1 INTRODUCTION

In the study of ship maneuvers, the development of

ship autopilots and ship simulators, the availability of

adequate mathematical models of the propulsive ship

complex, which would describe the dynamics of the

ship with high accuracy for the widest ranges of

changes in kinematic parameters, plays a decisive

role. The application of mathematical models of the

ship's movement to the study of various maneuvers of

the ship has recently become widely developed. The

construction and active use of such models began in

the middle of the twentieth century, these models are

highlighted, in particular, in [1 - 5]. A significant

contribution to the development and application of

such models was also made by Japanese scientists,

one of the first publications in this direction are works

[6, 7]. This approach, which is also covered in [8] and

is known as the MMG (Manoeuvring Modeling

Group) method, was widely used, in particular, in [9 -

16]. A significant contribution to the development and

application of mathematical models of ship dynamics

was also made by the Portuguese school of

researchers [17].

Mathematical models of the ship's propulsive

complex consist of mathematical models of inertial

and non-inertial forces and moments on the ship's

hull. The latter include hydrodynamic and

aerodynamic forces and moments on the ship's hull,

forces and moments caused by the operation of

propellers, auxiliary wind propulsors (sails), and

ship's rudders. For the adequacy of the general

mathematical model of ship dynamics, the adequacy

of the mathematical models of all components of non-

Construction and Analysis of New Mathematical Models

of the

Operation of Ship Propellers in Different

aneuvering Modes

. Kryvyi, M. Miyusov & M. Kryvyi

National University

“Odessa Maritime Academy”, Odessa, Ukraine

ABSTRACT: The influence of the curvilinear movement of the ship on the operation of the propeller was

studied. It is shown that even at small values of the drift angle and the angular velocity of the vessel, the

transverse component of the force on the propeller and the moment are non-zero and cannot be neglected. The

existing and proposed new effective mathematical models of the longitudinal and transverse components of

force and moment caused by the operation of the ship's propeller are analyzed. Simple expressions for the

coefficients of the propeller thrust and the moment on the propeller shaft, the wake fraction, the thrust-

deduction factor, and the flow straightening factor on the propeller at any drift angles and angular velocity are

obtained. Numerical analysis of the obtained dimensionless components of forces and moments caused by the

operation of the propeller is carried out, and their adequacy is shown. It is shown how the specified parameters

change for all possible values of the drift angle and angular velocity. For a few commercial vessels of various

types, technical characteristics and calculated dynamic parameters are given for the construction of

mathematical models of propeller operation during curvilinear movement of the vessel.

http://www.transnav

.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 17

Number 1

Marc

h 2023

DOI: 10.12716/1001.17.01.

inertial forces and moments on the ship's hull is

necessary.

Mathematical models of non-inertial forces and

moments are usually empirical in nature and are built

by processing data from experimental tests in ship

model basins or during natural experiments. In

particular, this applies to the forces and moments

caused by the operation of the propeller. To describe

the ship's maneuvers even in a horizontal plane

(without taking into account roll and heel), linear

mathematical models are mainly used. This is

explained by the difficulty of using non-linear models

and is argued by the small range of changes in the

ship's kinematic parameters, such as the speed v, drift

angle β, and angular speed

during "weak

maneuvers".

But the concept of "weak maneuver" is quite

conditional, and even with small drift angles at the

center of gravity of the vessel, the local drift angle at

the stern increases significantly due to the angular

velocity. Therefore, such important ship maneuvers as

circulation, Kempf's zigzag, and, even more so, sharp

evasion cannot be attributed to "weak" ship

maneuvers.

In the mathematical modeling of the specified

forces and moments, the rectilinear motion of the

vessel, or motion with small values of the drift angles,

is usually considered. At the same time, mainly, only

the longitudinal action of the propeller is considered,

and the transverse component of the force and

moment on the propeller is not justifiably neglected.

But experimental studies [18] show that even at small

drift angles, the transverse component of the force on

the ship's propeller takes on significant importance. In

addition, known mathematical models usually

contain coefficients and functions, which are defined

rather complicatedly using tables and graphs, which

is not convenient for numerical modeling. Among the

shortcomings of the existing models of non-inertial

forces should also be attributed their excessive, not

always justified, simplification. Therefore, the

construction of effective mathematical models of the

ship's propulsive complex for a wide range of changes

in kinematic parameters is not only an important

scientific problem, but also an urgent practical task. In

[19-24], new mathematical models of hydrodynamic

forces on the hull were obtained for a wide range of

changes in drift angle and angular velocity.

