Jurnal Mekanikal, Jilid II, /997
ON THE STRENGTH AND VIBRATION ANALYSIS OF A 8000
DWT CONTAINER SHIP PROPELLER
Mohd. Rarnzan Mainal
Department of Marine Technology
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
Ghazali Jaafar
ShipTech Pte Ltd
Singapore
ABSTRACT
This paper presents the investigation on the strength and vibration
analysis oj a propeller design Jor a 8000 deadweight container
ship which is based on the Bp
-
8 series. Such analysis is important
since the significant role oj the propeller to convert the greater
part of the power from an engine into thrust Jorce to propel a ship
and ensure that resonants ojpropeller induced vibration does not
coincide with the main hull vibration. Various methods of analysis
is studied in order to compare the results obtained and to justify
that such method comply with the classification societies.
1.0
INTRODUCTION
A marine propeller is a propulsion device, which converts the greater part of the
power from an engine into thrust force to propel a ship. The propeller is the most
common form of marine propulsion device. Therefore, it is essential to design the
29
Jurn al Mekan ikal, Jilid II. /997
Depth moulded
DM LD
Designed Draught
H
6500m
Loaded displacement
D
11151 .00 tonnes
Service Speed
V
Propeller Diameter
D
Number of Propeller
Number of blade
10.800 m
15.00 knots
3.90m
I
Z
4
Block coefficient
CB
0.705
Prismatic Coefficient
Cp
0.745
Midship coefficient
CM
0.947
Waterplane coefficient
CWL
0.831
CoefficientslRatios
LCB ( % aft of midship)
0.932
LCF ( % aft of midship)
2.075
Main Particulars of Propeller
Propeller diameter
Dia
3.900 m
Propeller boss diameter
Db
0.780 m
Blade area ratio
BAR
0.740m
Angle of rake
Rake
0.0 degrees
Length of blade section at 0.6R
1.580 m
Max . thickness at centre of shaft
0.176 m
31
Jumal Mekanikal, Jilid II, J997
propeller correctly. Nevertheless, two major criteria which often being neglected by
naval architects when designing the propeller is the strength and vibration aspect.
The strength of a marine propeller depends significantly on the blade
thickness. The determination of the blade thickness of a propeller is an important
aspect of the design. Its affects primarily the resistance of the propeller to failure and
damage. Its affect quite substantially the inertia, weight and thus the price. It affects
to a relative minor degree the efficiency, power absorption and cavitation
characteristics. Weight, price, cavitation and efficiency are all favourably influenced
by reduced thickness and thus the highest allowable mean stress is used consistent
with the achievement of all objectives.
In any ship, the major sources of vibration excitation are usually the main
engine and the propulsion train. Where the main engine is of a rotating type, the
propeller will probably be responsible for any significant vibration of the main hull.
Therefore, it is important to assess the amplitudes of vibration excited by the
propeller. Though these vibrations cannot be eliminated but it can be minimised. To
achieve this, ensure that resonants of propeller induced vibration does not coincide
with the main hull vibration.
This paper will discuss the strength and vibration analysis of the propeller
using various theories and compared those values that has been suggested by some
classification societies.
2.0
BASIS SHIP AND MAIN PROPELLER DIMENSIONS
The design calculation is based on a 8000 dwt Container Ship as shown in Fig. 1.
The principle particular of the ship are as follows:
Main Particulars of Ship
Length Overall
LOA
123.500 m
Length between perpendicular
Lap
115.450 m
Breadth moulded
B
20.800 m
30
Jumal Mekanikal, Jilid II, 1997
IIII
><
,
-'.
"5>-
x'
""'
'5<
Fig. 1 General Arrangement of 8000 Dwt Container Ship
32
Jumal Mekanikal, Jilid II, 1997
3.0
PROPELLER BLADE STRENGTH CALCULATIONS
The calculation, of the maximum mean stress of the propeller is usually based on the
the maximum tensile stress on the blade face at a prescribed position of the propeller
radius. However, prior to these calculations, it is necessary to first determine the
geometry of the propeller.
3.1
Propeller Dimensions
The offset for a 4 blade B Series propeller are given in Tables I and 2. The result of
the designed propeller using Series charts and polynomial equation are shown in
Tables 3 and 4. A propeller drawing, based on Series charts, is shown in Fig. 2.
