853
1 INTRODUCTION
When studying ship manoeuvres, such as circulation,
Kempf's zigzag, safe passage, etc., the presence of
adequate mathematical models of non-inertial forces
and moment on the ship's hull plays a very important
role. Many works are devoted to the construction of
mathematical models of hydrodynamic forces and
moment on the ship's hull [2–18][. At small drift
angles
15
, polynomial models are mainly used
to describe hydrodynamic forces and moments. The
numerical characteristics of such models are obtained,
as a rule (see, for example, [14–16]), based on
processing the data of full-scale and model tests in
wind tunnels, experimental basins, on rotary
installations and on planar mechanisms. A similar
approach to the construction of models of
hydrodynamic forces is also implemented in the
MMG (Maneuvering Modelling Group) method,
which is described in the works [2–6, 17, 18]. There
are also expressions for hydrodynamic derivatives up
to the fourth order for a large type of fishing ships,
dry cargo ships and tankers. One of the key stages of
the above-mentioned approaches is the choice of
methods for analysis and processing of experimental
data. So, in the works [4, 11], using the least squares
method with respect to one explanatory variable
(regressor), hydrodynamic force models for ships
with the value of the block coefficient
(0.49;0.7).
b
С
were constructed.
In the work [2] for ships with the value of the block
coefficient mainly from the range
(0.7;0.9)
b
С
models for the derivatives of longitudinal
hydrodynamic forces were obtained using several
regressors based on the minimum AIС (Akaike
Information Criterion).
This paper suggests a unified approach to the
construction of models of hydrodynamic forces and
moment, based on multivariate regression analysis,
using Fisher's and Student's criteria [1].
Construction and Analysis of Mathematical Models of
Hydrodynamic Forces and Moment on the Ship's Hull
Using Multivariate Regression Analysis
O. Kryvyi & M. Miyusov
National University „Odessa Maritime Academy”, Odessa, Ukraine
ABSTRACT: To analyse the existing mathematical models of hydrodynamic forces and moment on the ship's
hull and build new effective ones, an approach based on multivariate regression analysis is suggested. As
factors (regressors), various dimensionless ratios of the geometric parameters of the vessel, such as length,
breadth, draught, and block coefficient, were taken. When analysing existing mathematical models of
hydrodynamic derivatives and building new ones, the value of the multiple correlation coefficient R and the
value of standard errors were estimated. The significance of the models and the significance of all factors
(regressors) included in the model were assessed using Fisher's and Student's criteria. As a result, new adequate
mathematical models have been obtained for hydrodynamic constants with a high degree of correlation and an
excellent level of significance of regressors.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 15
Number 4
December 2021
DOI: 10.12716/1001.15.04.18
854
The analysis of existing models is carried out and
adequate models of the derivatives of hydrodynamic
forces and moment are obtained with a high level of
significance both for the models as a whole and for
each individual regressor for a wide range of values of
the block coefficient:
(0.49;0.9).
b
С
2 GENERAL REPRESENTATIONS OF
HYDRODYNAMIC FORCES
The projections
h
X
,
h
Y
of the hydrodynamic forces
on the coordinate axis associated with the ship and
the moment
h
M
around the axis are expressed as
follows:
2
,
x
hh
X v C=
2
,
y
h
h
Y v C=
2
,
m
hh
M v C=
where
( , ),
xx
hh
CC
=
( , ),
yy
hh
CC
=
( , )
mm
hh
CC
=
- the hydrodynamic characteristics of the ship's hull;
,,v
−
respectively, the magnitude of the resulting
velocity, the drift angle and the angular velocity of the
ship.
The solvability of the corresponding systems of
differential equations of the ship's motion [7–10, 12],
determines the sufficient smoothness of their right-
hand sides, which gives grounds to assume the
existence of Maclaurin series for the hydrodynamic
characteristics of the ship
( { , , })p x y m=
0
,
pp
jk
h jk
jk
CC
+=
=
0
0
1
.
( )!
p
jk
p
h
jk
jk
C
C
jk
+
=
=
=
+
(1)
p
jk
C
are called hydrodynamic constants (or
hydrodynamic derivatives) of forces and moment on
the ship's hull. Representations (1) make it possible to
approximate the hydrodynamic characteristics of the
ship's hull by polynomials at small angles of drift and
angular velocity.
For example, if we restrict ourselves in expansion
(1) to terms of the order not higher than third one and
take into account the features of hydrodynamic forces
[14–16], and the equality resulting from them
0
x
jk
C =
at
( , ) (0,0);(2,0);(1,1);jk
(0,2);(0,4)
and also
equalities
0,
y
jk
C =
0
m
jk
C =
at
( , ) (1,0);(3,0);(0,1);(1,2);(2,1);(0,3)jk
, then we
obtain the following representations
0
2 2 4
20 11 02 40
,
x x x x x
hx
C C C C C C
= − + + + +
(2)
3 2 2 3
10 01 30 21 12 03
,
y y y y y y y
h
C C C C C C C
= + + + + +
3 2 2 3
10 01 30 21 12 03
,
y y y y y y y
h
C C C C C C C
= + + + + +
where
0
x
C
is the coefficient of water resistance to the
straight-line motion of a vessel.
The hydrodynamic constants in representations (2)
are expressed through the geometric characteristics of
the vessel by processing experimental data.
3 METHOD FOR DETERMINING
HYDRODYNAMIC CONSTANTS
The following easily identifiable basic geometric
characteristics of the ship are usually used to
determine the hydrodynamic derivatives:
L −
length
on waterline,
B −
breadth on current waterline,
T −
the midship draught and block coefficient
.
b
C
From
these parameters, we compose the determining
regressors (factors):
1 2 3 4
, , , .
b
B T T
C
L L B
= = = =
(3)
We use factors (3) as basic ones in quasilinear
polynomial models (linear in coefficients) of
hydrodynamic derivatives. It should be noted that, as
a rule, basic regressors are used to build models of
hydrodynamic derivatives (3), this is primarily due to
their simplicity and availability. As the defining
regressors (explanatory parameters) of the models, we
will use the basic regressors (3) or their multipliers
(products of powers), i.e., we will look for the
hydrodynamic constants in the following form
( )
,,p x y z=
1
1
, , ( , 0, ,4)
j
l
p
j j j
jk l
j
l
C j k
=
=
= = =
. (4)
Indicator
, coefficients of the regression model
j
and indicators
j
,
j
of the regressions are
determined for each hydrodynamic constant
p
jk
C
.