The development of the theory and calculation of

the work of ship propellers is devoted, in particular,

to works [2, 25 - 30]. These studies mainly concern a

single propeller and the calculation of its parameters.

The paper [28] provides a large array of experimental

data for calculating the thrust coefficient of the

propeller. In the works [2, 25], some aspects of the

operation of the propeller during curvilinear

movement of the ship were investigated, but the

parameters of the models obtained there are

presented in a form that is not convenient for use. In

[3], the influence of auxiliary wind propulsors (sails)

on the operation of the ship's propulsion complex was

investigated.

There are known [18] experimental studies of the

influence of the curvilinear movement of the ship on

the operation of the propeller, which confirm the

occurrence of significant transverse forces during such

movement. Analysis of the latest research shows that,

despite the recent significant development of

mathematical models of the ship's propulsive

complex, many problems require further resolution.

In particular, the construction of adequate

mathematical models that would take into account all

components of the force and moment on the propeller

for a wide range of changes in the kinematic

parameters of the ship's movement and would be

convenient for numerical modeling.

The goal of this work is the construction and

numerical analysis of adequate mathematical models

of forces and moments caused by the operation of

ship propellers, which would, on the one hand, cover

the entire range of changes in the kinematic

parameters of the ship's motion, and on the other

hand, would be convenient for use in solving various

problems of the dynamics of the ship's propulsive

complex.

The mathematical model of the operation of the

ship's propeller is a multi-level model (as, after all,

any complex empirical model), it consists of several

dimensionless parameters, which are determined on

the basis of experimental studies and each of which is

important for the construction of the overall model.

Therefore, we will first determine how the curvilinear

movement affects the characteristics of the propeller,

then we will determine and perform research for each

parameter separately, and finally we will obtain a

general mathematical model of the operation of the

ship's propeller and conduct a numerical study of it as

a whole.

2 THE INFLUENCE OF THE SHIP'S CURVILINEAR

MOVEMENT ON THE CHARACTERISTICS OF

THE SHIP'S PROPELLER

2.1 Technical and kinematic parameters of the vessel

and propeller

The geometric and technical characteristics of the ship

and propeller will be denoted as follows: L – length of

the ship along the waterline; B – width of the vessel

along the current waterline; T – draft of the ship on

the midline,

– mass density of sea water, Cb – block

coefficient; W=C

bLBT – volume displacement of the

vessel;

S LT= σ

area of the underwater part of the

centerplane of the ship;

– reduced coefficient of

the underwater centerplane of the ship; n

p, Dp –

respectively, the rotation frequency and the diameter

of the propeller;



the relative position (in the stern,

x xL= <



) of the propeller from the center of

gravity of the vessel (for the main types of vessels, it is

assumed that

( . ; .)045 05

x ∈− −



);

a AD

−−

= π

–

blade area ratio of the propeller, the value of which is

within

..04 11÷

;

h HD

−

= ⋅

– pitch ratio of the

propeller, the values of which are, as a rule, within

..05 16÷

;

– the number of propeller blades; v – the

speed of the ship in the direction of movement;

–

the angular speed of rotation of the ship around the

axis Z;

/Fr v gL=

– Froude number; v0 – the value

of the ship's speed at the time of the start of the

maneuver (initial speed);

– local drift angle on

the propeller; v

P – speed of flow on the propeller. The

values of the main technical and kinematic parameters

of some types of merchant ships are given in Table 1.

