The 21st International Congress on Sound and Vibration
13-17 July, 2014, Beijing/China
VIBRATIONAL CHARACTERISTICS OF A MARINE
SHAFT COUPLING SYSTEM EXCITED BY PROPELLER
FORCE
Xincong Zhou,Li Qin, Kai Chen,Wanying Niu,Xinchen Jiang
School of Energy and Power Engineering, Wuhan University of Technology, 130 P.O.
Box, 1040 Heping Road, Wuhan, China 430063
Yeping Xiong
Faculty of Engineering and the Environment, University of Southampton, Highfield,
Southampton, UK
Li Qin, Jianzhou Quan, Xinping Wang
Huangpi NCO School of AFEWA,288 Huangpu Avenue, Wuhan, China 430345
e-mail: [email protected]
Marine propulsion shafting system subjects to various shocks and cycle exciting forces in actual operation. The propeller excitation is an important the marine shafting vibration excitation sources. In this paper, the finite element method is used to calculate a ship propulsion
shafting vibration responses under the propeller bearing forces. Propeller bearing forces can
be decomposed into components in three directions of forces and three directions of moments,
wherein the periodic propeller excitation will cause the longitudinal vibration, torsional vibration and whirling vibration of marine shafting. The mathematical equations governing the
dynamics of the circular shaft of regular constant section are derived. Compared with the free
vibration systems, it is found that adding the propeller exciting force can greatly change the
frequency band for marine shafting vibration isolation.
1.
Introduction
Marine shafting system is the indispensable component of a ship power plant. The primary
function of a marine shafting system is to carry out energy transfer from marine engine to propeller,
and at the same time to transmit axial thrust which is produced by the rotation of propeller to the
hull, and therefore drives the ship ahead. While sailing on the sea, different levels of vibration will
happen to the marine shafting system under the unbalanced force from internal and external. This
not only affects the normal life of mariners, but also will lead to different degrees of damage to the
components, even affects the normal navigation of the ship.
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
2.
The influence factors of Marine shafting vibration
Researches show that the various kinds of periodic exciting force or excitation torque is the
main reason why propulsion shafting vibrates continuously:
• Alternating radial force and tangential force: generated by gas pressure of diesel engine cylinder and inertia force of the moving parts like piston rod together effecting on the crankshaft and crank pin.
• Alternating longitudinal and lateral thrust and the torque：produced by propeller revolving
in uneven radial and circumferential three-dimensional flow field.
• Exciting force and excitation torque：produced by shafting parts while running.
3.
Analysis and calculation of propeller exciting force
The speed and direction of water will change when it flows into the propeller and encounters
the round disk. The uneven wake flow was mainly affected by the hull size and the structure of the
hull tail, such as the diameter, location, structure of accessory parts and installation site of the propeller.
3.1 Mechanics analysis of propeller excitation
As one of the main excitation sources of Marine shafting vibration， the propeller excitation
can be divided into two parts：The first part is shaft frequency vibration force. Shaft frequency
vibration force is caused by propeller manufacture and installation error, or uneven hydrodynamic
during operation. The second part is blade frequency exciting force.
Figure 1. Exploded view of the bearing force of propeller
Assume that the axial wake is stable, each round the blade rotates, a cyclical changing force
will turn out. Each round the propeller rotates, a cyclical changing force Z p will turn out. Therefore,
blade thrust force and the fundamental frequency of the vibration excitation torque of its horizontal
direction and vertical direction is Z pω . The high harmonic frequency is kZ pω , namely times the
pressure impulses.