When constructing dependencies (4), we will
evaluate both the significance level of the model as a
whole and the significance of each individual
regressor. Consequently, the model will be considered
adequate if the following criteria based on regression
analysis and analysis of variance are met.
1. The maximum possible value of the multiple
correlation coefficient
R
should be achieved:
00
, (0 1)R
(5)
1 where parameter
0
determines the level of
connection (correlation) of hydrodynamic
derivatives
p
jk
C
with regressors included in
representations (4). Moreover, if
0
(0.5 0.7)
,
the connection is considered average (satisfactory),
if
0
(0.7 0.8)
, the connection turns out to be
high (good) and if
0
(0.8 1)
, then the
connection is considered very high (excellent).
Otherwise, the connection cannot be considered
acceptable.
2. The statistical overall significance in the whole of
each model (4) will be determined based on the
Fisher criterion:
(1 , 1, ),
c nk F
F F m n m
− − −
(6)
1 where
2
2
1
1
c
R n m
F
m
R
−
=
−
−
is an observed statistics
with the following Fisher-Snedecor distribution (F-
distribution);
m −
the number of non-zero
coefficients of the models (4);
n −
sample size;
(1 , 1, )
nk F
F m n m
− − −
is a critical value of the F-
855
distribution for the significance level
F
. The less
F
is, at which inequality (3.4) is satisfied, the
higher the overall statistical significance of the
model is. Significance level
0.05.
F
is
considered excellent.
3. The level of significance of statistics
c
F
will be
determined using the probability
( (1 , 1, )).
F c nk F
P F F m n m
= − − −
Moreover, the
less
F
is, the higher the level of significance of
the statistics is. The level of significance can be
considered acceptable if the following condition is
fulfilled:
.
FF
(7)
4. Standard error
j
of the regressor
j
must
satisfy the condition:
.
jj
(8)
5. The statistical significance of each of the regression
coefficients is determined based on the Student's t-
test:
,
j
j kr j
j
t t t
=
(9)
1 where
(1 , )
js
t t n m
= − −
is a critical value of
the Student's distribution for the level of
significance
.
s
The less of
s
, at which
inequality (8) is satisfied, the higher is the overall
statistical significance of the coefficients of the
model is. The level of
0.05.
s
can be
considered excellent.
6. The significance level of the model regressors is
determined using the probability
( (1 , )).
j j kr s
P t t n m
= − −
In this case, the less
j
is, the higher the level of significance of the
corresponding regressor is. The level of
significance can be considered acceptable if the
following condition is fulfilled:
.
js
(10)
7. The absence of multicollinearity of the obtained
models, i.e., the absence of regressors with a high
pairwise correlation:
,
,,
jl
mk
R j l
(10*)
1 where
,
jl
R
is a coefficient of pair correlation
of regressors,
mk
is an indicator of the level of
correlation of regressors. It is believed that
multicollinearity is absent in the model if
mk
does not exceed
0.7 0.8.
When constructing models, the condition (5) is the
key one. However, the obtained dependencies must
be statistically significant with a sufficiently high level
of significance, i.e. conditions (6) and (7) must be
satisfied with a sufficiently small value of
.
F
The
condition (8) allows us to discard insignificant
regressors, conditions (9) and (10), with a sufficiently
small value of
s
, allow us to assess the significance
and significance level of each regressor in the model
respectively. Condition (10*) makes it possible to
exclude regressors leading to multicollinearity of the
obtained models.
To construct mathematical models of the
derivatives of hydrodynamic forces, we will apply the
procedure of adding regressors (explanatory
parameters). In this regard, firstly, the most significant
regressors of the model are determined (i.e., the
regressors with the highest values of the pair
correlation coefficients with the corresponding
hydrodynamic derivative). Then, starting with some
minimal regression model, with the most significant
regressors, we add new defining regressors until
criteria 1) - 6) are met. In this case, at each stage, we
check the fulfilment of condition (10*).
It should be noted that several adequate regression
models can be obtained in this way. In this case, we
will select those models for which the values of
,
Fs
and
is minimal, and the value of the
multiple correlation coefficient
R
is the maximum
possible.
4 ANALYSIS OF EXISTING MODELS OF
HYDRODYNAMIC CONSTANTS
Using the above-mentioned approach, we will analyse
the existing models of hydrodynamic forces and
moment on the ship's hull. To determine the
coefficients of the models, we will use the
experimental databases for hydrodynamic derivatives
of various types of vessels in deep water, given in the
works [2, 4, 18].
In particular, using the experimental data of works
[5, 18] (sample
14 12; (0.5;0.7) )
b
V n C= =
, and
work [2] (sample
15 18; (0.5;0.9) )
b
V n C= =
,
depending on the values of the block coefficient
b
C
,
for the derivatives of the longitudinal hydrodynamic
force, 3 samples (volume n) were compiled:
11 { 30; (0.5;0.9)}
12 { 14; (0.5;0.7)}
13 { 16; (0.7;0.9)}
b
b
b
V n C
V n C
V n C
= =
= =
= =
,
In works [4, 18], using the regressor
0 1 2
=
,
models of longitudinal hydrodynamic forces (models
A) were obtained. The coefficients of these models
will be determined using samples V14, V11, V12, V13.
