research article study on vibration of marine diesel...

Research ArticleStudy on Vibration of Marine Diesel-Electric HybridPropulsion System

Nengqi Xiao,1 Ruiping Zhou,1 Xiang Xu,2,3 and Xichen Lin1

1School of Energy and Power Engineering, Wuhan University of Technology, Wuhan 430063, China2College of Mechanical and Power Engineering, China Three Gorges University, Yichang 443002, China3State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Nengqi Xiao; [email protected]

Received 25 May 2016; Revised 29 September 2016; Accepted 11 October 2016

Academic Editor: Dane Quinn

Copyright © 2016 Nengqi Xiao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This study analyzes the characteristics of hybrid propulsion shafting and builds mathematical models and vibration equations ofshafting using the lumped parameter method.Main focus is on the asymmetric double diesel propulsion shafting operation processand the impact of the phase angle andmotor excitation on torsional vibration of shafting. Model result is validated by testing resultsconducted on double diesel propulsion shafting bench. Mathematical model and model-building methods of shafting are correct.

1. Introduction

With the rapid development of shipbuilding technology,marine power plant goes toward large-scale, complexity, andautomation development. As the provider of ship power, theship power plant plays an important role in the economy,maneuverability, safety, and reliability of the ship’s operation[1, 2]. With the introduction of the new standard of shipgas emission and ship vibration noise, the shipping indus-try has higher requirements for energy saving, emissionreduction, and low noise. The vibration of ship propulsionshafting directly affects the performance and safety of theship [3–5]. At present, the hybrid propulsion shafting iswidely used in various types of ships. Comparing with thetraditional propulsion shafting, hybrid propulsion shaftinghas multiple transmission branches of gearbox and multipleelastic coupling and the shaft. Its structure is more complex,and the shafting vibration is also more complex. The shiphybrid propulsion system is mainly composed of three parts,which are mechanical propulsion system, electric propulsionsystem, and integrated control system.

Ship propulsion shafting torsional vibration is one ofthe factors that affect the running safety and ship noiseand starts to receive extensive attention from the industry.Comparing with the current ship propulsion system, shippropulsion shafting structure is relatively simpler in the early

days. Shafting torsional vibration lumped parameter modelis usually simplified as straight branched-chain type. Straightbranched-chain type vibration model (undamped [6, 7] ordamped [8, 9]) is usually calculated by Holzer method [10,11]. With the development of shipbuilding technology, thepropulsion system of ship power plant is becoming moreand more complex, and the lumped parameter model thatdescribes the torsional vibration of the shaft system is inthe process of simplification. Torsional vibration model withbranching system has been solved with the system matrixeigenvalue extraction method or the trial-and-error searchmethod based on the transfer matrix [12–15]. With the con-tinuous improvement of the finite element software, the finiteelement method is used to calculate the torsional vibrationof the complex ship propulsion shafting [16, 17]. As theoperating conditions of hybrid propulsion system are morecomplex, the torsional vibrationmechanism ismore complex.At present, there are fewworks in literature about the researchon the torsional vibration of the hybrid propulsion systemin new ship propulsion plant. Hybrid propulsion system hasa more complex mathematical model of torsional vibration.The vibration excitation source includes the excitation torquegenerated by the gas pressure of the diesel engine, by thepropeller water power, by the electromagnetic field, and soon.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 8130246, 9 pageshttp://dx.doi.org/10.1155/2016/8130246

2 Mathematical Problems in Engineering

Table 1: Basic parameters of hybrid propulsion shafting.

Component Name Type Basic parameter3-cylinder diesel engine 1# TD226B-3C Power = 40Kw, speed = 1500 r/min, bore = 105mm, stroke = 120mm4-cylinder diesel engine 2# TD226B-4C Power = 60Kw, speed = 1500 r/min, bore =105mm, stroke = 120mmFlexible coupling HGTL1.8 Rated torque = 1.8 kNm, maximum torque = 4.5 kNm, relative damping = 1.1Gearbox 2GWH100 𝑖 = 1 : 1Motor shaft 2YVP200L Power = 30Kw, frequency = 50HzEddy current dynamometer DW250 Power = 250 kW

4-cylinder diesel engine 2#

Motor

GearboxShaft

dynamometerEddy current

engine 1#3-cylinder diesel

Figure 1: Three-dimensional model of hybrid propulsion shafting.