In making estimates for weight, moment of inertia and blade stresses, values
of the geometrical properties of blade sections are required. These comprise the
section area (As), distance of centroid from face chord
from leading edge
~), distance of centroid
(h), moment inertia about axis parallel to face chord (IN) and
moment of inertia about axis normal to face chord
(Ip), as shown in Fig. 3. The
formulation of these geometrical properties are given as by:
As
-
h
(1)
= 0.463t
(2)
= 0.455e
(3)
y
-
= 0.700et
IN
= 0.042et 2
(4)
Ip
= 0.040e 3t
(5)
Applying the above equations, the calculation of the propeller blade weight
is shown in Table 5 for series charts.
33
l
D/IUCftIJll 01
aor.cmON
LOR
~
s
..,R
O./IR
I:l
~
ir
;:s
Yr,
0.5.
0.511
O.4R
O.4R
.-s~
is:
-
-Q:.
E:::
'0
'0
'-l
~
SIDE ELEVA TlON
EXPANDED BLADE
PITCH DISTRIBUTION
TRANSVERSE VIEW
JUIN PARtlCl1L48S
PIfDP6~R
lJUJI6r6R
NUI/BU tJ' BUM
BUD• .tR..£j( RAno
P/tcH-DlAJl6r6R RArlO
ANOlA OF RUI'
.!IOO """'
•
0 .140
O.18B
o
MgTW.
UNIVERSITY OF GLASGO"
_._-.-
~
IWIJI'a,LD _ _ 111I
-.au """
3.911 PROPELLER
PLAN VIEW
t»r
,-* o-tJo)
-'==1::
(N
~
Fig. 2
Propeller Drawing
Jurnal Mekanikal, Jilid II. 1997
Table 1 Table of Ordinates of the B - Series Propeller
(Distance of Ordinates from Maximum Thickness)
Fromm;l ..., """" ,h.(lllCH
Ff'O'O'l .... ·" ....um lh,cl.Jocu
10 1I~ ,I"'1 cdJ':
'O~I"1cd'll:
1~'.J
l~~l
,I'
1110
,nOll
no
.noo
(,(1110
21' 00
n
(,(lOll
IInr'lfi
'JO no
'I)
9.. SO
1.1
(,J H
~(, ') ~
Qrd'"IIC1 fot Ihc bxl:
Sl H
11lU
'1'1. 110
l)U O
')01 00
"un
')7ral
'J' 2U
9) 2S
,·00
'>1 10
'1l "n
"IW
Q7, 00
'JllS
III 10
U lO
79 JS
I jOJO
'J7on
Jj
n ~s
161JC1
' JfIJ }
. . ..
O .lU
SOll S
7160
0.1 11
.47 10
u ) 11
J ] .10
060
.Ill ~u
10 lS
"' I J O
(, 7 IS
. '''
1610
16 >5
1''110
IS .Ill
UW
" " Ii )
(.J 10
Inn
'16lU
mno
" 00
'JJ UQ
"
"10
''mI Qj"'0
nm
011)
H IS
0 ''''
0');
I) ! O
;non
0 '0
U H
OlO
""
•••
o ~I
060
.0 "
.",n
11 10
U !O
• '0
10 ')0
., ,, 10
lit)
IH
".
r in
.''11 -2000
0nJ0Ntc.s tOrI~ h a:
,H
HO
""0
II.IS
nos
.II}
72 ~ 11
In J{)
ti7711
I S III
; <1 '/'0
"" "
sn n
~6
'0
HH
H ~o
);00
liB
jJ '5
to 10
1!?fJ
.' '''U
~'J
21 M
,r un
7000
OlIO
non
H lj
Jot IUJ
SO
:n )11
:r.20
lo x1
l foB
122 0
! 6j
,"0
I!
0 70
~
1
no
;'10
IJ I}
OJO
"'"
jQ
" AO
: 0 10
so
~ :'
NOle
J O IlO
H)S
111)11
j .l ;0
IJ ]0
i O .lO
.I H
,
/1 .111
2.1:'
e sn
Int1 0ll
" '"
~ 1 ~o
S1nu
&110
'n o
t.ll-oj
(!l(,O
JU
'"
.' 10
on
no
~o
1111111
~,
! .I ; 0
III ; n
" 0
('lorep.:rcbtI ~SC I of l"e 1,If"" ~1 rcl.alt'o .he:.nl\,"_" IIuc Lnc5' 01.,-: (onc., po,ond"' !J Ia llO"l
111:: cun o of lhocL.nc st bc1''1 n,..-.d lo be l.xt ll ~ n ... ( - ' :1" ' " I'ne, or. ~ pot"" ~ .. hl( h
~ . b-d. ~ bad Inscnal . culexh Ol ~ ~ 0 . ~ R
Table 2 Dimensions of the Four-Bladed Screws, Types B4, B4.55 and B4.70
".