Models А:
20 0
11 0
1.15 0.18
1.01 0.18
,
1.23 0.2
0.62 0.04
1.91 0.08
1.6 0.04
,
1.77 0.06
0.1 0.17
x
x
y
C
Cm
=−
−−
− = − +
−
856
02 0
40 0
0.09 0.008
0.07 0.017
,
0.03 0.012
0.65 0.098
6.68 1.1
5.41 1.11
7.5 1.18
0.43 0.43
x
Gy
x
C x m
C
−
−
+ = −
−
−
−
−
=+
−
In the work [2], with the use of the minimum AIC
(Akaike Information Criterion), models of
longitudinal hydrodynamic forces (models B) were
obtained. The coefficients of these models are
determined using samples V15, V11, V12, V13.
Models В:
20 1 2 1 2
2
40 1
7.14 38.4 46.6 5.94
0.63 2.48 3.1 0.55
,
1.41 4.94 7.35 1.01
15.67 75.31 91.19 12.91
0.0182 0.0826
0.0169
0.0159
0.0078
x
x
C
C
−
= + − −
− − − −
−
−
=+
02
0 3 0 3
0.701
0.139 0.021
,
0.21 0.032
0.242 0.679
5.2 14.7 107.8
0.087 0.672 3.093
,
0.075 0.952 3.972
5.139 15.36 113.94
x
Gy
C x m
+ = −
−
− − +
The upper coefficients in models A and B were
obtained, respectively, in the works [4, 18] and [2]
based on the experimental data presented there
(samples V14, V15, respectively). The second row of
coefficients corresponds to the V11 sample, the third
to the V12 sample and the fourth to the V13 sample.
Tables 1 and 2 show the correlation characteristics
of models A and B, respectively.
Table 1. Analysis of the model A [18]
_______________________________________________
R
F
Cond. (8)
s
_______________________________________________
20
x
C
V14 0.7 10
-2
+ 0.01
V11 0.55 2·10
-3
+ 2·10
-3
V12 0.72 4·10
-3
+ 5·10
-3
V13 0.16 0.55 - 0.8
11
x
y
Cm
−
V14 0.9 10
-4
+ 0.05
V11 0.81 10
-7
+ 0.2
V12 0.91 10
-5
+ 0.1
V13 0.03 0.91 - 0.2
02
x
Gy
C x m
+
V14 0.14 0.66 - 0.85
V11 0.12 0.55 - 0.55
V12 0.05 0.86 - 0.86
V13 0.34 0.2 + 0.2
40
x
C
V14 0.7 0.00 + 0.013
V11 0.41 0.03 + 0.025
V12 0.73 3·10-3 + 2·10
-3
V13 0.01 0.96 - 0.96
_______________________________________________
Analysis of the data given in Table 1 shows that for
the values of the block coefficient
(0.5;0.7),
b
С
model A establishes a good correlation between the
hydrodynamic derivatives
20
x
C
and
40
x
C
with the
regressor
( 0.7)R
, and for the hydrodynamic
derivative
11
x
y
Cm
−
this interconnection turns out to
be completely excellent
( 0.9)R
.
In all cases, there is a fairly high level of
significance of the models and regressor. For the
entire range of values of the block coefficient
(0.5;0.9),
b
С
the model for
20
x
C
establishes a
satisfactory correlation with the regressor, for
11
x
y
Cm
−
- excellent.
However, the significance of the regressor for
11
x
y
Cm
−
is not high:
0.2.
s
=
In all other cases,
model A turns out to be inadequate. In particular, for
the hydrodynamic derivative
02
x
Gy
C x m
+
turns out to
be inadequate for all ranges of variation of the block
coefficient.
Analysis of the data given in Table 2 shows that for
the hydrodynamic constants
20
x
C
and
40
x
C
models B
are not adequate for all ranges of change in the values
of the block coefficient. As for the hydrodynamic
constant
11
x
y
Cm
−
, a good correlation
( 0.71 0.92),R =
is observed for the range
(0.5;0.9),
b
С
for the V15 sample, however, there is
multicollinearity of the regressors:
0.92
mk
=
.
The latter leads to the fact that with an increase in
the sample size V11, the model turns out to be
inadequate with a low level of significance of the
regressors. The same is observed for the ranges
(0.7;0.9),
b
С
and
(0.5;0.7).
b
С
As for the
hydrodynamic constant
02
x
Gy
C x m
+
, model A can
only be used for
(0.7;0.9).
b
С
Table 2. Analysis of the model B [2]
_______________________________________________
R
F
Cond. (8)
s
mk
_______________________________________________
20
x
C
V15 0.59 0.12 + 0.05 0.9
V11 0.63 0.01 - 0.4 0.79
V12 0.75 0.07 - 0.5 0.98
V13 0.53 0.25 + 0.23 0.93
11
x
y
Cm
−
V15 0.71 0.05 + 0.05 0.92
V11 0.81 5·10-7 - 0.48 0.78
V12 0.61 0.06 + 0.05 0.9
V13 0.92 3·10-5 + 0.25 0.89
02
x
Gy
C x m
+
V15 0.52 0.03 + 0.03 0.64
V11 0.23 0.7 - 0.76 0.81
V12 0.33 0.75 - 0.93 0.99
V13 0.75 0.02 + 0.04 0.42
40
x
C
V15 0.30 0.23 - 0.87 -
V11 0.42 0.02 - 0.53 -
V12 0.51 0.07 - 0.41 -
V13 0.10 0.7 - 0.71 -
_______________________________________________
To analyse the existing models of the derivatives
of the transverse hydrodynamic forces, we used the
experimental data of the works [4, 18], from which,
depending on the values of the block coefficient
b
C
,
three samples were made for the derivatives of the
transverse hydrodynamic forces:
21 { 33; (0.49;0.9)}
22 { 20; (0.49;0.7)}
23 { 13; (0.7;0.9)}
b
b
b
V n C
V n C
V n C
= =
= =
= =
In the work [4], using regressors (3), models of
transverse hydrodynamic forces (models C) were
obtained. We will calculate the coefficients of these
models based on samples V21, V22, V23.
857
Models С
1 2 1 2
10 01
14
12
30
1.90 0.11 1.5
3.33 0.02 , 1.28
1.93 0.10 1.45
0.01 3.13 0.26
0.04 , 2.19 (1 ) 1.05 .