To sum up, this paper will study the characteristics of thehybrid propulsion system and its operating conditions andestablish a mathematical model of the torsional vibration ofthe hybrid propulsion system. Based on the mathematicalmodel of vibration, the free vibration and forced vibrationof the system are calculated using system matrix method.Through the research on the vibration mechanism and theexcitation torque of the hybrid propulsion system, the paperfocuses on the influence of the phase angle of twin-engineparallel operation and the electromagnetic excitation torqueon the propulsion shafting. By means of self-developedtest system and hybrid propulsion shafting test bench testand analysis, the correctness of the theoretical model andalgorithm is verified.

2. Establishment of Shafting TorsionalVibration Model and Equation

The hybrid propulsion shafting system is set up as follows:three-cylinder diesel engine and four-cylinder diesel engineare connected to electric eddy current dynamometer throughgearbox with transmission branch, the universal joint, anddrive shaft. Two diesel engines and the motor are connectedto the gearbox via the universal joint. The three-dimensionalmodel of the hybrid propulsion system is shown in Figure 1.The basic parameters of each component of the hybridpropulsion shafting are shown in Table 1.

The hybrid propulsion shafting has several operatingconditions as follows:

(1) Single machine condition: 3-cylinder diesel engine or4-cylinder diesel engine drives either eddy currentdynamometer or shaft generator or both of them.

(2) Twin diesel condition: 3-cylinder diesel engine andfour-cylinder diesel engine jointly drive either eddycurrent dynamometer or shaft generator or both ofthem.

(3) Single diesel hybrid conditions: 3-cylinder diesel en-gine or 4-cylinder diesel engine are combining withthe motor drive eddy current dynamometer.

(4) Twin diesel hybrid conditions: 3-cylinder diesel en-gine, 4-cylinder diesel engine, and the motor driveeddy current dynamometer.

2.1. Establishment of Shafting Vibration Model. Ship propul-sion shafting torsional vibration equivalent systemwas estab-lished by simplified principle and method of equivalentsystem. Each part of the mathematical model of torsionalvibration was modeled separately and then assembled. Thehybrid propulsion system can be simplified as a lumpedparameter torsional vibration model shown in Figure 2 andthe system equivalent parameters are shown in Table 2.The boundary condition of the mathematical model of thetorsional vibration of ship propulsion shafting is the sixdegrees of freedom of the two end points and the end pointof the branch in Figure 2. According to the length, diameter,and elastic modulus of the material, the stiffness of shaft canbe calculated in (1), but the stiffness of the elastic couplingis generally provided by the manufacturer. The dampingof the system can be divided into the external dampingproduced by the component and the external friction andthe hysteresis of the internal torsional deformation of thematerial in Figure 2. According to the specification, it ismainly to consider the external damping of the diesel enginemass point and propeller mass point as well as the internaldamping of the diesel engine crankshaft and the transmissionshaft in Table 2.The external damping coefficient of the dieselenginemass point and propellermass point is 0.0085 and 0.05in Figure 2. The internal damping coefficients of the dieselengine crankshaft and the drive shaft are 0.01 and 0.0063in Figure 2. The internal damping coefficient of the flexiblecoupling is provided by the manufacturer. In this paper, theinternal damping coefficient of elastic coupling is 1.13.

𝑘 = 𝜋32 𝐺𝐿 (𝐷4 − 𝑑4) , (1)

where 𝑘 is stiffness (Nm/rad). 𝐷 is external diameter (m).𝑑 is internal diameter (m). 𝐿 is length (m). 𝐷 is external

Mathematical Problems in Engineering 3

Table 2: Equivalent parameters of torsional vibration.

Number Inertia (kgm2) Stiffness (MNm/rad) Identified1 0.101 -2 0.0002 1.13 0.123 1.1 Cylinder 14 0.123 1.1 Cylinder 25 0.123 1.1 Cylinder 36 0.123 1.734 Cylinder 47 1.1134 - Flywheel8 0.1954 0.0141 Coupling9 0.1247 -10 0.0062 0.679311 0.0398 - Clutch12 0.0032 2.6209 Clutch13 0.0448 - Gear14 0.0462 2.5755 Gear15 0.0463 - Gear16 0.0492 - Gear17 0.0448 2.6209 Gear18 0.005 - Clutch19 0.0451 0.5898 Clutch20 0.0088 -21 0.02 0.0336 Shafting22 0.02 - Dynamometer23 0.11 - Gear24 0.0492 - Gear25 0.0463 - Gear26 0.0462 2.5755 Gear27 0.0448 - Clutch28 0.0032 2.6209 Clutch29 0.0398 -30 0.0062 0.6793 Coupling31 0.1247 -32 0.1954 0.0141 Flywheel33 1.1134 - Cylinder 334 0.123 1.734 Cylinder 235 0.123 1.1 Cylinder 136 0.123 1.137 0.0002 1.1 Gear38 0.0448 - Clutch39 0.002 2.6209 Clutch40 0.0408 -41 0.0164 0.5536 Motor

