PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Study on the influence of kinetic parameters of propeller on
acoustic and vibration characteristics of propulsion shafting
Yangyang ZHANG; Jun SUO
Naval Research Academy, China
ABSTRACT
Propeller in most research on the vibration characteristics of propulsion shafting is simplified, in order tostudy the influence of kinetic parameters of propeller itself on acoustic and vibration characteristics ofpropulsion shafting, the propulsion shafting model which contains precise three-dimensional elastic propeller
is established, focusing on the mass effect and elastic effect of propeller on shafting longitudinal vibration
based on analytical method and finite element method. Results show that the simplified method of propeller
has great influence on the propulsion shafting vibration, and the influence of propeller on the acoustic andvibration characteristics of propulsion shafting includes not only mass effect, but also obvious elastic effect.
From the point of view of shafting vibration control, increasing propeller mass and blade elasticity are
beneficial to shafting vibration control.
Keywords: propeller, mass effect , elastic effect, longitudinal vibration

1. INTRODUCTION
At present, most ships are propelled by propellers. The research results at home and abroad showthat [1-3] , the longitudinal vibration of propulsion shafting is an important factor affecting hull
vibration and noise. In order to study the effect of longitudinal vibration of pr opulsion shafting onhull vibration and noise, many geometric models of shafting have been established [4-6] , including
lumped mass model and distributed mass model. Because of the complexity of propeller geometry,the lumped mass point or uniform mass disk are used to simulate propeller geometry [4,5]. Neither of
the two modeling methods takes the effect of propeller elasticity on the longitudinal vibration of
propulsion shafting into account, which is obviously different from the actual situation.
In order to make the research results of longitudinal vibration of propulsion shafting closer to theactual situation, a propulsion shafting model considering the elasticity of propeller blades isestablished. The longitudinal vibration characteristics of propulsion shafting considering theelasticity of propeller blades are studied by using transfer matrix method and finite element method,respectively. Then the influence of the simplified method of propeller on longitudinal vibration ofshafting is simulated and analyzed and the influence of propeller mass and blade elasticity is further
analyzed.

2. Transfer Matrix Method for Longitudinal Vibration of Shafting
2.1 Transfer Matrix Theory of Beams
The one-dimensional beam structure is divided into n elements. The number of elements isincreased from 1 to n, and the number of nodes between elements is increased from 0 to n. Each nodecorresponds to the point transfer matrix, and correspondingly, beam elements from 1 to n correspondto the field transfer matrix, as shown in Figure 1.

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Figure.1 Element division of the beam structure
For the beam element i , the state vector of the left end section is defined as
the right end section is defined aszi L , the state vector ofzi R . For the whole beam structure, the state vector of the left end section

is defined as z 0 , the state vector of the right end section is defined as z n .

The relationship between the state vector z n and the state vector z 0 shown in Fig. 1 is obtained bysubstitution from left to right.

zn

PTn n Pn1

PT
i i

PT
Tz0
1 1PT
0 0 z0

㸦1㸧

PT
i i

㸦2㸧

in whichTPT
n n Pn1PT
1 1P0

T is the field transfer matrix between the state vectors at both ends of the beam structure. It is
obtained by multiplying the point transfer matrix at each node and the field transfer matrix of each beam
element in turn.
2.2

Transfer Matrix Method for Longitudinal Vibration of Shafting Considering the

Elasticity of Propeller Blades
The state vector of the longitudinal vibration of the shafting is defined as Z
displacement and the stress of the cross section are defined as u and[u N ]T , in which the

N . According to the deduction of

the author's previous paper [6], the point transfer matrix of the centralized mass points is

P

0º
ª 1
« mZ 2 1 »¬¼

㸦3㸧The point transfer matrix of a massless linear spring is

P

ª
«1
«
¬0

1º
K»
»
1¼

㸦4㸧
For the convenience of calculation, the thrust shaft is simplified to a uniform section without
considering the change of section area. The field transfer matrix of its longitudinal vibration is as follows.
Tsin( E l ) ºª« cos( E l )
EAE »
«
»
«¬  EAE sin( E l ) cos( E l ) »¼

㸦5㸧
The propeller is composed of hub and blade. The blade has certain elasticity. The influence ofpropeller on the vibration characteristics of propeller-shaft coupling system is not only reflected in the masseffect of propeller, but also should include the elastic effect of propeller blade. Therefore, the action
diagram of propeller on the shafting system is shown in Figure 2.

