For investigating the characteristics of the resonance changer and its influence on the longitudinal vibration of marine propulsion shafting system, by the transfer matrix method, the model of the shafting system, including the resonance changer is established. The influences of the resonance changer with various designing parameters on the force transmissibility through the thrust bearing foundation are evaluated. The results show that, the introduction of the resonance changer eliminates the resonance at original 1st natural frequency and introduces another two resonance peaks, as well as significantly lowering the maximum response compared with the original peak. The right one is attributed to the resonance changer, while the other is due to excitation of the other shafting system components. The 2nd natural frequency moves towards the high frequency direction and the peak value nearly stay invariable. If the designing parameters of the resonance changer are well optimized, the objective of reducing the longitudinal vibration transmission through the shafting system to the hull could be achieved

Hanguo Cui

Jiangyang He

Lin He

Wei Xu

Zhengmin Li

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 65 
Study of the influence of the resonance changer on the 
longitudinal vibration of marine propulsion shafting 
system 
Zhengmin Li1, Lin He2, Hanguo Cui3, Jiangyang He4, Wei Xu5 
1, 2, 4, 5Institute of Noise and Vibration, Naval University of Engineering, Wuhan, China 
1, 2, 4, 5National Key Laboratory on Ship Vibration and Noise, Wuhan, China 
1, 3Department of Mechanical Engineering, College of Power Engineering,  
Naval University of Engineering, Wuhan, China 
5Corresponding author 
E-mail: 1lizhengm100@163.com, 2linhewh@163.com, 3cuihanguo@163.com, 4hjywuhan@sina.com, 
5xuwei216@126.com 
(Accepted 28 August 2015) 
Abstract. For investigating the characteristics of the resonance changer and its influence on the 
longitudinal vibration of marine propulsion shafting system, by the transfer matrix method, the 
model of the shafting system, including the resonance changer is established. The influences of 
the resonance changer with various designing parameters on the force transmissibility through the 
thrust bearing foundation are evaluated. The results show that, the introduction of the resonance 
changer eliminates the resonance at original 1st natural frequency and introduces another two 
resonance peaks, as well as significantly lowering the maximum response compared with the 
original peak. The right one is attributed to the resonance changer, while the other is due to 
excitation of the other shafting system components. The 2nd natural frequency moves towards the 
high frequency direction and the peak value nearly stay invariable. If the designing parameters of 
the resonance changer are well optimized, the objective of reducing the longitudinal vibration 
transmission through the shafting system to the hull could be achieved. 
Keywords: resonance changer, longitudinal vibration, shafting system, transfer matrix method. 
1. Introduction 
For marine vessels, when the blade passing frequency of the propeller is equal or close to the 
natural frequency of the shaft longitudinal vibration, there will be excessive longitudinal vibration 
of the shafting system, which is not only the reason of the fatigue damage of the shaft and the 
wear of the thrust bearings, but also the main source for low frequency shipboard noise and 
underwater sound radiation from a submarine hull. For reducing the propeller-induced 
longitudinal vibration, several methods are presented [1-4], i.e. increasing the number of propeller 
blades, moving the thrust bearing further afterward, and adopting new thrusters such as water jet 
propulsion device. Such method is difficult to implement and is not always effective. The 
resonance changer (also called RC) [4] proposed by GOODWIN is a new option in reducing the 
propeller-induced vibration, and since then it has received much more attentions. 
Fig. 1 shows a thrust bearing which is a combination of a fluid-equalized thrust bearing [5] 
and a RC, and it could help reduce shaft longitudinal vibration. The thrust pads are supported by 
pistons which are in turn supported by hydraulic fluids. The piston cylinders are all interconnected 
and the supported fluids are connected to an oil reservoir through an oil pipe. The axial load is 
carried by the supported fluids and the disturbance forces are transmitted to the oil reservoir 
through the oscillatory oil pressure in the oil pipe. In this procedure, the virtual stiffness ܭ௛ is 
produced by the compression and expansion of the oil in the reservoir, and the virtual mass ܯ௛ 
and virtual damping ܥ௛  are produced by the viscous oil flow in the oil pipe. So, the piston  
cylinders, the supported fluids, the oil pipe as well as the oil reservoir work together as a RC. A 
replenishment hydraulic fluid system is also provided to compensate for shaft movement due to 
fluid leakage and the compression or expansion of the oil in the reservoir. Fig. 2 shows the main 
designing parameters of a thrust bearing with a RC, including ݎଵ, ݎଶ, ߠ, ݎ௣, ߠ௣, ݉, ଵܸ, ݀଴, ܮଵ, ݀ଵ, 
STUDY OF THE INFLUENCE OF THE RESONANCE CHANGER ON THE LONGITUDINAL VIBRATION OF MARINE PROPULSION SHAFTING SYSTEM.  
ZHENGMIN LI, LIN HE, HANGUO CUI, JIANGYANG HE, WEI XU 
66 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
which represent the inner radius of pad, the outer radius of pad, the angular of pad, the radius of 
supported point, the angular of the supported point, the number of thrust pads and oil cylinders, 
the volume of the oil reservoir, the diameter of the piston cylinder, the length and diameter of the 
oil pipe, respectively. 
 