diameter (m). 𝐺 is shear modulus of materials (N/m2),nodular graphite cast iron𝐺 = 6.76659×1010 (N/m2), or steel𝐺 = 8.14993 × 1010 (N/m2).2.2. Establishment of Shafting Torsional Vibration Equa-tion. Hybrid propulsion shafting torsional vibration lumpedparameter model mainly includes straight branched-chain

type parts and branch type parts, as shown in Figures 3 and4.

For straight branched-chain type parts, differential equa-tion of motion of 𝑘 mass point can be obtained according tobasic vibration theory as follows:

𝐽𝑘��𝑘 + 𝐶𝑘��𝑘 + 𝐶𝑘−1,𝑘 (��𝑘 − ��𝑘−1) + 𝐶𝑘,𝑘+1 (��𝑘 − ��𝑘+1)+ 𝐾𝑘−1,𝑘 (𝜙𝑘 − 𝜙𝑘−1) + 𝐾𝑘,𝑘+1 (𝜙𝑘 − 𝜙𝑘+1)

= 𝑇𝑘 (𝑡) .(2)

Equation (2) can be converted into

𝐽𝑘��𝑘 + (𝐶𝑘−1,𝑘 + 𝐶𝑘,𝑘+1) ��𝑘 − 𝐶𝑘−1,𝑘��𝑘−1 − 𝐶𝑘,𝑘+1��𝑘+1+ (𝐾𝑘−1,𝑘 + 𝐾𝑘,𝑘+1) 𝜙𝑘 − 𝐾𝑘−1,𝑘𝜙𝑘−1 − 𝐾𝑘,𝑘+1𝜙𝑘+1+ 𝐶𝑘��𝑘 = 𝑇𝑘 (𝑡) .

(3)

For branch type parts, 𝑚 + 1 mass point’s motiondifferential equation is as follows:

𝐽𝑚+1��𝑚+1 + 𝐶𝑚,𝑚+1 (��𝑚+1 − ��𝑚)+ 𝐶𝑚+1,𝑚+2 (��𝑚+1 − ��𝑚+2)+ 𝐾𝑚,𝑚+1 (𝜙𝑚+1 − 𝜙𝑚) + 𝐾𝑚+1,𝑚+2 (𝜙𝑚+1 − 𝜙𝑚+2)+ 𝐶𝑚+1��𝑚+1𝐶𝑚+1,𝑛 (��𝑚+1 − ��𝑛)+ 𝐾𝑚+1,𝑛 (𝜙𝑚+1 − 𝜙𝑛) = 𝑇𝑚+1 (𝑡) .

(4)

Equation (4) can be converted into

𝐽𝑚+1��𝑚+1 + (𝐶𝑚,𝑚+1 + 𝐶𝑚+1,𝑚+2 + 𝐶𝑚+1,𝑛) ��𝑚+1− 𝐶𝑚,𝑚+1��𝑚 − 𝐶𝑚+1,𝑛��𝑛 − 𝐶𝑚+1,𝑚+2��𝑚+2+ (𝐾𝑚,𝑚+1 + 𝐾𝑚+1,𝑚+2 + 𝐾𝑚+1,𝑛) 𝜙𝑚+1− 𝐾𝑚,𝑚+1𝜙𝑚 − 𝐾𝑚+1,𝑚+2𝜙𝑚+2 − 𝐾𝑚+1,𝑛𝜙𝑛+ 𝐶𝑚+1��𝑚+1 = 𝑇𝑚+1 (𝑡) ,

(5)

where 𝐽𝑘 is 𝑘 mass point’s inertia (kg⋅m2). 𝐶𝑘 is 𝑘 masspoint’s damping coefficient (N⋅m⋅s/rad). 𝐶𝑘,𝑘+1 is internaldamping coefficient between 𝐾 mass point and 𝐾 + 1 masspoint (N⋅m⋅s/rad). 𝐾𝑘,𝑘+1 is stiffness between 𝐾 mass pointand 𝐾 + 1 mass point (N⋅m/rad). 𝑇𝑘(𝑡) is quality excitingmoment (N⋅m) acting on 𝑘 mass point. ��𝑘, ��𝑘, and 𝜙𝑘 aretorsional angular displacement, angular velocity, and angularacceleration of 𝑘 mass point, respectively.

