Basic Design of a Series Propeller With Vibration Consideration

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ORIGINAL ARTICLE

J Mar Sci Technol (2007) 12:119–129

DOI 10.1007/s00773-007-0249-6

J.-H. Chen (*) · Y.-S. ShihDepartment of Systems and Naval Mechatronic Engineering,National Cheng Kung University, 1 Ta-Hsueh Rd., Tainan70101, Taiwane-mail: [email protected]

Basic design of a series propeller with vibration consideration bygenetic algorithm

Jeng-Horng Chen · Yu-Shan Shih

D propeller diameter

F X vibratory force in the x direction

F Y vibratory force in the y direction

F Z vibratory force in the z direction

J advance coefficient

K coefficient of ship type

K T thrust coefficient

K Q torque coefficient

Ln(r) the nth harmonic of the sectional lift force as a

function of radius

N propeller rotating speed (revolutions per

minute)

n propeller rotating speed (revolutions per

second)

P propeller blade pitch

Ps shaft horse power per blade

Q torque

QX vibratory torque in the x direction

QY vibratory torque in the y direction

QZ vibratory torque in the z direction

R propeller radius

Rn Reynolds number

rh radius of the hub

rt radius of the blade tip

S c

maximum allowable stress per square inch

S Total design objective function

S h subobjective function for hydrodynamic

efficiency

S VF subobjective function for vibration force

S VM subobjective function for vibration moment

sn exponent of J

T thrust

t blade thickness, time

tn exponent of P/D

un exponent of AE /AO

Received: February 15, 2006 / Accepted: May 2, 2007© JASNAOE 2007

Abstract Genetic algorithms (GAs) can powerfully

search for parameters in a large multidimensional design

space. Thus, the principle can be applied to preliminary

series propeller design problems with multiple consider-

ations. In the present study, B-series propeller design

was conducted using a GA for both hydrodynamic effi-

ciency and vibration consideration. The objective func-

tion was set by users who could freely weight the relative

importance of efficiency and vibration. GAs were suc-

cessfully shown to be able to obtain an optimal set of

parameters leading to efficient performance and low

vibration.

Key words Series propeller design · Genetic algorithm ·

Propeller vibration

List of symbols

AE propeller expanded area

AO propeller disk area

C 0.75R chord length at 0.75R

C Tn regression coefficient of thrust coefficient

C Total total score

C Qn

regression coefficient of torque coefficient

C ∆Tn regression coefficient for thrust coefficient

correction

C ∆Qn regression coefficient for torque coefficient

correction



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120 J Mar Sci Technol (2007) 12:119–129

V A advance velocity

VF magnitude of vibration forces of the propeller

VM magnitude of vibration torque of the propeller

vn exponent of Z

Z number of blades

b g geometric pitch angle

∆K T correction of thrust coefficient

∆K Q correction of torque coefficienth propeller efficiency

hv minimum propeller efficiency from which the

propeller’s vibration is considered

q s(r) projected skew angle

r density of water

n kinematic viscosity of water

Ω propeller rotating speed (rps)

1 Introduction

The theoretical propeller design method based on lifting-

line/surface theories is well established and is often used.1

With such design tools, naval architects can easily design

an optimized propeller using a computer without the

geometry constraints seen in series propellers, which

have relatively few choices of geometry. However, series

propellers still have their value. They are still widely used

in the preliminary design of light or moderately loaded

propellers, as described by Benini.2 Moreover, for those

who cannot afford lifting surface software, traditional

series propellers are still good choices.

The genetic algorithm (GA) has proved a powerful

tool for many difficult optimization problems in various

applications.3 It is especially useful in problems with

multiple objectives. It is robust in problems with many

local maximums because of its global search capability,

and this is rare compared with other optimization

methods. The disadvantage of the GA approach may lie

in its relatively slow speed for simple optimization prob-

lems. However, this will gradually become less impor-

tant with the rapidly increasing speed of computers.

Moreover, its global searching capability that leads to

robust results is sometimes considered more important

in complicated applications such as in the propeller

design problem. For example, Benini2 has demonstrated

that GAs can be successfully used in the multiobjective

design optimization of marine propellers.

It is a tedious job to design a series propeller by the

traditional calculation and chart method, without the use

of optimization methods or computers, due to the multi-

ple parameters involved and the many additional con-

straints such as cavitation and material strength.

Fortunately, optimization methods such as Bp-d dia-

grams have been developed to design a series propeller.

Nevertheless, the complicated nature of the challenge is

not a problem for GAs, even if there are additional objec-

tives on top of hydrodynamic efficiency.