0.03 5.36 0.01
354
256
503
yy
x
y
y
C C m m
C
C
= + − − = − +
+ − = − − +
−
=
−
2
1 4 1 4
2
1 2 1 2
21
86.3 2.47
((1 ) ) 52.1 (1 ) 0.31 ,
34.6 1.17
202.5 69.5 5.48
160.1 ( ) 57 4.78 ,
96.6 22.8 1.53
y
C
−
− + − +
−
= − + −
− − −
The upper coefficients in models C and D
correspond to the V21 sample, the second and third
respectively to the V22, V23 samples.
On the same samples, the coefficients of the
models of transverse hydrodynamic forces (model D),
proposed in the works [3–6, 18], were calculated:
Models D
1 2 2 1 4
10 12
12
01
1
2
30
21
1.74 2.68 5.16
1.17 3.84 , 5.12 (1 ) ,
2.12 1.47 5.48
1.46
1.31 ,
1.63
0.5 0.98
0.58 0.87 ,
0.08 0.52
4.66
8.
yy
y
x
y
y
CC
C m m
C
C
−
= + = −
− − = −
−
= + −
=
12
1.2
24 2.2 .
0.28 0.32
−
−−
Tables 3 and 4 show the main correlation
characteristics of the dependences C and D.
Table 3. Analysis of the model C
_______________________________________________
R
F
Cond. (8)
s
_______________________________________________
10
y
C
V21 0.72 3·10
-6
+ 0.007
V22 0.66 0.002 - 0.85
V23 0.71 0.007 + 0.25
01
y
x
C m m
−−
V21 0.78 10
-16
+ 0.85
V22 0.73 2·10
-4
- 0.9
V23 0.91 1·10
-5
- 0.4
30
y
C
V21 0.58 2·10
-3
+ 4·10
-2
V22 0.44 0.16 - 0.94
V23 0.86 1·10
-3
+ 0.29
21
y
C
V21 0.78 10
-6
+ 4·10
-4
V22 0.68 5·10
-3
+ 0.06
V23 0.67 5·10
-2
- 0.49
12
y
C
V21 0.52 2·10
-3
+ 0.02
V22 0.35 0.14 + 0.14
V23 0.74 5·10
-3
- 0.94
_______________________________________________
Analysis of models C shows that not all these
models of the derivatives of transverse hydrodynamic
forces are adequate.
Mathematical models have good correlation
characteristics with good regression indicators only
for constant
10
y
C
when
(0.49;0.9)
b
С
and
(0.7;0.9),
b
С
for constant
30
y
C
when
(0.7;0.9)
b
С
and for constant
12
y
C
when
(0.49;0.9)
b
С
and
(0.49;0.7)
b
С
.
Table 4. Analysis of the model D
_______________________________________________
R
F
Cond. (8)
s
mk
_______________________________________________
10
y
C
V21 0.99 9·10
-29
+ 5·10
-7
0.27
V22 0.99 7·10
-17
+ 0.01 0.64
V23 0.99 8·10
-11
+ 0.07 0.4
01
y
x
C m m
−−
V21 0.97 2·10
-21
+ 10
-21
-
V22 0.95 6·10
-11
+ 10
-16
-
V23 0.99 10
-14
+ 10
-15
-
30
y
C
V21 0.4 0.02 - 0.41 -
V22 0.52 0.02 - 0.51 -
V23 0.1 0.78 - 0.8 -
21
y
C
V21 0.42 0.02 + 0.02 -
V22 0.64 2·10
-3
+ 2·10
-3
-
V23 0.19 0.7 - 0.7 -
12
y
C
V21 0.91 9·10
-14
+ 10
-13
-
V22 0.91 3·10
-8
+ 2·10
-8
-
V23 0.97 4·10
-8
+ 1·10
-8
-
_______________________________________________
For models D mathematical models of constants
10
y
C
,
01
y
x
C m m
−−
and
12
y
C
have excellent regression
indicators for the entire range of values of the block
coefficient
b
C
. Mathematical models for the rest of
the hydrodynamic constants are inadequate.
To analyse the mathematical models of the
derivatives of the moment of hydrodynamic forces,
we used the experimental data of the works [4, 18]
and made three samples for the derivatives of the
moment depending on the values of the block
coefficient:
31 { 33; (0.49;0.9)}
32 { 20; (0.49;0.7)}
33 { 13; (0.7;0.9)}
b
b
b
V n C
V n C
V n C
= =
= =
= =
In the work [4], using regressors (3), models of
transverse hydrodynamic forces (models E) were
written. We will calculate the coefficients of these
models based on samples V31, V32, V33.
Models E
2
10 3 01 3 3
2
21 1 2 1 2
03
1.32 5.26 1.14
1.23 , 5.85 1.25 ,
2.09 25.36 2.13
52.39 16.02 1.46
3.24 ( ) 1.24 0.58 ,
21.37 7.75 0.50
3.91
3.17
6
mm
G
m
m
C C x m
C
C
= − = −
= − + −
− − −
=−
2
1 2 1 2
1.36 0.13
( ) 1.18 0.12 .
0.6 19.32 0.71
+−
The upper coefficients in the models E correspond
to the V21 sample, the second and third correspond to
the V22, V23 samples respectively.
858
On the same samples, the coefficients of the
models of transverse hydrodynamic forces (model F),
suggested in the works [3–6, 18], were calculated:
Models B
30 1
21 1 2
1
12 1 2
03 1 2
0.9 0.72
,
1.79 1.22
1.51 0.54
,
0.32 0.52
0.023 0.039
(1 ) ,
0.32 0.013
0.28 0.056
.
0.28 0.06
m
m
m
m
C
C
C
C
−
= − −
=−
−
= − +
=−
The upper coefficients in the F model are obtained
on the V31 sample while the second ones are obtained
on the V32 sample.