Mathematical models of forces and moments

caused by the operation of the ship's propeller will be

built relative to dimensionless kinematic phase

coordinates: speed:

v vv







, drift angle

and

angular velocity

ωω ω

0ZZ

Lv v Lv







Table 1. Technical and kinematic parameters of ships

________________________________________________

Ship DTC, KCS, KVLCC2 VLCC, VLGC, LPG,

contain. contain. tanker tanker tanker tanker

ship ship ESS0

________________________________________________

№ 1 2 3 4 5 6

________________________________________________

L [m] 355 230 320 325 226 147

B [m] 51 32.2 58 53 36.6 25.5

T [m] 14.5 10.8 20,8 21.79 11.8 8.8

0.661 0.651 0,809 0.831 0.72 0.740

[]

8.911 7.9 9.86 9.1 7.4 5.7



-0.45 -0.46 -0.48 -0.46 -0.47 -0.47

5 5 4 5 4 4

а 0.8 0.8 0.431 0.682 0.42 0.601

h 0.959 0.997 0.721 0.715 0.905 0.784

0.376 0.3365 0.4495 0.473 0.382 0.373

0.281 0.2474 0.351 0.374 0.288 0.2796

0.456 0.4749 0.292 0.319 0.3616 0.337

-0.266 -0.264 -0.211 -0.251 -0.186 -0.248

-0.265 -0.266 -0.218 -0.261 -0.218 -0.229

0.085 0.0845 0.053 0.076 0.050 0.0723

0.051 0.0530 0.028 0.031 0.0369 0.035

-0.036 -0.037 -0.022 -0.027 -0.024 -0.029

-0.033 -0.034 -0.009 -0.021 -0.01 -0.020

-0.048 -0.055 -0.031 -0.022 -0.051 -0.024

________________________________________________

2.2 Influence of the curvilinear movement of the ship on

the dynamic parameters of the propeller

The operating parameters of a ship's propeller during

curvilinear movement with a certain drift angle and

angular velocity change significantly compared to

straight-line movement. This is due to the following

factors.

First, the local drift angle will change on the

propeller, which will be determined by using the

equality of the projections of the velocity vector at the

center of gravity and on the propeller, respectively, on

the X axis and the Y axis:

cos cos ,

sin sin .

P P PZ

v vx

 β= β





β = β− ⋅ω





(1)

On the right side of the last equality, the second

term arose due to the rotation of the vessel around the

Z axis. Further, after dividing the first equality by the

second in relations (1), we represent the local drift

angle as follows

β ctg(tgβ

ar )

cos

PP P

=κ−



(2)

Here, by introducing a flow straightening factor

, the change in flow due to the influence of the

ship's hull during curvilinear motion is taken into

account. The calculation of the coefficient will be

discussed below.

Secondly, the speed of the flow on the propeller

will change, which, using relation (1), will be given as

follows:

( ψ)1

P PP

v kv= −

(3)

Where

is a dimensionless coefficient that takes

into account the influence of the ship's curvilinear

movement on the velocity of flow on the propeller

and can be calculated by the formula:

1 2 sin

PPP

kxx= − ω β+ ω



. (4)

In representation (3), the influence of the ship's

hull on the speed of flow on the propeller is

determined by the wake fraction

during

curvilinear movement, the calculation of which will

be discussed below.

Thirdly, during curvilinear movement, the

advance ratio of the propeller will also change, which

in this case will be given as follows

( ψ)

PP PP

kvv

−

= =



. (5)

Fourthly, the longitudinal component of the force

()

P PP

= −ζ

, where TP – the thrust of the propeller

will change due to the change in the dimensionless

thrust-deduction factor

on the propeller and the

advance ratio J

P. In addition, as a result of a violation

of the symmetry of the component forces acting in the

plane of the propeller disk, a transverse force Y

P will

be formed, which acts on the ship's propeller, as well

as a moment

P PP

M YL=

where

Lx=

– the

distance from the center of gravity of the ship to the

propeller. Due to the change in the advance ratio J

P ,

the moment on the propeller shaft Q

P will also change.

Dimensionless load coefficients [2, 24] correspond to

the specified components of forces, in terms of the

thrust of the propeller and in terms of the transverse

force:

22 22

P yP

PP PP

σ= σ =

ρπ ρπ

, (6)

as well as dimensionless coefficients of propeller:

thrust, transverse component of force and torque:

DDD

24 24 25

P PP

T Py Q

PP PP PP

T YQ

KK K

nnn

= = =

ρρρ

. (7)

3 MATHEMATICAL MODELS OF DYNAMIC AND

KINEMATIC PROPELLER PARAMETERS AND

THEIR NUMERICAL ANALYSIS

We will analyze existing and consider new

mathematical models of dynamic and kinetic

parameters of a ship's propeller. In particular, a

number of studies [2], [25-29] show that the

coefficients of propeller thrust and transverse force

are functions of the advance ratio J

P, blade area ratio a,

pitch ratio h and the number of propeller blades

( ,,,)

T TP

K K J ah= κ

( , , , ).