Suen and Kouh4 successfully applied the GA approach

to B-series propeller design considering regular design

parameters and cavitation limits. Their results were

comparable to those from the traditional computation

method. Thus, GA methods were shown to be able toreplace the traditional computation method and design

charts. Karim and Ikehata5 also used GAs to demon-

strate the design process of B-series propellers. They

considered more aspects than Suen and Kouh, including

Reynolds number correction of design parameters and

material strength, and therefore made the method more

complete. Recently, Benini2 also used a GA to design

an optimized B-series propeller by maximizing both the

efficiency and the thrust coefficient with cavitation

constraints.

However, both the traditional design chart method

and previous GA methods were limited in considering

and/or maximizing hydrodynamic aspects alone, i.e., the

hydrodynamic efficiency, the thrust coefficient, Reyn-

olds number effects, and cavitation. Other aspects, such

as material strength or vibration, were considered merely

hurdles to be jumped later or were not considered at all.

In other words, the propeller design problem was not

considered as a multiple-objective problem whose objec-

tives include more than hydrodynamic aspects. In the

practical design process, a naval architect has to consider

the propeller vibration problem as well. Thus, vibration

and hydrodynamic problems are usually handled sepa-

rately: A propeller vibration problem involves not only

the propeller itself, but also the wake flow of the ship

using the propeller. Assuming that the wake flow is

known to the propeller designer, the traditional approach

of propeller design has been to find the most efficient

propeller and then to check for vibration problems later.

If the propeller vibration is tolerable, then the design is

complete. Thus, unlike material strength and cavitation,

which are parts of the optimization problem of propeller

design, vibration has usually been treated separately.

Unlike material strength or cavitation constraints, the

levels of various vibration modes are not only required

to be less than a certain value, but also need to be con-

sidered in a more subtle way. Some vibration modes are

more tolerable than others, and some are troublesome.

Therefore, a designer needs a tool to evaluate various

vibration modes as early as during the initial design

phase. This is usually impossible when using the tradi-

tional design method. Thus, if vibration is included in

the propeller design process, a two- or multiple-objective

optimization problem results. The GA approach can

therefore demonstrate its advantages in such multiple-



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J Mar Sci Technol (2007) 12:119–129 121

objective optimization problems. Moreover, it is not the

purpose of the present research to make comparisons

with a sophisticated lifting line/surface theory-based

design method. This GA method was developed for the

traditional series propeller design method only.

Hence, the goals of the present research are: (1) to

develop an integrated method to consider vibration

in the typical series propeller design process, becausevibration has traditionally been a separate part of series

propeller design; (2) to demonstrate the possibility of

applying GA optimization techniques with multiple goal

functions to the propeller design problem, especially the

consideration of vibration, which has been handled in

previous similar studies; and (3) to integrate all aspects

considered in previous studies2,4,5 applying GA to propel-

ler design together with consideration of vibration in

order to form a complete description of this methodol-

ogy of series propeller design.

2 Genetic algorithms

The GA was invented by Holland6 and was inspired by

Darwin’s theory of biological evolution. The genetic

algorithm performs a parallel, noncomprehensive search

for the global maximum performance of the design

parameters. The flow chart of a typical GA is shown in

Fig. 1. The algorithm starts with a set of randomly gen-

erated solutions (design parameters). These design

parameters are represented as “artificial chromosomes”

in the GA and each solution forms the “population” of

a “generation.” Every solution is encoded by a string of

bits, 0 or 1. The strings are manipulated in processes

analogous to crossover and mutation to generate

the population of the next generation. In every genera-

tion, each solution’s fitness (score) is evaluated by

the objective(s) set by the designer according to his/her

goal or design philosophy. See Karim and Ikehata5 for

a detailed description.

The result is the solution with the highest score in the

final generation. Therefore, the number of generations

is an important parameter in GA. To obtain better (truly

optimized) results, the number of generation must be

suitably large. However, the number of generations

required varies with each individual problem and there

are no criteria with which to determine this number. The

mutation process also plays an important role in GA

optimization. It provides a chance to jump to any random

area in the design parameter space; thus, it provides a

way to avoid local maxima. The mutation probability is

usually a small value. The population size plays another

important role in the GA approach. The population size

cannot be too small for avoiding local maxima, too.

Unfortunately, a large number of generations and a

large population size mean an increased computation

time. If a problem is too complicated, it might take a

long time to reach a satisfactory result.