:u
II JU
u au
" - ~I-I
' ,',{I
11 m
U lIti
II \jill
11 111
! 'J I 'II
n :2
11
~O
111 1'
I ~ 'I~
J r, r••
Jl I(
J' un
! U 1.1
II
r","" o:m~
...c1'!lhnr 1he btlcit
"«' .",. asl"tt~ ll ,"
"' t~
·"~'.~_
""h
"'-'e~oa_
rtf
lIot fi q~
*
h oc I() u ....
Lcn l;lh 0 1 hl~ ~«1 1(ln
"... at"
FIOf" «"l1'(
linc lo
k'ad .
.If . '.l1I
; .! (.I
· f, .I!
- 1 f.lt
:'.l>.
; I JII
J I I. ;
.
i,. .1111
'I ~ 'If'l
~I J
"t ).
11101111
'J II Ilfl
'JIl l '"
11 .: :-'
\
~
..
r "'a1~h
~"'!: cd,e 1I1 pnc. cflU F of ~"s.h
flJ (1
{je '1("I;llh
;" al «
lI'IIcLI"CSI UhO "
pct«:ftt.oltJrtclhMlC'lCl'
)c. I.. oceor ......._ _ lhoc\;.ncn h ......
,, 11•. 1l; " 1 ! 11I70
,rte'''''''
(0', " ' 1 ".l 1:-\ I Aa,l.o l
: r,/l
1 4' 111
] !.&
) 4
lin
'.2
'"
''''
.. I'll'
111 11
~
1 'llIl
I.'"
'11,,,
U 1"
' I'
Illl
II ,iO
1,1.1'l:"!IU"III,, (LII(' ~ ~
J l CC!1 l1e ror, h.lr1
~1
'''I
;;0 flO
~ t1 I ~ ·n
( , t-e t« h OM
35
Jumal Mekanikal, Jilid II, 1997
5~CT'ON
c.HO-'Il.::.o......:;c
--J
SECTION MAXIMUM THY. t ,
T
+
Fig. 3 Geometrical Properties of the Blade Sections
Table 3 Dimensions of 4-Bladed Screw
r1R
0.20
0.25
IUO
0.40
050
060
070
ORO
461
494
526
589
644
694
738
741
7K6
832
890
910
886
1202
1280
1];8
1479
1554
Blade thickness
Distance of maximum thickness
143
135
126
L10
from leading edge
421
448
475
518
90
095
100
764
J·n
319
31R
8(}
658
401
1580
1550
1422
1143
94
77
61
44
)9
12
12
552
615
687
681
572
I)
From centre
line (Q
Ltal-
Length of the blade
ling edge
From centre
sections
hue to leadin~ edee
Toret lenarh
Note
All dnnenstoos act 1fI rmlhmetres.
36
Jurnal Mekanikal, Jilid II, J997
Table 4 Table of Ordinates of the B - Series Propeller
(Distance of Ordinates from Maximum Thickness)
From maximum thickness
10 tral l lO ~ edge (mm)
100
30 T 60
r/ R
From maximum thic kness
I
I 40
20
20
I
10 lead ing edg e (nun )
60
30 I 90
40
100
95
I
r/ R
Ordinates for Ihc: back
020
7(,
104
030
(,4
040
51
90
77
050
0 60
41
070
H
31
124
110
95
64
52
31
(,6
51
3K
25
II
0 30
U
41
30
1I91l
13
20
0')5
5
I)
0 20
43
26
16
3
o .1IJ
32
7
2
040
20
15
7
IU O
1
'J
o su
4
133
141
135
124
106
124
119
10K
92
92
79
81
69
010
122
107
103
103
93
77
e.G
57
OAO
91
92
77
61
46
53
40
050
76
59
43
63
49
45
·75
59
43
86
70
54
33
31
17
IS
23
23
25
20
I)
II
l)
IJ
;.
