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The Radii of Gyration of Merchant Ships
TECHNISCHE UNIVERSITEIT
Laboratorium vooc
Scheepshydromechanlcs
Archief
R W PEACH, MSE, PE, CEng. Member
Na va/Architect and Marine Engineer, USA
A K BROOK, BSc, MSc, CEng, British Maritime Technology Ltd
Report on a presentation and discussion
Mekelweg 2,2628 CD D&ft
1 6th February 1987
TeL: 015- 786873 - Fax: 015- 781836
Predicting the motion response of a ship at sea requires,
So far all the equations refer to radius of gyration for
among other things, values for the respective radii of
roll. The computer program was also used to obtain values
of radii of gyration associated with pitch and yaw, and Mr
Peach went on to describe the computations in some detail.
Three ships, each of a characteristically different type,
were used in the investigation: a Roll On/Roll Off (Ro/Ro),
gyration with reference to the three principal co-ordinate
axes through the centre of gravity of the ship. Mr Peach
began his presentation by pointing out that, until the
development of modern computers, direct calculations for
radii of gyration were considered to be too time consuming
in relation to the value of their application. Instead, various
empirical formulae were developed and have been used for
many years. In fact, as late as 197.7 Mr Peach discovered,
a Lighter Aboard Ship (LASH), and an LNG carrier.
Particulars of the ships are given in Table i and the
calculated radii of gyration are given in Table 2. Mr Peach
from an approach to the US Maritime Administration
Table I-Ship characteristics
(MarAd), that no directly calculated values were available.
He suggested that a computer program should be developed
for this purpose and MarAd agreed, adding two requirements. The program was to be produced as an entity i.e. not
trying to add radii of gyration calculations to any existing
SHIP TYPE
Builder
Design
Feet
La,
Metres
L, Feet
program for weights and centre of gravity. They also
wanted the program to produce values for weight per foot of
ship length as a longitudinal distribution.
Apart from the use of the program to perform direct
B,
consider the results in comparison with some existing
D.
calculations, Mr Peach thought that it was useful to
empirical formulae. First he presented the simplest type
Displ. Tons
where K is radius of gyration for roll
B is breadth of ship
C1 is a coefficient dependent upon ship type.
Traditionally, values have been quoted for C between 0.40
and 0.44. References i and 2 give corresponding values in
LCG Feet
T=2
15805
16059
382.20
116.49
38.57
15113
400.94
122.21
36.59
11.15
11.76
-0.16
-0.049
14.84
52.20
0.14
0.043
58.30
15.91
13.63
17.77
21.71
4.154
6.617
0.18
0.055
118.24
36.04
69.56
21.20
8.81
6.70
7TKr
/g GM
10.89
Not including cantilevers at stern
At side
La = Length overall
= Length between perpendiculars
*
Table 2-Radii of gyration
= Deck area coefficient = (Deck Area)/LB
= Equivalent hull depth = D + A/Lp
= Profile projected areas of erections and deck-
SHIP
houses
F
931.50
283.92
887.00
270.36
140.50
42.82
94.00
28.65
30298
30784
477.05
145.40
48.68
Metres
Feet
KM
Metres
Feet
GM
Metres
Roll Period, Seconds
(HIT -2.20) + (H/B)2 J
where
Cb = Block coefficient
= Beam
= Constant for the ship type
= 0.125 for passenger, passenger and cargo, and
772.00
235.31
724.00
220.68
100.00
30.48
60.00
18.29
14874
Metres
(K/B)2=Cl2=F[CbCU+ l.IOCU(l-Cb)
= Draft
684.00
208.48
640.00
195.07
102.00
31.09
69.50
21.18
TCG Feet
or
C1 = (1 + B/D)/AJ12
This form led him to suggest that perhaps (B + D) may be a
better parameter than the frequently used (B2 + D2)+
Another equation given by K ato3 is
B
Avondale
LG9-5-107a
Metres
K = (B+D)/jT
T
Bath
C8-5-8lb
VCG Feet
radius of gyration
LNG
Avondale
Tonnes
the form of C = l.108C1. He then considered the radii of
gyration for a thin-walled rectangular tube, which has a
LASH
C7-5-95A
Metres
Feet
Metres
Feet
Metres
K =C1 B
C
H
A
Ro/Ro
RO/RO
LASH
LNG
cargo ships
= 0.1.33 for tankers
= 0.177 for whalers
Kr (Roll)
Feet
36.2732
37.0488
50.4408
Metres
11.06
11.29
15.37
K (Pitch)
Feet
187.8276
205.0905
239.2502
Metres
57.25
62.5 I
72.92
K (Yaw)
Feet
187.3549
204.9517
239.9572
Metres
57.11
62.47
73.14
explained that the calculation used relevant information for
every individual piece of structure, outfit and equipment,
involving between 20000 and 40000 items. The results are
converted into coefficient form associated with various
parametric terms, and presented in Table 3. For comparison, values of Kr/B estimated from Kato's equation are
Finally, Mr Peach quoted from his own report4 to MarAd:
K = 0.30 (B2 + D2)+
based on the results of the full-length calculations by
included in Table 3. These are considerably higher than the
computed values and can only be correlated by using much
computer.