3 B-series propeller

3.1 Performance computation

Choosing from a propeller series is a convenient method

to design a propeller. Among the series propellers, the

B-series is one of the most often used and studied. B-

series propellers were developed in the Netherlands Ship

Model Basin,7 and the section of the blade was improved

later. The thrust and torque coefficients can be expressed

Fig. 1. Flowchart of the genetic algorithm



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122 J Mar Sci Technol (2007) 12:119–129

as functions of the blade number, blade area ratio, pitch

ratio, and advance coefficient8:

K C J P

D

A

AZ T T

s

t

E

o

u

v

nn

n

n n

n=

=∑

1

39

(1)

K C J P

D

A

AZ Q Q

s

t

E

o

u

v

nn

n

n n

n=

=∑

1

47

(2)

where C Tn and C Qn are the regression coefficients of the

thrust and torque coefficients, respectively; sn, tn, un, and

vn

are the exponents of J , P/D, AE /A

O, and Z , respec-

tively. The values of these coefficients are all shown in

Table 1. The open water efficiency h, the thrust T , and

torque Q can, thus, be computed using these values and

Eqs. 1 and 2 as follows:

ηπ

=JK

K T

Q2 (3)

T K n DT = ρ 2 4 (4)

Q K n DQ= ρ 2 5 (5)

n C Tn sn tn un vn n C Qn sn tn un vn

1 0.00880496 0 0 0 0 1 0.00379368 0 0 0 02 −0.204554 1 0 0 0 2 0.00886523 2 0 0 03 0.166351 0 1 0 0 3 −0.032241 1 1 0 04 0.158114 0 2 0 0 4 0.00344778 0 2 0 05 −0.147581 2 0 1 0 5 −0.0408811 0 1 1 06 −0.481497 1 1 1 0 6 −0.108009 1 1 1 0

7 0.415437 0 2 1 0 7 −0.0885381 2 1 1 08 0.0144043 0 0 0 1 8 0.188561 0 2 1 09 −0.0530054 2 0 0 1 9 −0.00370871 1 0 0 1

10 0.0143481 0 1 0 1 10 0.00513696 0 1 0 111 0.0606826 1 1 0 1 11 0.0209449 1 1 0 112 −0.0125894 0 0 1 1 12 0.00474319 2 1 0 113 0.0109689 1 0 1 1 13 −0.00723408 2 0 1 114 −0.133698 0 3 0 0 14 0.00438388 1 1 1 115 0.00638407 0 6 0 0 15 −0.0269403 0 2 1 116 −0.00132718 2 6 0 0 16 0.0558082 3 0 1 017 0.168496 3 0 1 0 17 0.0161886 0 3 1 018 −0.0507214 0 0 2 0 18 0.00318086 1 3 1 019 0.0854559 2 0 2 0 19 0.015896 0 0 2 020 −0.0504475 3 0 2 0 20 0.0471729 1 0 2 021 0.010465 1 6 2 0 21 0.0196283 3 0 2 0

22 −0.00648272 2 6 2 0 22 −0.0502782 0 1 2 023 −0.00841728 0 3 0 1 23 −0.030055 3 1 2 024 0.0168424 1 3 0 1 24 0.0417122 2 2 2 025 −0.00102296 3 3 0 1 25 −0.0397722 0 3 2 026 −0.0317791 0 3 1 1 26 −0.00350024 0 6 2 027 0.018604 1 0 2 1 27 −0.0106854 3 0 0 128 −0.00410798 0 2 2 1 28 0.00110903 3 3 0 129 −0.000606848 0 0 0 2 29 −0.000313912 0 6 0 130 −0.0049819 1 0 0 2 30 0.0035985 3 0 1 131 0.0025983 2 0 0 2 31 −0.00142121 0 6 1 132 −0.000560528 3 0 0 2 32 −0.00383637 1 0 2 133 −0.00163652 1 2 0 2 33 0.0126803 0 2 2 134 −0.000328787 1 6 0 2 34 −0.00318278 2 3 2 135 0.000116502 2 6 0 2 35 0.00334268 0 6 2 136 0.000690904 0 0 1 2 36 −0.00183491 1 1 0 2

37 0.00421749 0 3 1 2 37 0.000112451 3 2 0 238 5.65229E−05 3 6 1 2 38 −2.97228E−05 3 6 0 239 −0.00146564 0 3 2 2 39 0.000269551 1 0 1 2

40 0.00083265 2 0 1 241 0.00155334 0 2 1 242 0.000302683 0 6 1 243 −0.0001843 0 0 2 244 −0.000425399 0 3 2 245 8.69243E−05 3 3 2 246 −0.0004659 0 6 2 247 5.54194E−05 1 6 2 2

Table 1. Regression coeffi-cients and exponents of K T andK Q



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J Mar Sci Technol (2007) 12:119–129 123

where r is the density of water; n is the revolutions per

second, and D is the propeller diameter.