21
II
6
0.60
0 70
33
J5
21
4
3
12
11
Ordina tes for the
2
030
O KO
0 '10
0 ');
(OlCC
1
3
3
19
29
37
S7
o 2U
0
2
0
6
14
n
47
o JO
3
1
9
4
1
21
14
10
12
3K
23
0 40
U SO
h
I ')
II hU
I
10
0 70
3
lI KO
1
3
.1
0 70
I) 30
0
Table 5 Determination of Blade Centroid and Weight
Mil ll \ p..i rtll.;uIOll l
"11'1""Ik' 1.1,.11 111,;"'':'
D
Itu'.JII"ll...I ' I-.:.:·1
Nurn l-.:: r lIl" bbJ.,;
"
riR
ell"....J
_~" ut l
I
c
/\rea
m
l Ull
11012
0 90
00l')
(I SO
070
060
050
0 '0
03 0
010
DOH
0061
00 77
oOY4
U 110
0 116
O tH
OIJOU
0013
O.04l
0066
008 5
O. \07
011 4
0<.00
I 143
I ~22
I ;;U
I
sao
I.SS..I
1 ·.&7 lJ
I 118
0120
1 202
0110
Vol u n lc p u bl.de
0.120 m"J
W ci Khf per bl"d,.
Momenl o( u ,lumc
0.115 m
Ca:ncr uid
D,\lS J m
M.. nllOlrk:~ ": l h un....:
I"nll.:·)frn.. IC:f1..a 1
SM
le\ ,,'"
ltV)
IlMV I
('>'~ta, ~ .N ,,:
Fece I.:hord
v
[m ]
m" l
m
KlOll " tUm"l
MoIk:f1"I .J.:II..
·Uly
•
thl
u U.:i
I'JI )'11n\,l
19m
210 ' plll
,1
I
vso
1
1 7li
I S60
~
1.16j
Z
I 170
0975
07110
0585
03 'lO
,
1
•
I
0 000
OOY3
0038
0 .16 ;
0 170
0.0<1
o Z18
0<79
OWO
0. 163
0 137
0.361
0 199
0.39'/
01 78
OI ZO
02110
U.04 7
1 I II
\ 764
O(J()(.
0 0 13
DOlO
Om8
00 36
0040
001l
001 8
0066 .
! nJ
",o\4l'1c,,1 . .. . IH~
tt1i'um
Lead ed ge
paral1dllu
n O.Tll UI IU
h
Iace ch"l(c.J
face chord
(InJ
IIUlJO
(m" 41
~ nl" .& )
o UlJvE., ,,,,
u()()t Jl·. ...t ....
t71~ ·lJ6
I 7)2E-o.\
5 U6 IE.())
o Hu
I
0 .647
U7Q;
071 9
0 707
50!l8 E~)6
(1 67)
826KE.()5
0.61 8
U 547
I l'I~.().4
I ~ ~,lE-o l
I 162E4.z
1 <7610<>4
Y 934E.()J
lo&18E-4Ji
) OM , ,, 5
541IL '() ~
~0Il6 E~.3
I 215~.(Il
I 41 J~ .() Z
I lunntl
A
'
37
Jurnal Mekanikal, Jilid II, 1997
3.2
Blade Strength Calculations
The blade strength calculations are done based on Taylor's method, Keryser and
Arnoldus's method and compared with those provided by the classification societies.
It should be borne that the designed propeller has zero rake.
The propeller is
assumed to have sufficient hull clearances. Therefore, there is no necessity to have a
rake angle .
3.2.1
Taylor's Method
D. W. Taylor derived formula of determining the maximum compressive stress and
tensile stress of a propeller blade. The strength calculation is carried out at section
O.2R.
The maximum compressive stress occurring at the position of maximum
thickness is :
(6)
The values of coefficient C1 can be read off from Fig. 4. The product cb can
also be given as:
(7)
The maximum tensile stress which in general, is smaller in absolute value
than the maximum compressive stress and is given by;
( 8)
The factor 1.17 ~!... can be obtained from Table 6.
Cl
Equations 6 and 8 for calculating the maximum stress take no account of
those due to centrifugal force, since the rake angle is zero. If the propeller is raked
aft, these extra stresses may become considerable magnitude and cannot be ignored.