115
Table 3-Radii of gyration coefficients
SHIP TYPE
Kr/B
K/(B2 +
F
0.125
0.133
Kato's
K /B
0.17 7
Required F
K p/La
KI(La2 + D2)+
K /(L2 + D2)+
Ky/La
L
Ro/Ro
LASH
LNG
0.3556
0.2939
0.3705
0.3177
0.3590
0.2979
0.4278
0.4413
0.5090
0.0483
0.3400
0.3508
0.4046
0.0707
0.6014
0.6938
0.0336
0.2746
0.2732
0.2935
0.2657
0.2649
0.2833
0.2823
0.2655
0.2633
0.2831
0.2804
0.2568
0.2555
0.2697
0.2682
0.2576
0.2547
0.2705
0.2672
0.29 18
+ B2)+
0.2739
0.2709
0.2827
Ky/(Lp2 + B2)+
0.2891
K/(L
showed that, although a 10% variation in radius of gyration
caused a significant change in the natural period of roll, the
roll amplitude did not have a direct correlation. This was
detailed calculations were not essential, although logically
0.5831
the best possible estimate should be used. He was more
concerned about roll damping coefficients and considered
these to be a more serious problem than radii of gyration
because damping coefficients are difficult to estimate from
available data. In his opinion, more experimental work was
necessary in this respect. Accurate radii of gyration can be
calculated if required, but the same is not true for damping
coefficients at the present time.
THE DISCUSSION
Dr I L Buxton, opening the discussion, referred to the
long history of estimation of radii of gyration and reiterated
= Length between perpendiculars
Mr Brook's warning about the misuse of empirical data
La = Length overall
especially, for example, Kato's equation based upon
smaller values of the coefficient F, compared with those
recommended by Kato.
From the same table it can also be seen that the
calculated values of radii of gyration Kp and Ky, for pitch
and yaw, are significantly higher than the frequently used K
=0.25L.
Mr Peach concluded by emphasising that the computed
values did not take into account the effect of entrained
water in the context of added virtual mass.
A complementary presentation was then given by Mr
Keith Brook of British Maritime Technology. He started
distinct from an analyst such as Mr Peach. His experience
values of radii of gyration in computer calculations predicting ship motion response and his comments were largely in
relation to rolling. Mr Brook had tried Kato's equation but
was more confident using that proposed by Bureau Ventas
for radius of gyration in roll
'lumping' was not possible, i.e. using blocks or groups of
items, without significant loss of accuracy while achieving a
worthwhile reduction in data preparation. In reply, Mr
Peach simply reiterated that no lumping of data had been
considered.
Mr D Brown referred to the importance of good values of
roll radius of gyration for warships and fleet auxiliaries
because MOD (Navy) ask for design estimates of the
motions of specific locations such as helicopter decks. Dr
Buxton added a comment in a similar context, referring to
the importance of all three radii of gyration in the current
debate on the 'short fat' versus 'long thin' warship hull.
It was then pointed out by Professor J B Caidwell that
though roll amplitude may not be very sensitive to accuracy
of radius of gyration, rolling accelerations were likely to be
more relevant because they are dependent upon radius of
gyration squared. This could be important in relation to
acceleration forces on such items as containers stowed on
been involved, Mr. Brook presented information which
Table 4-Effect of error in roll radius of gyration on roll
response
RMS ROLL (DEGS)
SIG.WAVEHEIGHT(m)
5
8
8.3
7.9
7.4
10.7
10.6
10.3
13.0
12.0
10.6
OFFSHORE
SUPPLY
0.36
6.8
0.40
0.44
7.5
5.5
5.1
FISHERY
PROTECTION
0.36
0.40
0.44
0.36
8.2
9.4
4.6
4.8
10.5
11.5
3.7
2.7
8.6
7.3
5.8
12.7
1.1
2.2
4.5
14.1
0.8
0.6
1.7
1.3
3.5
2.7
CONTAINER
0.40
0.44
15.5
been examined and was informed that they had not been
included in the study. The calculations were for 'empty'
ships. This aspect was referred to by other contributors
Mr Whatmore's query and asked if some degree of
range 0.35 to 0.40 for a loaded cargo ship. The significance
of this was raised in the subsequent discussion.