According to (3) and (5), the differential equation offorced vibration of hybrid propulsion system is derived asfollows:

[𝐽] {��} + [𝐶] {��} + [𝐾] {𝜙} = {𝑇 (𝑡)} , (6)


1 23 4 5 6

7

8 9 1011

12

13

14

15

37 38

16

17

23

24

1819

20

26

25

27 2122

4-cylinder diesel engine

Flexible

Flexible

coupling

Gearbox

Shaft

Eddy current dynamometer

36

35 34 33 32

3-cylinder diesel engine

31 30 2928

coupling

3940 41

Motor

Figure 2: Lumped parameter model of hybrid propulsion shafting.

Jk−1 Jk Jk+1

Ck−1,k

Kk−1,k

Ck−1 Ck Ck+1

k − 1 k k + 1

Ck,k+1

Kk,k+1

Figure 3: Torsional vibration model of straight chain.

Table 3: Free vibration natural frequency.

Wn/harmonic 1 2 3 4 5r/min 834.9 1734.1 5046.4 10478.6 17314.0Hz 13.9 28.9 84.1 174.6 288.6

where inertia matrix [𝐽] is diagonal matrix. Stiffness matrix[𝐾] is three-diagonal matrix. Damping matrix [𝐶] is the sumof the internal damping matrix (three-diagonal matrix) andouter damping matrix (diagonal matrix). {𝑇(𝑡)} is vector ofthe excitation torque acting on the system.

According to (6) and the shafting torsional vibrationparameters shown in Table 2, the natural frequencies of thefree vibration system without damping can be obtained, asshown in Table 3.

Jm

Cm

m m + 1 m + 2

Jm+1Jm+2

Cm,m+1 Cm+1,m+2

Km,m+1 Km+1,m+2

Km+1,n

Cm+1,n

Cm+1Cm+2 Jn Jn+1

nCn

Cn+1

n + 1

Cn,n+1

Kn,n+1

Figure 4: Torsional vibration model of branch chain.

3. Research on Key Technology of HybridPropulsion Shafting Torsional Vibration

3.1. Influence of Phase Angle on Torsional Vibration. With thechange of ship sailing condition, the operation mode of hy-brid propulsion system will change. The phase angle is not afixed value in the process of twin-engine parallel operation.Phase angle is the angle between the fuel injection startingpoints of the first cylinder of each diesel engine. Torque ofeach cylinder of the diesel engine is a complex harmonictorque consisting of several harmonic moments. Its expres-sion is

𝑀∗𝑘 = 𝑀𝑘𝑒𝑖𝜉𝑘 , (7)

where𝑀𝑘 is 𝑘mass point’s harmonic excitation torque ampli-tude. 𝜉𝑘 is 𝑘 mass point’s harmonic excitation torque phase.

𝜉𝑘 = ] (𝜉1,𝑘 + 𝜃) , (8)

Mathematical Problems in Engineering 5Sy

nthe

tic am

plitu

de(∘)

800

800

700600

600

500400300200100 1000 1200 1400 1600 1800Phase angle ( ∘) Speed (r/min)

0

0

0.16

0.32

0.48

Figure 5: Amplitude of shaft with speed and phase difference.

Synt

hesis

ampl

itude

(∘)

100 200 300 400 500 600 7000Phase angle (∘)

0.33

0.34

0.35

0.36

Figure 6: 𝑉 = 1100 r/min synthesis of amplitude with phase angle.

where ] is order. 𝜉1,𝑘 is the firing angle between 𝑖 cylinder andthe first cylinder. 𝜃 is phase angle of twin-engine.

Equations (7) and (8) can be obtained by transformation.