Fig. 1. A schematic diagram of a thrust  
bearing with a RC 
 
Fig. 2. Specifications of a thrust  
bearing with a RC 
2. Model of longitudinal vibration of propulsion shafting system 
The main propulsion shafting system of a marine vessel can be simplified into the key features 
shown in Fig. 3 as far as the axial vibration is concerned, where ௜ܷ௟ , ܨ௜௟ , ௜ܷ௥ , ܨ௜௥  represent the 
amplitude of displacement and force of component ݅, and superscripts ‘ݎ’ and ‘݈’ denote the right 
end and the left end of component ݅ . Subscript ‘ ݅ ’ denotes different component. In detail, 
‘1’,‘2’,‘3’,‘4’,‘5’ denote the propeller, the stern shaft, the thrust bearing, the intermediate shaft 
and the coupling, respectively, and ‘ݐ’, ‘ℎ, ‘ܾ’ denote the thrust collar, the RC, the foundation, 
respectively. ௜ܶ represents the transfer matrix of component ݅. 
 
Fig. 3. A model of longitudinal vibration of propulsion shafting system 
2.1. Model of the thrust bearing with a resonance changer 
The simplified model of the RC is shown in Fig. 3. As described in literature [4], a 
mathematical analysis can be made as followings: 
ܨ௛௟ = ܣ଴ܲ =
ߩଵܮଵܣ଴ଶ
ܣଵ ൫
ሷܷ௛௥ − ሷܷ௛௟ ൯ + 8ߨߤଵܮଵ
ܣ଴ଶ
ܣଵଶ ൫
ሶܷ௛௥ − ሶܷ௛௟ ൯ +
ܣ଴ଶܤଵ
ଵܸ
൫ܷ௛௥ − ܷ௛௟ ൯, (1)
where, ܣ଴ = ݉ߨ(݀଴ଶ 2⁄ )ଶ is the total cross-sectional area of the supported fluids, ܣଵ = ߨ(݀ଵଶ 2⁄ )ଶ 
is the cross-sectional area of the oil pipe, and ܲ, ߩଵ , ߤଵ, ܤଵ are the oil pressure in the piston 
cylinder, the density of the oil, the oil viscous and the bulk modulus of the oil, respectively. Then, 
STUDY OF THE INFLUENCE OF THE RESONANCE CHANGER ON THE LONGITUDINAL VIBRATION OF MARINE PROPULSION SHAFTING SYSTEM.  
ZHENGMIN LI, LIN HE, HANGUO CUI, JIANGYANG HE, WEI XU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 67 
ܯ௛ , ܭ௛ , ܥ௛ , and ܨ௛  can be given by Eq. (2). ܨ௛  is the natural frequency of the RC. The 
transmission matrix of the RC is respectively expressed as Eq. (3), in which ߱ is the angular 
frequency of the disturbance forces: 
ܯ௛ =
ߩଵܮଵܣ଴ଶ
ܣଵ ,    ܭ௛ =
ܣ଴ଶܤଵ
ଵܸ
, ܥ௛ = 8ߨߤଵܮଵ
ܣ଴ଶ
ܣଵଶ ,   ܨ௛ =
ඨ (ܤଵܣଵ)(ߩଵܮଵ ଵܸ),
(2)
௛ܶ = ൥1
1
−ܯ௛߱ଶ + ܭ௛ + ݆߱ܥ௛
0 1
൩. (3)
The transmission matrix of the thrust bearing with a RC and its foundation is shown as: 
൜ܷ௕
௥
ܨ௕௥ ൠ = ൥
1 1ܭ௕
0 1
൩ ൤ 1 0−ܯ௕߱ଶ 1൨ ൥
1 1−ܯ௛߱ଶ + ܭ௛ + ݆߱ܥ௛
0 1
൩ ൥1
1
ܭ௢ + ݆߱ܥ௢
0 1
൩ ൜ ௧ܷ
௥
ܨ௧௥ ൠ, (4)
where, ܭ௕, ܯ௕ are the stiffness and mass of the thrust bearing foundation(including the supporting 
and surrounding structures), and ܭ௢, ܥ௢ are the axial stiffness and damping of the lubricated oil 
film. The procedure of calculating the dynamic coefficients described in literature [6], which is 
based on the thermal hydrodynamic lubrication theory and the perturbation equations formulated 
from a first order Taylor expansion of the oil film and pressure, is adopted in this paper. The hull 
of the ship or submarine could be regard as a rigid boundary condition, which means ௧ܷ௥ = 0. 
Then, Eq. (4) can be deduced as: 
ܨ௧௥ = ܭ௘ ௧ܷ௥ 
     =  (ܭ௕ − ܯ௕߱
ଶ)(ܭ௢ + ݆߱ܥ௢)(−ܯ௛߱ଶ + ܭ௛ + ݆߱ܥ௛)
(ܭ௕ − ܯ௭߱ଶ + ܭ௢ + ݆߱ܥ௢)(−ܯ௛߱ଶ + ܭ௛ + ݆߱ܥ௛) + (ܭ௕ − ܯ௭߱ଶ)(ܭ௢ + ݆߱ܥ௢) ௧ܷ
௥, (5)
