

4.28-FT; MODEL 1699B (YACHT)

IN

WAVES

4.28 FT.

X

.09 FT.

Z

w

>

w

I

2 o w

o

e' cl -J

200

t 00

1 o o

DOUBLE

I ¡

AMPLITUDE.S

HEAVE

o o

w

o

I

o

I

a-O

5L

.

T ¡ DOU BLE AMP L IT UDE S H E AV E

PITCH

s

.

s I I I I 2 3 s 4 w

o

o

4

J

w u,

4

r

a-200

o

200-

I 00

MODEL SPEED

,

FT/SEC.

PHASE

LAGS

HEAVE

- OLD

o o

o

-- o

'-NEW CALCULATIONS

PITCH

s I

I-s

0.

I I O I 2 3 4

FIG.

5

-MOTIONS

OF

4.28- FT.

MODEL 1699B

(YACHT)

IN WAVES

5.35

FT. X .09 FT.

I

EXPERIMENT I 00c o o §

.2

r

w > w

I

O

DOUBLE AMPLITUDES

HEAVE

2 4 O 8 0

MODEL SPEED,

FT./SEC,

o_

'EXPERIMENT

o

-NEW CALCULATIONS

PITCH

-SI S 2 4

r-T

PHASE LAGS

HEAVE

S

.

S FIG

L - MOTIONS

OF

5.71 - FT.

MODEL 1699 D

(LENGTHENED YACHT)

IN

WAVES

5.71 FT.

X

.12

FT.

o o -J w u, c

PITCH

O

200L

I

Q-.

.

I

.

.

L) 1

Q-loo

.

o o o o ( w

L

6

z

w

>

w

I

o O

.

0 2

DOUBLE AMPLITUDES

HEAVE

o

.

PITCH

L ₄ s o s o

.

o

I

6 8

MODEL SPEEDS

FT/SEC.

i

PHASE

LAGS

o

I

s OLD s I I 2 4 6 EXPERIMENT 8 FIG.

7

-MOTIONS

OF

5.71- FT.

MODEL 1699D (LENGTHENED YACHT)

IN

WAVES

7.14 FT.

X .12

FT.

200

o o

--HE AVE

w

o

o

----o

o CALCULATIONS NEW -J w O (I)

I

PiTCH

Q-200 -

I 00

-s

.

.

s

I

I

o

1.

Two-Dimensional Damping Coefficients

from Thin-Ship Theory

2.

Theoretical Motions of Two Yacht Models

In Regular I-lead Seas on the Basis of Damping Coefficients

-

Derived for Wide V-Forms

Two Notes by Paul Kaplan and Winnifred R. Jacobs