T E C H N I S C H E H O G E S C H O O L DELFT
A F D E L I N G DER M A R I T I E M E T E C H N I E K L A B O R A T O R I U M V O O R S C H E E P S H Y D R O M E C H A N I C AThe d i s t r i b u t i o n o f t h e hydrodynami
f o r c e s on a h e a v i n g and p i t c h i n g
s h i p m o d e l i n s t i l l w a t e r
P r o f . i r . J . G e r r i t s m a
and
Ing.W. Beukelman
R e p o r t n r . -22 -P
5 t h ONR Symposium 1964, B e r g e n .
Delft University of Technology S h i p H y d r o m e c h a n i c s L a b o r a t o r y M e k e l w e g 2
THE DISTRIBUTION O F THE H Y D R O D Y N A M I C
F O R C E S O N A H E A V I N G A N D PITCHING
SHIPAAODEL IN STILL WATER
J. G e r r i t s m a and W. Beukelman Technological University Delft, Netiierlands A B S T R A C T F o r c e d o s c i l l a t i o n t e s t s a r e c a r r i e d out w i t h a s e g m e n t e d s h i p m o d e l to i n v e s t i g a t e the d i s t r i b u t i o n of the h y d r o d y n a m i c f o r c e s a l o n g t h e h u l l f o r h e a v i n g a n d p i t c h i n g m o t i o n s . T h e v e r t i c a l f o r c e s on e a c h of the s e v e n s e c t i o n s of the s h i p m o d e l a r e m e a s u r e d a s a f u n c t i o n of f o r w a r d s p e e d a n d f r e q u e n c y . B y u s i n g the i n - p h a s e a n d q u a d r a t u r e c o m p o n e n t s of t h e s e f o r c e s , a n a n a l y s i s i s m a d e of t h e i r d i s t r i b u t i o n a l o n g t h e l e n g t h of t h e s h i p m o d e l . T h e e x p e r i m e n t a l r e s u l t s a r e c o m p a r e d w i t h the r e s u l t s of a s i m p l e s t r i p t h e o r y , t a k i n g i n t o a c c o u n t the e f f e c t o f f o r w a r d s p e e d . T h e c o m p a r i s o n s h o w s a s a t i s f a c t o r y a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t . INTRODUCTIONThe calculation of shipmotions i n regular head waves by using a s t r i p theory, has been discussed i n a number of papers. Recent contributions were given by Korvin-Kroukovsky and Jacobs [ l ] , Fay [ 2 ] , Watanabe [3] and Fukuda [ 4 ] .
In tliese papers the influence of f o r w a r d speed on the hydrodynamic forces is considered and dynamic cross-coupling terms are included i n the equations of motion, which are assumed to describe the heaving and pitching motions.
In earlier w o r k [5] i t was shown that a r e l a t i v e l y small influence of speed exists on the damping coefficients, the added mass and the exciting f o r c e s , at least f o r the case of head waves and f o r speeds which are of p r a c t i c a l interest. On the other hand, f o r w a r d speed has an important effect on some of the dynamic cross-coupling coefficients. Although, at a f i r s t glance these terms could be regarded as second order quantities, i t was pointed out by Korvin-Kroukovsky [ l ] and also by Fay [2] that they can be v e r y important f o r the amplitudes and phases of the motions. This has been c o n f i r m e d i n [5] where the coupling terms
are neglected i n a calculation of the heaving and pitching motions. In this calcu-lation we used coefficients of the motion equations, which were determined by forced oscillation tests. In comparison with the calculation where the cross-coupling terms are included and also in comparison with the measured motions, an important influence is observed, as shown i n F i g . 1, which is taken f r o m Ref. [5]. Further analysis showed that the discrepancies between the coupled and uncoupled motions were mainly due to the damping cross-coupling terms.
The influence of f o r w a r d speed has been discussed to some extent i n Voss-ers' thesis [6]. F r o m a f i r s t order slender body theory i t was found that the distribution of the hydrodynamic forces along an oscillating slender body is not influenced by f o r w a r d speed. Vossers concluded that the inclusion of speed dependent damping cross-coupling t e r m s is not i n agreement with the use of a
D i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
s t r i p theory. In view of the above mentioned results such a s i m p l i f i c a t i o n does not hold f o r actual shipforms.
For s y m m e t r i c a l shipforms at f o r w a r d speed, i t was shown by Timman and Newman [7] that the damping cross-coupling coefficients f o r heave and pitch are equal in magnitude, but opposite i n sign. Their conclusion is valid f o r thin or slender submerged or surface ships and also f o r non-slender bodies.
Golovato's work [8] and some of our experiments [5] on oscillating ship-models confirmed this fact f o r actual surface ships to a certain extent.
The effects of f o r w a r d speed are indeed v e r y important f o r the calculation of shipmotions in waves. The two-dimensional solutions f o r damping and added mass of oscillating cylinders on a f r e e surface, as given by G r i m [9] and Tasai [10] show a very satisfactory agreement with experimental results. When the effects of f o r w a r d speed can be estimated w i t h sufficient accuracy, such two-dimensional values may be used to calculate the total hydrodynamic forces and moments on a ship, provided that integration over the shiplength is p e r m i s s i b l e .
In order to study the speed effect on an oscillating shipform i n more detail, a series of forced oscillating experiments was designed. The main object of these experiments was to f i n d the distribution of the hydrodynamic forces along the lengtli of the ship as a function of f o r w a r d speed and frequency of oscillation.
THE EXPERIMENTS
The oscillation tests were c a r r i e d out with a 2.3 meter model of the Sixty Series, having a block coefficient Cg = 0.70. The main dimensions are given i n Table 1. The model is made of polyester, r e i n f o r c e d with fibreglass, and con-sists of seven separate sections of equal length. Each of the sections has two end-bulkheads. The width of the gap between two sections is one m i l l i m e t e r . Tlie sections are not connected to each other, but they are kept in their position by means of stiff strain-gauge dynamometers, which are connected to a longitu-dinal steel box g i r d e r above the model. The dynamometers are sensitive only f o r forces perpendicular to the baseline of the model.
By means of a ScotchYoke mechanism a harmonic heaving or pitching m o -tion can be given to the combina-tion ol the seven sec-tions which f o r m the shipmodel. The total forces on each section could be measured as a function of f r e -quency and speed.
A non-segmented model of the same f o r m was also tested i n the same con-ditions of frequency and speed to compare the f o r c e s on the whole model with the sums of the section r e s u l t s . A possible effect of the gaps between the sections could be detected i n this way. The arrangement of the tests with the segmented model and with the whole model is given i n F i g . 2.