Q QP

K K J ah= κ

Usually, these

functions are approximated by polynomials:

( ,,, )

TP p k P

K J ah A a h J

ζν

κ= κ

∑

, (8)

( ,,, )

QP p k P

K J ah B a h J

ζν

κ= κ

∑

, (9)

where the number of terms

, exponents

,,,

kkkk

qgζν

and coefficients Ak, Bk, depend on the

design features of the propellers and are determined

by processing experimental data [27, 28] based on

correlation analysis. In particular, for values of

27κ= ÷

, it is not difficult to obtain the following

representations:

01 2 3

( ,,,)

TP P P P

K J ah J J Jκ =µ +µ +µ +µ

, (10)

** * *

( ,,,)

01 2 3QP P P P

K J ah J J Jκ =µ +µ +µ +µ

. (11)

Table 1 shows the values of the coefficients of

expressions (10) and (11), as well as the values of

other technical and dynamic parameters necessary for

the construction of mathematical models of forces and

moment components for some types of commercial

vessels.

The load coefficient on the thrust of the propeller

can be easily expressed through the dimensionless

coefficient of thrust of the propeller (10) and the

advance ratio of the propeller (5):

σ=

. (12)

For the transverse force load factor, we will use the

presentation of work [2], and replacing there

with

sin

, we will generalize them to the entire

range of changes in the local drift angle

β 90

°°

≤≤

( )sin

12 0yP P P

σ = ϑ +ϑ σ β

, (13)

. . ( . );

0 177 0 087 0 55haϑ= + −

( . . ( . )) ( . . ( . ));

0 275 0 233 0 55 0 067 0 018 0 55

ah aϑ=+− +−−

;

β= ω=

σ=

– the value of the thrust load factor

of the propeller during rectilinear movement.

The calculation of the wake fraction for curvilinear

motion is based on the wake fraction for rectilinear

motion. The values of the latter are calculated by

empirical formulas obtained on the basis of

experimental data. The most acceptable

representations here are [25]:

ε1

ψ 0.11 0.16( ) ε ψ

P B PP

C WD

−

′

= + −∆

, (14)

. ( . ),

0 if 0.2,

0 3 0 2 if 0.2,

C Fr Fr

≤



′

∆=



−>



where

ε 1

, if the propeller is located in the

diametrical plane and

ε2



for the side propellers.

The following formula is usually used to calculate

the wake fraction during curvilinear movement

ψ ψ ()

0PP P

= ⋅β

, (15)

where

()

– the function of the local drift angle

on the propeller

, which is determined by

processing the results of experiments, in particular, in

[7] the following representation is given

()

−β

β=

. (16)

At the same time, it should be noted that in

representation (15) in [7], a simplified representation

for the local drift angle is used: a linear dependence is

used and the flow slope at the stern of the vessel is not

taken into account. Representation (16), in addition to

the indicated simplification, has the disadvantage that

the values of the function decrease too slowly with an

increase in the local drift angle and have a non-zero

value in the region of the boundary angles. In work

[2], a dependence is proposed

()



 β



− β ≤β



β=







β >β



(17)

where

the limit value of the local drift angle on

the propeller, at which the wake becomes zero, its

value for commercial ships is:

β 

, the value of

the indicator is selected from the range:

56n = ÷

Representation (17) contains a non-differentiable

function with respect to the drift angle, which is

inconsistent with the physical meaning of the

coefficient

. Therefore, it is suggested to use the

following improved representation:

()







− β ≤β





β=







β >β





(18)

Figure 1 compares all three relationships for ship 1

(DTC, container ship) at initial speed and relative

speed

.[ . [

7 7 m/s] 14 97 knots].v = =

Graphs 1-3

correspond to the value of the relative angular

velocity

0ω=

, graphs 4-6 correspond to the relative

angular velocity

..09ω=

Figure 1. Comparison of the wake fractions

Solid red plots show representation (17), dashed

black plots (18), and dashed blue curves (16). The

calculation results show that dependence (18) is more

accurate than dependence (16), approaches

representation (17), while remaining a differentiated

function.