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Figure.2 The mechanical model of shafting longitudinal vibration considering the blade elasticity
Among them, m y and k y is the equivalent mass and stiffness of the blades; mg is the mass ofpropeller without the equivalent mass of the blades, m y and mg is regarded as mass point. E 㸪 U 㸪 A
and L are elastic modulus, material density, cross-section area and total length of the uniform shaft; k1 㸪

k 2 㸪 k 3 and k 4 are lateral support stiffness of rear stern bearing, middle stern bearing, front stern bearing
and thrust bearing. k t is the longitudinal support stiffness of thrust bearing.
Thus, the total transfer matrix of shafting longitudinal vibration considering blade elasticity can be
expressed asTPT
3 1P2 PP
1 0

㸦6㸧The left end of the propeller is free, and the cross-section stress is exciting force of the propeller.

N1

f

㸦7㸧
The right end of the shaft is rigidly fixed and its longitudinal displacement is zero.

u5

0

㸦8㸧
When the longitudinal excitation force f is applied to the propeller end, the relationship between thestate vector of the right end and the left end of the propulsion shafting is as follows.

>uN @T
5

>T @ x >uN @

T

1

㸦9㸧
Obviously, T is a 4 u 4 square matrix, so formula (9) can be expressed as

ªu º
«N »
¬ ¼5

ªT11 T12 º ªu º
«T T » « N »¬ 21 22 ¼ ¬ ¼1

㸦10㸧
Substituting formula (7) and (8) into formula (10) can obtain
T11u1  T12 f

0

㸦11㸧
Order

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f

0 , the modal frequency of the shafting longitudinal vibration can be obtained by solving

equation (11).
Solve equation (11), we can obtain the left-most state vector of the shafting.
T12
­
f
°u1 
T11
®
°N
¯ 1 f

㸦12㸧
The longitudinal vibration response of shafting at any section can be obtained by substituting formula(12) into formula (10) and selecting corresponding transfer matrix according to different response
positions.
In order to verify the correctness of the above theoretical analysis, the vibration acceleration level at
the thrust bearing of the shafting is solved by using MATLAB programming, and the corresponding finite
element model is established in Abaqus for simulation analysis. The finite element model is shown in
Figure.3, and the specific parameters of the propulsion shafting are shown in table.1.

Figure.3 The finite element model of shafting
Table.1 Parameters of shaftingparameters

unit

value
14.92

shaft lengthLshaft outer radiusR

shaft inner radius

r

m
m
m

my

kg

2626

mg

kg

1314

Young modulusEPa

2.1e11densityU

kg / m3

7800

equivalent mass ofpropeller blade
equivalent mass ofpropeller hub

parameters

unit

value

0.135

ky

N /m

1.28e8

0.06

k1

N /m

3.5e7

bearing

k2

N /m

2.0e7

stiffness

k3

N /m

2.0e7

k4

N /m

1.0e8

kt

N /m

1.8e9

Poissonratio

0.3

The above transfer matrix method and finite element method are used to calculate the vibration
acceleration of the propulsion shafting at the thrust bearing, respectively. The results are shown in Figure. 4.
It can be seen from the figure that the calculation results of transfer matrix method and finite element
method are basically the same, which shows the accuracy of finite element method in calculating theinfluence of propeller on shafting vibration. The vibration acceleration spectrum at thrust bearing have two
peaks at the first longitudinal vibration frequency (63.56 Hz) of the shaft and the first longitudinal vibration
frequency (30.57 Hz) of the blade, which indicates that the elasticity of propeller blade has great influenceon shafting vibration.

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Figure.4 The results contrast of analytical method and finite element method

3. Influence of Propeller on Longitudinal Vibration of Shafting System
3.1

Influence of Propeller Simplification on Longitudinal Vibration of Shafting System

In the related research of shafting vibration, because of the complexity of the actual shafting structure,most models simplify the propeller by using lumped mass points or homogeneous disks of equal massinstead. Because of the different simplification methods, the results are also different. The finite element
model of the propulsion shafting with an accurate three-dimensional elastic propeller is shown in Figure. 5.
The effect of propeller modeling on the longitudinal vibration modes of shafting is shown in Table.2.


Figure.5 Finite element model of propulsion shafting
Table.2 The longitudinal vibration modal frequency of shafting
modal orderno propeller

lumped mass

homogeneous diskelastic propeller

155.38 Hz41.09Hz

39.68Hz

41.02Hz2

106.08 Hz

94.04Hz89.44Hz

99.50 HzIt can be seen from the table that after installing propeller, the first and second order longitudinal
vibration modal frequencies of the shafting system decrease, and the longitudinal vibration modal
frequencies of the shafting system are different under the three modeling modes of the propeller. The firstorder frequency of lumped mass model is closer to the elastic propeller model, and the shafting longitudinal
modal frequencies are lower than those of the other two simplified modes.