where ܭ௘ represents the equivalent stiffness. ܯth is the mass of the thrust collar. Then, the transfer 
matrix of the thrust bearing with a RC and the foundation can be expressed as: 
ଷܶ = ൤ 1 0−ܯ௧௛߱ଶ + ܭ௘ 1൨. (6)
2.2. Model of longitudinal shaft vibration and responds calculation 
The transfer matrix of the shafting system ܶ is shown as Eq. (7), in which, ଵܶଵ, ଵܶଶ, ଶܶଵ, ଶܶଶ 
are the elements of ܶ. The transfer matrices of the propeller and the coupling are expressed as 
Eq. (5) and the transfer matrix of the uniform shaft undergoing longitudinal vibration is expressed 
as Eq. (6).Where, ܯ௣ , ܯ௖ , ݈ , ܧ , ߩ௦ , ܣ are the lumped mass of propeller, the lumped mass of 
coupling, the length of shaft, Young’s modulus, the density of the shaft, the cross-sectional area 
of the shaft, respectively: 
ܶ = ହܶ ସܶ ଷܶ ଶܶ ଵܶ = ൤ ଵܶଵ ଵܶଶଶܶଵ ଶܶଶ൨, (7)
ଵܶ,ହ = ൤ 1 0−ܯ௣,௖߱ଶ 1൨, (8)
ଶܶ,ସ = ቎ cos(݈߬)
sin(݈߬)
ܧܣ߬
−ܧܣ߬ sin(݈߬) cos(݈߬)
቏ , ߬ = ߱ඥܧ ߩ௦⁄
. (9)
STUDY OF THE INFLUENCE OF THE RESONANCE CHANGER ON THE LONGITUDINAL VIBRATION OF MARINE PROPULSION SHAFTING SYSTEM.  
ZHENGMIN LI, LIN HE, HANGUO CUI, JIANGYANG HE, WEI XU 
68 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
When the load ଴݂ is acted on the propeller, the relationship between state vectors of the right 
and left ends of the shafting system is presented in Eq. (10). Substituting the free-free boundary 
conditions (ܨଵ௟ = ܨହ௥ = 0) into Eq. (10), the state vector of the propeller end can be obtained by 
solving the Eq. (10), shown as Eq. (11). Utilizing the state vector of the propeller end and the 
accumulated transmission matrix from propeller to the component ݅ , the response at any 
component along the shaft could be obtained: 
൜ܷହ
௥
ܨହ௥ ൠ = ൤
ଵܶଵ ଵܶଶ
ଶܶଵ ଶܶଶ
൨ ቊ ଵܷ
௟
ܨଵ௟ − ଴݂ቋ, (10)
ቊ ଵܷ
௟
ܨଵ௟ ቋ = ൝
− ଶܶଶ
ଶܶଵ
଴݂
0
ൡ. (11)
The force transmissibility through the thrust bearing foundation could be easily obtained by 
ܨ௕௥/ ଴݂. 
3. Discussions of results 
The corresponding configuration data of the thrust bearing are load ܨ = 200 kN, shaft speed 
݊ = 220 r/min, ݎଶ = 0.24 m, ݎଵ = 0.125 m, ݉ = 8, ߠ଴ = 0.611 rad, ݎ௣ = 0.186 m, ߠ௣ = 0.3543 rad. 
The corresponding configuration data of the shaft are ݈ଵ = 14 m, ݈ଶ = 2 m, the diameter of the 
shaft ݀ = 0.16 m, ܯ௣ = 7000 kg, ܯ௖ = 1000 kg, ߩ௦ = 760 kg/m3, ܧ = 200 GPa. The 
corresponding configuration data of the RC are ݀଴ = 60 mm, ݀ଵ = 5 mm, ଵܸ = 1.6 L, ܮଵ = 2 m, 
ߩଵ = 760 kg/m3, ܤଵ = 1940 MPa, ߤଵ = 0.23 pa·s. 
Fig. 4 is the contrast figure of the force transmissibility through the thrust bearing foundation 
before and after a RC is attached to the thrust bearing. Fig. 4(a), Fig. 4(b), Fig. 4(c) and Fig. 4(d) 
exhibit the influences of the volume of the oil reservoir ଵܸ, the diameter of the piston cylinder ݀଴, 
the length ܮଵ and diameter ݀ଵ of the oil pipe, respectively.  
Compared with the situation in absence of the RC, the introduction of the RC results in the 
elimination of the dominant peak at the original 1st natural frequency and alteration of the 2nd 
natural frequency much more towards the high frequency direction. Simultaneously, it also 