3.2 The experience formula of the propeller exciting force
Propeller exciting force is defined as the value under its rated speed. When the actual speed is
lower than the rated speed, the propeller exciting force will decrease with the rotational speed by
square of the speed ratio. When propeller is running in the uneven flow field, the frequency of exciting force “ f x ”is:
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
fs =
nzr
Hz
60
（1）
n —The time of propeller exciting force（ n = 1, 2,3,... ）
z —The number of propeller blades
r —The revolutions of propeller， r / min
The propeller exciting force is the sum of each blade vibration force. Respectively，the six
components of the exciting force are:
Fx= Z p Fx 0 + Z p Fx 0
ET 0
1 − ω0
∞
∑ω
k =1
kZp
（2）
cos(kZ pωt + η kZp + kZ pθ1 )
Z p M x 0 EQ 0
Z M
E
ω1 cosη1 + p x 0 Q 0
Fy =
2 0.7 R 1 − ω0
2 0.7 R 1 − ω0
∞
∑ [ω
k =1
kZp −1
cos(kZ pωt + η kZp −1
（3）
+ kZ pθ1 ) + ωkZp +1 cos(kZ pωt + η kZp +1 + kZ pθ1 )]
Z p M x 0 EQ 0
Z M
E
ω1 sin η1 + p x 0 Q 0
Fz =
2 0.7 R 1 − ω0
2 0.7 R 1 − ω0
∞
∑ [ω
k =1
kZp −1
sin(kZ pωt + η kZp −1
（4）
+ kZ pθ1 ) − ωkZp +1 sin( kZ pωt + η kZp +1 + kZ pθ1 )]
M=
Z p M x0 + Z p M x0
x
EQ 0
1 − ω0
∞
∑ω
k =1
kZp
cos(kZ pωt + η kZp + kZ pθ1 )
Zp
Z
E
E
0.7 RFx 0 T 0 ω1 cosη1 + p 0.7 RFx 0 T 0
My =
2
1 − ω0
2
1 − ω0
∞
∑ [ω
k =1
kZp −1
（5）
cos(kZ pωt + η kZp −1
+ kZ pθ1 ) + ωkZp +1 cos(kZ pωt + η kZp +1 + kZ pθ1 )]
（6）
Zp
Zp
E
E
0.7 RFx 0 T 0 ω1 sin η1 +
0.7 RFx 0 T 0
Mz =
2
1 − ω0
2
1 − ω0
∞
∑ [ω
k =1
kZp −1
sin(kZ pωt + η kZp −1
+ kZ pθ1 ) − ωkZp +1 sin( kZ pωt + η kZp +1 + kZ pθ1 )]
（7）
Make a tuning analysis of the propeller bearing force，and discuss the frequency components
of propeller bearing force ,then the bearing force can be expressed as:
∞
T=
A0 + ∑ [ An cos(2π fnt ) + Bn sin(2π fnt ) ]
（8）
Of which A0 represents volumes do not change over time, that is ,the average volume; An , Bn
are the harmonic components relative to n times fundamental frequency; f = 1/ T is the fundament
al frequency.
Constant thrust T 0 （kN）is：
T0 = kt 0 ρ n 2 D 4
（9）
Constant torque Q 0 （kN·m）is：
Q0 = kq 0 ρ n 2 D 5
（10）
n =1
Of which ρ Is the density of water，kg/m3; n is the rotate speed of propeller, r/min.
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
3.3 The theoretical calculation method of propeller exciting force
Propeller runs in an uneven velocity field around the stern, so the shafting inherits and passes
on constant or alternating reaction components caused by torsion, as well as the gravity of the
shaft. The changing frequency of these forces is equal to the speed of propeller multiplied by the
number of blades; the usual amplitude range is shown in table 1:
Table 1.
load on the propeller
alternate torque
alternate bending
mo-
ment
alternate thrust
Number of blades Z
4
（0.07~0.10）M KP
5
（0.03~0.05）M KP
（0.1~0.15）M KP
（0.2~0.3）M KP
（0.07~0.10）P 0
（0.02~0.04）P 0
M KP ：Constant torque；P 0 ：Constant thrust component。
The published data is the result of regression analysis according to different blades number. The regression polynomial expression is as follows：
n1
n2
n3
K T = ∑∑∑ Aijk (P / D ) (J ) ( AE / AO )
i
j
k
i =0 j =0 k =0
n1
n2
n3
K Q = ∑∑∑ Bijk (P / D ) (J ) ( AE / AO )
i
j
（11）
k
i =0 j =0 k =0
Of which: P / D —pitch ratio；
J —advance coefficient；Identified as 0.5 in this calculation.
AE / AO —disk-area ratio.
The odd number times of blade frequency are the main harmonic components of propeller exciting force and moment on the Y/Z direction, with a mean value which is zero.
Table 2. Main parameters of 8530 TEU container ship propeller
items
parameters
-1
rotate speed/r·min
104
material
CuAl10Ni
diameter/mm
8800
Number of blades
6
pitch ratio
0.98
Disk-area ratio
0.906
mean pitch/mm
8 662
Quality of propeller (air) /kg
92 580
Quality of propeller (with water) /kg
106 467
The moment of inertia (air) /(kg·m2)
363 400
2
The moment of inertia (with water) /(kg·m )
ICSV21, Beijing, China, 13-17 July 2014
494 380
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Take the propeller of 8530TEU container ship as an example, the type of main engine is B&W
12K98MC-C，the rated speed is 104r/min, the rated power is 68520kW，the net tonnage is 57364,
the design speed is 25.8kn，and the propulsion type is Single machine single oar. Take the related
data into formula (11) to calculate the ship propeller exciting force and vibration torque as listed in
table 3.
φ1420
φ795
φ800
φ795
φ1420
φ977
φ850
φ977
φ977
12880
φ975
φ975
φ975
φ871.4
φ790
14180
1036 557 1035 1035
450
200
5916
450
165
520
165
755 1000
630
5215
7335
165
150
后尾管轴承
3#中间轴轴承
前尾管轴承
φ
φ
φ
φ
φ
φ
φ
φ
φ
Figure 2. The shafting of 8530TEU container ship（Part 1）
2#中间轴轴承
1#中间轴轴承
Figure 3. The shafting of 8530TEU container ship（Part 2）
Table 3. Feature values of each direction when propeller rotates for one revolution under rated speed
direction
X
Y
Z
Sine vibration force / kN
（frequency / Hz）
Sine excitation torque /
kN·m
（frequency / Hz）
3.27
45.64
（63.59）
（10.6）
8.38
68.0
（10.6）
（10.6）
8.45
68.17
（10.6）
（10.6）
4. Analysis of shafting vibrational characteristic under propeller exciting force
The bearing can be simplified as a rigid model, a linear elastic model or a nonlinear model
considering the impact of the oil film. In this paper, a linear elastic model is chosen to be the simplified model of bearing.