𝑀∗𝑘 = 𝑀𝑘 cos [] (𝜍1,𝑖 + 𝜃)] + 𝑖 ⋅ 𝑀𝑘 sin [] (𝜍1,𝑖 + 𝜃)] . (9)

Phase angle of twin-engine parallel operation is changedwithin the range of 0∘ to 720∘. This paper uses 1∘ as intervalto calculate middle shaft’s synthetic amplitude changing withthe speed and phase according to (6) and (9), as shownin Figure 5. Calculation results at 1000 r/min are shown inFigure 6. Figure 5 shows the relationship between the phaseangle of the diesel engine, the speed of the diesel engine, andthe amplitude of the torsional vibration of the shaft. Whenthe diesel engine speed is a fixed value, the torsional vibrationamplitude of the shaft can be obtained with the change of thephase angle of the diesel engine by Figures 5 and 6.When thephase angle of two diesel engines is a fixed value, the curveshows the torsional vibration characteristics of the shaft in therange of diesel engine speed.

The calculation results show that synthetic amplitudevalue periodically fluctuates with changing phase angle underthe same speed; thus the phase angle has a certain effect onthe calculation results of torsional vibration in twin-engineparallel operation.

3.2.Motor VibrationMechanism andMotor Excitation Torque.In an ideal state, the air gap magnetic field of the motor is in

q

A

dC

B

Figure 7: Motor physical model.

sinusoidal distribution. Because of themachining error of themotor in themanufacturing process, themagnetic field of themotor must be nonsinusoidal. In order to study the vibrationmechanism of the motor and obtain the analytical formula ofthe electromagnetic torque, the physical model of 𝐴, 𝐵, and𝐶 phase motor is established as shown in Figure 7.

According to 𝐴, 𝐵, and 𝐶 phase flux linkage, the rela-tionship between the fluxes can be set up as shown in thefollowing three phase matrix equations:

𝜓𝐴𝐵𝐶 = [[[𝜓𝑎 (𝜃)𝜓𝑏 (𝜃)𝜓𝑐 (𝜃)

]]]

=[[[[[[[

𝜓𝑎 (𝜃)𝜓𝑏 (𝜃 − 2𝜋3 )𝜓𝑐 (𝜃 + 2𝜋3 )

]]]]]]]

=[[[[[[[[[[[

∞∑𝑘=1

𝜓(2𝑘−1) cos [(2𝑘 − 1) 𝜃]∞∑𝑘=1

𝜓(2𝑘−1) cos [(2𝑘 − 1) (𝜃 − 2𝜋3 )]∞∑𝑘=1

𝜓(2𝑘−1) cos [(2𝑘 − 1) (𝜃 + 2𝜋3 )]

]]]]]]]]]]]

.

(10)

Equation (10) can be transformed to 𝑑𝑞0 coordinates fluxequation through coordinate transformation:

𝜓𝑑𝑞 =[[[[[[[[[[

∞∑𝑘=1

{𝜓1 + [𝜓(6𝑘−1) + 𝜓(6𝑘+1)] cos (6𝑘𝜃)}∞∑𝑘=1

{[−𝜓(6𝑘−1) + 𝜓(6𝑘+1)] cos (6𝑘𝜃)}∞∑𝑘=1

𝜓(6𝑘+3) cos (6𝑘 + 3) 𝜃

]]]]]]]]]]

. (11)

Motor flux produced by fundamental sinusoidal currentis

𝜓𝐹 = [[[𝐿𝑑 0 00 𝐿𝑞 00 0 𝐿0

]]]

[[[𝑖𝑑𝑖𝑞𝑖0]]]

, (12)


where 𝐿𝑑, 𝐿𝑞, and 𝐿0 represent 𝑑, 𝑞, and 0 under 𝑑𝑞0 coor-dinate axis of the stator inductance, respectively. 𝑖𝑑, 𝑖𝑞, and 𝑖0represent 𝑑, 𝑞, and 0 under 𝑑𝑞0 coordinate axis of the current,respectively.

According to (11) and (12) total flux is

𝜓toal = [[[𝐿𝑑 0 00 𝐿𝑞 00 0 𝐿0

]]]

[[[[

𝑖𝑑𝑖𝑞𝑖0]]]]

+[[[[[[[[[[

∞∑𝑘=1

{𝜓1 + [𝜓(6𝑘−1) + 𝜓(6𝑘+1)] cos (6𝑘𝜃)}∞∑𝑘=1

{[−𝜓(6𝑘−1) + 𝜓(6𝑘+1)] sin (6𝑘𝜃)}∞∑𝑘=1

𝜓(6𝑘+3) cos (6𝑘 + 3) 𝜃

]]]]]]]]]]

.

