chinese journal of ship research vol.14 no

CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.1，FEB 2019

Received：2017 - 10 - 02Supported by: Fund Project of National Key Laboratory on Ship Vibration & Noise (9140C280301)Author(s): Hu Zechao, male, born in 1991, Ph.D. candidate. Research interest: vibration noise control of ship power plant.

E-mail: [email protected] Lin, male, born in 1957, professor, doctoral supervisor, academician of Chinese Academy of Engineering. Researchinterest: ship stealth technology. E-mail: [email protected] Wei, male, born in 1980, Ph.D., associate professor. Research interest: vibration noise control of ship power plant.E-mail: [email protected]

*Corresponding author：Xu Wei

To cite this article：Hu Z C, He L, Xu W, et al. Optimization design of resonance changer for marine propulsion shafting inlongitudinal vibration[J/OL]. Chinese Journal of Ship Research, 2019, 14(1). http://www.ship-research.com/EN/Y2019/V14/I1/107.

DOI：10.19693/j.issn.1673-3185. 01077

Optimization design of resonancechanger for marine propulsionshafting in longitudinal vibration

Hu Zechao1，2，He Lin1，2，Xu Wei*1，2，Li Zhengmin1，2，Zhao Xingqian1，2

1 Institute of Noise & Vibration，Naval University of Engineering，Wuhan 430033，China2 National Key Laboratory on Ship Vibration & Noise，Wuhan 430033，China

Abstract：［Objectives］The transmission path of the longitudinal vibration of propeller shafting can be changed andthe base response attenuated via a Resonance Changer（RC） integrated in the thrust bearing，enabling the naturalfrequency of the shafting to be avoided by the propeller's blade frequency and multiplier frequency excitation force. Inthis way，the purposes of vibration reduction and frequency adjustment can be achieved.［Methods］In this paper，the mechanical model of longitudinal vibration of propeller shafting is established and the vibration model of thepropeller shaft system is calculated on the basis of the transfer matrix method. The influence of the main parameters ofthe RC on the vibration isolation effect of propeller shafting is analyzed by taking the force transmission rate as theindex. The methods of the minimization of maximum value and parameter correction of the curve area are used tooptimize the main parameters of the RC.［Results］The results show that the vibration isolation effect of propellershafting is significantly improved when the RC is installed，and the vibration reduction effect of the RC can beimproved by using the design method of the parameter correction of the curve area.［Conclusions］The rational designof the RC's parameters can produce a vibration isolating system with excellent vibration isolation effects.Key words：Resonance Changer（RC）；propulsion shafting；transfer matrix method；longitudinal vibrationCLC number: U664.2

0 Introduction

Under the uneven wake field, the pulse excitationforce generated by the periodic operation of the pro⁃peller is the main noise source when the ship is sail⁃ing at a medium or high speed. The longitudinal exci⁃tation force is transmitted to the hull by the thrustbearing, which causes the vibration of the shaftingand the hull, affects the operation safety of the ship,and reduces the acoustic performance of the hull. Inorder to reduce the transmission of the longitudinalexcitation force to the hull, a shock absorber can beinstalled on the shafting. Considering that the trans⁃mission of thrust bearing has characteristics of large

thrust and small displacement, it is necessary to de⁃sign a vibration isolating device with low stiffnessand high resistance. The hydraulic damping deviceadjusts the stiffness and damping of the systembased on the compressibility of fluid. Through ratio⁃nal design, the dynamic characteristics of propulsionshafting can meet requirements. The installation ofResonance Changer (RC) at the thrust bearing not on⁃ly can adjust the longitudinal natural frequency ofpropulsion shafting to deviate from the pulse excita⁃tion frequency of the propeller, but also can reducethe longitudinal vibration response of the stern of thehull to achieve vibration isolation. Goodwin [1] be⁃lieved that RC can be equivalent to a

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mass-spring-damper unit, and designed a hydraulicdamping device which can reduce the longitudinal vi⁃bration of shafting in a specific frequency band.Dylejko et al. [2] and Li et al. [3] used the transfer ma⁃trix method to establish a mathematical model of thepropeller-shaft-hull system, and analyzed the influ⁃ence of the main parameters of RC on the force trans⁃mission rate of propeller shafting. Li et al. [4] andWang et al. [5] optimized the RC design using the dy⁃namic harmonic vibration elimination theory, and ob⁃tained the optimal natural frequency ratio and damp⁃ing ratio of RC. However, the analysis model is sim⁃ple, and the application range is limited.

