E s t i m a t i o n o f Resistance, T r i m and D r a f t of P l a n i n g C r a f t
T a m o t s u N a g a i , Kanaga.wa Inst, of Technology 'Y a s u s h i Y o s h i d a , Techn. Res. and Development lust., .Jap. Defense Agency
1 I n t r o d u c t i o n
For estimating the resistance and floating position of planing craft with wave- and V-shaped bottom systematical towing tests were conducted with 13 models of 2.5 m length w i t h
wave-l aped ho oms Each modewave-l was towed in 6 conditions differing with respect to dispwave-lacement a rest and longitndn.al centre of mass; thus altogether 78 resistance curves were measured and used for the regression analysis.
lieiJli?' ! ' ' ; ° 7 Td"^"^ defines 11 characteristic shape parameters at 4 stations: chine heights to h,; keel heights and H,; a measure for hull length L; lialfl)readth of chine b,
ta b,; a measure for transom curvature /; and trim at rest 9. L is measured from tlie crossing
between chme and keel line to the transom; thus it does not generally coincide with wetted ength at rest. Initial trim 0 is the angle in minutes between waterUne at rest and keel line f r o m stations .5 to 10. A l l lengths are non-dimensionalized by division with V ' / ^ where V is he displacement volume o f t h e model at rest. The 11 non-dimensional characteristics will be designated A ^ , j = l , 2 , . . . l l .
A t first the pnncipal components Z, were determined using the correlation matrix A Its elements are the correlations between pairs A'„ X , . The four largest eigenvectors of A\ until /U, and the corresponding eigenvalues A, until (decreasing monotonously in size) were computed The eigenvectors are used to define the principal components z / a s Unear combinations ol the A'^-:
'- = 1,2,3,4. (1) Here A", and a, designate mean and standard deviation of A',- for the 78 model conditions.
Due to the orthogonahty of the eigenvectors, the four Z, are independent of each other and will be used m the regression analysis. Fig. 2 shows the 13 models plotted at positions l ü u s t r a t i n g the two most important principal components Z, and Z.. Due to the different loading conditions, each of the 13 models stands for a cluster of 6 (Z^, Z , ) combinations.
2 R e g r e s s i o n f o r m u l a e u s i n g t h e p r i n c i p a l c o m p o n e n t s
Table 1 gives the correlation coefficients between parameters Z, and Y,- exceeding 0 60 I t shows that Z, is mainly influenced by the chine hight at sections 0, 2 and 5 and the halfwidth at section 5; Z , depends mainly on huU length, transom curvature and halfwidth at the transom-Zs contains the halfwidth at sections 2 and 5, whereas Z, corresponds to the initial t r i m angle.'
The total resistance coefficient C, =R,esistance/0.5/9V'2V2/3 is approximated as
C, = Co -I- CiZi -F- coZo -f C3Z3 -f C4Z4. (2)
For each velocity tested, the regression coefficients c,, . = 0...4 are determined from the 78 sets of measurements by applying the least-squares method. Corresponding formulae are used lor the t r i m change M (positive bow up; in minutes) and for the non-dimensional draft change at transom A D (positive for decreasing draft; made uoii-dimensional by division with V ' / ^ ) .
'Prof, Dl-., Dept. of Mecii. Eng., 1030, Sinioogino, Atsugi Chy, Kanagawa Pref., 243-02 Ja pan
Z2 Z3 h.iv'i^ = X. 0.87 hs/V'/' = .Y3 0.93 Hs/V'/' = X , H./V'l^ = A's 0.96 LIV'I^^ A'e -0.61 b./V"^ = X j 0.63 -0.65 Ö3/V'/3 = A's -0.78 / / V ' / 3 = A's 0.89 0 = Xio -0.65 ö , / V ^ / 3 = A ' u 0.80
Table 1. Correlation between Z, and
A',-.Ord.10 , p r d . 5 I O r d . 2 j^Ord.O
B a s e Line-p
Fig. 1. Ciiaracteristic shape parameters
A D C t 0 . 1 0 0 . 0 5 2 0 0 I 0 0 - 0 . 0 5 - 0 . 1 0 1.0 2.0 3.0 0 ].G 20 30 F'; 1.0 2.0 30 ; F V I ' l ' c 0 . 1 0 0 . 0 5
Fig. 3. Test results and results of regression formulae (2), (3), (4) for a model used to derive the regression coef-ficients Ae AD 200 1 0 0 • O 0 ° - 0 . 0 5 - 0 . 1 0 1.0 2.0 30 .0 20 3.0 Fv o experiment • eq. '2 A eq. 3 • eq. 4 1.0 20 30 » F , - I— ^ o — , — T o I
Fig. 4. Test results and results of regression formulae (2), (3), (4) for a model not used to derive the regression coeffi-cients
3 S i m p l e e m p i r i c a l f o r m u l a e
To avoid the necessity of determining the principal components Z,- before applying the regression formula (2), as an alternative, four nondimensional characteristics each of which is strongly correlated to one of the principal components were selected (see Table 1): the non-dimensional equivalents oiho^L^bo, and 6. The deadrise angle at midship, which is one of the most interesting terms for planing hull design, is included implicitely as the ratio between ho and bo. Using these parameters, Ct is approximated by tlie least-squares method as
Ct = do + dyho/V"^ + doL/V"^ + d^bo/V'l^ + d,0. (3)
Changes in floating position AO and AD are expressed correspondingly.