When constructing models of hydrodynamic
derivatives
03
y
C
,
30
m
C
,
12
m
C
, in the work [4], the
parameter
1
(1 )(1 )
a wa pa
CC
−
= − −
is also used as the
basic regressor, where
wa
C
and
pa
C
are the
waterplane area coefficient and prismatic coefficient
of aft half hull between APP and ship station five. The
coefficients are calculated as:
1
( ) ,
wa wa a a
C A L B
−
=
1
( ) ,
pa a a a
C A L
−
=
where
wa
A
is the water plane
area of the aft section,
a
A
is the cross-sectional area
equal to the largest underwater section of the aft hull,
a
is the displacement of the aft hull,
a
B
is the
vessel’s breadth of the aft hull and
a
L
is its length.
However, the analysis of these dependencies
showed their poor correlation features. Moreover, the
data for calculating the parameter
a
are not always
available in the reference literature.
Tables 5 and 6 show the correlation characteristics
of models E and F, respectively.
For models E mathematical models of
hydrodynamic constants
10
m
C
,
01
m
G
C x m
−
and
03
m
C
have excellent regression indicators for the entire
range of values of the block coefficient
b
С
. The
mathematical model for
21
m
C
is inadequate.
Analysis of the F models shows that not all these
models of the derivatives of the transverse
hydrodynamic forces are adequate. Mathematical
models only for the hydrodynamic constant
03
m
C
have quite good correlation characteristics and good
regression indicators.
Table 5. Analysis of the model E
_______________________________________________
R
F
Cond. (8)
s
_______________________________________________
10
m
C
V21 0.91 2·10
-13
+ 8·10
-14
V22 0.91 2·10
-08
+ 9·10
-09
V23 0.99 1·10
-14
+ 10
-14
01
m
G
C x m
−
V21 0.88 3·10
-9
+ 8·10
-5
V22 0.99 5·10
-14
+ 4·10
-10
V23 0.77 8·10
-3
+ 0.07
21
m
C
V21 0.55 0.005 + 0.003
V22 0.11 0.91 - 0.9
V23 0.72 0.03 - 0.4
03
m
C
V21 0.84 10
-8
+ 5·10
-5
V22 0.88 4·10
-6
+ 0.004
V23 0.91 2·10
-4
+ 0.04
_______________________________________________
Table 6. Analysis of the model F
_______________________________________________
R
F
Cond. (8)
s
_______________________________________________
10
m
C
V21 0.4 0.02 + 0.02
V22 0.34 0.14 + 0.14
01
m
G
C x m
−
V21 0.28 0.11 + 8·10
-5
V22 0.1 0.67 - 0.67
21
m
C
V21 0.25 0.16 + 0.26
V22 0.06 0.8 - 0.83
03
m
C
V21 0.7 6·10
-6
+ 6·10
-6
V22 0.79 4·10
-5
+ 4·10
-5
_______________________________________________
Thus, the analysis of the existing models for the
derivatives of hydrodynamic forces and moment
shows that many of them cannot be used for the entire
range of variation of the values of the block coefficient
b
C
. Only some of them can provided a fairly good
correlation on limited ranges. Obviously, a univariate
correlation analysis cannot provide the construction of
adequate models with a high level of significance for
the entire range of change in values
b
C
As for the
approach of the work [2] for mathematical models of
longitudinal hydrodynamic forces, then, it is obvious
that the use of the minimum criterion AIC only cannot
ensure the fulfilment of criteria 1) - 7).
5 CONSTRUCTION OF NEW MATHEMATICAL
MODELS OF HYDRODYNAMIC FORCES AND
MOMENTS
The analysis of the known models indicates that there
is a need to build new adequate models of the
derivatives of the longitudinal hydrodynamic forces
on the ship's hull with a high level of significance that
meet the criteria 1) - 7). The standard scheme of
multivariate regression analysis [1], and the method
described in the second section, made it possible to
construct several new adequate models of the
derivatives of longitudinal hydrodynamic forces and
moment with high correlation indicators. Some of
these models having the highest level of correlation
and levels of significance as well as the standard
errors of the regressors of which satisfy condition (7)
are given below.
To construct the hydrodynamic derivatives of
transverse forces, we use samples V11, V12, V13.
In particular, for the constant
20
x
C
for the entire
range of variation of the block coefficient
(0.5;0.9)
b
С
the following representations should be
highlighted:
20 1 1 4 1 2 3
0.086 0.389(1 ) 5.599 ,
x
C
= − − − +
(11)
20 1 1 2 3
0.173 3.74 .
x
C
= − +
(12)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide excellent
correlation:
20 1 1 2 3
0.173 4.855 ,
x
C
= − +
(13)
859
20 1 1 2 3
0.528 6.088 ,
x
C
= − +
(14)
20 1 2 1 4
2.529(1 ) 1.714(1 ) ,
x
C
= − − −
(15)
For the range of values of the block coefficient
(0.7;0.9)
b
С
, the following model can be also used:
20 1 4 1 2 3
2.529(1 ) 15.086 .
x
C
= − −
(16)
Table 7 shows the correlation characteristics of
models (11) - (16).
Table 7. Analysis of the model for
20
x
C
.
_______________________________________________
R
F
s
mk
_______________________________________________
(11) V11 0.8 5·10
-5
0.02 0.5
(12) 0.75 3·10
-4
3·10
-4
0.5
(13) V12 0.81 0.003 0.003 0.04
(14) 0.84 0.003 7·10-4 0.6
(15) 0.85 0.003 5·10-4 0.79
(16) V13 0.8 0.004 0.14 0.59
_______________________________________________
For the constant
11
x
y
Cm
−
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
the
following representations should be highlighted:
11 0 3
0.978 0.603 ,
x
y
Cm
− = − −
(17)
11 1 4 2 3
0.504 3.086 .
x
y
Cm
− = − −
(18)
For the range of values of the coefficient of total
completeness
(0.5;0.7)
b
С
the following
dependence also provides excellent correlation
11 1 4 2 3
0.396 3.634 .
x
y
Cm
− = − −
(19)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following model can be also used
2 2 2
11 0 0 3
6.344 3.634 .
x
y
Cm
− = − −
(20)
Table 8 shows the correlation characteristics of
models (17) - (20).