In this study, the mechanical model of longitudi⁃nal vibration of propeller shafting is proposed, andthe vibration response of propeller excitation forcetransmitted to shell is calculated by the transfer ma⁃trix method. With the force transmission rate as theindex, the influences of the piston cylinder diameterd0 , connecting pipe length l1 , connecting pipe diam⁃eter d1 and tank volume V1 on the vibration isola⁃tion effects of the propeller shafting are analyzed.Moreover, the main parameters of RC are optimizedusing the method of the minimization of maximumvalue of force transfer rate and the correction methodof minimum area enclosed by the force transfer rateand coordinate axis respectively.1 Dynamic model of RC

RC consists of a tank filled with oil, an externalpipe system and a piston cylinder. The working fluidin the device can change the longitudinal stiffnessand damping of the shafting [6]. Fig. 1 shows the prin⁃ciple model of RC, in which P indicates the pres⁃sure difference between the two sides of the piston,and x0 and x1 indicate the displacements at bothends of the piston cylinder.

In order to facilitate the derivation of the dynamicequation of RC, the following assumptions should bemade [7]:

1) The tank wall is rigid and all the compressionof the fluid occurs in the tank.

2) The fluid in the connecting pipe is in a laminarflow state.

3) The fluid in the pipe can be regarded as thelumped mass.

4) The hydraulic effective length of the fluid isequal to the actual length of the connecting pipe.

5) Compression effect in the pipeline is not consid⁃ered.

According to these assumptions and the D'Alem⁃bert principle, it can be known that the pressure act⁃ing on the connecting pipe in the piston cylinder isequal to the sum of the inertia force of the oil in thepipe, the damping force in the pipe and the force re⁃quired to compress the oil in the tank. Then the dy⁃namic equation of RC can be described as follows:

A1P = ρ1 A1l1

(x1 - x0)A0

A1

+ 8πμ1l1

(x1 - x0)A0

A1

+

A1B1

(x1 - x0)A0

V1

（1）Where A0 = π(

d0

2)2 and A1 = π(

d1

2)2 respectively in⁃

dicate the sectional areas of piston cylinder and con⁃necting pipe; B1 indicates the bulk modulus of theoil; μ1 and ρ1 respectively indicate the viscosityand density of the oil; x0 , x0 and x1 , x1 respective⁃ly indicate the speeds and accelerations at both endsof the piston cylinder. The Eq. (2) can be obtainedby multiplying the two ends of Eq. (1) by A0 /A1 .

A0 P =ρ1 A2

0l1

A1

(x1 - x0) +8πμ1l1 A2

0

A21

(x1 - x0) +

A20 B1

V1

(x1 - x0) （2）Assuming that

Mh =ρ1 A2

0l1

A1

Kh =8πμ1l1 A2

0

A21

Ch =A2

0 B1

V1

F0 = A0 P

Where Mh ，Kh and Ch represent the mass, stiff⁃ness and damping of RC, respectively, Eq. (2) can beconverted into a mathematical model ofmass-spring-damping:

F0 = Mh(x1 - x0) + Ch(x1 - x0) + Kh(x1 - x0) （3）Where F0 refers to the external force acting on thepiston.2 Mathematical model of longitu-

dinal vibration of propellershafting

The mechanical model of longitudinal vibration of

Fig.1 Structure diagram of RC

l1

d0

d1

x1

x0x0

P

V1

Piston cylinder Connecting pipe

PistonOil Tank

Hu Z C, et al. Optimization design of resonance changer for marine propulsion shafting in longitudinal vibration 79


CHINESE JOURNAL OF SHIP RESEARCH，VOL.14，NO.1，FEB 2019propeller shafting is shown in Fig. 2. The model canbe decomposed into five subsystems, and each sub⁃system can use the transfer matrix to represent thetransfer relation of longitudinal vibration at the leftand right ends of the element. In Fig. 2, the sub⁃scripts p, t, c, b, h respectively indicate the propel⁃ler, thrust collar, coupling, base and RC; Mp ，M t ，