To calculate the thrust-deduction factor

for

propellers located in the diametrical plane, it is

advisable to use the following generalized

representations

. ψ( . ψ),if 1,

. ψ( . ψ),if ,

06 1 067

08 1 025 2



 









 



(19)

Table 1 shows the values of the wake fraction

and the thrust-deduction factor

during straight-

line movement for some types of vessels.

To calculate the slope of the flow coefficient

on the propeller during curvilinear movement, it is

advisable to use the representation:

() , ;

00 0







κ + −κ β ≤β





κ=







β >β





(20)

where

– the slope angle for small local drift

angles, for commercial vessels, with a propeller in the

diametrical plane, is usually chosen

0.8κ=

; to

calculate

, you need to use formula (1), putting

there

κ=

Figures 2 - 5 show three-dimensional graphs of the

coefficients

and

for all possible

values of the drift angle

and angular velocity

for ship 1 from table 1, at the initial speed

7.7[m/s] 14.97[knots]= =v

and relative speed

1.2.=



Figure 2. The wake fraction

Figure 3. The thrust-deduction factor

Figure 4. Dimensionless coefficient

100

Figure 5. The slope of the flow coefficient

The results of the calculations show the adequacy

of the given dependencies, their good consistency

with the physical meaning of the coefficients for the

entire possible range of changes in motion

parameters, and prove the possibility of their

application for solving any problems of ship

maneuvering.

4 MATHEMATICAL MODELS OF FORCES AND

MOMENTS CAUSED BY A PROPELLER DURING

CURVILINEAR MOVEMENT AND THEIR

NUMERICAL ANALYSIS

The expressions obtained above for dimensionless

coefficients make it possible to determine the

projections of forces and moments on the propeller

during curvilinear movement. In particular, the

longitudinal component of the force on the propeller

and the moment on the propeller shaft are given

through the dimensionless coefficients (10) and (11) as

follows:

()()D

PPPPPPT

X T nK= −ζ = −ζ ρ

P PPQ

Q nK= ρ

. (21)

The transverse component of the force and

moment that arise on the propeller and act on the hull

of the vessel is given through the dimensionless load

factor (13) as follows:

p P P yP P P P

Y v M YL= ρπ σ =

. (22)

When studying various maneuvers of the ship, in

mathematical models of the dynamics of the

propulsive complex usually pass to dimensionless

phase coordinates. To do this, dividing the first two

differential equations of ship motion [4, 5] by the

expression

05 Sv

, and the third by the expression

05 SLv

, we proceed to the dimensionless

components of the inertial and non-inertial forces and

moment acting on the ship. In particular, the

longitudinal and transverse components of the

dimensionless forces and the moment caused by the

operation of the propeller during curvilinear motion

can be represented as follows:

P X T p Y yP P m yP

X GK Y G M G

= =σ=σ





, (23)

24 22 2 2

22 2

PP P PP

XY m

nD vD vLD

GG G

S SL

ππ

= = =



Figures 6 - 9 show three-dimensional graphs of the

advance ratio of the propeller J

P, the longitudinal

component of the force



, the transverse

component of the force



and the relative moment



for the widest range of changes in the drift angle

and angular speed, respectively. Calculations were

carried out for vessel 1 from table 1, at initial speed

7.7[m/s] 14.97[knots]= =v

and relative speed

1.2.=



The calculated results confirm that even at

small values of the drift angle and angular velocity,

the transverse component of the force



and the

moment on the ship's propeller



are not zero,

and as the drift angle increases, their values approach

the longitudinal component of the force



, and near

the drift angle

β≈±

have values of the same

order (modulo). These results are in good agreement

with the experimental studies of work [18].

Figure 6. The advance ratio of the propeller JP

Figure 7. Longitudinal component of force



101

Figure 8. Transverse component of the force



Figure 9. Relative moment



It should also be noted that the transverse

component of force



and moment



reach their

largest values (modulo) at

β≈±

, while the signs

of the transverse component of force and moment

coincide with the sign of the local drift angle. As for

the longitudinal component of the force



, its

greatest value is reached during rectilinear motion

( 0)

β=

, and when

β→±

and

2ω→±

approaches zero, when

β≈±

and

2ω≈±

takes a

negative value, this is explained by the fact that at

these values of the drift angle and angular velocity,

the local drift angle

on the propeller takes the

value:

β >±

, and therefore the direction of

action of forces caused by the operation of the

propeller and the direction of movement of the vessel

form an obtuse angle.