According to Taylor, the extra compressive stress due to centrifugal action is
greatest at the position of maximum ordinate on the back of the blade element and
38
Jurnal Mekanikal, Jilid II, 1997
the greatest extra tensile stress occurs at trailing edge. The computation of these
extra compressive and tensile stresses due to the centrifugal force is mentioned in
[1].
,
1900
i
i
\
lBOO
\
1700
:
,
1500
i
1400
i
I
I
\:
~
I
I
!,
1
900
U
BOO
i
I
I
I
I
I
,
,
700
D,S
0,6
0.7
i
I
!
i
I
I
O.B
i
II
I
:
,
-, !
i
I
I
I
i
i
I H/o!
I
i
I
i
I
!
I
I
\
I
I
I
\
II
I
I
!
II
!
I
I
I
I
I
I
I
I
\
i
i
I
I
1000
i
i
I
!
1100
,
I
I
i
1300
1200
"
: -------;
I
;\
1600
!
I
i
!
I
-.-.
i
!
I
I
I
I
I
i
I
I
I
I
I
I
i
f
I
i
I
I
I
I
I
II
I
i
I
I
I
iI
,:
:
,
I
Ki
,
'
,
I
'
,
,
I
i
i
I
I
!
i
I
II
0,9
1.0
1.1
1.2
1.3
I
;
\,4
Fig. 4 Relation Between Coefficient C\ and Pitch Ratio [19]
Table 6 Values of 1.17 LlC for Blade Elements at 0.2R [19]
Pitch
Ratio
1.17L1C
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
0.650
0.710
0.754
0.784
0.804
0.817
0.823
0.20
Table 7 Summary of Stresses at 0.2R using Taylor's Method
Compressive stress
Tensile stress
Kg/em'"
Lb/in'"
Kg/em'"
Lblin'"
515
7327
388
5524
39
Jurnal Mekanikal, Jilid II. 1997
Applying Eqs. 6 and 8, summary of the maximum stress occurs at 0.20R is
shown in Table 7. The limitation in applying this method is that the maximum pitch
diameter ratio is 1.4 for determining C J value and 1.3 in determining 1.17LlC value.
3.2.2
Keyser & Aarnoldus 's Method
In reference 11, the moments M bs and M bt , as shown in Fig. 4, may be written as:
(9)
(10)
Hence,
M',
= M bs cosa +M b, sina
(11)
The trust and torque force distributions are largely governed by the radial
wake distribution behind the vessel and by the pitch distribution of the propeller.
These can be categories as follows:
a. Propellers with constant pitch working in a homogeneous velocity field.
b. Propellers with constant pitch working in unequal velocity field.
c. Propellers with variable pitch working in an unequal velocity field (pitch
reduction approximately 20% towards the boss)
For a twin screw vessels, it fall into the first category. Whilst, single screw
vessels fall into the last two categories. The coefficients of
Is and It
in Eqns. 9 and
10 .is given in Table 8 for the three categories.
40
Jumal Mekanikal, Jilid II. /997
Fig.5 Decomposition of Bending Moments
Table 8 Factors for the Determination of Bending Moments
HO"'(t,r n~O\I'
.
.r«
C O"ltl ""C
fi. ld
C O""Il I ""1
o-
t,
f
f,
I,
0
0.2
0 .1
0 .4
0 .1
0.6
0.7
0 .8
0 .9
Vu u b'r fid d
puch
V n i,blr 'irld
V 1ru blr pIech
pil ch
t,
t,
0.444
0.148
0.2 l7
0.176
0 .10 8
O.Olll
0 .0225
0.00 40
0.3 7 8
0 .281
0.202
0 .116
0 .0810
0.0 440
0.0 17 1
0. 00 10
0
0.48 I
0.H4
0.29 1
0 .201
0. 110
0.0700
0.0 100
0.0 08 0
0.42 I
0.H6
0.2lS
0. 164
0.1 01
0 .0 111
0.021. 1
0 .00 6 0
0.464
0.164
0. ~ 0 6
0.2i1
0.109
0.22J
0.191
0. 120
0.0644
0.0214
0.0048
0.089 9
0.04 64
0.0182
0.00 12
0 .1~9
I-
r,
0.6ZIR
"
0 .606R
0. l i8R
The compressive stress in a section of a propeller blade nearly always larger
than the tensile stress. The calculation of the compressive stress is given by:
(J'=
M
b
awds2/cos 2 e
(12)
41
Jurnal Mekanikal, Jilid II, 1997
where a wd is the coefficient of section modulus for maximum compressive stress or
maximum tensile stress with a straight neutral axis or coefficient ofsection modulus
for maximum tensile and compressive stress with s curved neutral line as shown in
Fig. 6.