M b ，Mc respectively indicate the mass of the pro⁃peller, thrust collar, base and coupling; K0 and Kb

respectively indicate the stiffness of oil film andbase; C0 indicates the damping of oil film; L s and

L se respectively indicate the actual length and the ef⁃fective length of stern shaft; and L indicates thelength of intermediate shaft. Units 1-5 are respec⁃tively located from the propeller to the coupling. Ti

( i = 1, 2, 3, 4, 5) represents its corresponding unittransfer matrix; U L

j and U Rj are respectively the

displacement response of the left and right end facesof unit j; F L

j and F Rj are respectively the force re⁃

sponse of the left and right end faces of unit j; andsubscript j can be replaced by b, h, c, t, p.

PropellerT1

Stern shaftT2 T3

Integrated RC thrust bearingT4 T5

Intermediate shaft Coupling

U Lj U R

j

Mp

L s

L seF L

j F Rj

M t

U Rt

F Rt

U Lh

F Lh

U Rh

F Rh

U Lb

F Lb

U Rb

F Rb

Thrustcollar

Oil film RC Base Shell

K0Kh

ChMh

C0

MbKb

Mb

Mc

L

Fig.2 Mechanical model of longitudinal vibration of propeller shafting

2.1 Transfer matrix of longitudinal vi-bration points of shafting

Considering the effect of the propeller pulse exci⁃tation force Fp , the longitudinal transfer matrix ofpropeller shafting should be rewritten to the form ofT3 ´ 3 , and Eqs. (4)-(7) represent the transfer matrixof each subsystem of propeller shafting:

T1 =é

ë

êê

ù

û

úú

1 0 0-Mpω

2 1 Fp

0 0 1

（4）T2 =

é

ë

ê

ê

êêêê

ê

êù

û

ú

ú

úúúú

ú

újω cos k(L s - L se) jωcos k(L s - L se) - cos kLs cos kLse

EAk sin kLs

0

-EAk sin kLs

cos k(L s - L se)cos kL se 0

0 0 1

（5）T4 =

é

ë

êêêê

ù

û

úúúú

cos kL sin kLEAk

0

-EAk sin kL cos kL 00 0 1

（6）

T5 =é

ë

êê

ù

û

úú

1 0 0-Mcω

2 1 00 0 1

（7）Where k = ω c refers to the longitudinal wave num⁃ber of the shafts (the material properties and section⁃al areas of stern shaft and intermediate shaft are thesame), in which ω refers to the angular frequency

and c = E ρ ( ρ is the density of shafts) refers tothe longitudinal wave velocity of the shafts. Besides,E and A refer to the elastic modulus and sectionalarea of stern shaft and intermediate shaft, respective⁃ly. Propeller and coupling can be considered as thelumped mass blocks. Due to the long stern shaft, theeffective length should be generally taken into ac⁃count during calculation, and the correspondingtransfer matrix is T2 .

The integrated RC thrust bearing can be furtherdecomposed into three units: thrust collar, oil filmand RC. The transfer matrix equation from the rightend face of thrust collar to the hull can be expressedas Eq. (8):

é

ë

ê

êêê

ù

û

ú

úúú

U Rb

F Rb

1

=é

ë

ê

êêê

ù

û

ú

úúú

1 1Kb

0

0 1 00 0 1

é

ë

ê

êêê

ù

û

ú

úúú

1 0 0

-M bω2 1 0

0 0 1

é

ë

ê

êêê

ù

û

ú

úúú

1 1-Mhω

2 + Kh + jωCh

0

0 1 00 0 1

é

ë

ê

êêê

ù

û

ú

úúú

1 1K0 + jωC0

0

0 1 00 0 1

é

ë

ê

êêê

ù

û

ú

úúú

U Rt

F Rt

1（8）

The shell connected with the base has a large stiff⁃ness, which can be regarded as the stiffness bound⁃ary condition of propeller shafting (namely U R

b = 0) [8].Then the relationship between F R

t and U Rt can be

deduced by substituting U Rb into Eq. (8):

80


ì

í

î

ïï

ïï

F Rt = KeU

Rt

Ke =(Kb - M bω

2)(K0 + jωC0)(-Mhω2 + Kh + jωCh)

(Kb - M bω2 + K0 + jωC0)(-Mhω

2 + Kh + jωCh) + (Kb - M bω2)(K0 + jωC0)

（9）

Where Ke refers to the equivalent stiffness betweenthrust collar and shell.