As a third alternative, quadratic terms in non-dimensional L und & were added:
Ct = fo + fyhojv'i' + hL/v"^ + hbojv'i^ Ar ue + M L / v ' ' y + ue\ (4)
Again for Ad and A D corresponding formulae were used.
4 C o m p a r i s o n w i t h t o w i n g test d a t a
To demonstrate the accuracy of the regression formulae, Fig. 3 compares test results for 6';,
A9 and A D with values obtained from equations (2) to (4) for a model with the following data:
^ 0.174, /z./V^/^ ^ 0.276, / i g / V ' / ^ = 0.404, = 0.000,
/ f 4 / V ' / 3 = 0.583, X / V ' / - ' ' = 6.682, Ö./V'/^ = 0.797, Ös/V^/^ = 0.581, (5) / / V i / 3 = -0.024, e = 69, = 0.609.
The accuracy is estimated as quite good. The errors are smallest for formula (4) in this case. For another model (no. 14) which was not included in the regression analysis, five model conditions were tested to reconfirm the formulae. As an example Fig. 4 shows results for this model with the following set of parameters:
= 0.034, / i . / V ' / ^ = 0.209, / ï a / V ' / ^ = 0.355, i^a/V^/^ ^ 0.023,
5"4/V^/3 ^ 0.571, Z / V ' / 3 ^ 5.793, Ö./V^/^ ^ 0.664, Öa/V'/^ = 0.508, (6) / / V ' / 3 = 0.000, 0 = 0, ^ Q_g57_
Results are estimated to be reasonable and to confirm the applicabihty of the simpler formulae (3) and (4). Although (3) and (4) give almost the same accuracy, designers prefer to use (4) which, contrary to (3), contains also quadratic terms. The coefficients in (4) are indicated in Table 2 for various Froude numbers.
5 A p p l i c a t i o n t o a t o r p e d o b o a t
For a triple-screw torpedo boat of 35 m length with displacement 133 t (Raymond and
Blackman 1975), 21 trial tests were carried out in good weather conditions using various engine
ratings. In some trials only the center propeller was powered. Engine power and trim angle were measured (Figs. 5, 6).
To compare the measured power in full scale with our regression for the resistance coefficient
in model .scale, at first the resistance coefficient is changed from model scale (C^n,) to the fuU
scale value [Ct,):
6V, = C , „ , 4-(CV. - C V „ J . S ' / V ^ / ^ (7) For the plate friction coefficients Cj, and C;,„ Schoenherr's formula was used with a Reynolds
number based on the wetted length of the boat in planing condition. The wetted surface S was estimated depending on the Froude number Dy = ^Is/^'^^Q as follows:
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 5/221.87712 1.00 1.00 1.06 1.01 0.88 0.82 0.73 0.64 The effective power (Resistance times speed) is obtained as
PE = a,-O.F>p,V'V'" (8)
and indicated in Fig. 5 together with the measured shaft power
PQ-Dividing the effective power PE (obtained by application of the regression formula for the resistance) by the measured shaft power Fp gives an estimation of the propulsive efficiency
T]!}-. At 30 (40) knots we obtain qp = 0.50 (0.49). This corresponds well to the generally
assumed range for r//? of 0.50 to 0.55 for a boat without appendages. Thus, a reasonable power prediction is obtained in this example.
As shown i n Fig. 6, the trim change was underestimated here, but the dependence on speed is quite similar.
References
Yoshida, Y. and Nagai, T. (1976), Principal component analysis and factor analysis referring to model test results in still water of high speed craft, .] „Soc.NAJapan 40, 58-66
Raymond, V. and Blackman, B.(ed.) (1975-76), Jane's fighting ships, London, p.213
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@ ® N o t e ( 1 ) M o d e l ' s s c a l e 0 5 1 0 15 ( 2 ) [ n ( h i s s c a l e a l l m o d e l ' s b . e q u a l t oFig. 2. Classification of models
A e 4 ° 1" N o t e T r i a l T e s t £ q . ( 3 ) E q . ( 4 ) -CO o o o Po , P = ( l<W ) 5 0 0 0 i O 5 0 S p e e d ( k I )
Fig. 5. Measured Pp and estimated PE of a torpedo boat
Fig. 6. Measured and estimated trim change
AO of a torpedo boat
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