In applying Eqns. 9 and 12, summary of the maximum stresses occurs at
O.20R in shown in Table 9.
-
').1.
<1_,
~., ~
V-
»>
~'.I
//
Q-14
"
'"
k::::
-,
a. .. c
,
,
•
d
I
--
-.
0
-
je
"".
-o
Y>
Fig. 6 o.; Coefficient [II]
Table 9 Stresses at O.2R by Keryer and Arnoldus's Method
Compressive stress
3.2.3
Tensile stress
Kg/em"
Lb/in"
Kg/em"
Lb/in"
539
7664
455
6471
Classification Rules
Despite the calculations as mentioned in earlier sections, it is necessary to satisfy the
Classification Society's requirement. In general, these requirements are based on
minimum thickness at O.25R, and O.6R for a fixed propeller. A summary of the
formulation by the Classification Societies is shown in Table 10.
42
Jurnal.Mekanikal, Jilid II, 1997
The results of minimum thickness of the designed propeller at O.25R and
O.6R based in the formulations is given in Table 11.
Table 10 Extract of Classification Societies Guidance
Ctassificauon
Minimum thickness a! () 25R & O.60R
Society
Lloyd's
Register of
Shippuig
( =
r
America
Bureau of
Shipping
t ..
= ,')
KCA + 100
I:Fr II-N
AH
089 )
-
( 'n
Nippon
tr =
Kaiji
Kyokai
eRN
I 3150 Ml?
~ /:FI?Uf.N
.: .£.:'J[ BK) 1
-l
c.,) 4 C J
KI~SW
K, ZN/
Table 11 Comparison of Minimum BladeThickness at O.25R and O.6R
Actual thickness
·O.Z5R
135
ByLRS
ByABS
ByNK
O.6R
O.25R
O.6R
O.25R
O.6R
O.25R
O.6R
77
133
68
84
76
99
54
43
Jurnal.Mekanikal, JiIid II, /997
The results of minimum thickness of the designed propeller at 0.25R and
0.6R based in the formulations is given in Table 11.
Table 10 Extract of Classification Societies Guidance
Classtfication
Society
Minimum thickness a! () 25R & O.60R
Lloyd's
Register of
KCA
00 13150 tl,d?
/ = I:FU/,N + 1
r
Shippuig
America
Bureau of
Shipping
fr
=,')
289
y
AH ± (~ BK )
CnCRN
C,) '+C J
l
1
= IK,~SW
Nippon
f
Kaiji
Kyokai
J
V1:'FI?ULV
r
~ K:. ZNI
Table 11 Comparison of Minimum Blade Thickness at 0.25R and 0.6R
Actual thickness
·0.Z5R
135
ByLRS
ByABS
ByNK
0.6R
0.25R
0.6R
0.25R
0.6R
0.25R
0.6R
77
133
68
84
76
99
54
43
Jurna/ Mekanikal, Ji/id II. /997
4.0
PROPELLER INDUCED VIBRATION
Three distinguished methods will be used to determine the main hull vibrations
which include Todd's, British Maritime Technology (BMT) Design and Erich
Danckwardt , s formulae.
4.1
Todd's Formula
The first of these empirical formulae to be commonly used was that due to Schlick
[18]. It is modified form ofthe ordinary beam formula;
(13)
where, I
= C 2BD 3
Todd proposed to replace I by the expression BD 3 and let the value of C2 be
merged in an overall coefficient,
(14)
An empirical formulae of Cargo Ship for 2 node vertical vibration is given as;
(15)
where
~l = i\(1.2 + 3~ J
It should be highlighted that Todd's formula is in imperial units.