(13)

Electromotive force can be derived from the total mag-netic chain derivative:

𝐸𝑑𝑞0 = [[[𝐸𝑑𝐸𝑞𝐸0

]]]

=[[[[[[[[[[[

−𝜔𝑟𝐿𝑑𝑖𝑑 − 𝜔𝑟 ∞∑𝑖=1

{[(6𝑘 − 1) 𝜓(6𝑘−1) + (6𝑘 + 1) 𝜓(6𝑘+1) sin 6𝑘𝜃]}𝜔𝑟𝐿𝑑𝑖𝑑 + 𝜔𝑟 ∞∑

𝑘=1

{𝜓1 + [− (6𝑘 − 1) 𝜓(6𝑘−1) + (6𝑘 + 1) 𝜓(6𝑘+1)] cos (6𝑘𝜃)}−𝜔𝑟 ∞∑𝑘=1

[(6𝑘 + 3) 𝜓(6𝑘+3) sin (6𝑘 + 3) 𝜃]

]]]]]]]]]]]

. (14)

According to the principle of the motor, the electromag-netic torque is

𝑇𝑒 = 1𝜔𝑚 ⋅ 32𝐸𝑇𝑖𝑑𝑞0 = 𝑝𝜔𝑟 ⋅32𝐸𝑇𝑖𝑑𝑞0, (15)

where 𝜔𝑚 is mechanical angular velocity of the rotor. 𝜔𝑟 isrotor electrical angular velocity.

By (14) and (15), the analytical solution of the electromag-netic torque is

𝑇𝑒 = 32𝑝 [(𝐿𝑑 − 𝐿𝑞) 𝑖𝑑𝑖𝑞 + 𝜓1𝑖𝑞] − 32⋅ 𝑝 [∞∑𝑘=1

{[(6𝑘 − 1) 𝜓(6𝑘−1) + (6𝑘 + 1) 𝜓(6𝑘+1)]

⋅ sin (6𝑘𝜃)} 𝑖𝑑] + 32𝑝∞∑𝑘=1

{[− (6𝑘 − 1) 𝜓(6𝑘−1)+ (6𝑘 + 1) 𝜓(6𝑘+1)] cos (6𝑘𝜃)} 𝑖𝑞.

(16)

The second and third terms in (16) are 6𝑘 order torquecaused by the magnetic field harmonic. Therefore, in theprocess of torsional vibration calculation of the hybridpropulsion shafting, 6𝑘 order torque caused by the fieldharmonic is needed to be considered at the quality pointof the motor. According to the torsional vibration model ofthe hybrid propulsion system as shown in Figure 2, we canknow that the second torque and third torque in (16) are theexcitation torque at the 41st mass point of the motor.

According to the forced vibration differential equation(4), the amplitude of the twenty-first mass point drive shaftcan be calculated under the condition of only consider-ing the motor excitation torque, as shown in Figure 6. InFigure 8, the result of the forced vibration calculation of the

6 order12 order18 order

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Am

plitu

de(∘)

600 800 1000 1200 1400 1600 1800400Engine speed (r/min)

Figure 8: The 21st mass (shaft) of amplitude values. Only considermotor excitation.

twenty-first mass point is given. The forced vibration ampli-tude curve includes 6 harmonic amplitude curve, 12 har-monic amplitude curve, and 18 harmonic amplitude curve.

To sum up, according to the diesel engine exciting forceequation (9), motor electromagnetic harmonic exciting forcecaused by (16) and the forced vibration equation (6) of hybridpropulsion shafting vibration response is calculated. As thephase angle of twin-engine parallel operation changes inthe range of 0 degrees to 720 degrees, only the calculationresult of the vibration amplitude of the transmission shaftat 𝜃 = 120∘ is shown in Figure 9. From Figure 9, we cansee that the resonance amplitude of the transmission shaft is


Syn

400 600 800 1000 1200 1400 1600 1800 2000200Engine speed (r/min)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Am

plitu

de(∘)

V = 1.5

V = 2

V = 3

V = 4

Figure 9: The 21st mass (shaft) of amplitude values.

Figure 10: Experiment platform of hybrid propulsion shafting.

0.394 degrees when the resonance speed of 𝑉 = 1.5 order is1060 r/min.