With the influence of inertial load, the Y displacement vector diagram and torque vector diagram of 8530TEU Container ship shafting are as follows:
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 4. Y displacement diagram with the influence of propeller exciting force
Figure 5. Torque diagram with the influence of propeller exciting force
A shaft system model is established with ANSYS, the shaft system is constrained by boundary
conditions, and then modal analysis is conducted. Free vibration calculation results of the shaft system are shown in table below.
Table 4. Free vibration frequency of 8530 TEU container ship
Longitudinal
vibration
torsional
vibration
cyclotron frequency /min-1
Vibration type
frequency /min-1
frequency /min-1
1 step vibration
2304.308
126.034
349.129
2 steps vibration
5304.701
1807.763
418.496
3 steps vibration
8109.236
3434.232
925.816
4 steps vibration
11226.334
4780.889
1266.169
There is no forced vibration happen to the bearings of propulsion shafting under radial vibration force、vertical bending moment、horizontal bending moment and axial vibration force of the
propeller. The amplitude-frequency response curves are as follows:
Figure 6. The vertical amplitude frequency
Figure 7. The vertical amplitude frequency
curve of rear tube bearing
curve of front tube bearing
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 8. The vertical amplitude frequency
curve of NO.3 neck bearing
Figure 10. The vertical amplitude frequency
curve of NO.1 neck bearing
Figure 9. The vertical amplitude frequency
curve of NO.2 neck bearing
Figure 11. The torsional amplitude frequency
curve of rear tube bearing
Figure 12. The torsional amplitude frequency
Figure 13. The torsional amplitude frequency
curve of front tube bearing
curve of NO.3 neck bearing
Figure 14. The torsional amplitude frequency
Figure 15. The torsional amplitude frequency
curve of NO.2 neck bearing
curve of NO.1 neck bearing
ICSV21, Beijing, China, 13-17 July 2014
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21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
5.
Conclusion
This paper has introduced the torsional vibration, longitudinal vibration and whirl vibration of
marine propulsion shafting, calculated the natural frequency of propulsion shafting free vibration
respectively by using modal analysis in ANSYS, and the results show that there is no resonance
phenomenon appears to shafting under the first order natural frequency. This paper has also emphatically discusses the forced vibration response of propulsion shafting under propeller exciting
force, and the calculated results are analyzed in comparison.
ACKNOWLEDGMENT
This project is sponsored by the grants from the National Natural Sciences Foundation of
China (No. 51139005), the Ministry of Transport Applied Basic Research Project (Key Platform No.
2013329811360), and Innovation Groups Project of Hubei Province Natural Science Foundation
(No. 2013CFA007).
REFERENCES
1
2
3
4
5
6
7
8
9
10
11
Lyon, R. H. 1975 Statistical energy analysis of dynamic systems. Cambridge, MA: MIT Press.
Lyon, R. H. & Maidanik, G. 1962 Power flow between linearly coupled oscillators. J.
Acoust.Soc. Am. 34, 623-639.
Fahy, F. J. Statistical energy analysis: a critical overview. Phil. Trans. R. Soc. Lon d.
A346,431-447.
Fahy, F. J. & Price, W. G. 1998 IUTAM Symp. on Statistical Energy Analysis, Southhampton,
UK. Dordrecht: Kluwer.
Fung, Y. C. A first course in continuum mechanics. New York: McGraw-Hill. 1977.
Y.P. Xiong, J.T. Xing, W.G. Price. Interactive power flow characteristics of an integrated equipmentnonlinear isolator-travelling flexible ship excited by sea waves[J]. Journal of Sound and Vibration 287
(2005) 245–276.
Y.P.Xiong, J.T.Xing, W.G.Price, Power flow analysis of complex coupled systems by progressive approaches, J. SoundVib. 239(2001)275–295.
Goyder, H. G. D. & White, R. G. 1980c Vibrational power flow from machines into buildup
structures. III. Power flow through isolation systems. J. Sound Vib. 68, 97-117.
Xing, J. T. & Price, W. G. The energy-flow equation of continuum dynamics. In IUTAM
Symp. Statistical Energy Analysis, Southampton, UK.
Crocker, M. J. Ed., Handbook of Noise and Vibration Control, John Wiley & Sons, Hoboken,
NJ, 528–545, (2007).
Migdalovici, M., Sireteanu, T. and Videa, E. M. Control of Vibration of Transmission Lines,
International Journal of Acoustics and Vibration, 15 (2), 65–71, (2010).
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