For higher nodes of vertical and horizontal vibration, the relationship with 2
node vertical vibration is given by:
44
Jurnal Mekanikal, Jilid II, 1997
Horizontal Vibration
Vertical Vibration
Average limits of the above frequencies are as follows;
4.2
2 node vertical
1.5%
2 nodes horizontal
2.5%
3 node vertical
3.0%
3 nodes horizontal
9 .0%
4 node vertical
8.0-10.0%
4 nodes horizontal
8.0-10.0%
BMT Design Formula
BMT design formula derived from measured natural frequency data for warship are
represented in a graphical format with appropriate base function. An example of this
graph is shown in Fig. 7, where only vertical node frequencies have been
considered, as the horizontal node frequencies do not have significant effect on hull
vibration. The base function is given by;
~
x=v~
(16)
where, I v = second moment of area amidships
(17)
These formula are in metric units.
4.3
Erich Danckardt Formula
Erich Danckwardt presented an empirical formula for determining hull frequency
based on Todd's formula, He simplifies the formulation for determining vertical
4S
Jurnal Mekanikal, Jilid II. /997
second moment of area with coefficient C v ' for different type of merchant ship.
Two node frequency is given by:
N 2V
f5
v
= 200000
3
M 6 1L
cv = 1.11 x 10-
3
+ 28
(18)
for container ship
(19)
iOCD
eoo
_
6 00
.,
200.
I
100
Fr~q llcnc _'
~N V tc p rni
80
( ,j
60
=- - --'
- -::::::-::=.
==~-=.-:-.
--~ =---~ ~~ -~ :-~::--=-~
-
i
=:~
.. - -
-
.:: ==---==---
---,~---.:..===~ - - - --_.
__ .
=-=--
- . _ - - - -- - - - - - - _._-- - -
~.:..::.~~_
_ . 1
-.::: <:~ -~-- ~ - j~ : -= ~:
•
_
~
'
. _
.
•
- ------ - ._-j -
j--
---- -~~ ::=:
. _
.
_
10
100
10
,
=
.10 '
(I;"
V ~·
Fig.7 Two Node Vertical Hull Natural Frequency [3J
46
Ju rnal Mekanikal. Ji/id 11, /99 7
Higher nodes of vertical and horizontal frequency are given as;
Horizontal vibrat jon
Vertical vibration
N 2H = 1.50N2v
Frequency bands for the above nodes are:
2 node
± 2.5%
3 node
± 5.0%
4 node
± 7.5%
5 node
± 10.0%
Summary of the different nodes ofvertical and hull frequenc ies is shown in
Table 12. For 2 .node vertical frequency , Todd's formula differ by + 7% from BMT
Design formula and -64% from Erich Danckwardt formula. In fact, there is wide
variation in the results for other nodes for Erich Dan ckwardt as compared with
Todd's and BMTs.
Table 12 Compar ison Tabulation of the 3 methods of
Vibration Analsysis
Methods used are
I Tod d's fo rm ula
) BMTs design Cannula
3 Erich Danckwardt lo rm uln
I
"lodes
Ve rt ica l
1
I
Met ho d 1
')2
Frequency (c pm )
Me thod J
Met hod 2
YX
IX]
"
.'-'
(}O
-+
IX<1
27)
~55
)
3(,7
no
X7
lOll
.
10)
.
213
,
Horizontal
2
13x
J
27)
-+
5
4 13
50
15x
47
Jurnal Mekanikal, Jilid II, 1997
5.0
DISCUSSIONS
5.1
Strength Analysis
According to Taylor, the minimum tensile and compressive stresses for manganese
bronze propeller is 6000 lbs/inch". From the results in Section 3, it is found that the
designed propeller stresses are greater than the minimum requirement as shown in
Tables 7 and 9. The compressive stress is approximately 23-27% greater than
minimum stress. And the tensile stress is approximately 7-8% greater than minimum
stress. However, it is observed that by Taylor's method, as shown in Table 7, the
tensile stress is 8% less than the minimum stress. This implies that blade thickness
need to be increased in order to meet the minimum stress requirement if calculations
are done by Taylor's method.
The thickness of the designed propeller as found in section 8.2 at 0.25R and
0.60R also exceed the minimum thickness as specified by the Classification
Societies as shown in Table 11. According to the Series Charts method, Lloyd's Rule
seem to give a optimistic value of 1% difference of thickness at 0.25R and 0.60R.
But ABS and NK are rather pessimistic with value of 3% difference of thickness at
0.25R and 0.60R.