The mechanical oscillator and the measuring system is shown in F i g . 3. In principle the measuring system is s i m i l a r to the one described by Goodman [ l l ] : the measured f o r c e signal is multiplied by c o s <jJt and s i n oit and a f t e r
Table 1
Main Particulars of tlie Shipmodel
Length between perpendiculars 2.258m
Length on the waterline 2.296 m
Breadth 0.322 m
Draught 0.129 m
Volume of displacement 0.0657 m^
Block coefficient 0.700
Coefficient of mid-length section 0.986
Prismatic coefficient 0.710
Waterplane area 0.572m2
Waterplane coefficient 0.785
Longitudinal moment of i n e r t i a of waterplane 0.1685 m'»
L . C . B . f o r w a r d of Lpp/2 0.011m
Centre of e f f o r t of waterplane a f t e r Lpp/2 0,038 m
Froude number of service speed 0.20
integration the f i r s t harmonics of the in-phase and quadrature components can be found with distortion due to vibration noise. In some details the electronic circuit d i f f e r s somewhat f r o m the description i n [ l l ] . In particular synchro,re-solvers are used instead of sine-cosine potentiometers, because they allow higher rotational speeds.
The accuracy of the instrumentation proved to be satisfactory which is i m -portant f o r the determination of the quadrature components, which are s m a l l i n comparison with the in-phase components of the measured f o r c e s .
Throughout the experiments only f i r s t harmonics were determined. I t should be noted that non-linear effects may be important f o r the sections at the bow and the stern where the ship is not w a l l - s i d e d . The forced oscillation tests were c a r r i e d out f o r frequencies up to w = 14 rad/sec and four speeds of ad-vance were considered, namely: F n = 0.15, 0.20, 0.25 and 0.30. Below a f r e -quency of w = 3 to 4 rad/sec the experimental results are influenced by w a l l effect due to reflected waves generated by the oscillating model.
The motion amplitudes of the shipmodel covered a sufficiently large range to study the linearity of the measured values (heave ~4 cm, pitch - 4 . 6 degrees). An example of the measured forces on section 2, when the combination of the seven sections p e r f o r m s a pitching motion, is given i n F i g . 4,
co z= r sinuit r I I I ( ! I " I I I I F j s i n ( i i ) t + a i ) 1 2 3 4 5 6 7 H E A V I N G T E S T ViïITH S E G M E N T E D MODEL UI F j S i n ( u t + a , ) a z - r s i n u i t l 61 y F , s i n ( ü ) t + a , ) Vït I Vil s i n 0) t F * s i n ( u t + 6 i » ) 1 2 3 4 5 6 7 PITCHING T E S T W I T H S E G M E N T E D M O D E L sin (i)t X
I
O O . 3 n O m O 05 3-•O 3 O H E A V I N G T E S T W I T H W H O L E M O D E L PITCHING T E S T WITH W H O L E M O D E L F i g . 2 - A r r a n g e m e n t of o s c i l l a t i o n t e s t s1 1 1 MODULATED C A R R I E R G E A R E D MOTOR S C O T C H Y O K E I ¥ \ P H A S E S H I F T E R 1 r ^ ^ — [ S T E E L BOX GIRDER ] 1-S T R A I N GAUGE DYNAMOMETER
V V
DEMODULATOR INTEGRATOR IN P H A S E COMPONENT Q U A D R A T U R E COMPONENTD i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s on a S h i p m o d e l . 2 0 0 + 100 + 200 +300 - I 1 1 1 r PITCH SECTION 2 ( A F T ) F n = .20 T 1 1 1 1 1 : 1 r QUADRATURE COMPONENT ( j J C I R C U L A R FREQUENCY
^Qdƒsec IN PHASE COMPONENT
• r = 1 c m • r = 2 cm A r = 3 c m J 1 I 1 I L . 50 40 o " 30 » , l i . CD 20 10 0 F i g . 4 - C o m p o n e n t s of f o r c e o n s e c t i o n 2 , p i t c h i n g m o t i o n
PRESENTATION OF THE RESULTS Whole Model
It is assumed that the force F and the moment M acting on a forced heaving or pitching shipmodel can be described by the following equations:
Heave Pitch (a + pv) i'o + b i „ + C2^ (A+kyypv) ë + B0 + ce dd + eè + eO s i n (oJt + a) -M^ s i n (<^+'fi) Kg s i n (ojt + 7 ) -Fg s i n (oit + S ) (1) (2)
For a given heaving motion z = z s i n cjt, i t follows that:
F„ s i n a C2_ - F_ cos a E = PV D = - M , s i n /S gz^ + c o s /3 (3) 221-249 O - 66 - 16 225
Similar expressions are valid f o r the pitching motion. The determination of the damping coefficients b and B and the damping cross-coupling coefficients
e and E is straightforward; f o r a given frequency these coefficients are
propor-tional to the quadrature components of the forces or moments f o r unit amplitude of motion. For the determination of the added mass, the added mass moment of inertia, a and A, and the added mass cross-coupling coefficients d and D i t is necessary to know the restoring force and moment coefficients c and C , and the statical cross-coupling coefficients g and G ,
The statical coefficients can be determined by experiments as a function of speed at zero frequency. For heave the experimental values show very l i t t l e variation with speed; they were used i n the analysis of the test results.
In the case of pitching there is a considerable speed effect on the restoring
moment coefficient C . c decreases approximately 12% when the speed increases
f r o m F n = 0.15 to 0.30. This reduction is due to a hydrodynamic l i f t on the hull when the shipmodel is towed w i t h a constant pitch angle. Obviously this l i f t ef-fect also depends on the frequency of the motion. Consequently, the coefficient of the restoring moment, as determined by an experiment at zero frequency, may d i f f e r f r o m the value at a given frequency.
As i t is not possible to measure the r e s t o r i n g moment and the statical cross-coupling as a function of frequency, i t was decided to use the calculated values at zero speed. This is an a r b i t r a r y choice, which affects the coefficients of the acceleration terms: f o r harmonic motions a decrease of c by AC results i n an increase of A by Ac/w^ when C is used in the calculation.
The results f o r the whole model are given i n the Figs. 5 and 6. The results f o r the heaving motion were already published i n [13]; they are presented here f o r completeness.