4. Test and Analysis ofHybrid Propulsion Shafting

4.1. Hybrid Propulsion Shafting Experiment Platform. Inorder to verify the correctness of the calculation result, hybridpropulsion shafting experiment platform is used for torsionalvibration measurement and analysis, as shown in Figure 10.The key parameters of hybrid propulsion system experimentplatformare listed inTable 1.Thegearwheel is installed on thetransmission shaft to facilitate the installation of the vibrationtest fixture, as shown in Figure 11.

This test system uses eddy current sensor.The noncontactelectrical measuring method is adopted to improve theshafting torsional vibration testing, as shown in Figure 12.

4.2. Data Analysis. In the experimental bench, the test oftorsional vibration of the transmission shaft is carried out in

Figure 11: Measuring point of hybrid propulsion shafting.

the 800 r/min–1500 r/min range by using the self-developedYDZT-2013 vibration test system. Through the analysis ofthe data, it is found that the torsional vibration amplitudeof the transmission shaft has resonance point in the 1.5harmonic times, as shown in Figures 13 and 14. We can seethat the resonance amplitude of the transmission shaft is0.4040 degrees when the resonance speed of 𝑉 = 1.5 orderis 1079.53 r/min.

Comparison and analysis of the data from the test and thetheoretical calculation are as below:

(1) The natural frequency of the system can be calcu-lated as 1619.29 r/min from the test values shownin Figure 13. Comparing with the 2-order naturalfrequency 1734.1 r/min shown in Table 3, the errorbetween the test value and the theoretical value is6.6%.The theoretical value is basically consistent withthe test value.

(2) From the calculation results of Figure 9, the resonanceamplitude of the transmission shaft is 0.394 degreeswhen the resonance speed of 𝑉 = 1.5 order is1060 r/min in theoretical calculation. The resonanceamplitude of the transmission shaft is 0.4040 degreeswhen the resonance speed of 𝑉 = 1.5 order is1079.53 r/min in vibration test and analysis, as shownin Figure 13. The error between the test value andthe theoretical value is 2.4%. The theoretical value isbasically consistent with the test value.

5. Conclusion

(1) With the same speed and different phase angle oftwin-engine parallel operation, hybrid propulsionsystem of shafting torsional vibration amplitudechanges periodically with the phase angle as shown inFigure 6.The significance of the research is to providethe theoretical basis of the ship’s torsional vibrationto the design professionals during the design stage ofship propulsion shafting.

(2) Due to the motor’s nonsinusoidal magnetic field dis-tribution, 6𝑘 order torque caused by electrical har-monic has been analyzed. The general analytical for-mula of electromagnetic torque excitation force is


Gear

Preamp One shot Low pass filter

Magnetic sensors

Points enlarge Recorder

Capacitance

Figure 12: Test principles of noncontact measuring method.

amplitude: 0.4040Channel: 1, order: 1.5, number: 23, speed (r/min): 1079.53,

1385

.16

1059

.24

1140

.72

1222

.2

1303

.68

977.

76

1466

.64

896.

28

(r/min)

0.19460.22240.2502

0.2780.30580.33360.36140.3892

0.4170.44480.47260.5004

(deg

)

×100

Figure 13: 𝑉 = 1.5-order amplitude value of the drive shaft.

(deg

)

×100

−0.7189

−0.3279

0.06310.45410.84511.23611.6271

360 720 1080 14400(deg)

Figure 14: Speed = 1079.53 r/min time domain waveform of driveshaft.

derived. The significance of the research is to lay atheoretical foundation and more accurate theoreticalcalculation for the forced vibration calculation of themarine hybrid propulsion system.

(3) The hybrid propulsion system is tested using the testbench and the self-developed YDZT-2013 type vibra-tion instrument. The theoretical data and the testingdata are compared and analyzed, and the mathemati-cal model and the algorithm are verified.

Competing Interests

The authors declare that they have no competing interests.

References

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[15] S.-C. Hsieh, J.-H. Chen, and A.-C. Lee, “A modified transfermatrix method for the coupling lateral and torsional vibra-tions of symmetric rotor-bearing systems,” Journal of Sound&Vibration, vol. 289, no. 1-2, pp. 294–333, 2006.

[16] Y. Kaneko, T.Momoo, andY.Okada, “Torsional vibration analy-sis of blade-disk-shaft system by the finite-element method andthe transfer matrix method,” Transactions of the Japan Society ofMechanical Engineers C, vol. 61, no. 586, pp. 2210–2215, 1995.

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