Table 8. Analysis of the model for
11
x
y
Cm
−
.
_______________________________________________
R
F
s
mk
_______________________________________________
(17) V11 0.98 10
-17
0.04 0.68
(18) 0.98 10
-16
5·10
-5
0.14
(19) V12 0.99 10
-8
5·10
-5
0.56
(20) V13 0.98 10
-12
0.14 0.38
_______________________________________________
For a constant
02
x
Gy
C x m
+
for the range of
variation of the block coefficient
(0.5;0.7)
b
С
the
following representations should be highlighted:
02 1 1 2 1 4
0.07 0.34(1 ) 0.37 ,
x
Gy
C x m
+ = + − −
(21)
02 1 2 1 4
0.303(1 ) 0.166 .
x
Gy
C x m
+ = − −
(22)
The following model is also adequate for the range
(0.7;0.9)
b
С
:
02 1 0 1 3
0.49 9.68 0.484.
x
Gy
C x m
+ = + −
(23)
Table 9 shows the correlation characteristics of
models (21) - (23).
Table 9. Analysis of the model for
02
x
Gy
C x m
+
_______________________________________________
R
F
s
mk
_______________________________________________
(21) V12 0.68 0.07 0.24 0.586
(22) 0.68 0.05 0.08 0.378
(23) V13 0.66 0.02 0.03 0.271
_______________________________________________
For the constant
40
x
C
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
the
following representations should be highlighted:
40 1 4 1 2 3
2.85 33.225 ,
x
C
=−
(24)
40 1 1 2 3
0.899 26.105 ,
x
C
=−
(25)
40 1 4 1 3
4.89 17.463 .
x
C
=−
(26)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following model provides excellent
correlation:
40 1 2 3 4
1.78 35.83 ,
x
C
=−
(27)
40 1 4 1 3
5 18.057 .
x
C
=−
(28)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following model can be also used:
40 1 2 1 2 3
26.973(1 ) 167.485 .
x
C
= − − +
(29)
Table 10 shows the correlation characteristics of
models (24) - (29).
Table 10. Analysis of the model for
40
x
C
_______________________________________________
R
F
s
mk
_______________________________________________
(32) V11 0.8 10
-6
10
-4
0.25
(33) 0.79 10
-6
0.002 0.28
(34) 0.79 10
-6
10
-3
0.55
(35) V12 0.79 0.004 0.003 0.75
(36) 0.75 0.009 0.006 0.75
(37) V13 0.86 10
-4
0.12 0.73
_______________________________________________
To construct the hydrodynamic derivatives of
transverse forces, we use samples V21, V22, V23.
For the constant
10
y
C
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
the
following representations should be highlighted:
2 1 2 3
10
1.36 16.79 .
y
C
=+
(30)
860
1 4 1 2 3
10
0.94(1 ) 39.2 ,
y
C
= − +
(31)
1 1 2 3
10
0.33 19.69 ,
y
C
=+
(32)
12
10
0.13 1.39 .
y
C
=+
(33)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following model provides excellent
correlation:
2 1 2 3
10
1.12 24.15 ,
y
C
=+
(34)
1 4 1 2 3
10
1.27(1 ) 28.2 ,
y
C
= − +
(35)
1 1 2 3
10
0.24 34.02 .
y
C
=+
(36)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following models can be also used:
2 1 2 3
10
1.61 11.23 ,
y
C
=+
(37)
1 4 1 2 3
10
2.75(1 ) 47.4 ,
y
C
= − +
(38)
1 1 2 3
10
0.38 10.87 .
y
C
=+
(39)
Table 11 shows the correlation characteristics of
models (30) - (39).
Table 11. Analysis of the model for
10
y
C
_______________________________________________
R
F
s
mk
_______________________________________________
(30) V21 0.99 10
-28
10
-5
0.65
(31) 0.97 10
-19
10
-3
0.06
(32) 0.99 10
-24
10
-4
0.55
(33) 0.99 10
-24
0.1 0.32
(34) V22 0.99 10
-16
10
-4
0.72
(35) 0.99 10
-15
10
-3
0.57
(36) 0.98 10
-14
10
-4
0.34
(37) V21 0.99 10
-9
0.04 0.51
(38) 0.97 10
-6
0.04 0.38
(39) 0.99 10
-10
0.02 0.72
_______________________________________________
For the entire range of variation of the block
coefficient
(0.5;0.9)
b
С
the following
representations should be highlighted
1 1 2 3
01
0.21 4.76 ,
y
x
C m m
− − = − −
(40)
13
01
0.21 0.8 ,
y
x
C m m
− − = − −
(41)
12
01
0.12 0.54 .
y
x
C m m
− − = − −
(42)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following model provides excellent
correlation:
1 1 2 3
01
0.16 5.85 ,
y
x
C m m
− − = − −
(43)
13
01
0.09 1.37 .
y
x
C m m
− − = − −
(44)
Table 12 shows the correlation characteristics of
models (40) - (44).
Table 12. Analysis of the model for
01
y
x
C m m
−−
_______________________________________________
R
F
s
mk
_______________________________________________
(40) V21 0.97 10
-18
10
-4
0.04
(41) 0.96 10
-16
10
-2
0.12
(42) 0.96 10
-17
10
-2
0.07
(43) V22 0.96 10
-9
10
-4
0.15
(44) 0.95 10
-17
0.04 0.21
_______________________________________________
For the entire range of variation of the block
coefficient
(0.5;0.9)
b
С
the following
representation should be highlighted:
1 4 1 2 3
30
24.09(1 ) 99.72 .
y
C
= − −
(45)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide excellent
correlation:
1 4 1 2 3
30
24.82(1 ) 116.47 ,
y
C
= − −
(46)
13
30
6.19 17.62 .
y
C
=−
(47)
For the range of values of the coefficient of total
completeness
(0.7;0.9)
b
С
the following models
can be also used:
1 4 1 2 3
30
10.82(1 ) 64.99 ,
y
C
= − +
(48)
3
30
22.2 ,
y
C
=
(49)
2
1 2 3 1 4
30
64.43 166.62((1 ) ) .
y
C
= − −
(50)
Table 13 shows the correlation characteristics of
models (48) - (52).