5.2
Vibration Analysis
Careful scrutiny of Erich Danckwardt formula reveals that there is an error in the
overall coefficient in Eqn. 18. Since this equation is based on Todd's formulation,
the overall coefficient should be:
From Todd's formula:
48
Jurnal Mekanikal, Jilid II, 1997
Converting imperial units to metric
f
= 46750 0.3048 x
1.016
r;; = 25606 r;;
ft
(20)
VI
Hence, the overall constant for Erich Danckwardt formula should be 25606
instead of 200000.
A graph showing frequency of hull vibration versus propeller shaft
revolutions for the designed ship is shown in Fig. 8.
The propeller blade frequency is 840 rpm at the propeller shaft revolution as
shown in Fig. 8. This means that propeller resonants are clearly away from hull
resonants. Hence, the designed propeller has minimum implication to hull vibration.
For these exercise, Todd's formula provides a simplified and good estimate
for determining the hull vibration frequency at the preliminary design stage.
I
~ao1
+ - - , 1_ _
-+--~--\-.
__
I
_J
I
'"j
<,
100
f==t==~--==='9======'~ No.
'0-
~r- ~. ~-l ~---=-- -r..-
RPM 0,. P/lOP8LLEII SHAFT
Fig.8 Frequency of Hull Vibration vs Propeller Shaft Rpm (Todd's Method)
49
Jumal Mekanikal, Jilid IL 1997
6.0
CONCLUSION
A marine propeller not only should have a shape that engine power is converted into
trust at an efficiency as high as possible, but it should also be capable of sustaining
the attending loads without fracture. This implies the possibilities of the stresses in
the propeller blades being calculated, and these stresses should not exceed a certain
maximum value. The same implies to the vibration induced by the propeller. From a
designer's point of view, during the preliminary design stage, it is necessary to use a
simple formula but yet accurate results to calculate the strength and vibration of the
required propeller.
REFERENCES
1.
American Bureau of Shipping, Rules for Building and Classing - Steel
Vessels', 1977.
2.
Bureau veritas, The Rules and Regulations for the Construction and
Classi fication of Steel Vessels, 1977.
3.
British maritime Institute, BMT's Ship Design Manual - Vibration, London,
1989.
4.
Danish Maritime Institute, Towing Tank Tests with a 500 TED Container
Ship, 1990.
5.
Det Norske Veritas, Rules for Classification of Ships', 1977.
6.
Danckwardt, E., Vibration, University of Rostock, Germany, 1980.
7.
Holtrop, 1., A Statistical Re-Analysis of Resistance and Propulsion Data,
International Shipbuilding Progress, Vol. 31, 1981.
8.
Holtrop, J. and Mennen, G.GJ., A Statistical Power Prediction Method,
International Shipbuilding Progress, Vol. 25, 1977.
9.
Harvald, S. A., Resistance and Propulsion of Ships, Wiley-Interscience
Publication, New York, 1983.
50
Jurnal MekanikaI, Jilid II, 1997
10.
Kuiper, G., The Wageningen Propeller series', MARIN Publication, 1992.
11.
Keyser, R. and Arnoldus, W., Strength Calculation of Marine Propellers',
International Shipbuilding Progress, Vol. 6, 1957.
12.
Lloyd's Register of Shipping, Rules and Regulations for Classification of
Ships, London, 1992.
13.
Nippon Kaiji Kyokai, Rules and Regulations for Classification of Ships,
Tokyo, 1986.
14.
O'Brien, T.P., The Design of Marine Screw Propellers, Hutchison Scientific
and technical, New York, 1962.
15.
Sabit, A., Optimum Efficiency Equations of the NMSB Propeller Series 4
and 5 Blades, International Shipbuilding Progress, Vol. 23, 1976.
16.
Sugai, K.,Comparison of Propeller Performance Between Propeller Open test
and Calculation Based on Lifting Surface Theory', University of Glasgow,
Dept. of naval Architecture and Ocean Engineering, Glasgow, 1960.
17.
Ship Structure Committee, Ship Vibration Guide', SSC-350, 1990.
18.
Todd, F. H., Ship Hull Vibration, Erward Arnold Ltd., London, 1961.
19.
Van Lammeren, W. P.A., Troost, L. and Koning, J. G., Resistance,
Propulsion and Steering of Ships, Technical Publishing Co., 1948.
20.
Yide, S. and Marchal, J. L. J., Expressions of the B, - 8 Diagrams
III
Polynomial for Marine Propeller Series, Royal Institution of Naval
Architects, London, 1993.
51