According to Eq. (9), the thrust bear⁃ing-base-shell model can be simplified to the equiv⁃alent mechanical model shown in Fig. 3, and thetransfer matrix T3 corresponding to the integratedRC thrust bearing can be simplified to Eq. (10):

T3 =é

ë

êê

ù

û

úú

1 0 0-M tω

2 + Ke 1 00 0 1

（10）

2.2 Calculation of longitudinal vibrationresponse of shafting

After the transfer matrices of T1 - T5 are ob⁃tained, the vibration response of propeller shaftingcan be solved according to the boundary conditions.Eq. (11) refers to the transfer matrix from the inputend of the propeller to the output end of the cou⁃pling. In the equation, Trq represents the elementof T in the r-th row and the q-th column (1≤r, q≤3). When pulse excitation force Fp acts on the pro⁃peller, the left end face of propeller and the right endface of coupling can be regarded as the free bound⁃ary conditions (namely, F L

1 = F R5 = 0 ), and Eq. (12)

can be obtained by substituting the constraint condi⁃tions into Eq. (11):

T = T5T4T3T2T1 （11）U L

1 = -T23

T21

Fp （12）Eq. (13) and Eq. (14) indicate the transfer matrix

T ′ of the input end of propeller from base to shell,where T ′rq represents the element of T ′ in the r-throw and q-th column (1≤r, q≤3). After Eq. (14) issubstituted into Eq. (9), the response force Fb of theexcitation force transmitted to the shell by the propel⁃ler shafting is obtained, from which the force trans⁃mission rate Tf = Fb Fp of propeller shafting can becalculated.

T ′ = T3T2T1 （13）U R

t = T ′11UL

1 + T ′13 Fp （14）Fb = F R

t = KeUR

t （15）3 Optimization design method of

RC structure

In order to make the propeller shafting have agood vibration isolation effect, it is necessary to de⁃sign the parameters of RC. The rational design ofstructure parameters enables the RC device to ab⁃sorb most of the vibration energy of propeller shaft⁃ing, which not only limits the force transmission rateto a certain range and makes the resonance peak be⁃come smaller, but also adjusts the natural frequencyof propeller shafting, so as to avoid the propeller'sblade frequency excitation and achieve the purposeof vibration reduction and frequency adjustment. Bytaking the parameters of l1 ，d0 ，d1 , V1 of RC asthe design objectives, we optimize the RC structureusing the method of the minimization of maximumvalue of force transmission rate [9] and the correctionmethod of the minimum curve area of force transmis⁃sion rate [10].3.1 Optimization method of minimiza-

tion of maximum value

The optimization of RC structural parameters canbe expressed as the design problem of minimizingthe maximum value of peak value of the force trans⁃mission rate curve, namely that through some algo⁃rithm, a set of design variables (l1d0d1V1) aresearched to minimize the multiple peak values ofT a

f (l1d0d1V1) of force transmission rate of propel⁃ler shafting within the analysis frequency range. Thisoptimization method can be described as Eq. (16):

obj1 = min{max1 a m

T af (l1d0d1V1)} （16）

Where a represents the order of longitudinal mode;m represents the number of peaks of the force trans⁃mission rate in the analyzed frequency band.

The optimization method of minimization of maxi⁃mum value focuses more on the variation of thepeaks of force transmission rate, and is suitable forthe optimization of constraint conditions with strictlimits on the magnitude of forces transmitted to theshell or on the strength of shafting. Therefore, it is es⁃sentially a local optimization method.