introduces another two dominant resonance peaks around the original 1st natural frequency. The 
amplitudes and locations of the two peaks are related to the different parameters of RC, but both 
of the amplitudes are lower than the original one. Off resonance frequency, there is little difference 
between the situations with and without a RC, which proves that the RC only exhibits good 
performance over a narrow frequency bandwidth. 
As shown in Fig. 4(a), with the increasing of ଵܸ , ܭ௛  and ܨ௛  decrease while ܯ௛ , ܥ௛  stays 
invariable. The peak on the right side moves towards the original 1st natural frequency and the 
amplitude of the peak increases. As shown in Fig. 4(b), with the increasing of ݀଴, ܯ௛, ܭ௛ and ܥ௛ 
decrease while ܨ௛ stays invariable. The two peaks move together towards the original 1st natural 
frequency and the amplitudes of the two peaks stay nearly equal and decrease gradually. As shown 
in Fig. 4(c), with the increasing of ܮଵ , ܯ௛  and ܥ௛  increase and ܨ௛  decreases while ܭ௛  stays 
invariable. The peak on the right side moves towards the original 1st natural frequency and the 
amplitude of the peak increases. As shown in Fig. 4(d), with the increasing of ݀ଵ, ܯ௛ and ܥ௛ 
decrease and ܨ௛ increases while ܭ௛ stays invariable. The peak on the right side moves towards the 
high frequency direction and the amplitude of the peak decreases. 
From above analysis, it is obvious that the peak on the right side is attributed to the RC, while 
the peak on the left side is due to excitation of the original shafting system. It could be concluded 
that, with the decreasing of ܨ௛ (increasing of ܯ௛ or decreasing of ܭ௛), the influences of the RC 
on the longitudinal vibration of the shafting system is more and more significant. And that is why 
the two peaks moves and changes synchronously when ܨ௛ stays invariable. It could be foreseen 
STUDY OF THE INFLUENCE OF THE RESONANCE CHANGER ON THE LONGITUDINAL VIBRATION OF MARINE PROPULSION SHAFTING SYSTEM.  
ZHENGMIN LI, LIN HE, HANGUO CUI, JIANGYANG HE, WEI XU 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533 69 
that, when ଵܸ is small enough, ܭ௛ will be infinite, so the RC could be regarded as a rigid support. 
As a result, the peak on the left side overlaps the original peak at 1st natural frequency, while the 
peak on the right side disappeared. When ܮଵ is long enough or ݀ଵ is small enough, ܯ௛ will be 
infinite. As a result, the peak on the right side will overlap the original peak at 1st natural  
frequency, while the peak on the left side disappeared. Be careful, because of the limitation of the 
axial shaft movement, ܭ௛ should not be too small. 
Considering the mass, stiffness and the 1st natural frequency of the shafting systems, 
optimizing ଵܸ, ݀଴, ܮଵ, ݀ଵ to acquire a natural frequency of the RC which is close to the original 
1st natural frequency of the shafting system, a appropriately large virtual mass, a appropriately 
small virtual stiffness and a appropriate virtual damping, there will be a good effect of vibration 
reduction when the RC is attached to the shafting system. 
 