3.2 Cavitation constraint

Cavitation could affect a propeller’s performance and so

needs to be considered in the propeller design process.A simple way to avoid cavitation is to increase blade

area ratio. The minimum blade area ratio required to

avoid cavitation was suggested by Keller9 as:

A

A

Z T

P P DK E

o v

=

+( )

−( )+

min

. .1 3 0 3

02

(6)

where (AE /AO)min is the expanded area ratio, Z is the

number of blades, P0 is the static pressure at the center

line of the propeller shaft (kgw/m2), Pv is the vapor pres-

sure (kgw/m2), and D is the propeller diameter (m). The

coefficient K is 0 for fast twin-screw ships, 0.1 for othertwin-screws ships, and 0.2 for single-screw ships.

3.3 Strength constraint

To achieve adequate blade thickness and thus ensure

material strength, a simple formula to determine the

minimum ratio of blade thickness at 0.7R to the dia-

meter has been proposed by Oostervelt and van

Oossanen10:

tD

P D P

ND S R

S

C D N

min

. .

. ..

= + −( )

+0 73

12

0 0028 0 212375 1125

4 1232 2

7788

3

( )

(7)

where (tmin/D)0.7R is the minimum blade thickness ratio

at 0.7R, Ps is the shaft horse power per blade (hp), N is

the revolutions per minute, and S c is the maximum

allowable stress per square inch (psi). According to the

B-series propeller geometry, the ratio of maximum

thickness of blade at 0.7R to the diameter is 0.015, i.e.

(t/D)0.7R = 0.015. Using Eq. 7 and the geometry of

B-series propellers, the required blade thickness can be

obtained as follows:

tD R

t

D R

S

C

P D P

ND S

( ) ≥ ( )

+−( )

0 7 0 7

30 0028 0 21

2375 1125

4 123

. .

min

. ..

or

++( )

≤D N 2 2

12788

3 0 015.

.

(8)

3.4 Reynolds number correction

The Reynolds number correction for the polynomial

thrust and torque coefficients of propellers was obtained

by Oosterveld and van Oossanen,11 as described briefly

in the following. The Reynolds number of a propeller at

0.75R can be calculated by:

RC V nD

n R

R A( ) = + ( )0 75

0 752 20 75

.

. . π

υ (9)

where C 0.75R is the chord length at 0.75R (m), V A is the

advance velocity (m/s), and n is the kinematic viscosity

of water (m2/s). If the Reynolds number (Rn) is greater

than 2 × 106, the corrections to the thrust and torque

coefficients given by Eqs. 10 and 11 are suggested by

Oosterveld and van Oossanen10,11:

∆ ∆K C LA

A

P

DJ Z T T

s E

o

t u

v w

nn

n

n n

n n=

=∑

1

9

(10)

∆ ∆K C LA

A

P

DJ Z Q Q

s E

o

t u

v w

nn

n

n n

n n=

=∑

1

13

(11)

where L = logRn − 0.301. The regression coefficients C ∆Tn

and C ∆Qn and exponents sn, tn, un, vn, and wn are shown in

Table 2.

n C ∆Tn sn tn un vn wn n C ∆Qn sn tn un vn wn

1 0.000353485 0 0 0 0 0 1 −0.000591412 0 0 0 0 02 −0.00333758 0 1 0 2 0 2 0.00696898 0 0 1 0 03 −0.00478125 0 1 1 1 0 3 −6.66654E−05 0 0 6 0 14 0.000257792 2 1 0 2 0 4 0.0160818 0 2 0 0 05 6.43192E−05 1 0 6 2 0 5 −0.000938091 1 0 1 0 06 −1.10636E−05 2 0 6 2 0 6 −0.00059593 1 0 2 0 07 −2.76315E−05 2 1 0 2 1 7 7.82099E−05 2 0 2 0 08 0.0000954 1 1 1 1 1 8 5.2199E−06 1 1 0 2 19 3.2049E−06 1 1 3 1 2 9 −8.8528E−07 2 1 1 1 1

10 2.30171E−05 1 0 6 0 111 −1.84341E−06 2 0 6 0 112 −0.00400252 1 2 0 0 013 0.000220915 2 2 0 0 0

Table 2. Regression coeffi-cients and exponents forchanges in the thrust and

torque coefficient



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124 J Mar Sci Technol (2007) 12:119–129

4. Propeller vibration

The propeller is one of the main sources of vibration in

a ship. Two major sources of propeller vibration are

cavitation and an uneven load of hydrodynamic forces

on the blade/shaft. However, if cavitation is well under

control, the only main source of propeller vibration is

the circumferential variation of wake flow behind a ship,which produces the six components of propeller excita-

tion forces and moments. Moreover, the radial variation

of the harmonics of the circumferential variation of the

wake is important.12

In the analysis of vibration problems, the mass and

moment of inertia of a propeller can be calculated

directly given the propeller’s material and geometry. The

added mass and damping coefficients of B-series propel-

lers needed for analysis can be obtained from Parsons’

work,13 in which the lifting line/surface theory was used

to evaluate the values of added mass and damping coef-

ficients. Propeller vibration forces and moments due to

wake variation can be obtained using the theoretical

approach of Tsakonas et al.14 as shown in Eqs. 12–17.