Fig.3 Equivalent mechanical model of thrust bearing and base

U RtU L

t

F Lt F R

t

Thrust collar

Equivalent spring

Ub

Ke

Fb

Shell




3.2 Minimum area correction method

For each set of design variables (l1d0d1V1) thearea enclosed by the force transmission rate curveTf (l1d0d1V1) and the coordinate axis f is denotedas Aa

f (l1d0d1V1) and the method can be describedas Eq. (17).

obj2 = min1 a m

Aaf (l1d0d1V1) （17）

In other words, for the minimum value of Aaf (l1

d0d1V1) in the whole variable range, the corre⁃sponding (l1d0d1V1) is searched. This method opti⁃mizes the force transmission rate curve in the entireanalyzed frequency band. Although the local peakvalues of the curve may be increased, the overall vi⁃bration isolation effect of propeller shafting can beimproved in the analyzed frequency band, so it is aglobal optimization method.4 Case analyses

Taking the propeller shafting of a certain type ofship as an example, the calculated parameters are asfollows: Mp = 7 000 kg, Mc = 1 000 kg, M b = 4 000 kg,M t = 500 kg, ρ1 = 860 kg/m3, ρ = 7 850 kg/m3, E =200 GPa, A = 0.02 m2, L s = 14.6 m, L se = 14 m,L = 2 m, Kb = 5 × 109 N/m, B1 = 1.38 GPa, μ1 =0.23 Pa·s, and number m of propeller blades being 7.

According to the experimental data, the oil filmstiffness K0 and the damping C0 are related to therotating speed n and the load F acting on the pro⁃peller. When F = 200 kN, the variation curve of K0

and C0 with the rotating speed is shown in Fig. 4.Assuming that n = 220 r/min (the correspondingblade frequency is 25.7 Hz), the corresponding K0 =1.4 × 1010 N/m, and C0 = 6.5 × 108 N·s/m. The pre⁃liminarily designed RC parameters are l1 = 1 m,d0 = 0.06 m, d1 = 0.01 m and V1 = 1.6 L.

The influence of RC parameters ( l1 ，d0 ，d1 ，V1 )

on vibration isolation effect of propeller shafting isshown in Fig. 5. This figure shows that when RC isnot installed, the resonance peaks of first two ordersof propeller shafting appear near the 30 and 132 Hz,and the first-order resonance peak is large. Com⁃pared with the force transmission rate curve in thecase without installing the RC system, the first-ordermode peaks of propeller shafting after the install⁃ment of RC are converted into the low order modeswith smaller second-order peaks, which are respec⁃tively located at both sides of the first-order modefrequency when the RC system is not installed. Obvi⁃ously, in the analyzed frequency band of 0-200 Hz,RC has a great influence on the modes of the firsttwo orders in propeller shafting.

According to the variation trends of the Mh ，Kh ，

Ch , ωh frequencies of RC with the l1 ，d0 ，d1 ，V1 ,the following conclusions can be drawn:

1) As l1 increases, the RC equivalent damping Ch

increases, and the peak of the force transmission rateof propeller shafting decreases accordingly. If l1 istoo long, the mass of RC will increase. If l1 is tooshort, the peak of the first-order mode of propellershafting is large. Therefore, the value range of l1

must be constrained, and it can be constrained asl1Î[0.5 m 10 m] .

Fig.4 Variation curve of oil film stiffness and damping withrotating speed

1.61.41.21.00.80.60.40.2

0

K0

/（N·m

-1 ）

20 40 60 80 100 120 140 160 180 200 220Rotating speed n/（r·min-1）

876543210

C0

/（N·s·

m-1 ）

×1010 ×108

Oil film stiffnessOil film damping

（a）Connecting pipe length l1

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200f/Hz

2.52.01.51.00.50-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0-4.5

Tf

/dB

Without RCl1 =1 m，Mh =5 602 kg，Kh =4.4×108 N/m，Ch =1.5×105 N·s/m，ωh =44.7 Hzl1 =5 m，Mh =28 012 kg，Kh =4.4×108 N/m，Ch =7.5×105 N·s/m，ωh =20.0 Hzl1 =10 m，Mh =56 024 kg，Kh =4.4×108 N/m，Ch =1.5×106 N·s/m，ωh =14.1 Hz

82


2) When ωh is constant, as d0 increases, Kh

and Mh increase at the same proportion. If d0 ®¥

RC can be regarded as a rigid body, and the forcetransmission rate of RC will be close to the case with⁃out RC. Therefore, the value of d0 should not be toolarge when the RC is designed. To achieve good vi⁃bration isolation effect, we can set d0Î[d1 0.1 m] .