Describing a recent sensitivity study with which he had
3
which demonstrated that speed loss at sea could be a
consequence of particular longitudinal weight distributions.
Dr Buxton asked Mr Peach whether loaded conditions had
less detail than others? Mr Peach indicated that all items had
been treated in full detail. However, Mr W Hills reworded
K/B = 0.289-/ i + 4 (KG/B)2
He expressed the view that it should be remembered that
Kato's equation was derived some years ago and it was
probably unfair to expect it to relate to modern vessels of
special types; the limitations of empirical formulae are not
always appreciated. In comparison with the range of K/B
from 0.40 to 0.44 given in References I and 2, Mr Brook
mentioned that values used in the UK were usually in the
T
(SECS)
important, for example, in the way that pitching affects ship
performance. He referred to a paper by Swaan and Rijker5
Mr J Whatmore, doing post-graduate work on midship
section scantlings by computer at Newcastle University,
indicated that he expected the work to progress towards the
inclusion of radius of gyration. He asked Mr Peach if his
computation had indicated whether the results were sensitive to particular items, i.e. could some items be treated in
has been particularly in the context of using estimated
K/B
information from the 1950s. He considered it timely to
update such information because of the increasing awareness of the need to predict ship motions. This is very
during the discussion and Mr O M Clemmetsen pointed out
that typically the cargo weight was about two thirds of the
total loaded ship weight.
by identifying himself as a user of this type of data as
VESSEL
based upon results from calculations giving RMS roll
amplitude in various sea states as shown in Table 4.
Following this, he suggested that highly accurate and
deck.
Mr J Davison suggested to Mr Peach that while (B + D)
may be a relevant parameter for the radius of gyration of a
hollow rectangular shell, it might be more appropriate to use
(B2 + D2) for loaded ships by analogy with the radius of
gyration of a solid rectangle. Referring to added virtual
mass, he pointed out that it was not just radius of gyration
which was affected, but also the total moving mass. it
116
would seem preferable, he suggested, to treat added virtual
mass separately as in vibration calculations. Both Mr Peach
Dr T Svensen, in proposing a vote of thanks to Mr Peach
and Mr Brook, mentioned that he would treat radii of
and Mr Brook agreed that there was scope for more
gyration with much greater respect in future, and summed
up the discussion by referring to the general consensus that
the presentations had re-emphasised the need for further
research on this important topic, especially in relation to
experimental work in this respect.
The President, Dr Mime, asked if any data had been
collected from the actual behaviour of ships at sea. Mr
Brook replied that consideration had been given by the
increasing interest in the prediction of slamming, deck
wetness and similar seakeeping problems.
offshore industry to continuous monitoring of the metacentric height of a vessel by measuring a vessel's natural roll
REFERENCES
period, i.e. using an estimated radius of gyration in the
equation T = 2 r KA/g GM. However, the difficulties of
Principles of Naval Architecture, SNAME, 1941
Principles of Naval Architecture, SNAME., 1967
KATO, H. On the Approximate Calculation of Ship's Rolling Period,
JSNAJ, Vol 89, 1956
determining the vessel's natural period accurately mean this
method is problematic. Mr Whatmore asked if Mr Peach
was able to provide, for publication, information on the
PEACH. R. W. ENGINEERING ASSOCIATES, FinaiReport on Study of
Ship Radii of Gyration for US Department of Commerce, Maritime
Administration, April 1979
SWAAN, W. A. and RIJKER. H. Speed Loss at Sea as a Function of
Longitudinal Weight Distribution, Trans. NECIES, 25 Jan 1963
weight per foot of length distribution which had been
produced as a supplement to the calculations for radii of
gyration. Mr Peach readily agreed and Fig. 1 is included for
this purpose.
TONS PER FXT
- 220
- 200
- 180
- 160
- 140
-120
CONVENTIONAL TRAPEZOIDAL
WEIGHT DISTRIBUTION CURVE
- 100
-80
-60
i
20
0
h50
0O
350
DISTANCE FROM FORE PERPENDICULAR IN FEET
Fig. 1Distribution of weight per foot of length for LNG (LG9S-107a)
117
300