Results f o r the Sections
The components of the forces on each of the seven sections were determined in the same way as f o r the whole model. As only the forces and no moments on the sections were measured two equations remain f o r each section:
Heave
( a * + P V * ) z'o + t)*z^ + c*z^ = F ' s i n (wt + a ' ) ,
Pitch (4)
( d * + P V * X j ) Ö + e é + g0 = - F ^ s i n (wt + S*) ,
where p v ' x j is the mass-moment of the section i with respect to the pitching axis. The star (*) indicates the coefficients of the sections. The section efficients divided by the length of the sections give the mean cross-section co-efficients, thus:
D i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l HEAVING MOTION 0 5 10 15 0 5 10 15 Ü) , . (1) ^ Fn = .15 Fn = .20 F n = .25 F n = 30 F i g . 5 - E x p e r i m e n t a l r e s u l t s f o r w h o l e m o d e l 227
E t n I 0.5 PITCHING MOTION \ \ S 10 Ê 8 \
\
\
\ \\
10 15 . rod ƒ sec 10 . rod^see f\
V
/
/
-— 10 rad/sec 15 . F n = .15 - F n = .20 - F n =.25 . F n =.30 rad/si F i g . 6 - E x p e r i m e n t a l r e s u l t s f o r w h o l e m o d e lD i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
and so on. Assuming that the distributions of the cross-sectional values of the coefficients a ' , b ' , etc., are continuous curves, these distributions can be de-termined f r o m the seven mean cross-section values. In the Figs. 7, 8, 9 and 10 the distributions of the added mass a , the damping coefficient b and the c r o s s -coupling coefficients d and e are given as a function of speed and frequency. Numerical values of the section results, a * , b * , etc., are summarized i n the Tables 2, 3, 4 and 5.
Table 2
Added Mass f o r the Sections and the Whole Model kg sec V m F n = 0.15 r a d / sec a* a r a d / sec 1 2 3 4 5 6 7 Sum of Sections Whole Model 4 6 8 10 12 -1.21 0.31 0.24 0.20 0.18 0.59 0.66 0.60 0.69 0.78 1.08 1.09 1.29 1.40 0.54 1.38 1.37 1.48 1.60 0.87 1.26 1.28 1.34 1.45 0.41 0.65 0.76 0.85 0.90 -0.17 0.02 0.10 0.14 0.17 5.36 5.44 5.99 6.48 1.84 5.37 5.26 5.91 6.39 F n = 0.20 4 0.59 0.83 1.29 1.59 1.15 0.22 -0.27 5.40 5,63 6 0.32 0.65 1.00 1.40 1.23 0.64 0 5.24 5.19 8 0.21 0.55 1.08 1.38 1.21 0.75 0.12 5.30 5.18 10 0.19 0.65 1.23 1.49 1.33 0.83 0.14 5,86 5.78 12 0.20 0.77 1.37 1.60 1.45 0.88 0.17 6,44 6.32 F n = 0.25 4 0,86 1,09 1.26 1.66 1.20 0,16 -0,32 5.91 4,99 6 0,33 0,65 1.01 1.38 1,19 0,55 -0.02 5.09 4.89 8 0,20 0,54 1.03 1.39 1.26 0.68 0.08 5.18 5,13 10 0,18 0,62 1.19 1,48 1.34 0.77 0.12 5.70 5,65 12 0.20 0,76 1.37 1,60 1.45 0.83 0.16 6,37 6.21 F n = 0,30 4 0.70 0,91 1.49 1,58 1,07 -0.10 -0.22 5.43 5.59 6 0.25 0,44 1.15 1.39 1.07 0,45 0.07 4,82 4.51 8 0.16 0,42 1.14 1.45 1,08 0,58 0.13 4,96 4,93 10 0,15 0,55 1.26 1.47 1,22 0,68 0,17 5.50 5.48 12 0.17 0.69 1.41 1.57 1,35 0,81 0.19 6,19 6.18 229
F n = .15 F n = . 2 0 ' r T - . - 1 - ^ - m ^ ^ r 1 - r f 7 7 W l j r l 3 [ . T T T T T 7 7 F n = . 2 5 F n = . 3 0
r
e 0 4 0 • 4 Ct) = 4 r a d / s e c U) - 6 r a d / s e c 01 = 8 r a d / s e c 0 ) = l O r o d / s e c 2 1¬ 0 C J ; l 2 r a d / s o c F i g . 7 - D i s t r i b u t i o n of a o v e r the l e n g t h of the s h i p m o d e lD i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l F n = . 1 5 F F n = . 2 0 2 0 0 2 0 10 0 : 2 0 ' 10 0 2 0 10 0 20 W = 4 r a d / s e c —t-(il - 6 r o d / i o c 01 = 8 r a d / s e c I u l = 10 r a d / s e c Ul -12 r a d / s e c 20 10 . 0 I 1 20 . 10 ^ 0 S20 a* " i o 0 I 20 0 2 0 10 0 CO = 4 r a d / s e c = 6 r a d / s e c UJ = 8 r a d / s e c c ü = 10 r a d / s e c CÜ = I 2 r a d / s e c F n = . 2 5 1 1 3 I 1 I 5 — r i F n = . 3 0 ' i t n I 2 I 3 I 4 I 5 I 7 T r 7 F i g . 8 - D i s t r i b u t i o n of b o v e r the l e n g t h of the s h i p m o d e l 231
In F i g . 8 i t is shown that the d i s t r i b u t i o n of the damping coefficient b de-pends on f o r w a r d speed and frequency of oscillation. The damping coefficient of the f o r w a r d part of the shipmodel increases when the speed is increasing. At the same time a decrease of the damping coefficient of the afterbody is noticed. For high frequencies negative values f o r the crosssectional damping c o e f f i -cients are found.