3) When Kh is constant, the equivalent mass Mh

of RC increases as d1 decreases. When RC is de⁃signed, its equivalent mass should not be designedas too large, and we can set d1Î[ ]0.005 m d0 .

4) When Mh is constant, the increase of V1 isequivalent to the thickening of oil layer in the oiltank, the stiffness Kh of RC will decrease according⁃ly, and the vibration isolation effect will be en⁃

hanced. However, the excessive V1 will cause thedisplacement response of the coupling end to exceedits allowable range and thus affect the normal opera⁃tion of the motor, while the too small V1 will makethe force transmission rate curve close to the curvewithout RC, resulting in RC failure. In order to avoidthe above two phenomena, we can set V1Î[0.8 L

2.4 L] .After the constraint conditions of RC parameters

are determined, the objective functions obj1 andobj2 are used to optimize RC parameters, and the op⁃timization results are shown in Fig. 6. The followingscan be seen from the figure.

1) The optimized force transmission rate of RC issignificantly reduced in the analyzed frequency band

（b）Piston cylinder diameter d0

（c）Connecting pipe diameter d1

（d）Oil tank capacity V1

Fig.5 Influence of RC's main parameters on vibration isolation effect of propeller shafting system

2.52.01.51.00.50-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0-4.5

Tf

/dB

2.52.01.51.00.50-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0-4.5

Tf

/dB

2.52.01.51.00.50-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0-4.5

Tf

/dB

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200f/Hz

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200f/Hz

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200f/Hz

Without RCd0 =0.04 m，Mh =1 106 kg，Kh =8.7×107 N/m，Ch =3.0×104 N·s/m，ωh =44.7 Hzd0 =0.06 m，Mh =5 602 kg，Kh =4.4×108 N/m，Ch =1.5×105 N·s/m，ωh =44.7 Hzd0 =0.10 m，Mh =43 228 kg，Kh =4.4×109 N/m，Ch =1.2×106 N·s/m，ωh =44.7 Hz

Without RCd1 =0.005 m，Mh =22 410 kg，Kh =4.4×108 N/m，Ch =2.4×106 N·s/m，ωh =22.3 Hzd1 =0.01 m，Mh =5 602 kg，Kh =4.4×108 N/m，Ch =1.5×105 N·s/m，ωh =44.7 Hzd1 =0.02 m，Mh =1 401 kg，Kh =4.4×108 N/m，Ch =9.4×103 N·s/m，ωh =89.3 Hz

Without RCV1 =0.8 L，Mh =5 602 kg，Kh =8.8×108 N/m，Ch =1.5×105 N·s/m，ωh =63.2 HzV1 =1.6 L，Mh =5 602 kg，Kh =4.4×108 N/m，Ch =1.5×105 N·s/m，ωh =44.7 HzV1 =2.4 L，Mh =5 602 kg，Kh =2.9×108 N/m，Ch =1.5×105 N·s/m，ωh =36.5 Hz




compared with that of the initially designed RC, andthe peaks in the modes of the first three orders are re⁃duced.

2) The method of the minimization of maximumvalue can minimize the peaks of the force transmis⁃sion rate, but cannot guarantee that RC has excellentvibration isolation performance in the entire ana⁃lyzed frequency band.

3) The minimum area correction method can makeRC have the best vibration isolation performance inthe analyzed frequency band, but it does not rule outthat there are large peaks at some frequencies.

For this example, according to the optimization re⁃sults, the objective function obj2 should be used tooptimize the RC parameters. Although the peaks ofthe force transmission rate curve is slightly increasedrelative to obj1, the vibration isolation effect of theoptimized RC in the frequency range of 0– 200 Hzis obviously improved. Compared with the initial de⁃sign, the optimization method of obj2 reduces thepeaks at the first-order mode by about 5.3 times,and avoids the blade frequency excitation force of25.7 Hz, which obtains an evident effect of vibrationreduction and frequency adjustment, and satisfiesthe use requirements.5 Conclusions