5 CONCLUSIONS

Thus, the influence of the curvilinear movement of the

ship on the operation of the propeller was

investigated. It is shown that even at small values of

the drift angle and the angular velocity of the vessel,

the transverse component of the force on the propeller

and the moment are non-zero and cannot be

neglected. The existing and proposed new effective

and convenient mathematical models of the

longitudinal and transverse components of force and

moment caused by the operation of the ship's

propeller in dimensional and dimensionless forms are

analyzed. In particular, simple representations are

obtained for the dimensionless coefficients of the

propeller thrust and the moment on the propeller

shaft, the wake fraction, the thrust-deduction factor

and the flow straightening factor on the propeller for

any drift angles and angular velocity. Numerical

analysis of the obtained dimensionless components of

forces and moments caused by the operation of the

propeller is carried out and their effectiveness is

shown. The specified parameters for all possible

values of the drift angle and angular velocity were

studied and their adequacy and applicability were

shown. For a number of commercial vessels of various

types, technical characteristics and calculated

dynamic parameters are given for the construction of

mathematical models of propeller operation during

curvilinear movement of the vessel.

The obtained results will make it possible to build

general adequate mathematical models of the ship's

propulsive complex.

REFERENCES

[1] R. Y. Pershytz, Dynamic control and handling of the

ship. L: Sudostroenie, 1983.

[2] Gofman A. D.: Propulsion and steering complex and ship

maneuvering. Handbook. L.: Sudostroyenie.1988.

[3] Miyusov M. V.: Modes of operation and automation of

motor vessel propulsion unit with wind propulsors.

Odessa, 1996

[4] Kryvyi O. F.: Methods of mathematical modelling in

navigation. ONMA, Odessa, 2015.

[5] Kryvyi O. F, Miyusov M. V.: Mathematical model of

movement of the vessel with auxiliary wind-propulsors,

Shipping & Navigation, v. 26, pp.110-119, 2016.

[6] Inoe S., Hirano M., Kijima K.: Hydrodynamic derivatives

on ship maneuvering, Int. Shipbuilding Progress, v. 28,

№ 321, pp. 67-80, 1981.

[7] Kijima K.: Prediction method for ship maneuvering

performance in deep and shallow waters. Presented at

the Workshop on Modular Maneuvering Models, The

Society of Naval Architects and Marine Engineering,

v.47, pp.121 130, 1991.

[8] Yasukawa H., Yoshimura Y.: Introduction of MMG

standard method for ship manoeuvring predictions, J

Mar Sci Technol, v. 20, 37–52pp, 2015. DOI

10.1007/s00773-014-0293-y

[9] Yoshimura Y., Masumoto Y.: Hydrodynamic Database

and Manoeuvring Prediction Method with Medium

High-Speed Merchant Ships and Fishing, International

Conference on Marine Simulation and Ship

Maneuverability (MARSIM 2012) pp.494-504

[10] Yoshimura Y., Kondo M.: Tomofumi Nakano, et al.

Equivalent Simple Mathematical Model for the

Manoeuvrability of Twin-propeller Ships under the

102

same propellers, Journal of the Japan Society of Naval

Architects and Ocean Engineers, v.24, №.0, p.157. 2016,

https://doi.org/10.9749/jin.133.28

[11] Wei Zhang, Zao-Jian Zou: Time domain simulations of

the wave-induced motions of ships in maneuvering

condition, J Mar Sci Technol, 2016, v. 21, pp. 154–166.