a) Variations in the volume of the oil reservoir 
 
b) Variations in the diameter of the piston cylinder 
 
c) Variations in the length of the oil pipe 
 
d) Variations in the diameter of the oil pipe 
Fig. 4. Effect of RC with variations in designing parameters on force transmissibility 
4. Conclusions 
In this paper, force transmissibility through the thrust bearing foundation for different cases is 
employed to study the dynamic behavior of the RC as well as its influences on the longitudinal 
vibration of the propulsion shafting system. The conclusions can be summarized as follows: 
The introduction of the RC eliminates the resonance at low frequency suffered previously and 
introduces another two resonance peaks around original 1st natural frequency, as well as 
significantly lowering the maximum response compared with the original peak. The peak on the 
right side is attributed to the RC, while the other is due to excitation of the other shafting system 
components. Furthermore the resonance frequency of the 2nd natural frequency moves towards 
the high frequency direction and the peak value nearly stay invariable. Such variations in force 
transmissibility characteristics are linked with the variation of the designing parameters of the RC, 
STUDY OF THE INFLUENCE OF THE RESONANCE CHANGER ON THE LONGITUDINAL VIBRATION OF MARINE PROPULSION SHAFTING SYSTEM.  
ZHENGMIN LI, LIN HE, HANGUO CUI, JIANGYANG HE, WEI XU 
70 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2015, VOLUME 5. ISSN 2345-0533  
such as the diameter of the piston cylinder, the diameter of the oil pipe, the length of the oil pipe, 
the volume of the oil reservoir. By optimizing above parameters, the purpose of reducing severe 
longitudinal vibration and its transmission into the hull through a thrust bearing over a specified 
frequency range would be realized. 
References 
[1] Zhang G. B., Zhao Y., Li T. Y. Propeller excitation of longitudinal vibration characteristics of marine 
propulsion shafting system. Shock and Vibration, 2014, p. 1-19. 
[2] Pan J., Farag N., Lin T., Juniper R. Propeller induced structural vibration through the thrust bearing. 
Proceedings of Acoustics, Australia, 2002, p. 390-399. 
[3] Dylejkoa P. G., Kessissogloua N. J., Tsob Y., Norwoodb C. J. Optimization of a resonance changer 
to minimize the vibration transmission in marine vessels. Journal of Sound and Vibration, Vol. 300, 
Issues 1-2, 2007, p. 101-116. 
[4] Goodwin A. J. H. The design of a resonance changer to overcome excessive axial vibration of 
propeller shafting. Transactions of Institute of Marine Engineers, Vol. 72, 1960, p. 37-63. 
[5] Shapiro W., Graham R., Anderson G. Predicted performance characteristics of hybrid 
fluid-equalized tilting-pad thrust bearings. Journal of Lubrication Technology, Vol. 105, 1983, 
p. 476-483. 
[6] Jang G. H., Kim Y. J. Calculation of dynamic coefficients in a hydrodynamic bearing considering 
five degrees of freedom. Journal of Tribology, Vol. 121, 1999, p. 499-505. 


Study of the influence of the resonance changer on the longitudinal vibration of marine propulsion shafting system

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Study of the influence of the resonance changer on the longitudinal vibration of marine propulsion shafting system

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