Table 3
Damping Coefficients f o r the Sections and the Whole Model kg s e c / m Fn = 0.15 CO b ' b r a d / sec 1 2 3 4 5 6 7 Sum of Sections Wliole Model 4 6 8 10 12 2.03 1.82 1.61 1.36 0.95 9.78 4.42 2.31 1.08 0.47 4.55 2.26 0.76 0.44 5.78 4.58 2.75 1.39 0.87 3.80 4.52 3.35 2.36 1.89 4.80 4.78 3.94 3.43 3.09 2.00 1.67 1.53 1.49 1.50 26.34 17.75 11.87 9.21 35.63 26.53 17.49 11.63 8.54 Fn = 0.20 4 6 8 10 12 1.53 1.95 1.50 1.10 0.74 4.53 3.95 1.91 0.37 -0.15 5.08 4.32 2.25 0.62 0.21 5.05 4.45 2.81 1.54 1.01 5.73 4.52 3.49 2.70 2.18 6.63 5.07 4.38 4.01 3.84 2.50 2.07 1.94 1.90 1.93 31.05 26.33 18.28 12.24 9.76 31.33 26.15 17.78 12.14 9.03 F n = 0.25 4 6 8 10 12 2.13 1.97 1.48 0.95 0.52 4.80 3.43 1.58 -0.06 -0.58 5.38 4.17 2.28 0.60 -0.03 5.20 4.23 2.83 1.68 1.03 5.98 4.62 3.68 3.00 2.63 7.63 5.68 5.21 4.96 4.74 2.85 2.35 2.19 2.20 2.29 33.97 26.45 19.25 13.33 10.62 35.88 27.63 18.75 12.69 9.78 , F n = 0.30 4 6 8 10 12 1.78 1.75 1.21 0.64 0.42 4.40 2.77 0.99 -0.87 -0.56 4.40 3.50 1.70 0.17 -0.63 5.15 4.10 2.81 1.88 1.37 6.78 5.18 4.50 4.07 3.72 7.60 6.32 5.73 5.42 5.28 2.98 2.55 2.51 2.59 2.66 33.09 26.17 19.45 13.90 11.26 38.10 28.45 20.40 13.95 10.42
D i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
The added mass distribution, as shown i n F i g . 7, changes v e r y l i t t l e with f o r w a r d speed but there is a s h i f t f o r w a r d of the distribution curve f o r increas-ing frequencies.
Negative values f o r the cross-sectional added mass are found f o r the bow sections at low frequencies. For higher frequencies the influence of frequency becomes v e r y s m a l l .
Table 4
Added Mass Cross-Coupling Coefficients f o r the Sections and the Whole Model
kg sec 2 Fn = 0.15 Ü3 d* d r a d / sec 1 2 3 4 5 6 7 Sum of Sections Whole Model 4 ..
-
_ _ +0.59 +0.28 _ _ _ 6 -0.42 -0.47 -0.33 +0.02 +0.46 +0.57 +0.13 -0.04 +0.09 8 -0.27 -0.44 -0.40 -0.01 +0.38 +0.50 +0.13 -0.11 -0.16 10 -0.19 -0.43 -0.40 -0.01 +0.37 +0.49 +0.15 -0.02 -0.10 12 -0.19 -0.45 -0.40 -0.01 +0.40 +0.51 +0.15 +0.01 -0.04 Fn = 0.20 4 -0.57 -0.67 _ _ _ +0.78 +0.32 6 -0.39 -0.52 -0.34 +0.01 +0.46 +0.59 +0.13 -0.06 -0.06 8 -0.24 -0.45 -0.40 -0.01 +0.39 +0.51 +0.11 -0.09 -0.14 10 -0.20 -0.45 -0.40 -0.01 +0.38 +0.51 +0.13 -0.04 -0.08 12 -0.20 -0.47 -0.41 -0.01 +0.40 +0.53 +0.14 -0.02 -0.03 Fn = 0.25 4 -0.62 -0.59 -0.01 +0.12 +0.72 +0.86 +0.21 +0.69 +0.15 6 -0.39 -0.50 -0.32 +0.02 +0.46 +0.59 +0.13 -0.01 0.00 8 -0.23 -0.48 -0.40 -0.01 +0.39 +0.52 +0.14 -0.07 -0.13 10 -0.18 -0.46 -0.42 -0.01 +0.38 +0.51 +0.13 -0.05 -0.08 12 -0.20 -0.46 -0.42 -0.01 +0.40 +0.51 +0.15 -0.03 -0.05 Fn = 0.30 4 -0.62 -0.61 +0.13 +0.08 +0.64 +0.93 +0.20 +0.75 +1.09 6 -0.29 -0.47 -0.36 +0.01 +0.43 +0.59 +0.21 +0.12 +0.01 8 -0.21 -0.47 -0.44 -0.01 +0.38 +0.53 +0.16 -0.06 -0.11 10 -0.19 -0.46 -0.44 -0.02 +0.38 +0.51 +0.15 -0.07 -0.10 12 -0.20 -0.46 -0.44 -0.02 +0.39 +0.52 +0.16 -0.05 -0.06 233F n = . 1 5 F n = . 2 0 F n = . 2 5
1 N I , I ,TTTrT77
1.0 0 . 5 0 0 . 5 1.0 0 . 5 0 0 . 5 1.0 0 . 5 0 0 . 5 1.0 as 0 0 . 5 1.0 as 0 as / . d / s e c j W - 6 r a d / s e c ] / 2 ^ OJ - 8 r o d / s e c W -10rad/sec, X W = 12 r a d / s e c F n = . 3 0 F i g . 9 - D i s t r i b u t i o n of d ' o v e r the l e n g t h of the s h i p m o d e lD i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s on a S h i p m o d e l F n = .15 1 I i i 3 I 1 I 5 I 6 Fn = . 2 0 5 0 - 5 - 1 0 5 0 - 5 - 1 0 5 0 - 5 - 1 0 5 0 - 5 - 1 0 5 0 - 5 - 1 0 C j = 4 r o d / s e c £i) = 6 r a d / s e c OJ = 8 r o d / s e c t" W = I O r a d / s e c 5 0 - 5 -10 5 [- 6iJ = 6 r a d / s e c 0 5 Y C i J = 8 r o d / s e c 0 S - 5 - 1 0 - « 5 0 - 5 - 1 0 ' I - 5 10 — I 1 1— 0 J = 4 r o d / s e c a l = I O r a d / s e c 1 a i = 12 r a d / s e c Fn = . 2 5 Fn = . 3 0 - ° 5 ^ - 1 0 < 5 " - 5 ? - I O 0 ^ - 5 -10 t 5 0 - 5 - 1 0 "1 1 1 OJ = 4 r a d / s e c W = 6 r a d / s e c OJ = 8 r o d / s e c a ) = I O r a d / s e c OJ -12 r a d / s e c F i g . 1 0 - D i s t r i b u t i o n of e o v e r the l e n g t h of the s h i p m o d e l 235
Table 5
Damping Cross-Coupling Coefficients f o r the Sections and the Whole Model
kg sec F n = 0.15 J / * e e r a d /
sec 1 2 3 4 5 6 7 Sum of Whole
Sections Model 4
-
-
-
_ +1.63 +1.34 _ -2.43 6 -1.65 -2.58 -2.12 -1.19 -0.09 +1.70 +1.21 -4.72 -5.32 8 -1.71 -2.49 -2.45 -1.81 -0.68 +1.20 +1.09 -6.84 -6.75 10 -1.40 -2.01 -2.43 -2.10 -1.21 +0.88 +1.05 -7.22 -7.04 12 -1.07 -1.55 -2.28 -2.39 -1.52 +0.63 +1.05 -7.13 -6.88 F n = 0.20 4 -1.22 -3.07-