In this paper, the mathematical model of longitudi⁃nal vibration of propeller shafting is established, andthe influence of RC parameters on the vibration isola⁃tion performance of propeller shafting is analyzed us⁃ing the transfer matrix method with the force trans⁃mission rate as the index and based on the actualship data. The constraint conditions of design param⁃eters are given by theoretical analysis, and the RCstructure is optimized based on the method of theminimization of maximum value and the minimumcurve area correction method. The following conclu⁃sions are obtained:

1) The introduction of RC eliminates the first-or⁃der mode of propeller shafting, but brings the low or⁃der modes with smaller second-order peaks, whichare respectively located on both sides of the first-or⁃der mode frequency in the case without RC. The sec⁃ond-order mode frequency moves in the high-fre⁃quency direction, but the peak does not changemuch and has no effect on the third-order mode.

2) After the installation of RC, the vibration isola⁃tion effect of the shafting has been significantly im⁃proved.

3) RC parameters have great influence on the vi⁃bration isolation performance of propeller shafting,and the rational design of RC structure can make thesystem obtain a good vibration isolation effect.

4) Compared with the initial design scheme of RC,both the optimization methods can improve the vibra⁃tion isolation performance of propeller shafting andkeep its natural frequency away from the propeller'sblade frequency and multiplier frequency excitationforce. The optimization method of the minimizationof maximum value can minimize the peaks of theforce response transmitted to the shell, and the de⁃sign method of the minimization of curve area canmake the vibration reduction effect of the RC devicebetter in the whole analyzed frequency band.References［1］ Goodwin A J H. The design of a resonance changer to

overcome excessive axial vibration of propeller shafting［J］. Transactions of the Institute of Marine Engineers，1960，72：37-63.

［2］ Dylejko P G，Kessissoglou N J，Yan T，et al. Optimi⁃zation of a resonance changer to minimize the vibrationtransmission in marine vessels［J］. Journal of Soundand Vibration，2007，300（1/2）：101-116.

［3］ Li Z M，He L，Cui H G，et al. Study of the influenceof the resonance changer on the longitudinal vibrationof marine propulsion shafting system［C］//InternationalConference on Vibroengineering. Nanjing， China：

Fig.6 Optimal results for RC parameters

2.52.01.51.00.5

0-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0-4.5

Tf

/dBobj1：（ l1 , d0 , d1 , V1）=（1.994 2 m，0.041 7 m，0.005 m，0.8 L）

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200f/Hz

Without RCInitial design Mh =5 602 kg，Kh =4.4×108 N/m，Ch =1.5×105 N·s/m，ωh =44.7 Hzobj1 Mh =10 427 kg，Kh =2.1×108 N/m，Ch =1.1×106 N·s/m，ωh =22.4 Hzobj2 Mh =2 374 kg，Kh =9.9×107 N/m，Ch =2.6×105 N·s/m，ωh =32.6 Hz

obj2：（ l1 , d0 , d1 , V1）=（0.500 4 m，0.040 7 m，0.005 m，1.5 L）

84


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船舶推进轴系纵向振动共振转换器的优化设计

胡泽超 1，2，何琳 1，2，徐伟*1，2，李正民 1，2，赵兴乾 1，2

1 海军工程大学振动噪声研究所，湖北武汉 4300332 船舶振动噪声重点实验室，湖北武汉 430033

摘要：［目的目的］在推力轴承上集成共振转换器（RC）可以改变轴系纵向振动的传递路径，衰减传递到基座的响

应使轴系的固有频率避开螺旋桨叶频及其倍叶频激励力，从而实现减振、调频的目的。［方方法法］为此，建立推

进轴系纵向振动的力学模型，基于传递矩阵法计算桨轴系统的振动响应，以力传递率为指标，分析 RC的主要参

数对推进轴系隔振效果的影响，分别采用最大值最小化方法和曲线面积最小的参数修正方法，对 RC 的主要参

数进行优化设计。［结果结果］研究结果表明：加装 RC 后，轴系的隔振效果得到了明显的改善，采用曲线面积最小修

正的优化设计方法可使 RC 的减振调频效果更佳。［结论结论］通过对 RC 结构参数的合理设计能使减振系统获得优

良的隔振效果。

关键词：共振转换器；推进轴系；传递矩阵法；纵向振动



chinese journal of ship research vol.14 no

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