F Ze L r e r drX j Z t

Z j Z r

g

r

r

S

h

t

==

∞−∑ ∫ Re ( ) cos ( )( )λ

λ λ

λ θ β Ω

1 (12)

F Z

e

L r L r e r dr

Y j Z t

Z Z j Z r

g S

=

× +[ ]

=

∞

− +−

∑Re

( ) ( ) sin ( )( )

21

1 1

λ

λ

λ λ λ θ β

Ω

rr

r

h

t

∫

(13)

F Z j

e

L r L r e r d

Z j Z t

Z Z j Z r

g S

=

× −[ ]

=

∞

− +−

∑Re

( ) ( ) sin ( )( )

21

1 1

λ

λ

λ λ λ θ β

Ω

rrr

r

h

t

∫

(14)

Q Ze L r e r drX j Z t

Z j Z r

g

r

r

S

h

t

= −=

∞−∑ ∫ Re ( ) sin ( )( )λ

λ λ

λ θ β Ω

1 (15)

QZ

e

L r L r e r dr

Y j Z t

Z Z j Z r

g S

=

× +[ ]

=

∞

− +−

∑Re

( ) ( ) cos ( )( )

21

1 1

λ

λ

λ λ λ θ β

Ω

rr

r

h

t

∫

(16)

QZ

j e

L r L r e r d

Z j Z t

Z Z j Z r

g S

=

× −[ ]

=

∞

− +−

∑Re

( ) ( ) cos ( )( )

21

1 1

λ

λ

λ λ λ θ β

Ω

rrr

r

h

t

∫

(17)

where Ω is the revolutions per second; t is the time; b g is

the geometric pitch angle; rh is the radius of the hub; rt

is the radius of the blade tip; q s(r) is the projected skew

angle; Ln(r) is the nth harmonic of the oscillatory lift due

to the nth harmonic of the oscillatory normal inflow

velocity, which is perpendicular to the average inflow

direction; and j is the root of −1. Usually, the lifting

forces in these equations can be obtained using a com-

puter program based on lifting surface theory.13 Such

programs treat propeller blades as vortex sheets and

calculate the forces generated.

Hence, given a propeller’s geometry, material, and theship’s wake data, one can compute the vibration in six

degrees of freedom. In the present study, the vibration

forces and moments of each propeller were computed by

a modified version of Propex, a propeller vibratory exci-

tation program developed by Vorus et al.15 at the Uni-

versity of Michigan.

5. Series propeller design with vibration consideration

by genetic algorithm

As described above, the GA approach solves problems

by computing the objective functions through encoded

parameters under some constraints with the parameters

generated both continuously (from the parent genera-

tion) and randomly (from “mutation”) instead of calcu-

lating the values or the trends of the objective function.

Thus, the GA approach can be applied to problems with

multiple objectives and constraints, i.e., it can be applied

to optimize series propeller design when hydrodynamic

efficiency is not the only objective considered. Although

lifting line/surface theory is used to evaluate hydrody-

namic coefficients and vibrating lifting forces in the

present study, a GA can still be helpful in designing the

propeller. The reason is that even though a computer

program using lifting line/surface theory can obtain a

propeller design with optimized hydrodynamic efficiency

or with vibration consideration, it usually requires more

complicated programming to combine both functions.

Moreover, when more constraints or objectives are

added, such as cavitation or structural strength, addi-

tional evaluation tools are required, and the whole design

process needs to be divided into several parts and thus

becomes less integrated. For series propellers, since their

hydrodynamic performances are well documented with

regression data, the computation of hydrodynamic per-

formance can be replaced by regression equations instead

of lifting line/surface theory. But more importantly, GAs

can easily include any consideration in the optimization

process. Thus, it is a better tool when an integrated

design process is desired.

5.1 Objective function and constraints

The objective function in this dual-purpose optimization

process may have many kinds of different definitions,



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J Mar Sci Technol (2007) 12:119–129 125

according to the design philosophy applied, i.e., the

balance of the requirements for both efficiency and

vibration. Therefore, here we just show one of the many

possibilities for the definition of the objective function.

Since our objective is to find the optimized design param-

eters (P/D, AE /AO, and D) not only to maximize the effi-

ciency but also to minimize the vibration, the objective

function, S Total , is a function of both efficiency and vibra-tory excitations, and is defined as:

S S S S Total VF VM = + +η (18)

There are three parts to the objective function, i.e., the

subobjective functions for hydrodynamic efficiency,

vibration force, and vibration moment, respectively.