Figure.6 Longitudinal vibration transfer function of shafting under different simplified methods of propeller

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The frequency spectrum of longitudinal vibration acceleration transfer function of shafting under
different propeller simplified modes is shown in Figure. 6. It should be pointed out that, because the shaft
itself has less contact with water and the effect of the attached ripple water on the shaft is small, the effectof the attached ripple water on the propeller is mainly considered, so the effect of the attached ripple water
is not considered when the centralized mass and homogeneous disk simplification are adopted.
Compared with the pure shafting without propeller, the peak value of the longitudinal vibration transfer
function of the shafting moves to the low frequency when the simplified method of lumped mass points andequal mass homogeneous disk is adopted, and the mass effect of the propeller has a greater impact on thelongitudinal vibration of the shafting; when the precise three-dimensional elastic propeller is used, there are
two peaks in the longitudinal vibration transfer function. The first peak corresponds to the first-order
longitudinal vibration of the shaft and the second peak corresponds to the first-order bending vibration of
the propeller blade in the same direction. It can be seen that the influence of actual propeller on the shafting
vibration not only include mass, but also include the elasticity of the propeller blade. The mass effect andblade elasticity effect must be taken into account simultaneously in research of the influence of propeller on
shafting vibration.
3.2

Influence of Propeller Mass on Longitudinal Vibration of Shafting

In order to study the influence of propeller mass on the longitudinal vibration of shafting independently,
only the propeller material density is changed. For the exact three-dimensional elastic propeller-shaft
system, the vibration acceleration and sound radiation transfer functions of shafting at the thrust bearing
position under unit longitudinal excitation of propeller and different propeller mass are shown in Figure. 7.

(a) acceleration transfer function(b) acoustic radiation transfer function

Figure.7 Influence of propeller mass on acceleration and acoustic radiation transfer function of shafting
In Figure 7, the first peak corresponds to the first-order longitudinal vibration of the shaft and thesecond peak corresponds to the first-order bending vibration of the blade in the same direction. It can beseen from the figure that with the increase of propeller mass, the peak value frequency of the transfer
function of the shafting at the first-order longitudinal vibration of the shaft and the first-order bending
vibration of the propeller blade in the same direction moves to the low frequency, and the amplitudedecreases. The influence of propeller mass on the peak value of the first-order bending vibration of thepropeller blade in the same direction is greater, and the maximum difference of the sound radiation transfer
function at the second peak is 5.82 dB. It can be seen that without changing the elasticity of propeller blade,4697

properly increasing the mass of propeller is beneficial to reduce the vibration and sound radiation ofshafting.
3.3

Influence of Propeller Blade Elasticity on Shafting Longitudinal Vibration

In order to study the influence of propeller blade elasticity on the longitudinal vibration of shafting
independently, only the elastic modulus of propeller material is changed. For an exact three-dimensionalelastic propeller-shaft system, the vibration acceleration and sound radiation transfer functions of the
shafting at the thrust bearing position under the unit longitudinal excitation of propeller and different
propeller blade elasticity are shown in Figure. 8.

(a) acceleration transfer function(b) acoustic radiation transfer function

Figure.8 Influence of blade elasticity on acceleration and acoustic radiation transfer function of shafting
It can be seen from the figure that with the increase of propeller blade elasticity (the elastic modulus Eof the propeller material decreases), the peak value frequency of the transfer function of shafting at thefirst-order longitudinal vibration of the shaft and the first-order bending vibration of the propeller blade in
the same direction moves to the low frequency, and the amplitude decreases. The elasticity of the propeller
blade has little effect on the peak value and frequency shift of the first order longitudinal vibration of the
shaft. The elasticity of the propeller blade mainly affects the peak value of the first bending vibration
frequency of the blade in the same direction. The maximum difference of the amplitude of the soundradiation transfer function at the second peak value is 9.12 dB. It can be seen that the longitudinal vibrationof shafting can be effectively reduced by increasing propeller blade elasticity without changing propellermass.

4. Conclusions
In this paper, the influence of propeller dynamic characteristics on acoustic and vibration characteristics
of shafting is studied, and the following conclusions are drawn:(1) The influence of propeller on shafting vibration is not only reflected in its mass effect, but also in
its elastic effect. The blade elasticity is evidently reflected in the longitudinal vibration of shafting. It is
inaccurate to simplify the propeller into a lumped mass or homogeneous disk in the study of shafting
longitudinal vibration. The influence of the blade elasticity must be considered.
(2) From the point of view of shafting vibration control, increasing propeller mass and blade elasticity
are beneficial to shafting vibration control.

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