DOI 10.1007/s00773-015-0340-3

[12] Wei Zhang, Zao-Jian Zou, De-Heng Deng: A study on

prediction of ship maneuvering in regular waves, Ocean

Engineering, v. 137, pp 367-381, 2017,

http://dx.doi.org/10.1016/ j.oceaneng. 2017.03.046

[13] Erhan Aksu, Ercan Köse, Evaluation of Mathematical

Models for Tankers' Maneuvering Motions, JEMS

Maritime Sci, v.5 №1, pp. 95-109, 2017. DOI:

10.5505/jems.2017.52523

[14] Kang D., Nagarajan V., Hasegawa K. et al:

Mathematical model of single-propeller twin-rudder

ship, J Mar Sci Technol, v. 13, pp. 207–222, 2008,

https://doi.org/10.1007/s00773-008-0027-0

[15] Shang H., Zhan C., Liu Z.: Numerical Simulation of

Ship Maneuvers through Self-Propulsion, Journal of

Marine Science and Engineering, 9(9):1017, 2021.

https://doi.org/10.3390/ jmse 9091017

[16] Ni. Shengke, Zhengjiang Liu & Yao Cai.: Ship

Manoeuvrability-Based Simulation for Ship Navigation

in Collision Situations, J. Mar. Sci. Eng. 2019, 7, 90;

doi:10.3390/jmse7040090

[17] Sutulo S. & C. Soares G.: Mathematical Models for

Simulation of Maneuvering Performance of Ships,

Marine Technology and Engineering, (Taylor & Francis

Group, London), p 661–698, 2011.

[18] Lebedeva M. P., Vishnevskii L. I.: Forces on the

maneuvering ship propeller. Vestnik GUMiRF im. ad. S.

O. Makarova, v. 11, №3, 554–564, 2019. DOI:

10.21821/2309-5180-2019-11-3-554-56.

[19] Kryvyi O. F, Miyusov M. V.: “Mathematical model of

hydrodynamic characteristics on the ship's hull for any

drift angles”, Advances in Marine Navigation and Safety

of Sea Transportation. Taylor & Francis Group, London,

UK., pp. 111-117, 2019.

[20] Kryvyi O. F, Miyusov M. V.: “The Creation of

Polynomial Models of Hydrodynamic Forces on the Hull

of the Ship with the help of Multi-factor Regression

Analysis”, 8 International Maritime Science Conference.

IMSC 2019. Budva, Montenegro, pp.545-555 http://www.

imsc2019. ucg.ac.me/IMSC2019_ BofP. pdf

[21] Kryvyi O., Miyusov M.V.: Construction and Analysis of

Mathematical Models of Hydrodynamic Forces and

Moment on the Ship's Hull Using Multivariate

Regression Analysis, TransNav, the International

Journal on Marine Navigation and Safety of Sea

Transportation, Vol. 15, No. 4,

doi:10.12716/1001.15.04.18, pp. 853-864, 2021

[22] Kryvyi O. F, Miyusov M. V.: Mathematical models of

hydrodynamic characteristics of the ship’s propulsion

complex for any drift angles, Shipping & Navigation, v.

28, pp. 88-102, 2018. DOI: 10.31653/2306-5761.27.2018.88-

102

[23] Kryvyi O. F, Miyusov M. V.: New mathematical models

of longitudinal hydrodynamic forces on the ship’s hull,

Shipping & Navigation, v. 30, pp. 88-89, 2020. DOI:

10.31653/2306-5761. 30. 2020.88-98

[24] Kryvyi O. F, Miyusov M. V., Kryvyi M. O.:

Mathematical modelling of the operation of ship's

propellers with different maneuvering modes, Shipping

& Navigation, v. 32, pp. 71-88, 2021. DOI: 10.31653/2306-

5761.32.2021.71-88

[25] Basin M., Miniovich I. Ja.: Theory and calculation of

propellers. GSISP, L. 1963.

[26] Turbal V. K., Shpakov V. S., Shtumpf V. M.: Design of

merchant ships form and propulsors. L: Sudostroenie,

1983.

[27] Kuiper G.: The Wageningen Propeller Series. MARIN

Publication 92-001, 1992.

[28] Oosterveld M.W.C., van Oossanen P.: Further

computer-analyzed data of the Wageningen B-screw

series, Int Shipbuilding Prog., vol. 22, N 251, pp. 251-262,

1975.

[29] Biven R.: “Interactive Optimization Programs for Initial

Propeller Design”, University of New Orleans Theses

and Dissertations. 1009. 2009.

https://scholarworks.uno.edu/td/1009

[30] Carlton J.: Marine propellers and propulsion. Oxford:

Butterworth-Heinemann, 2012