_ +2.39 +1.77 -6.63 6 -1.68 -2.43 -2.40 -2.06 -0.68 +1.52 +1.42 -6.31 -6.65 8 -1.59 -2.36 -2.83 -2.50 -1.25 +1.11 +1.32 -8.10 -8.23 10 -1.29 -2.04 -3.02 -2.87 -1.75 +0.82 +1.29 -8.86 . -8.86 12 -0.98 -1.65 -2.99 -2.97 -2.06 +0.61 +1.30 -8.74 -8.75 F n = 0.25 4 -1.52 -3.04 -3.47 -3.03 -0.96 +2.16 +1.91 -7.95 -6.70 6 -1.50 -2.21 -2.85 -2.66 -1.36 +1.47 +1.61 -7.50 -7.38 8 -1.50 -2.26 -3.21 -2.97 -1.79 +1.11 +1.51 -9.11 -9.30 10 -1.22 -2.14 -3.56 -3.39 -2.27 +0.86 +1.49 -10.23 -10.18 12 -0.85 -1.81 -3.66 -3.58 -2.53 +0.66 +1.47 -10.30 -10.31 Fn = 0.30 4 -1.37 -2.82 -3.61 -3.06 -1.22 +2.19 +1.98 -7.91 -7.55 6 -1.23 -1.93 -3.16 -3.06 -1.84 +1.43 +1.72 -8.07 -7.95 8 -1.30 -1.96 -3.55 -3.42 -2.32 +1.03 +1.67 -9.85 -9.81 10 -1.19 -2.06 -3.94 -3.90 -2.70 +0.76 +1.67 -11.36 -11.25 12 -0.91 -1.97 -4.08 -4.19 -2.97 +0.56 +1,69 -11.87 -11.84The d i s t r i b u t i o n of the damping cross-coupling coefficient e varies w i t h speed and frequency as shown in F i g . 10. F r o m F i g . 9 i t can be seen that the added mass cross-coupling coefficient depends v e r y l i t t l e on speed. For higher frequencies the influence of frequency is s m a l l .
As a check on the accuracy of the measurements the sum of the results f o r the sections were compared with the results f o r the whole model. The following relations were analysed:
D i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l la* = a d ' X dx = A 2 b * = b ƒ ^ ' " L S d * = d dx = B a ' x d x = D S e * = e
ƒ
b ' x d xThe results are shown i n F i g . 11 f o r a Froude number F n = 0.20, For the other Speeds a s i m i l a r result was found. A numerical comparison is given in the T a -bles 2, 3, 4 and 5. I t may be concluded that the section results are i n agreement with the values f o r the whole model. No influence of the gaps between the sec-tions could be found.
E
1
\ 1 r 1—
\ \ X 0 5 10 15 r a d / s e c 5 10 w — » • r a d / s e c 15 a 0 5 10 01 r a d / s e c 15 I 0 i \ y-* 0 - I • 0 4 0 I O V S.30 2 0 XI 10 ' O 5 10 0 1 — ^ r a d / s e c ' 0 5 10 15 ul — ^ r a d / s e c 0 5 10 0 ) — • • r o d / s e c a. 4 0 5 10 15 w — r a d / s e c 15 0K
• -e * - 8 - 1 0 0 5 10 10 r a d / s e c • S U M O F S E C T I O N S o W H O L E M O D E L 15 F i g . 1 1 - C o m p a r i s o n o f t h e s u m s of s e c t i o n r e s u l t s a n d t h e w h o l e m o d e l r e s u l t s f o r F r o u d e n u m b e r F n = 0 . 2 0237
ANALYSIS OF THE RESULTS
The experimental values f o r the hydrodynamic forces and moments on the oscillating shipmodel w i l l now be analysed by using the s t r i p theory, taking into account the effect of f o r w a r d speed. For a detailed description of the s t r i p theory the reader is r e f e r r e d to [1], [2] and [3]. For convenience a short d e s c r i p -tion of the s t r i p theory is given here. The theoretical estima-tion of the hydro-dynamic forces on a cross-section of unit length is of particular interest w i t h regard to the measured distributions of the various coefficients along the length of the shipmodel.
Strip Theory
A right hand coordinate system x^y^z^ is f i x e d i n space. The z ^ - a x i s is v e r t i c a l l y upwards, the x^-axis is i n the direction of the f o r w a r d speed of the vessel and the o r i g i n lies i n the undisturbed water surface. A second r i g h t hand system of axis xyz is fixed to the ship. The o r i g i n is i n the centre of g r a v i t y . In the mean position of the ship the body axis have the same directions as the fixed axis.
Consider f i r s t a ship p e r f o r m i n g a pure harmonic heaving motion of s m a l l amplitude in s t i l l water. The ship is piercing a thin sheet of water, n o r m a l to the f o r w a r d speed of the ship, at a f i x e d distance f r o m the o r i g i n .
At the time t a s t r i p of the ship at a distance x f r o m the centre of g r a v i t y is situated i n the sheet of water. F r o m x^ = Vt + x i t follows that x = - V , where
V is the speed of the ship.
The v e r t i c a l velocity of the s t r i p with regard to the water is i^, the heav-ing velocity. The o s c i l l a t o r y part of the hydromechanical f o r c e on the s t r i p of unit length w i l l be
where m' is the added mass and N ' is the damping coefficient f o r a s t r i p of unit length and y is the half width of the s t r i p at the waterline. Because
i t follows that
( m ' i „ ) - N ' z „ - 2 p g y z „ .
(5)
For the whole ship we f i n d , because
D i s t r i b u t i o n of H y d r o d y n a n n i c F o r c e s o n a S h i p m o d e l
F „ = - ^ J m'dxj 2„ - ^J^ N'dxJ i „ - P g A „ 2 „ (6)
where A^, is the waterplane area. The moment produced by the f o r c e on the s t r i p is given by
MH = - ^ K - i'o + (N'X - V x ^ ) + 2 p g x y z „
Because
r X ^ dx = -m , clx
we f i n d f o r the whole ship
" H = ^1 x m ' d x j + ^1 N ' x d x + V m j i „ + p g S „ z ^
where is the statical moment of the waterplane area.