Their definitions are based on normalized values of their

performance, and their relative importance in the total

result is set by the designer as shown here:

S C

C Total

η

ηη= ×

(19)

S C

C

VF

T APT VF

VF

Total

= × −×

1

(20)

S C

C

VM

Q APQVM

VM

Total

= × −×

1

(21)

where h is the hydrodynamic efficiency of the propeller;

VF is the magnitude of the vibration forces of the propel-

ler, defined as:

VF F F F X Y Z = + +2 2 2 (22)

VM is the magnitude of the vibration moments of the

propeller, defined as:

VM Q Q QX Y Z = + +2 2 2 (23)

APT is the maximum acceptable vibration force as a

percentage of the thrust and is set according to the design

requirement. APQ is the maximum acceptable vibration

moment as a percentage of the torque, and is also set by

the designer. We represent the relative importance of the

efficiency, vibration forces, and vibration moments by

introducing a total score C Total defined as:

C C C C Total VF VM = + +η (24)

where C h, C VF , and C VM can be any real number repre-

senting the relative importance of these three subobjec-

tive functions. They are set by the designer. Note that

the actual values of C h, C VF , and C VM are not important

because they are merely to show the relative importance

among these items and to generate the value of C h/C Total .

Many different combinations could lead to the desired

relative importance of C h and the desired value of C h/

C Total . For example, when C h, C VF , and C VM are equal to

5, 3, and 2, respectively, the result is the same as when

C h, C VF , and C VM are equal to 50, 30, and 20 (both result

in a C h/C Total value of 0.5). The optimization process also

operates under the following constraints:

1. Cavitation constraint: the expanded area ratio

should be larger than a minimum value in order

to avoid cavitation,A

A

A

AE

O

E

O≥ ( )

minwhere

A

AE

O( )

mincan

be calculated using Eq. 6

2. Strength constraint: to ensure adequate material

strength, a minimum blade thickness is required, as

shown in Eq. 8, tD R

t

D R( ) ≥ ( )0 7 0 7. .

min

3. Thrust requirement: the thrust has to match the

design requirement, T calculated > T required

4. Power constraint: the delivered power has to be

adequate.

5.2 Design example

The design parameters for this example are shown in

Table 3. The parameters relating to the GA method, the

crossover probability and the mutation probability, are

chosen according to general GA experience. Their effects

on the optimization results have already been discussed

by Karim and Ikehata,5 and thus will not be discussed

here. The blade number is chosen by the designer directly

and thus is not part of the GA computation. The wake

Table 3. Parameters of the design example

Z (number of blades) 6Shaft depth (m) 3Type of ship Single–screw shipPD (delivered power) (kW) 440V A (advance velocity) (m/s) 14.42rps (revolutions per second) 12Material of propeller NylonInitial tip skewa (%) 150Initial slopeb 2Wake inflow Model 4282 as shown in

Table 4APT (%) 10APQ (%) 10Pcross (crossover probability) 0.6Pmutate (mutation probability) 0.02

APT , maximum acceptable vibration force as a percentage of thrust; APQ, maximum acceptable vibration moment as a percent-age of torquea 100% is where the tip of one blade is over the root of the nextblade.b Initial chord line slope at the hub, non-dimensional, unit equalto radians per nondimensional radius



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126 J Mar Sci Technol (2007) 12:119–129

inflow data are from a Model 4282 V-shaped-stern ship,

as shown in Table 4.

6 Results and discussion

6.1 Optimized propeller without vibration

consideration

First, we considered the optimization process of a B-

series propeller design without vibration in order to

compare the present method to previous similar studies.

Thus, we set C VF and C VM to zero for an efficiency-only

objective. Since the computing parameters of the GA

will influence the result, it is necessary to check the algo-

rithm’s performance using different computing parame-

ters. The population size for each generation is the first

to be examined. Figure 2 shows the influence of popula-

tion size on the convergence process of the objective

function, efficiency. It is obvious that a larger population

size converged to a better result in earlier generations.

For example, a population size of 50 converged at a

lower efficiency and took six more generations (the 24th

generation) to converge than for a population size of 200

(the 18th generation). This result is similar to previ-

ous studies on series propeller design using the GA

approach.4,5

Table 5 summarizes the results of geometry and per-

formance using different population sizes. The results

for a population size of 50 reached a lower efficiency, as

seen in Fig. 2, and also led to different geometry of the

propeller blade than for the other two cases, with a

clearly smaller pitch ratio (P/D) and expanded area ratio

(AE /AO) and a larger diameter (D). This implies that

selecting too small a population size may not result in

optimized propeller geometry.