Table 13. Analysis of the model for
30
.
y
C
_______________________________________________
R
F
s
mk
_______________________________________________
(53) V21 0.84 10
-7
0.005 0.57
(54) V22 0.84 10
-4
0.02 0.77
(55) 0.88 10
-5
0.04 0.21
(56) V23 0.96 10
-5
0.081 0.38
(57) 0.96 10
-6
10
-06
-
(58) 0.98 10
-7
0.004 0.36
_______________________________________________
For the entire range of variation of the block
coefficient
(0.5;0.9)
b
С
the following
representation should be highlighted:
1 1 2
21
3.26 8.92 3.52,
y
C
= + −
(51)
861
12
21
4.87 5.14 4.44.
y
C
= + −
(52)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide excellent
correlation:
1 1 2
21
3.12 10.3 ,
y
C
= − +
(53)
1 1 4
21
4.83 9.77 .
y
C
= − +
(54)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following models can be also used:
1 1 2
21
0.93 7.15 ,
y
C
= − +
(55)
1 2 1 4
21
3.74 4.27(1 ) .
y
C
= − −
(56)
Table 14 shows the correlation characteristics of
models (51) - (56).
Table 14. Analysis of the model for
21
.
y
C
_______________________________________________
R
F
s
mk
_______________________________________________
(51) V21 0.75 10
-5
0.003 0.46
(52) 0.75 10
-5
0.005 0.08
(53) V22 0.78 10
-2
0.004 0.63
(54) 0.78 10
-2
0.003 0.63
(55) V23 0.91 10
-2
0.07 0.72
(56) 0.91 10
-4
0.04 0.12
_______________________________________________
For the entire range of variation of the block
coefficient
(0.5;0.9)
b
С
the following
representation should be highlighted:
3 1 4
12
6.96 9.07(1 ) .
y
C
= − + −
(57)
22
1 1 4
12
0.52 16.7(1 ) .
y
C
= + −
(58)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide excellent
correlation:
12
12
1.78 1.47 ,
y
C
=−
(59)
1 1 2
12
1.98 3.55 .
y
C
=−
(60)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following models can be also used:
12
12
1.55 4.95 ,
y
C
=−
(61)
1 1 2
12
1.25 4.69 .
y
C
=−
(62)
Table 15 shows the correlation characteristics of
models (57) - (62).
Table 15. Analysis of the model for
12
.
y
C
_______________________________________________
R
F
s
mk
_______________________________________________
(65) V21 0.95 10
-15
10
-4
0.71
(66) 0.89 10
-10
10
-4
0.62
(67) V22 0.96 10
-9
10
-2
0.28
(68) 0.96 10
-9
0.07 0.03
(69) V23 0.97 10
-6
0.01 0.46
(70) 0.96 10
-5
0.03 0.72
_______________________________________________
For this hydrodynamic derivative it was possible
to obtain the following models with satisfactory
statistical characteristics:
1 1 4
03
0.1 0.22
0.23 0.49 .
0.07 0.22
y
C
= − +
(63)
2
1 4 1 4
03
0.6 2.06
0.86 3.07 ( ) .
0.31 1.12
y
C
= − +
(64)
The upper lines in the dependencies (63) and (64)
were obtained for the V21 sample; the second and
third ones were obtained for the V22 and V23 samples
respectively.
Table 16 shows the correlation characteristics of
models (63) - (64).
Table 16. Analysis of the model for
03
.
y
C
_______________________________________________
R
F
s
mk
_______________________________________________
(63) V21 0.54 5·10
-3
0.09 0.4
V22 0.74 9·10
-4
0.02 0.63
V23 0.62 0.08 0.05 0.28
(64) V21 0.65 3·10
-4
5·10
-4
-
V22 0.71 3·10
-3
6·10
-3
-
V23 0.64 0.06 0.03 -
_______________________________________________
To construct the hydrodynamic derivatives of the
moments we will use the samples V31, V32, V33.
The following models have excellent regression
characteristics for the constant
10
m
C
for all samples:
10 1 3
2.15
1.91 ,
2.64
m
C
=
(65)
10 1 3
0.08 0.66 0
0.09 0.59 0 .
0.11 1.96 0.08
m
C
= + −
(66)
Table 17 shows the correlation characteristics of
models (65) and (66).
862
Table 17. Analysis of the model for
10
m
C
_______________________________________________
R
F
s
mk
_______________________________________________
(65) V31 0.96 10
-18
10
-18
-
V32 0.93 10
-8
10
-9
-
V33 0.99 10
-12
10
-12
-
(66) V31 0.96 10
-16
10
-4
0.12
V32 0.95 10
-8
3·10
-3
0.22
V33 0.99 10
-8
0.04 0.62
_______________________________________________
For the constant
01
m
G
C x m
−
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
the
following representations have excellent correlation
characteristics:
01 1
0.06 ,
m
Gx
C x m
− = −
(67)
01 4
0.13 ,
m
G
C x m
− = −
(68)
01 1 4
0.37(1 ) ,
m
G
C x m
− = − −
(69)
22
01 1 3 1 3
3.14(1 ) 34.93(1 ) .
m
G
C x m
− = − − + −
(70)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide good
correlation:
01 1
0.08 ,
m
G
C x m
− = −
(71)
22
01 1 3 1 3
2.92(1 ) 31.62(1 ) .
m
G
C x m
− = − − + −
(72)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following dependence can be also
used:
01 1
0.05 .
m
G
C x m
− = −
(73)
Table 18 shows the correlation characteristics of
the models (67) - (73).