(7)
(8)
For a pitching ship the v e r t i c a l speed of the s t r i p at x with regard to the water w i l l be -xè + vö, and the acceleration is -x'é+ 2vé. The v e r t i c a l f o r c e on the s t r i p w i l l be
Fp = - m ' ( - x ö + VÖ) - N'C-xè + VÖ) - a p g y x Ê » ,
or
F p = m'xë + ^N'x - 2 V m ' - x V 6 + ^ 2 p g y x + v2 ^ - N ' v j ö . (9)
The total hydromechanical force on the pitching ship w i l l be
F p = f f m ' x d x ^ ö + / [ N ' x d ; ; - V m ] ö + (pgS^-V f N ' d x ^ ö . (10)
\ L J / \ •'l J
The moment produced by the f o r c e on the s t r i p is given by
"p = - XF ; = - n - ' x ^ ö - ( N ' x 2 - 2 V r a ' x - x 2 v ^ ) ^ - ( s p g y x ^ + V^x ^ - N ' V x ) ö . (11)
The total moment on the pitching ship w i l l be
O
m ' x ^ d x ' j ö - / f N ' x ^ d x ' l ö - f p g l ^ - V ^ m - v f N ' x d x ^ . (12) Mpbecause
x ^ V dm'
dx dx = - 2 V X dx .
A summary of the expressions f o r the various coefficients f o r the whole ship according to the notation in Eqs. (1) and (2) i s given in Table 6.
Table 6
Coefficients f o r the Whole Ship According to the Strip Theory
a =
j
m'dx L d =ƒ
m'x dx L b = [ N'dx e = ƒ N ' x d x - Vm L c = g = / ' g S w A = f m'x^dx + ^ D =ƒ
m'x dx B = j N ' x ^ d x E = ƒ N ' x d x + Vm C = G = P g S ^ (13)For the cross-sectional values of the coefficients s i m i l a r expressions can be derived f r o m the Eqs. (5) to (12). For the comparison with the experimental results two of these expressions are given here, namely:
Also i t follows that
h' = N' - V ^ . dx e' = N ' x - 2Vm' - x V ^ dx (14) and A = ƒ d ' x d x B = e ' x d x . (15)
D i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
Comparison of Theory and Experiment
For a number of cases the experimental results are compared with theory. F i r s t of a l l the damping cross-coupling coefficients are considered. F r o m Eqs. (13) i t follows that:
E = ƒ N ' x d x + Vm
(16)
e = f N ' x d x - Vm .
The f i r s t t e r m i n both expressions is the cross-coupling coefficient f o r zero forward speed. For a f o r e and a f t s y m m e t r i c a l ship this t e r m is equal to zero. For such a ship the resulting expressions are equal i n magnitude but have oppo-site sign, which is in agreement with the result found by T i m m a n and Newman [7]. The experiments c o n f i r m this fact as shown in F i g . 13 where e and E are plotted on a base of f o r w a r d speed as a function of the frequency of oscillation. The magnitude of the speed dependent parts of the coefficients is equal within very close l i m i t s . Extrapolation to zero speed shows that the e and E lines i n -tersect in one point which should represent the zero speed cross-coupling co-efficient.
Using G r i m ' s two-dimensional solution f o r damping and added mass at zero speed [9] the coefficients e and E were also calculated according to the Eqs. (16). The distribution of added mass and damping coefficient f o r zero speed i s given i n F i g . 12 and the calculated damping cross-coupling coefficients are shown i n F i g . 13.
E • o i
0 0.1 0.2 0.3
F n »
F i g . 13 - C o m p a r i s o n o f c a l c u l a t e d a n d m e a s u r e d v a l u e s f o r e a n d E
D i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
The calculated values are in line with the experimental results. The natu-ral frequencies f o r pitch and heave are respectively w = 7.0/6.9 rad/sec and i n tllis important region the calculation of the damping cross-coupling coefficients is quite satisfactory. The zero speed case w i l l be studied in the near f u t u r e by oscillating experiments in a wide basin to avoid w a l l influence.
Another comparison of theory and experiment concerns the d i s t r i b u t i o n along the length of the shipmodel of the damping coefficient and of the damping cross-coupling coefficient e . F r o m Eq. (14):
b ' = N ' - V ^ , dx
e ' = N'x - 2Vni' " x V ^ .
Again using G r i m ' s two-dimensional values f o r N' and m ' , these distributions could be calculated. An example is given i n F i g . 14. Also i n this case the agreement between the calculation and the experiment is good. For high speeds negative values of the cross-sectional damping i n the afterbody can be explained on the basis of the expression f o r b ' , because i n that region dm'/dx is a p o s i -tive quantity.
Finally the values f o r the coefficients A, B, a and b f o r the whole model, as given by the Eqs. (13) were calculated and compared with the experimental results. Figure 15 shows that the damping i n pitch is over-estimated f o r low frequencies. The other coefficients agree quite w e l l with the experimental r e -sults.
F i g . 1 4 - C o m p a r i s o n of t h e c a l c u l a t e d d i s t r i b u t i o n of e a n d b w i t h e x p e r i m e n t a l v a l u e s f o r F r o u d e n u m b e r 0 . 2 0
4
\
c kLCULATIOH \ E X P E R I M E N T \ V . V\
CALCU LATOM ^S^"^ EXPERIMENT 5 , 10 Ü) — raJ/sec F n . . l 5 F n . . 2 0 F » . . 2 S _ - F n . . 3 0 X C A L C ; L A T I O H EXPERtHEN 1 i D _ r o i y ' s F i g . 15 - C o m p a r i s o n of c a l c u l a t e d a n d m e a s u r e d v a l u e s f o r a , b, A a n d B ( w h o l e m o d e l ) LIST OF SYMBOLS^ • • • ^ "I coefficients of the motion equations (hydromechanical part),
A . . . G J .
^ • • • s I the same for a section of the ship,
A * . . . G * /
the same for a cross-section of the ship,
a . . . g A ' . . . G
C b Block coefficient,
Fn Froude number
D i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s on a S h i p m o d e l
Fn,Fp o s c i l l a t o r y part of the hydromechanical f o r c e on a heaving or pitching ship,
g acceleration of gravity,
kyy longitudinal radius of i n e r t i a of the ship, length between perpendiculars,
U^,Kg amplitude of moment on a heaving or pitching ship,
M„,Mp o s c i l l a t o r y part of the hydromechanical moment on a heaving or
pitching ship,
m' added mass of a cross-section (zero speed),
N' damping coefficient of a cross-section (zero speed),
t time,
V f o r w a r d speed of ship,
x y z r i g h t hand coordinate system, f i x e d to the ship,
Xg,y^,z^ r i g h t hand coordinate system, f i x e d i n space,
v e r t i c a l displacement of ship,
x j distance of centre of gravity of a section to the pitching axis, a. A r . S phase angles,
0 pitch angle, P density of water,
c i r c u l a r frequency,
V volume of displacement of ship, and V* volume of displacement of section.