6.2 Optimized propeller with vibration consideration

It is probably better to consider the vibration excitation

forces and moments in the propeller geometry design

process, if possible. In the past, it was usually impossible

to do this for a series propeller if its geometry was not

changeable; vibration analysis was done after the most

efficient propeller had been designed. In order to con-

sider the vibration in the design process, we have to

evaluate both the efficiency and vibration of each pro-

peller considered in the process. Therefore, the GA

method has to evaluate many propellers in the pro-

cess and thus will take a long time to finish the task.

x

q

vx/us (Axial inflow, positive aft) vt/ us (Tangential inflow, positive CCW)

0.3 0.5 0.7 0.9 0.3 0.5 0.7 0.9

0° 0.31 0.24 0.12 0.1 0 0 0 030° 0.35 0.37 0.46 0.53 0.047 −0.021 −0.081 −1.6660° 0.4 0.55 0.72 0.8 −0.063 −0.135 −0.193 −0.20790° 0.5 0.77 0.86 0.9 −0.12 −0.164 −0.17 −0.147

120° 0.55 0.86 0.91 0.92 −0.09 −0.105 −0.101 −0.091150° 0.42 0.79 0.9 0.91 −0.024 −0.041 −0.043 −0.039180° 0.38 0.45 0.57 0.72 0 0 0 0210° 0.42 0.79 0.9 0.91 0.024 0.041 0.043 0.039240° 0.55 0.86 0.91 0.92 0.09 0.105 0.101 0.091270° 0.5 0.77 0.86 0.9 0.12 0.164 0.17 0.147300° 0.4 0.55 0.72 0.8 0.063 0.135 0.193 0.207330° 0.35 0.37 0.46 0.53 −0.047 0.021 0.081 0.166360° 0.31 0.24 0.12 0.1 0 0 0 0

Table 4. Wake survey resultsfor model 4282 V-shaped stern

vx, axial velocity; us, ship veloc-ity; vt, tangential velocity;CCW, counter-clockwise

0 10 20 30 40 50

Generation

0

0.2

0.4

0.6

0.8

E f f i c i e n c y ,

η

Sym. Population/Gen.20010050

Fig. 2. Influence of population size (population per generation) onthe convergence process



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J Mar Sci Technol (2007) 12:119–129 127

Vibration analysis usually takes more time than

efficiency and geometry analysis because a computer

only needs to calculate simple regression equations to

obtain the performance–geometry relationship.

However, there are ways to increase the speed of the

whole process. For example, it is not necessary to evalu-

ate the vibration for those propellers with very low effi-

ciency. Hence, we do not waste resources on the more

time-intensive part of the evaluation. Thus, we consider

a minimum efficiency, hv, at which we start to evaluate

a propeller’s vibration. This minimum efficiency is again

set by the designer. The time saved using this method is

huge, as can be seen in Fig. 3. We used a personal com-

puter with an AMD Athlon 1GHz CPU. The CPU time

is shown for three different generation settings. If we

consider the vibration for each propeller, the total time

for 50 generations is more than 3 min, and for 100 gen-

erations it is more than 8 min. It can also be seen that a

rapid decrease in time occurred when the minimum effi-

ciency for vibration analysis approached the best effi-

ciency. Note that when hv was set to be greater than the

final efficiency, the algorithm did not consider vibration

at all, and thus the CPU time was the smallest, as can

be seen in Fig. 3.

It would be interesting to know the effect of different

weightings, or the relative importance of efficiency,

vibration forces, and vibration moments, on the conver-

gence process. Figure 4 shows that if efficiency is less

than about 20% of the total evaluation score, i.e., it

is relatively unimportant, then, the final efficiency is

roughly equal to hv. If efficiency contributes more than

20% of the total score, no matter what the value of hv,

the efficiencies converge. In Figs. 4–7, only the values of

C h/C Total are shown because the actual values of C h, C VF ,

and C VM are not important, as explained above. Figure

5 shows that different weighting and hv values generate

only about a 5% variation in the vibration results and

thus seem to have no significant effect. The effect of

population size on the results is summarized in Table 6.