Table 18. Analysis of the model for
01
.
m
G
C x m
−
_______________________________________________
R
F
s
mk
_______________________________________________
(67) V21 0.86 10
-9
0.08 -
(68) 0.85 10
-9
10
-9
-
(69) 0.85 10
-9
10
-9
-
(70) 0.88 10
-10
10
-5
-
(71) V22 0.97 10
-12
10
-12
-
(72) 0.98 10
-9
10
-9
-
(73) V23 0.75 10
-2
10
-2
-
_______________________________________________
For the constant
30
m
C
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
and
for values
(0.5;0.7)
b
С
the following models have
good correlation features (the upper coefficients are
the sample V31, the lower ones are V32):
30 1 2
2.27
(1 ) ,
2.45
m
C
=−
(74)
30 1 4
1.33
(1 ) ,
1.42
m
C
=−
(75)
30 1 3
5.68
(1 ) ,
5.83
m
C
=−
(76)
2
11
30 1
1 2 2
2 4 2 4
1.83 1.59 0.48
(1 ) (1 )
.
3.33 2.6 1.37
m
C
−−
−−
= − −
(77)
Table 19 shows the correlation characteristics of
models (74) - (77).
Table 19. Analysis of the model for
30
.
m
C
_______________________________________________
R
F
s
mk
_______________________________________________
(74) V21 0.60 2·10
-4
2·10
-4
-
V22 0.63 3·10
-3
3·10
-3
-
(75) V21 0.55 8·10
-4
8·10
-4
-
V22 0.57 8·10
-3
8·10
-3
-
(76) V21 0.55 9·10
-4
9·10
-4
-
V22 0.56 9·10
-3
9·10
-3
-
(77) V21 0.64 10
-3
0.02 0.31
V22 0.73 5·10
-3
0.08 0.31
_______________________________________________
For the constant
12
m
C
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
the
following representations have excellent correlation
characteristics:
21 1 4
3.33(1 ) ,
m
C
= − −
(78)
21 1 4 1 2 3
4.14(1 ) 9.8 .
m
C
= − − +
(79)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide good
correlation:
21 1 4
3.45(1 ) ,
m
C
= − −
(80)
21 1
0.83 ,
m
C
=−
(81)
2
44
21
2
2
2
0.59 0.17 .
m
C
= − +
(82)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following dependences should be
also highlighted:
21 1
0.2 ,
m
C
=−
(83)
2
44
21
2
2
2
0.59 0.17 .
m
C
= − +
(84)
Table 20 shows the correlation characteristics of
models (78) - (85).
863
Table 20. Analysis of the model for
21
.
m
C
_______________________________________________
R
F
s
mk
_______________________________________________
(78) V21 0.94 10
-14
10
-15
-
(79) 0.95 10
-15
5·10
-3
0.57
(80) V22 0.94 10
-9
10
-9
-
(81) 0.95 10
-9
10
-9
-
(82) 0.96 10
-9
0.02 0.77
(83) 0.97 10
-10
10
-3
-
(84) V23 0.97 10
-7
10
-7
-
(85) 0.95 10
-5
0.01 -
_______________________________________________
For the constant
12
m
C
for the entire range of
variation of the block coefficient
(0.5;0.9)
b
С
and
for the range
(0.5;0.7)
b
С
the following model has
satisfactory correlation characteristics:
12 1 4 1 4
0.76 0.34
(1 ) .
1 0.51
m
C
= − −
(85)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following dependence provide a
satisfactory correlation:
12 1 1 4
0.44 2.26 .
m
C
= − +
(86)
Table 21 shows the correlation characteristics of
models (85), (86).
Table 21. Analysis of the model for
12
.
m
C
_______________________________________________
R
F
s
mk
_______________________________________________
(85) V21 0.5 6·10
-3
2·10
-2
0.3
V22 0.54 2·10
-2
0.2 0.72
(86) V23 0.66 0.05 0.04 0.74
_______________________________________________
For the constant
03
m
C
for the entire range of
variation of the block coefficient:
(0.5;0.9)
b
С
the
following representations have excellent correlation
characteristics:
03 1 4 1 2 3
0.35(1 ) 1.47 ,
m
C
= − − +
(87)
2
44
03
2
2
2
0.03 0.01 .
m
C
= − +
(88)
For the range of values of the block coefficient
(0.5;0.7)
b
С
the following models provide good
correlation:
03 1 1 2
0.1 0.27 ,
m
C
= − +
(89)
2
44
03
2
2
2
0.05 0.02 ,
m
C
= − +
(90)
03 1 4 1 2 3
0.33(1 ) 1.69 .
m
C
= − − +
(91)
For the range of values of the block coefficient
(0.7;0.9)
b
С
the following dependences should be
also highlighted:
03 1 4 1 2 3
0.1(1 ) 0.96 ,
m
C
= − − −
(92)
4
03
2
0.01 .
m
C
=−
(93)
Table 22 shows the correlation characteristics of
models (89) - (93).
Table 22. Analysis of the model for
03
.
m
C
_______________________________________________
R
F
s
mk
_______________________________________________
(89) V21 0.91 10
-11
10
-4
0.57
(90) 0.86 10
-8
10
-3
-
(91) V22 0.93 10
-7
10
-3
0.31
(92) 0.93 10
-7
10
-4
-
(93) 0.93 10
-7
10
-4
0.77
(94) V23 0.92 10
-4
0.2 0.38
(95) 0.92 10
-5
10
-5
-
_______________________________________________
6 CONCLUSIONS
The results shown in Tables 7-21 confirm that almost
all the new models of hydrodynamic forces and
moment on the hull which have been obtained, in
contrast to the existing ones, establish a high degree of
correlation with an excellent level of significance of
the connection with regressors. The fact that there are
several adequate models that meet criteria 1) - 7) for
each hydrodynamic derivative allows to choose the
optimal model. If the manoeuvre for vessels with a
wide range of changes in the values of the block
coefficient
(0,5;0,9)
b
C
is studied, it is necessary to
use models are based on the samples V11, V21, V31.
For narrower ranges of change
b
C
, it is advisable to
use models that are based on the samples V12, V21,
V32 or V13, V23, V33.
The suggested approach allows to obtain new
adequate mathematical models of other non-inertial
forces on the hull, which will allow to build more
accurate mathematical models of the dynamics of the
ship’s propulsion complex.
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