REFERENCES
1. Korvin-Kroukovsky, B . V. and Jacobs, W. R., "Pitching and heaving motions of a ship i n regular waves," S.N.A.M.E., 1957.
2. Fay, J. A . , "The motions and internal reactions of a vessel i n regular waves," Journal of Ship Research, 1958.
3. Watanabe, Y . , "On the theory of pitch and heave of a ship," Technology Re-ports of the Kyushu University, V o l . 3 1 , No. 1, 1958, English translation by Y. Sonoda, 1963.
4. Fukuda, J., "Coupled motions and midship bending moments of a ship i n regular waves," Journal of the Society of Naval Architects of Japan, No. 112, 1962.
5. G e r r i t s m a , J., "Shipmotions i n longitudinal waves," International Shipbuild-ing Progress, 1960.
6. Vossers, G., "Some applications of the slender body theory i n ship h y d r o -dynamics," Thesis, Delft, 1962.
7. T i m m a n , R. and Newman, J. N . , "The coupled damping coefficient of a s y m -m e t r i c ship," Journal of Ship Research, 1962.
8. Golovato, P., "The forces and moments on a heaving surface ship," Journal of Ship Research, 1957.
9. G r i m , O., "A method f o r a more precise computation of heaving and p i t c h -ing motions both in smooth water and in waves," T h i r d Symposium of Naval Hydrodynamics, Scheveningen, 1960.
10. Tasai, F.,
a. "On the damping force and added mass of ships heaving and p i t c h i n g , " b . "Measurements of the waveheight produced by the f o r c e d heaving of the
c y l i n d e r s , "
c. "On the f r e e heaving of a cylinder floating on the surface of a f l u i d , " Reports of Research Institute f o r Applied Mechanics, Kyushu University, Japan, V o l . V I I I , 1960.
11. Goodman, A . , "Experimental techniques and methods of analysis used i n submerged body research," T h i r d Symposium on Naval Hydrodynamics, Scheveningen, 1960.
12. Zunderdorp, H . J . and Buitenhek, M . , "Oscillator techniques at the Ship-building L a b o r a t o r y , " Report No. I l l , ShipShip-building Laboratory, Technolog-i c a l UnTechnolog-iversTechnolog-ity, D e l f t , 1963.
13. G e r r i t s m a , J. and Beukelman, W., " D i s t r i b u t i o n of damping and added mass along the length of a shipmodel," International Shipbuilding Progress, 1963.
D i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
D I S C U S S I O N
E. V. Lewis
Webb Institute of Naval Architecture Glen Cove, Long Island, Netv York
This is a noteworthy paper i n an important series by Professor G e r r i t s m a and his colleagues that is of v i t a l importance to ship motion theory. This con-tinuing work has been characterized by u n e r r i n g choice of the r i g h t research subjects and by extraordinary experimental s k i l l . The results have served to c l a r i f y the so-called " s t r i p theory" of ship motion calculations and to provide step by step confirmation of the d i f f e r e n t elements of the theory. Thus the t r e -mendous power of this comparatively simple approach to the problems of ship motions is being reinforced and the value of the pioneering insight of K o r v i n -Kroukovsky and others c o n f i r m e d .
It may not be generally realized that this type of experiment, in which forces on seven d i f f e r e n t sections are measured, is of unusual d i f f i c u l t y , not only because of the many simultaneous readings to be taken, but i n the need f o r accurate determination of in-phase and out-of-phase force components i n spite of extraneous noise. The authors have mastered this d i f f i c u l t problem.
The particular value of the resulting research is in showing that when the ship velocity terms are included, excellent predictions of the longitudinal d i s tribution of damping forces are obtained. Furthermore, the nature of the c r o s s -coupling coefficients, E and e, has been c l a r i f i e d by the demonstration that they should be equal at zero speed and d i f f e r only by the t e r m ±Vm at f o r w a r d speeds. (Incidentally, m is not defined, but is apparently equal to - a.)
Incidental features of the paper are s i m p l i f i c a t i o n s i n the coefficients, which are not immediately obvious. It is mentioned that
and therefore the e coefficient i s also s i m p l i f i e d [Eq. (13)]. Hence, the simple relationship between e and E emerges i n Eq. (16) and F i g , 13.
It. is hoped that this important w o r k strengtliening the s t r i p theory approach
V i i l l be continued, including oscillation tests at zero speed and restrained tests
in waves. M y congratulations to the authors f o r a beautiful piece of research. which makes the B coefficient, Eq. (13), much simpler than given i n {l). Also
*
D I S C U S S I O N
J. N . Newman
David Taylor Model Basin Washington, D.C.
F i r s t of a l l l e t me congratulate the authors on yet another i n the series of excellent papers which we have come to expect f r o m D e l f t .
Certainly one of the most valuable results obtained recently is the v e r y simple f o r w a r d speed c o r r e c t i o n to the s t r i p theory, as outlined i n the s t r i p theory paragraph, and the c o r r e l a t i o n of this theory w i t h experiments. I t would seem that a l l important speed effects are taken into account simply by replacing the time derivative i n a f i x e d coordinate system by that f o r a moving coordinate system, o r
_d dt
_9_ Bt
As a result, the added mass coefficient contributes both to the acceleration and velocity terms of the equations of motion, since
4- ( m ' i ) - m ' ï - V ^ i .
d t ° ° dx °
However this process seems rather a r b i t r a r y ; why not repeat i t f o r the second time derivative, so that
d t ^ - - f N ' = dt N ' - 2V \ dx I 2pgyZo 2/0gy + V 2 d V dx^ d N ' dx z - ?
I t i s clear f r o m the experimental results that too much cross-coupling would, result, and thus that the last equation is ridiculous both i n appearance and i n p r a c t i c a l u t i l i t y , but I am l e f t wondering why the equation used i n the paper is so much better. Is i t possible to give any r a t i o n a l explanation f o r t l i i s ?
Finally, since Professor Vossers is not here to defend himself, l e t me point out that, i n general, f o r w a r d speed w i l l have an effect on the d i s t r i b u t i o n of hydrodynamic f o r c e s along an oscillating slender body. Vossers reached the opposite conclusion only f o r the special case of high frequencies of encounter and v e r y slow speeds.