It is similar to the case without vibration consideration:

a larger population generates better result in terms of

Table 5. Geometry and performance results using different popu-lation sizes without vibration consideration

Population per 200 100 50generation

P/D 1.383 1.385 1.144AE /AO 0.999 1.015 0.761D (m) 1.02 1.02 1.141J 1.178 1.178 1.053K T 0.148 0.148 0.088K Q 0.037 0.0371 0.0212Efficiency 0.75 0.75 0.653Thrust (N) 23098 23096 21508Torque (N-m) 5887.7 5892.7 5892.7Power (kW) 443.9 444.3 444.3

0 0.2 0.4 0.6 0.8 1

Minimum Efficiency to Consider Vibration(ηv)

0

200

400

600

800

C P U

T i m e ( s )

507

418

293

219209

74

140

34

Sym. Generation Final Eff.100 0.75150 0.75025 0.748

263

461

372

53

173

23

117

9

81

59

1153

21

Fig. 3. Effects of minimum propeller efficiency for which the pro-peller’s vibration is considered (hv) and generations on computerCPU time

0 0.2 0.4 0.6 0.8 1

Cη/CTotal

0

0.2

0.4

0.6

0.8

E f f i c i e n c y , η

Sym. ηv0.70.6

0.50.4

Fig. 4. Effects of hv and weighting of efficiency (C h/C Total ) on theefficiency with vibration consideration



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128 J Mar Sci Technol (2007) 12:119–129

efficiency and vibration. Figure 6 shows the effect of

population size on the vibration forces. If the population

size is not large enough, it is possible to get a slightly

higher efficiency with a much larger vibration force,

which may not be desirable.

By comparing the results of the GA method with and

without vibration consideration, we found that the con-

vergence processes are similar, but the results of effi-

ciency and vibration are different. Figure 7 shows an

example with vibration consideration for a population

size of 100, an hv of 0.6, and a C h/C Total ratio of 0.556

compared with the result without vibration consider-

ation (C h/C

Total =

1). When vibration is considered, the

efficiency is only slightly reduced (less than 1%), but the

vibration forces are reduced more than 3%. The detailed

results are shown in Table 7. Based on these results, it is

clear that the population size and generation number are

more important than other computational parameters

when using the GA method to obtain a good design. It

is also evident that a set of parameters for a B-series

propeller with good efficiency and low vibration is

obtainable using the GA approach in less than 5 min

when using an average PC.

0 0.2 0.4 0.6 0.8 1

Cη / CTotal

0

400

800

1200

1600

V i b r a t i o n F o r c e

( N ) Sym. ηv

0.7

0.60.50.4

Fig. 5. Effects of hv and C h/C Total on vibration forces

0 10 20 30 40 50

Generation

0

500

1000

1500

2000

V i b r a t i o n

F o r c e ( N )

ηv =0.6

Cη / CTotal=0.1

Sym. Population/Gen.200100

50

η=0.613

η=0.614

η=0.635

Fig. 6. Effects of the population size (population per generation)on vibration forces

0 10 20 30 40 50

Generation

0

0.2

0.4

0.6

0.8

E f f i c i e n c y

, η

0

400

800

1200

1600

2000

V i b r a t i onF or c e ( N

) ,V i b r a t i onM om en t ( N-M )

Cη/CTotal

1.0 0.556EfficiencyVibration ForceVibration Moment

Fig. 7. Influence of vibration consideration on the progress of the

genetic algorithm

Table 6. Optimized propellers using different population sizeswith vibration consideration

Population per generation 200 100 50

P/D 1.4 1.399 1.175AE /AO 0.451 0.451 0.826D (m) 0.992 0.992 1.131J 1.211 1.211 1.062K T 0.16 0.16 0.099K Q 0.0429 0.0428 0.0236

Efficiency 0.613 0.614 0.635Vibration force (N) 1121 1121 1910Vibration moment (N-m) 403 403 649Thrust (N) 22394 22326 23469Torque (N-m) 5943 5926 6291Power (kW) 448.1 446.8 474.4



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J Mar Sci Technol (2007) 12:119–129 129

Table 7. Comparison of optimized propellers with and withoutvibration consideration

Vibration consideration No Yes

P/D 1.385 1.4AE /AO 1.015 0.998D (m) 1.02 1.011J 1.178 1.189

K T 0.148 0.152K Q 0.0371 0.0382Efficiency 0.75 0.747Vibration force (N) 1295 1247Vibration moment (N-m) 424 409Thrust (N) 23096 22823Torque (N-m) 5892 5799Power (kW) 444.3 437.3

7 Conclusion

The present study showed that it is possible to use a

genetic algorithm to design a series propeller not onlywhen considering the optimization of hydrodynamic

efficiency with material strength and cavitation limi-

tations, but also when considering the optimization of

both efficiency and vibration, provided that the wake

information of the ship is known. When using the GA

approach, some techniques for parameter setting to

provide quick and correct results were discussed, along

with the influence of these parameters on the results.

Acknowledgments. The authors would like to thank David Carrollfor providing his free source code for the genetic algorithm, whichwas rewritten for the present study. The program adapted for cal-

culating propeller vibration, Propex, was originally developedby Professor Parsons and Professor Vorus at the University of Michigan.

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Basic Design of a Series Propeller With Vibration Consideration

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