D i s t r i b u t i o n o f H y d r o d y n a m i c F o r c e s o n a S h i p m o d e l
D I S C U S S I O N O F THE P A P E R S B Y G E R R I T S M A A N D
B E U K E L M A N A N D B Y V A S S I L O P O U L O S A N D M A N D E L
T . R. Dyer Technological University Delft, NetherlandsThe paper by Vassilopoulos and Mandei r i g o r o u s l y examined seakeeping theory, with valuable emphasis on p r a c t i c a l ship design. The paper by G e r r i t s m a and Beukelman contains significant experimental results and a clear concise s t r i p theory, thus relating theory and physical phenomena. However, the paper by Vassilopoulos and Mandei agrees only p a r t i a l l y with G e r r i t s m a and Beukel-man, and with Korvin-Kroukovsky.
The papers were examined by this discusser w i t h the following results: 1. Complete agreement exists as to (a) which motion derivatives appear i n each coefficient, and (b) the appearance of velocity dependent t e r m s a r i s i n g purely f r o m the mechanics of a f i x e d axis system.
2. Disagreement exists as to the importance of the effect of f o r w a r d speed on s t r i p theory, but this is the only point of disagreement.
' This disagreement led to d i f f e r e n t evaluations of some motion derivatives. Direct comparison of the coefficients i n the two papers does not reveal a l l d i s -agreement, because of the cancellation of terms due to s t r i p theory by terms due to the mechanics of a f i x e d axis system. The disagreement i n the s t r i p the-ory specifically arose i n two ways: (1) G e r r i t s m a and Beukelman consider sec-tional added mass to be a function of t i m e , as suggested by Korvin-Kroukovsky. This is a "three-dimensional c o r r e c t i o n " and is j u s t i f i e d experimentally by a velocity dependence i n the b ' t e r m f o r the three-dimensional end sections of Gerritsma and Beukelman's model. (2) G e r r i t s m a and Beukelman consider the distance x , between the body axis o r i g i n and the hypothetical sheet of water, to be a function of time. This is independent of dimensionality. The second d i f f e r -ence is confusing; f o r Vassilopoulos and Mandei do i m p l i c i t l y take x as function of time when converting f r o m movable to f i x e d axes, but do not when applying the s t r i p theory.
The s t r i p theory of G e r r i t s m a and Beukelman was r e - d e r i v e d , eliminating these disagreements. The results agreed completely with those of Vassilopoulos and Mandei. Application of integrals quoted by G e r r i t s m a and Beukelman showed agreement between that paper and Korvin-Kroulcovsky. This therefore showed no e r r o r s in Korvin-Kroukovsky's work, only disagreement with Vassilopoulos and Mandei as to the role of f o r w a r d speed on the s t r i p theory. Conversion of Gerritsma and Beukelman results to a movable axis system revealed no d i f f i -culties, but clearly showed which speed t e r m s r e s u l t f r o m mechanics and which f r o m s t r i p theory.
The differences, therefore, are seen to be completely a r e s u l t of a d i f f e r e n t assumption of the importance of f o r w a r d speed on s t r i p theory, independent of what axis system is used. The assumption of G e r r i t s m a and Beukelman seems to be justified by experiment. The derivation of the equations of motion by Vassilopoulos and Mandei, due to Abkowitz, seems the most rigorous and satis-f y i n g . However, the evaluation osatis-f the motion derivatives by G e r r i t s m a and Beukelman, due in part to Korvin-Kroukovsky, seems to yield better results.
This discusser therefore feels i t most p r a c t i c a l to use the f o r m e r w o r k to study the mathematics of motion and the latter to evaluate the motion derivatives.
* * *
REPLY T O T H E D I S C U S S I O N B Y E. V . L E W I S
J. G e r r i t s m a and W. Beukelman
Technological University Delft, Netherlands
The authors are g r a t e f u l to have Professor L e w i s ' comments on their paper. The definition of m, which is omitted i n the paper, is given by
m'dx = ra = a .
L
It Should be noted that
x d m ' = - m'dx
L L
and not
ƒ
x d m ' = m'dx ,L L
as suggested by Professor Lewis.
The work reported i n this paper was recently extended f o r the zero f o r w a r d speed case.
These tests were c a r r i e d out i n a wide basin to avoid w a l l influence, due to reflected waves. The results support the conclusions of the present paper.
Within the v e r y near f u t u r e the restrained tests i n waves with the segmented model w i l l be c a r r i e d out i n our Laboratory. The results w i l l be compared with calculated values.
D i s t r i b u t i o n of H y d r o d y n a m i c F o r c e s on a S h i p m o d e l
REPLY T O THE D I S C U S S I O N B Y J . N. N E W M A N
J. G e r r i t s m a and W. Beukelman
Technological University Delft, Netherlands
For a f u l l y submerged slender body of revolution in unsteady motion, the total hydrodynamic f o r c e on a transverse section is equal to the negative time rate of change of f l u i d momentum. By taking the time derivative i n the moving body axis system the expression
For tlie surface ship, i t is assumed that the flow over the submerged portion of the ship is s i m i l a r to the flow over the lower half of a f u l l y submerged body with circular cross sections.
Corrections are then necessary f o r the shape of the sections and f o r f r e e surface effects. I t is assumed that these corrections are introduced by using Grim's values f o r the sectional damping and added mass coefficients of c y l i n -ders having ship-like cross sections oscillating at a f r e e surface. I t is admitted that this assumption is more or less intuitive and i t was c l e a r l y necessary that the assumptions being made had to be v e r i f i e d by experiments, as shown i n the paper.
The authors cannot give a s i m i l a r physical interpretation of the procedure put f o r w a r d in D r . Newman's discussion; they have therefore no rational expla-nation why such an approach is not successful. In addition, the r e s u l t would certainly not agree w i t h the experiments.
Vossers' results are discussed too shortly in our paper, and the authors are grateful to D r . Newman f o r his additional comments.
However, f o r the actual ship f o r m , as tested in our case, the f o r w a r d speed effect cannot be neglected, even at quite low speeds, say Fn = 0.15.
For pitch, the method, as given i n our paper, is valid f o r such combinations of f o r w a r d speed and frequency that the motion of the ship i n the stationary sheet of water does not depart too much f r o m a harmonic motion (see Ref. [ 2 ] ) .
_ / •• dm' . m z „ - V - 7 — 2 , ° dx ° ' is found.
